Step | Hyp | Ref
| Expression |
1 | | bdayelon 27207 |
. . . . 5
⊢ ( bday ‘𝐴) ∈ On |
2 | | bdayelon 27207 |
. . . . 5
⊢ ( bday ‘𝐵) ∈ On |
3 | | naddcl 8661 |
. . . . 5
⊢ ((( bday ‘𝐴) ∈ On ∧ (
bday ‘𝐵)
∈ On) → (( bday ‘𝐴) +no ( bday
‘𝐵)) ∈
On) |
4 | 1, 2, 3 | mp2an 690 |
. . . 4
⊢ (( bday ‘𝐴) +no ( bday
‘𝐵)) ∈
On |
5 | | bdayelon 27207 |
. . . . . . 7
⊢ ( bday ‘𝐶) ∈ On |
6 | | bdayelon 27207 |
. . . . . . 7
⊢ ( bday ‘𝐸) ∈ On |
7 | | naddcl 8661 |
. . . . . . 7
⊢ ((( bday ‘𝐶) ∈ On ∧ (
bday ‘𝐸)
∈ On) → (( bday ‘𝐶) +no ( bday
‘𝐸)) ∈
On) |
8 | 5, 6, 7 | mp2an 690 |
. . . . . 6
⊢ (( bday ‘𝐶) +no ( bday
‘𝐸)) ∈
On |
9 | | bdayelon 27207 |
. . . . . . 7
⊢ ( bday ‘𝐷) ∈ On |
10 | | bdayelon 27207 |
. . . . . . 7
⊢ ( bday ‘𝐹) ∈ On |
11 | | naddcl 8661 |
. . . . . . 7
⊢ ((( bday ‘𝐷) ∈ On ∧ (
bday ‘𝐹)
∈ On) → (( bday ‘𝐷) +no ( bday
‘𝐹)) ∈
On) |
12 | 9, 10, 11 | mp2an 690 |
. . . . . 6
⊢ (( bday ‘𝐷) +no ( bday
‘𝐹)) ∈
On |
13 | 8, 12 | onun2i 6476 |
. . . . 5
⊢ ((( bday ‘𝐶) +no ( bday
‘𝐸)) ∪
(( bday ‘𝐷) +no ( bday
‘𝐹))) ∈
On |
14 | | naddcl 8661 |
. . . . . . 7
⊢ ((( bday ‘𝐶) ∈ On ∧ (
bday ‘𝐹)
∈ On) → (( bday ‘𝐶) +no ( bday
‘𝐹)) ∈
On) |
15 | 5, 10, 14 | mp2an 690 |
. . . . . 6
⊢ (( bday ‘𝐶) +no ( bday
‘𝐹)) ∈
On |
16 | | naddcl 8661 |
. . . . . . 7
⊢ ((( bday ‘𝐷) ∈ On ∧ (
bday ‘𝐸)
∈ On) → (( bday ‘𝐷) +no ( bday
‘𝐸)) ∈
On) |
17 | 9, 6, 16 | mp2an 690 |
. . . . . 6
⊢ (( bday ‘𝐷) +no ( bday
‘𝐸)) ∈
On |
18 | 15, 17 | onun2i 6476 |
. . . . 5
⊢ ((( bday ‘𝐶) +no ( bday
‘𝐹)) ∪
(( bday ‘𝐷) +no ( bday
‘𝐸))) ∈
On |
19 | 13, 18 | onun2i 6476 |
. . . 4
⊢ (((( bday ‘𝐶) +no ( bday
‘𝐸)) ∪
(( bday ‘𝐷) +no ( bday
‘𝐹))) ∪
((( bday ‘𝐶) +no ( bday
‘𝐹)) ∪
(( bday ‘𝐷) +no ( bday
‘𝐸)))) ∈
On |
20 | 4, 19 | onun2i 6476 |
. . 3
⊢ ((( bday ‘𝐴) +no ( bday
‘𝐵)) ∪
(((( bday ‘𝐶) +no ( bday
‘𝐸)) ∪
(( bday ‘𝐷) +no ( bday
‘𝐹))) ∪
((( bday ‘𝐶) +no ( bday
‘𝐹)) ∪
(( bday ‘𝐷) +no ( bday
‘𝐸))))) ∈
On |
21 | | risset 3230 |
. . 3
⊢ (((( bday ‘𝐴) +no ( bday
‘𝐵)) ∪
(((( bday ‘𝐶) +no ( bday
‘𝐸)) ∪
(( bday ‘𝐷) +no ( bday
‘𝐹))) ∪
((( bday ‘𝐶) +no ( bday
‘𝐹)) ∪
(( bday ‘𝐷) +no ( bday
‘𝐸))))) ∈
On ↔ ∃𝑥 ∈
On 𝑥 = ((( bday ‘𝐴) +no ( bday
‘𝐵)) ∪
(((( bday ‘𝐶) +no ( bday
‘𝐸)) ∪
(( bday ‘𝐷) +no ( bday
‘𝐹))) ∪
((( bday ‘𝐶) +no ( bday
‘𝐹)) ∪
(( bday ‘𝐷) +no ( bday
‘𝐸)))))) |
22 | 20, 21 | mpbi 229 |
. 2
⊢
∃𝑥 ∈ On
𝑥 = ((( bday ‘𝐴) +no ( bday
‘𝐵)) ∪
(((( bday ‘𝐶) +no ( bday
‘𝐸)) ∪
(( bday ‘𝐷) +no ( bday
‘𝐹))) ∪
((( bday ‘𝐶) +no ( bday
‘𝐹)) ∪
(( bday ‘𝐷) +no ( bday
‘𝐸))))) |
23 | | fveq2 6879 |
. . . . . . . . . . . 12
⊢ (𝑎 = 𝑔 → ( bday
‘𝑎) = ( bday ‘𝑔)) |
24 | 23 | oveq1d 7409 |
. . . . . . . . . . 11
⊢ (𝑎 = 𝑔 → (( bday
‘𝑎) +no ( bday ‘𝑏)) = (( bday
‘𝑔) +no ( bday ‘𝑏))) |
25 | 24 | uneq1d 4159 |
. . . . . . . . . 10
⊢ (𝑎 = 𝑔 → ((( bday
‘𝑎) +no ( bday ‘𝑏)) ∪ (((( bday
‘𝑐) +no ( bday ‘𝑒)) ∪ (( bday
‘𝑑) +no ( bday ‘𝑓))) ∪ ((( bday
‘𝑐) +no ( bday ‘𝑓)) ∪ (( bday
‘𝑑) +no ( bday ‘𝑒))))) = ((( bday
‘𝑔) +no ( bday ‘𝑏)) ∪ (((( bday
‘𝑐) +no ( bday ‘𝑒)) ∪ (( bday
‘𝑑) +no ( bday ‘𝑓))) ∪ ((( bday
‘𝑐) +no ( bday ‘𝑓)) ∪ (( bday
‘𝑑) +no ( bday ‘𝑒)))))) |
26 | 25 | eqeq2d 2743 |
. . . . . . . . 9
⊢ (𝑎 = 𝑔 → (𝑥 = ((( bday
‘𝑎) +no ( bday ‘𝑏)) ∪ (((( bday
‘𝑐) +no ( bday ‘𝑒)) ∪ (( bday
‘𝑑) +no ( bday ‘𝑓))) ∪ ((( bday
‘𝑐) +no ( bday ‘𝑓)) ∪ (( bday
‘𝑑) +no ( bday ‘𝑒))))) ↔ 𝑥 = ((( bday
‘𝑔) +no ( bday ‘𝑏)) ∪ (((( bday
‘𝑐) +no ( bday ‘𝑒)) ∪ (( bday
‘𝑑) +no ( bday ‘𝑓))) ∪ ((( bday
‘𝑐) +no ( bday ‘𝑓)) ∪ (( bday
‘𝑑) +no ( bday ‘𝑒))))))) |
27 | | oveq1 7401 |
. . . . . . . . . . 11
⊢ (𝑎 = 𝑔 → (𝑎 ·s 𝑏) = (𝑔 ·s 𝑏)) |
28 | 27 | eleq1d 2818 |
. . . . . . . . . 10
⊢ (𝑎 = 𝑔 → ((𝑎 ·s 𝑏) ∈ No
↔ (𝑔
·s 𝑏)
∈ No )) |
29 | 28 | anbi1d 630 |
. . . . . . . . 9
⊢ (𝑎 = 𝑔 → (((𝑎 ·s 𝑏) ∈ No
∧ ((𝑐 <s 𝑑 ∧ 𝑒 <s 𝑓) → ((𝑐 ·s 𝑓) -s (𝑐 ·s 𝑒)) <s ((𝑑 ·s 𝑓) -s (𝑑 ·s 𝑒)))) ↔ ((𝑔 ·s 𝑏) ∈ No
∧ ((𝑐 <s 𝑑 ∧ 𝑒 <s 𝑓) → ((𝑐 ·s 𝑓) -s (𝑐 ·s 𝑒)) <s ((𝑑 ·s 𝑓) -s (𝑑 ·s 𝑒)))))) |
30 | 26, 29 | imbi12d 344 |
. . . . . . . 8
⊢ (𝑎 = 𝑔 → ((𝑥 = ((( bday
‘𝑎) +no ( bday ‘𝑏)) ∪ (((( bday
‘𝑐) +no ( bday ‘𝑒)) ∪ (( bday
‘𝑑) +no ( bday ‘𝑓))) ∪ ((( bday
‘𝑐) +no ( bday ‘𝑓)) ∪ (( bday
‘𝑑) +no ( bday ‘𝑒))))) → ((𝑎 ·s 𝑏) ∈ No
∧ ((𝑐 <s 𝑑 ∧ 𝑒 <s 𝑓) → ((𝑐 ·s 𝑓) -s (𝑐 ·s 𝑒)) <s ((𝑑 ·s 𝑓) -s (𝑑 ·s 𝑒))))) ↔ (𝑥 = ((( bday
‘𝑔) +no ( bday ‘𝑏)) ∪ (((( bday
‘𝑐) +no ( bday ‘𝑒)) ∪ (( bday
‘𝑑) +no ( bday ‘𝑓))) ∪ ((( bday
‘𝑐) +no ( bday ‘𝑓)) ∪ (( bday
‘𝑑) +no ( bday ‘𝑒))))) → ((𝑔 ·s 𝑏) ∈ No
∧ ((𝑐 <s 𝑑 ∧ 𝑒 <s 𝑓) → ((𝑐 ·s 𝑓) -s (𝑐 ·s 𝑒)) <s ((𝑑 ·s 𝑓) -s (𝑑 ·s 𝑒))))))) |
31 | | fveq2 6879 |
. . . . . . . . . . . 12
⊢ (𝑏 = ℎ → ( bday
‘𝑏) = ( bday ‘ℎ)) |
32 | 31 | oveq2d 7410 |
. . . . . . . . . . 11
⊢ (𝑏 = ℎ → (( bday
‘𝑔) +no ( bday ‘𝑏)) = (( bday
‘𝑔) +no ( bday ‘ℎ))) |
33 | 32 | uneq1d 4159 |
. . . . . . . . . 10
⊢ (𝑏 = ℎ → ((( bday
‘𝑔) +no ( bday ‘𝑏)) ∪ (((( bday
‘𝑐) +no ( bday ‘𝑒)) ∪ (( bday
‘𝑑) +no ( bday ‘𝑓))) ∪ ((( bday
‘𝑐) +no ( bday ‘𝑓)) ∪ (( bday
‘𝑑) +no ( bday ‘𝑒))))) = ((( bday
‘𝑔) +no ( bday ‘ℎ)) ∪ (((( bday
‘𝑐) +no ( bday ‘𝑒)) ∪ (( bday
‘𝑑) +no ( bday ‘𝑓))) ∪ ((( bday
‘𝑐) +no ( bday ‘𝑓)) ∪ (( bday
‘𝑑) +no ( bday ‘𝑒)))))) |
34 | 33 | eqeq2d 2743 |
. . . . . . . . 9
⊢ (𝑏 = ℎ → (𝑥 = ((( bday
‘𝑔) +no ( bday ‘𝑏)) ∪ (((( bday
‘𝑐) +no ( bday ‘𝑒)) ∪ (( bday
‘𝑑) +no ( bday ‘𝑓))) ∪ ((( bday
‘𝑐) +no ( bday ‘𝑓)) ∪ (( bday
‘𝑑) +no ( bday ‘𝑒))))) ↔ 𝑥 = ((( bday
‘𝑔) +no ( bday ‘ℎ)) ∪ (((( bday
‘𝑐) +no ( bday ‘𝑒)) ∪ (( bday
‘𝑑) +no ( bday ‘𝑓))) ∪ ((( bday
‘𝑐) +no ( bday ‘𝑓)) ∪ (( bday
‘𝑑) +no ( bday ‘𝑒))))))) |
35 | | oveq2 7402 |
. . . . . . . . . . 11
⊢ (𝑏 = ℎ → (𝑔 ·s 𝑏) = (𝑔 ·s ℎ)) |
36 | 35 | eleq1d 2818 |
. . . . . . . . . 10
⊢ (𝑏 = ℎ → ((𝑔 ·s 𝑏) ∈ No
↔ (𝑔
·s ℎ)
∈ No )) |
37 | 36 | anbi1d 630 |
. . . . . . . . 9
⊢ (𝑏 = ℎ → (((𝑔 ·s 𝑏) ∈ No
∧ ((𝑐 <s 𝑑 ∧ 𝑒 <s 𝑓) → ((𝑐 ·s 𝑓) -s (𝑐 ·s 𝑒)) <s ((𝑑 ·s 𝑓) -s (𝑑 ·s 𝑒)))) ↔ ((𝑔 ·s ℎ) ∈ No
∧ ((𝑐 <s 𝑑 ∧ 𝑒 <s 𝑓) → ((𝑐 ·s 𝑓) -s (𝑐 ·s 𝑒)) <s ((𝑑 ·s 𝑓) -s (𝑑 ·s 𝑒)))))) |
38 | 34, 37 | imbi12d 344 |
. . . . . . . 8
⊢ (𝑏 = ℎ → ((𝑥 = ((( bday
‘𝑔) +no ( bday ‘𝑏)) ∪ (((( bday
‘𝑐) +no ( bday ‘𝑒)) ∪ (( bday
‘𝑑) +no ( bday ‘𝑓))) ∪ ((( bday
‘𝑐) +no ( bday ‘𝑓)) ∪ (( bday
‘𝑑) +no ( bday ‘𝑒))))) → ((𝑔 ·s 𝑏) ∈ No
∧ ((𝑐 <s 𝑑 ∧ 𝑒 <s 𝑓) → ((𝑐 ·s 𝑓) -s (𝑐 ·s 𝑒)) <s ((𝑑 ·s 𝑓) -s (𝑑 ·s 𝑒))))) ↔ (𝑥 = ((( bday
‘𝑔) +no ( bday ‘ℎ)) ∪ (((( bday
‘𝑐) +no ( bday ‘𝑒)) ∪ (( bday
‘𝑑) +no ( bday ‘𝑓))) ∪ ((( bday
‘𝑐) +no ( bday ‘𝑓)) ∪ (( bday
‘𝑑) +no ( bday ‘𝑒))))) → ((𝑔 ·s ℎ) ∈ No
∧ ((𝑐 <s 𝑑 ∧ 𝑒 <s 𝑓) → ((𝑐 ·s 𝑓) -s (𝑐 ·s 𝑒)) <s ((𝑑 ·s 𝑓) -s (𝑑 ·s 𝑒))))))) |
39 | | fveq2 6879 |
. . . . . . . . . . . . . 14
⊢ (𝑐 = 𝑖 → ( bday
‘𝑐) = ( bday ‘𝑖)) |
40 | 39 | oveq1d 7409 |
. . . . . . . . . . . . 13
⊢ (𝑐 = 𝑖 → (( bday
‘𝑐) +no ( bday ‘𝑒)) = (( bday
‘𝑖) +no ( bday ‘𝑒))) |
41 | 40 | uneq1d 4159 |
. . . . . . . . . . . 12
⊢ (𝑐 = 𝑖 → ((( bday
‘𝑐) +no ( bday ‘𝑒)) ∪ (( bday
‘𝑑) +no ( bday ‘𝑓))) = ((( bday
‘𝑖) +no ( bday ‘𝑒)) ∪ (( bday
‘𝑑) +no ( bday ‘𝑓)))) |
42 | 39 | oveq1d 7409 |
. . . . . . . . . . . . 13
⊢ (𝑐 = 𝑖 → (( bday
‘𝑐) +no ( bday ‘𝑓)) = (( bday
‘𝑖) +no ( bday ‘𝑓))) |
43 | 42 | uneq1d 4159 |
. . . . . . . . . . . 12
⊢ (𝑐 = 𝑖 → ((( bday
‘𝑐) +no ( bday ‘𝑓)) ∪ (( bday
‘𝑑) +no ( bday ‘𝑒))) = ((( bday
‘𝑖) +no ( bday ‘𝑓)) ∪ (( bday
‘𝑑) +no ( bday ‘𝑒)))) |
44 | 41, 43 | uneq12d 4161 |
. . . . . . . . . . 11
⊢ (𝑐 = 𝑖 → (((( bday
‘𝑐) +no ( bday ‘𝑒)) ∪ (( bday
‘𝑑) +no ( bday ‘𝑓))) ∪ ((( bday
‘𝑐) +no ( bday ‘𝑓)) ∪ (( bday
‘𝑑) +no ( bday ‘𝑒)))) = (((( bday
‘𝑖) +no ( bday ‘𝑒)) ∪ (( bday
‘𝑑) +no ( bday ‘𝑓))) ∪ ((( bday
‘𝑖) +no ( bday ‘𝑓)) ∪ (( bday
‘𝑑) +no ( bday ‘𝑒))))) |
45 | 44 | uneq2d 4160 |
. . . . . . . . . 10
⊢ (𝑐 = 𝑖 → ((( bday
‘𝑔) +no ( bday ‘ℎ)) ∪ (((( bday
‘𝑐) +no ( bday ‘𝑒)) ∪ (( bday
‘𝑑) +no ( bday ‘𝑓))) ∪ ((( bday
‘𝑐) +no ( bday ‘𝑓)) ∪ (( bday
‘𝑑) +no ( bday ‘𝑒))))) = ((( bday
‘𝑔) +no ( bday ‘ℎ)) ∪ (((( bday
‘𝑖) +no ( bday ‘𝑒)) ∪ (( bday
‘𝑑) +no ( bday ‘𝑓))) ∪ ((( bday
‘𝑖) +no ( bday ‘𝑓)) ∪ (( bday
‘𝑑) +no ( bday ‘𝑒)))))) |
46 | 45 | eqeq2d 2743 |
. . . . . . . . 9
⊢ (𝑐 = 𝑖 → (𝑥 = ((( bday
‘𝑔) +no ( bday ‘ℎ)) ∪ (((( bday
‘𝑐) +no ( bday ‘𝑒)) ∪ (( bday
‘𝑑) +no ( bday ‘𝑓))) ∪ ((( bday
‘𝑐) +no ( bday ‘𝑓)) ∪ (( bday
‘𝑑) +no ( bday ‘𝑒))))) ↔ 𝑥 = ((( bday
‘𝑔) +no ( bday ‘ℎ)) ∪ (((( bday
‘𝑖) +no ( bday ‘𝑒)) ∪ (( bday
‘𝑑) +no ( bday ‘𝑓))) ∪ ((( bday
‘𝑖) +no ( bday ‘𝑓)) ∪ (( bday
‘𝑑) +no ( bday ‘𝑒))))))) |
47 | | breq1 5145 |
. . . . . . . . . . . 12
⊢ (𝑐 = 𝑖 → (𝑐 <s 𝑑 ↔ 𝑖 <s 𝑑)) |
48 | 47 | anbi1d 630 |
. . . . . . . . . . 11
⊢ (𝑐 = 𝑖 → ((𝑐 <s 𝑑 ∧ 𝑒 <s 𝑓) ↔ (𝑖 <s 𝑑 ∧ 𝑒 <s 𝑓))) |
49 | | oveq1 7401 |
. . . . . . . . . . . . 13
⊢ (𝑐 = 𝑖 → (𝑐 ·s 𝑓) = (𝑖 ·s 𝑓)) |
50 | | oveq1 7401 |
. . . . . . . . . . . . 13
⊢ (𝑐 = 𝑖 → (𝑐 ·s 𝑒) = (𝑖 ·s 𝑒)) |
51 | 49, 50 | oveq12d 7412 |
. . . . . . . . . . . 12
⊢ (𝑐 = 𝑖 → ((𝑐 ·s 𝑓) -s (𝑐 ·s 𝑒)) = ((𝑖 ·s 𝑓) -s (𝑖 ·s 𝑒))) |
52 | 51 | breq1d 5152 |
. . . . . . . . . . 11
⊢ (𝑐 = 𝑖 → (((𝑐 ·s 𝑓) -s (𝑐 ·s 𝑒)) <s ((𝑑 ·s 𝑓) -s (𝑑 ·s 𝑒)) ↔ ((𝑖 ·s 𝑓) -s (𝑖 ·s 𝑒)) <s ((𝑑 ·s 𝑓) -s (𝑑 ·s 𝑒)))) |
53 | 48, 52 | imbi12d 344 |
. . . . . . . . . 10
⊢ (𝑐 = 𝑖 → (((𝑐 <s 𝑑 ∧ 𝑒 <s 𝑓) → ((𝑐 ·s 𝑓) -s (𝑐 ·s 𝑒)) <s ((𝑑 ·s 𝑓) -s (𝑑 ·s 𝑒))) ↔ ((𝑖 <s 𝑑 ∧ 𝑒 <s 𝑓) → ((𝑖 ·s 𝑓) -s (𝑖 ·s 𝑒)) <s ((𝑑 ·s 𝑓) -s (𝑑 ·s 𝑒))))) |
54 | 53 | anbi2d 629 |
. . . . . . . . 9
⊢ (𝑐 = 𝑖 → (((𝑔 ·s ℎ) ∈ No
∧ ((𝑐 <s 𝑑 ∧ 𝑒 <s 𝑓) → ((𝑐 ·s 𝑓) -s (𝑐 ·s 𝑒)) <s ((𝑑 ·s 𝑓) -s (𝑑 ·s 𝑒)))) ↔ ((𝑔 ·s ℎ) ∈ No
∧ ((𝑖 <s 𝑑 ∧ 𝑒 <s 𝑓) → ((𝑖 ·s 𝑓) -s (𝑖 ·s 𝑒)) <s ((𝑑 ·s 𝑓) -s (𝑑 ·s 𝑒)))))) |
55 | 46, 54 | imbi12d 344 |
. . . . . . . 8
⊢ (𝑐 = 𝑖 → ((𝑥 = ((( bday
‘𝑔) +no ( bday ‘ℎ)) ∪ (((( bday
‘𝑐) +no ( bday ‘𝑒)) ∪ (( bday
‘𝑑) +no ( bday ‘𝑓))) ∪ ((( bday
‘𝑐) +no ( bday ‘𝑓)) ∪ (( bday
‘𝑑) +no ( bday ‘𝑒))))) → ((𝑔 ·s ℎ) ∈ No
∧ ((𝑐 <s 𝑑 ∧ 𝑒 <s 𝑓) → ((𝑐 ·s 𝑓) -s (𝑐 ·s 𝑒)) <s ((𝑑 ·s 𝑓) -s (𝑑 ·s 𝑒))))) ↔ (𝑥 = ((( bday
‘𝑔) +no ( bday ‘ℎ)) ∪ (((( bday
‘𝑖) +no ( bday ‘𝑒)) ∪ (( bday
‘𝑑) +no ( bday ‘𝑓))) ∪ ((( bday
‘𝑖) +no ( bday ‘𝑓)) ∪ (( bday
‘𝑑) +no ( bday ‘𝑒))))) → ((𝑔 ·s ℎ) ∈ No
∧ ((𝑖 <s 𝑑 ∧ 𝑒 <s 𝑓) → ((𝑖 ·s 𝑓) -s (𝑖 ·s 𝑒)) <s ((𝑑 ·s 𝑓) -s (𝑑 ·s 𝑒))))))) |
56 | | fveq2 6879 |
. . . . . . . . . . . . . 14
⊢ (𝑑 = 𝑗 → ( bday
‘𝑑) = ( bday ‘𝑗)) |
57 | 56 | oveq1d 7409 |
. . . . . . . . . . . . 13
⊢ (𝑑 = 𝑗 → (( bday
‘𝑑) +no ( bday ‘𝑓)) = (( bday
‘𝑗) +no ( bday ‘𝑓))) |
58 | 57 | uneq2d 4160 |
. . . . . . . . . . . 12
⊢ (𝑑 = 𝑗 → ((( bday
‘𝑖) +no ( bday ‘𝑒)) ∪ (( bday
‘𝑑) +no ( bday ‘𝑓))) = ((( bday
‘𝑖) +no ( bday ‘𝑒)) ∪ (( bday
‘𝑗) +no ( bday ‘𝑓)))) |
59 | 56 | oveq1d 7409 |
. . . . . . . . . . . . 13
⊢ (𝑑 = 𝑗 → (( bday
‘𝑑) +no ( bday ‘𝑒)) = (( bday
‘𝑗) +no ( bday ‘𝑒))) |
60 | 59 | uneq2d 4160 |
. . . . . . . . . . . 12
⊢ (𝑑 = 𝑗 → ((( bday
‘𝑖) +no ( bday ‘𝑓)) ∪ (( bday
‘𝑑) +no ( bday ‘𝑒))) = ((( bday
‘𝑖) +no ( bday ‘𝑓)) ∪ (( bday
‘𝑗) +no ( bday ‘𝑒)))) |
61 | 58, 60 | uneq12d 4161 |
. . . . . . . . . . 11
⊢ (𝑑 = 𝑗 → (((( bday
‘𝑖) +no ( bday ‘𝑒)) ∪ (( bday
‘𝑑) +no ( bday ‘𝑓))) ∪ ((( bday
‘𝑖) +no ( bday ‘𝑓)) ∪ (( bday
‘𝑑) +no ( bday ‘𝑒)))) = (((( bday
‘𝑖) +no ( bday ‘𝑒)) ∪ (( bday
‘𝑗) +no ( bday ‘𝑓))) ∪ ((( bday
‘𝑖) +no ( bday ‘𝑓)) ∪ (( bday
‘𝑗) +no ( bday ‘𝑒))))) |
62 | 61 | uneq2d 4160 |
. . . . . . . . . 10
⊢ (𝑑 = 𝑗 → ((( bday
‘𝑔) +no ( bday ‘ℎ)) ∪ (((( bday
‘𝑖) +no ( bday ‘𝑒)) ∪ (( bday
‘𝑑) +no ( bday ‘𝑓))) ∪ ((( bday
‘𝑖) +no ( bday ‘𝑓)) ∪ (( bday
‘𝑑) +no ( bday ‘𝑒))))) = ((( bday
‘𝑔) +no ( bday ‘ℎ)) ∪ (((( bday
‘𝑖) +no ( bday ‘𝑒)) ∪ (( bday
‘𝑗) +no ( bday ‘𝑓))) ∪ ((( bday
‘𝑖) +no ( bday ‘𝑓)) ∪ (( bday
‘𝑗) +no ( bday ‘𝑒)))))) |
63 | 62 | eqeq2d 2743 |
. . . . . . . . 9
⊢ (𝑑 = 𝑗 → (𝑥 = ((( bday
‘𝑔) +no ( bday ‘ℎ)) ∪ (((( bday
‘𝑖) +no ( bday ‘𝑒)) ∪ (( bday
‘𝑑) +no ( bday ‘𝑓))) ∪ ((( bday
‘𝑖) +no ( bday ‘𝑓)) ∪ (( bday
‘𝑑) +no ( bday ‘𝑒))))) ↔ 𝑥 = ((( bday
‘𝑔) +no ( bday ‘ℎ)) ∪ (((( bday
‘𝑖) +no ( bday ‘𝑒)) ∪ (( bday
‘𝑗) +no ( bday ‘𝑓))) ∪ ((( bday
‘𝑖) +no ( bday ‘𝑓)) ∪ (( bday
‘𝑗) +no ( bday ‘𝑒))))))) |
64 | | breq2 5146 |
. . . . . . . . . . . 12
⊢ (𝑑 = 𝑗 → (𝑖 <s 𝑑 ↔ 𝑖 <s 𝑗)) |
65 | 64 | anbi1d 630 |
. . . . . . . . . . 11
⊢ (𝑑 = 𝑗 → ((𝑖 <s 𝑑 ∧ 𝑒 <s 𝑓) ↔ (𝑖 <s 𝑗 ∧ 𝑒 <s 𝑓))) |
66 | | oveq1 7401 |
. . . . . . . . . . . . 13
⊢ (𝑑 = 𝑗 → (𝑑 ·s 𝑓) = (𝑗 ·s 𝑓)) |
67 | | oveq1 7401 |
. . . . . . . . . . . . 13
⊢ (𝑑 = 𝑗 → (𝑑 ·s 𝑒) = (𝑗 ·s 𝑒)) |
68 | 66, 67 | oveq12d 7412 |
. . . . . . . . . . . 12
⊢ (𝑑 = 𝑗 → ((𝑑 ·s 𝑓) -s (𝑑 ·s 𝑒)) = ((𝑗 ·s 𝑓) -s (𝑗 ·s 𝑒))) |
69 | 68 | breq2d 5154 |
. . . . . . . . . . 11
⊢ (𝑑 = 𝑗 → (((𝑖 ·s 𝑓) -s (𝑖 ·s 𝑒)) <s ((𝑑 ·s 𝑓) -s (𝑑 ·s 𝑒)) ↔ ((𝑖 ·s 𝑓) -s (𝑖 ·s 𝑒)) <s ((𝑗 ·s 𝑓) -s (𝑗 ·s 𝑒)))) |
70 | 65, 69 | imbi12d 344 |
. . . . . . . . . 10
⊢ (𝑑 = 𝑗 → (((𝑖 <s 𝑑 ∧ 𝑒 <s 𝑓) → ((𝑖 ·s 𝑓) -s (𝑖 ·s 𝑒)) <s ((𝑑 ·s 𝑓) -s (𝑑 ·s 𝑒))) ↔ ((𝑖 <s 𝑗 ∧ 𝑒 <s 𝑓) → ((𝑖 ·s 𝑓) -s (𝑖 ·s 𝑒)) <s ((𝑗 ·s 𝑓) -s (𝑗 ·s 𝑒))))) |
71 | 70 | anbi2d 629 |
. . . . . . . . 9
⊢ (𝑑 = 𝑗 → (((𝑔 ·s ℎ) ∈ No
∧ ((𝑖 <s 𝑑 ∧ 𝑒 <s 𝑓) → ((𝑖 ·s 𝑓) -s (𝑖 ·s 𝑒)) <s ((𝑑 ·s 𝑓) -s (𝑑 ·s 𝑒)))) ↔ ((𝑔 ·s ℎ) ∈ No
∧ ((𝑖 <s 𝑗 ∧ 𝑒 <s 𝑓) → ((𝑖 ·s 𝑓) -s (𝑖 ·s 𝑒)) <s ((𝑗 ·s 𝑓) -s (𝑗 ·s 𝑒)))))) |
72 | 63, 71 | imbi12d 344 |
. . . . . . . 8
⊢ (𝑑 = 𝑗 → ((𝑥 = ((( bday
‘𝑔) +no ( bday ‘ℎ)) ∪ (((( bday
‘𝑖) +no ( bday ‘𝑒)) ∪ (( bday
‘𝑑) +no ( bday ‘𝑓))) ∪ ((( bday
‘𝑖) +no ( bday ‘𝑓)) ∪ (( bday
‘𝑑) +no ( bday ‘𝑒))))) → ((𝑔 ·s ℎ) ∈ No
∧ ((𝑖 <s 𝑑 ∧ 𝑒 <s 𝑓) → ((𝑖 ·s 𝑓) -s (𝑖 ·s 𝑒)) <s ((𝑑 ·s 𝑓) -s (𝑑 ·s 𝑒))))) ↔ (𝑥 = ((( bday
‘𝑔) +no ( bday ‘ℎ)) ∪ (((( bday
‘𝑖) +no ( bday ‘𝑒)) ∪ (( bday
‘𝑗) +no ( bday ‘𝑓))) ∪ ((( bday
‘𝑖) +no ( bday ‘𝑓)) ∪ (( bday
‘𝑗) +no ( bday ‘𝑒))))) → ((𝑔 ·s ℎ) ∈ No
∧ ((𝑖 <s 𝑗 ∧ 𝑒 <s 𝑓) → ((𝑖 ·s 𝑓) -s (𝑖 ·s 𝑒)) <s ((𝑗 ·s 𝑓) -s (𝑗 ·s 𝑒))))))) |
73 | | fveq2 6879 |
. . . . . . . . . . . . . 14
⊢ (𝑒 = 𝑘 → ( bday
‘𝑒) = ( bday ‘𝑘)) |
74 | 73 | oveq2d 7410 |
. . . . . . . . . . . . 13
⊢ (𝑒 = 𝑘 → (( bday
‘𝑖) +no ( bday ‘𝑒)) = (( bday
‘𝑖) +no ( bday ‘𝑘))) |
75 | 74 | uneq1d 4159 |
. . . . . . . . . . . 12
⊢ (𝑒 = 𝑘 → ((( bday
‘𝑖) +no ( bday ‘𝑒)) ∪ (( bday
‘𝑗) +no ( bday ‘𝑓))) = ((( bday
‘𝑖) +no ( bday ‘𝑘)) ∪ (( bday
‘𝑗) +no ( bday ‘𝑓)))) |
76 | 73 | oveq2d 7410 |
. . . . . . . . . . . . 13
⊢ (𝑒 = 𝑘 → (( bday
‘𝑗) +no ( bday ‘𝑒)) = (( bday
‘𝑗) +no ( bday ‘𝑘))) |
77 | 76 | uneq2d 4160 |
. . . . . . . . . . . 12
⊢ (𝑒 = 𝑘 → ((( bday
‘𝑖) +no ( bday ‘𝑓)) ∪ (( bday
‘𝑗) +no ( bday ‘𝑒))) = ((( bday
‘𝑖) +no ( bday ‘𝑓)) ∪ (( bday
‘𝑗) +no ( bday ‘𝑘)))) |
78 | 75, 77 | uneq12d 4161 |
. . . . . . . . . . 11
⊢ (𝑒 = 𝑘 → (((( bday
‘𝑖) +no ( bday ‘𝑒)) ∪ (( bday
‘𝑗) +no ( bday ‘𝑓))) ∪ ((( bday
‘𝑖) +no ( bday ‘𝑓)) ∪ (( bday
‘𝑗) +no ( bday ‘𝑒)))) = (((( bday
‘𝑖) +no ( bday ‘𝑘)) ∪ (( bday
‘𝑗) +no ( bday ‘𝑓))) ∪ ((( bday
‘𝑖) +no ( bday ‘𝑓)) ∪ (( bday
‘𝑗) +no ( bday ‘𝑘))))) |
79 | 78 | uneq2d 4160 |
. . . . . . . . . 10
⊢ (𝑒 = 𝑘 → ((( bday
‘𝑔) +no ( bday ‘ℎ)) ∪ (((( bday
‘𝑖) +no ( bday ‘𝑒)) ∪ (( bday
‘𝑗) +no ( bday ‘𝑓))) ∪ ((( bday
‘𝑖) +no ( bday ‘𝑓)) ∪ (( bday
‘𝑗) +no ( bday ‘𝑒))))) = ((( bday
‘𝑔) +no ( bday ‘ℎ)) ∪ (((( bday
‘𝑖) +no ( bday ‘𝑘)) ∪ (( bday
‘𝑗) +no ( bday ‘𝑓))) ∪ ((( bday
‘𝑖) +no ( bday ‘𝑓)) ∪ (( bday
‘𝑗) +no ( bday ‘𝑘)))))) |
80 | 79 | eqeq2d 2743 |
. . . . . . . . 9
⊢ (𝑒 = 𝑘 → (𝑥 = ((( bday
‘𝑔) +no ( bday ‘ℎ)) ∪ (((( bday
‘𝑖) +no ( bday ‘𝑒)) ∪ (( bday
‘𝑗) +no ( bday ‘𝑓))) ∪ ((( bday
‘𝑖) +no ( bday ‘𝑓)) ∪ (( bday
‘𝑗) +no ( bday ‘𝑒))))) ↔ 𝑥 = ((( bday
‘𝑔) +no ( bday ‘ℎ)) ∪ (((( bday
‘𝑖) +no ( bday ‘𝑘)) ∪ (( bday
‘𝑗) +no ( bday ‘𝑓))) ∪ ((( bday
‘𝑖) +no ( bday ‘𝑓)) ∪ (( bday
‘𝑗) +no ( bday ‘𝑘))))))) |
81 | | breq1 5145 |
. . . . . . . . . . . 12
⊢ (𝑒 = 𝑘 → (𝑒 <s 𝑓 ↔ 𝑘 <s 𝑓)) |
82 | 81 | anbi2d 629 |
. . . . . . . . . . 11
⊢ (𝑒 = 𝑘 → ((𝑖 <s 𝑗 ∧ 𝑒 <s 𝑓) ↔ (𝑖 <s 𝑗 ∧ 𝑘 <s 𝑓))) |
83 | | oveq2 7402 |
. . . . . . . . . . . . 13
⊢ (𝑒 = 𝑘 → (𝑖 ·s 𝑒) = (𝑖 ·s 𝑘)) |
84 | 83 | oveq2d 7410 |
. . . . . . . . . . . 12
⊢ (𝑒 = 𝑘 → ((𝑖 ·s 𝑓) -s (𝑖 ·s 𝑒)) = ((𝑖 ·s 𝑓) -s (𝑖 ·s 𝑘))) |
85 | | oveq2 7402 |
. . . . . . . . . . . . 13
⊢ (𝑒 = 𝑘 → (𝑗 ·s 𝑒) = (𝑗 ·s 𝑘)) |
86 | 85 | oveq2d 7410 |
. . . . . . . . . . . 12
⊢ (𝑒 = 𝑘 → ((𝑗 ·s 𝑓) -s (𝑗 ·s 𝑒)) = ((𝑗 ·s 𝑓) -s (𝑗 ·s 𝑘))) |
87 | 84, 86 | breq12d 5155 |
. . . . . . . . . . 11
⊢ (𝑒 = 𝑘 → (((𝑖 ·s 𝑓) -s (𝑖 ·s 𝑒)) <s ((𝑗 ·s 𝑓) -s (𝑗 ·s 𝑒)) ↔ ((𝑖 ·s 𝑓) -s (𝑖 ·s 𝑘)) <s ((𝑗 ·s 𝑓) -s (𝑗 ·s 𝑘)))) |
88 | 82, 87 | imbi12d 344 |
. . . . . . . . . 10
⊢ (𝑒 = 𝑘 → (((𝑖 <s 𝑗 ∧ 𝑒 <s 𝑓) → ((𝑖 ·s 𝑓) -s (𝑖 ·s 𝑒)) <s ((𝑗 ·s 𝑓) -s (𝑗 ·s 𝑒))) ↔ ((𝑖 <s 𝑗 ∧ 𝑘 <s 𝑓) → ((𝑖 ·s 𝑓) -s (𝑖 ·s 𝑘)) <s ((𝑗 ·s 𝑓) -s (𝑗 ·s 𝑘))))) |
89 | 88 | anbi2d 629 |
. . . . . . . . 9
⊢ (𝑒 = 𝑘 → (((𝑔 ·s ℎ) ∈ No
∧ ((𝑖 <s 𝑗 ∧ 𝑒 <s 𝑓) → ((𝑖 ·s 𝑓) -s (𝑖 ·s 𝑒)) <s ((𝑗 ·s 𝑓) -s (𝑗 ·s 𝑒)))) ↔ ((𝑔 ·s ℎ) ∈ No
∧ ((𝑖 <s 𝑗 ∧ 𝑘 <s 𝑓) → ((𝑖 ·s 𝑓) -s (𝑖 ·s 𝑘)) <s ((𝑗 ·s 𝑓) -s (𝑗 ·s 𝑘)))))) |
90 | 80, 89 | imbi12d 344 |
. . . . . . . 8
⊢ (𝑒 = 𝑘 → ((𝑥 = ((( bday
‘𝑔) +no ( bday ‘ℎ)) ∪ (((( bday
‘𝑖) +no ( bday ‘𝑒)) ∪ (( bday
‘𝑗) +no ( bday ‘𝑓))) ∪ ((( bday
‘𝑖) +no ( bday ‘𝑓)) ∪ (( bday
‘𝑗) +no ( bday ‘𝑒))))) → ((𝑔 ·s ℎ) ∈ No
∧ ((𝑖 <s 𝑗 ∧ 𝑒 <s 𝑓) → ((𝑖 ·s 𝑓) -s (𝑖 ·s 𝑒)) <s ((𝑗 ·s 𝑓) -s (𝑗 ·s 𝑒))))) ↔ (𝑥 = ((( bday
‘𝑔) +no ( bday ‘ℎ)) ∪ (((( bday
‘𝑖) +no ( bday ‘𝑘)) ∪ (( bday
‘𝑗) +no ( bday ‘𝑓))) ∪ ((( bday
‘𝑖) +no ( bday ‘𝑓)) ∪ (( bday
‘𝑗) +no ( bday ‘𝑘))))) → ((𝑔 ·s ℎ) ∈ No
∧ ((𝑖 <s 𝑗 ∧ 𝑘 <s 𝑓) → ((𝑖 ·s 𝑓) -s (𝑖 ·s 𝑘)) <s ((𝑗 ·s 𝑓) -s (𝑗 ·s 𝑘))))))) |
91 | | fveq2 6879 |
. . . . . . . . . . . . . 14
⊢ (𝑓 = 𝑙 → ( bday
‘𝑓) = ( bday ‘𝑙)) |
92 | 91 | oveq2d 7410 |
. . . . . . . . . . . . 13
⊢ (𝑓 = 𝑙 → (( bday
‘𝑗) +no ( bday ‘𝑓)) = (( bday
‘𝑗) +no ( bday ‘𝑙))) |
93 | 92 | uneq2d 4160 |
. . . . . . . . . . . 12
⊢ (𝑓 = 𝑙 → ((( bday
‘𝑖) +no ( bday ‘𝑘)) ∪ (( bday
‘𝑗) +no ( bday ‘𝑓))) = ((( bday
‘𝑖) +no ( bday ‘𝑘)) ∪ (( bday
‘𝑗) +no ( bday ‘𝑙)))) |
94 | 91 | oveq2d 7410 |
. . . . . . . . . . . . 13
⊢ (𝑓 = 𝑙 → (( bday
‘𝑖) +no ( bday ‘𝑓)) = (( bday
‘𝑖) +no ( bday ‘𝑙))) |
95 | 94 | uneq1d 4159 |
. . . . . . . . . . . 12
⊢ (𝑓 = 𝑙 → ((( bday
‘𝑖) +no ( bday ‘𝑓)) ∪ (( bday
‘𝑗) +no ( bday ‘𝑘))) = ((( bday
‘𝑖) +no ( bday ‘𝑙)) ∪ (( bday
‘𝑗) +no ( bday ‘𝑘)))) |
96 | 93, 95 | uneq12d 4161 |
. . . . . . . . . . 11
⊢ (𝑓 = 𝑙 → (((( bday
‘𝑖) +no ( bday ‘𝑘)) ∪ (( bday
‘𝑗) +no ( bday ‘𝑓))) ∪ ((( bday
‘𝑖) +no ( bday ‘𝑓)) ∪ (( bday
‘𝑗) +no ( bday ‘𝑘)))) = (((( bday
‘𝑖) +no ( bday ‘𝑘)) ∪ (( bday
‘𝑗) +no ( bday ‘𝑙))) ∪ ((( bday
‘𝑖) +no ( bday ‘𝑙)) ∪ (( bday
‘𝑗) +no ( bday ‘𝑘))))) |
97 | 96 | uneq2d 4160 |
. . . . . . . . . 10
⊢ (𝑓 = 𝑙 → ((( bday
‘𝑔) +no ( bday ‘ℎ)) ∪ (((( bday
‘𝑖) +no ( bday ‘𝑘)) ∪ (( bday
‘𝑗) +no ( bday ‘𝑓))) ∪ ((( bday
‘𝑖) +no ( bday ‘𝑓)) ∪ (( bday
‘𝑗) +no ( bday ‘𝑘))))) = ((( bday
‘𝑔) +no ( bday ‘ℎ)) ∪ (((( bday
‘𝑖) +no ( bday ‘𝑘)) ∪ (( bday
‘𝑗) +no ( bday ‘𝑙))) ∪ ((( bday
‘𝑖) +no ( bday ‘𝑙)) ∪ (( bday
‘𝑗) +no ( bday ‘𝑘)))))) |
98 | 97 | eqeq2d 2743 |
. . . . . . . . 9
⊢ (𝑓 = 𝑙 → (𝑥 = ((( bday
‘𝑔) +no ( bday ‘ℎ)) ∪ (((( bday
‘𝑖) +no ( bday ‘𝑘)) ∪ (( bday
‘𝑗) +no ( bday ‘𝑓))) ∪ ((( bday
‘𝑖) +no ( bday ‘𝑓)) ∪ (( bday
‘𝑗) +no ( bday ‘𝑘))))) ↔ 𝑥 = ((( bday
‘𝑔) +no ( bday ‘ℎ)) ∪ (((( bday
‘𝑖) +no ( bday ‘𝑘)) ∪ (( bday
‘𝑗) +no ( bday ‘𝑙))) ∪ ((( bday
‘𝑖) +no ( bday ‘𝑙)) ∪ (( bday
‘𝑗) +no ( bday ‘𝑘))))))) |
99 | | breq2 5146 |
. . . . . . . . . . . 12
⊢ (𝑓 = 𝑙 → (𝑘 <s 𝑓 ↔ 𝑘 <s 𝑙)) |
100 | 99 | anbi2d 629 |
. . . . . . . . . . 11
⊢ (𝑓 = 𝑙 → ((𝑖 <s 𝑗 ∧ 𝑘 <s 𝑓) ↔ (𝑖 <s 𝑗 ∧ 𝑘 <s 𝑙))) |
101 | | oveq2 7402 |
. . . . . . . . . . . . 13
⊢ (𝑓 = 𝑙 → (𝑖 ·s 𝑓) = (𝑖 ·s 𝑙)) |
102 | 101 | oveq1d 7409 |
. . . . . . . . . . . 12
⊢ (𝑓 = 𝑙 → ((𝑖 ·s 𝑓) -s (𝑖 ·s 𝑘)) = ((𝑖 ·s 𝑙) -s (𝑖 ·s 𝑘))) |
103 | | oveq2 7402 |
. . . . . . . . . . . . 13
⊢ (𝑓 = 𝑙 → (𝑗 ·s 𝑓) = (𝑗 ·s 𝑙)) |
104 | 103 | oveq1d 7409 |
. . . . . . . . . . . 12
⊢ (𝑓 = 𝑙 → ((𝑗 ·s 𝑓) -s (𝑗 ·s 𝑘)) = ((𝑗 ·s 𝑙) -s (𝑗 ·s 𝑘))) |
105 | 102, 104 | breq12d 5155 |
. . . . . . . . . . 11
⊢ (𝑓 = 𝑙 → (((𝑖 ·s 𝑓) -s (𝑖 ·s 𝑘)) <s ((𝑗 ·s 𝑓) -s (𝑗 ·s 𝑘)) ↔ ((𝑖 ·s 𝑙) -s (𝑖 ·s 𝑘)) <s ((𝑗 ·s 𝑙) -s (𝑗 ·s 𝑘)))) |
106 | 100, 105 | imbi12d 344 |
. . . . . . . . . 10
⊢ (𝑓 = 𝑙 → (((𝑖 <s 𝑗 ∧ 𝑘 <s 𝑓) → ((𝑖 ·s 𝑓) -s (𝑖 ·s 𝑘)) <s ((𝑗 ·s 𝑓) -s (𝑗 ·s 𝑘))) ↔ ((𝑖 <s 𝑗 ∧ 𝑘 <s 𝑙) → ((𝑖 ·s 𝑙) -s (𝑖 ·s 𝑘)) <s ((𝑗 ·s 𝑙) -s (𝑗 ·s 𝑘))))) |
107 | 106 | anbi2d 629 |
. . . . . . . . 9
⊢ (𝑓 = 𝑙 → (((𝑔 ·s ℎ) ∈ No
∧ ((𝑖 <s 𝑗 ∧ 𝑘 <s 𝑓) → ((𝑖 ·s 𝑓) -s (𝑖 ·s 𝑘)) <s ((𝑗 ·s 𝑓) -s (𝑗 ·s 𝑘)))) ↔ ((𝑔 ·s ℎ) ∈ No
∧ ((𝑖 <s 𝑗 ∧ 𝑘 <s 𝑙) → ((𝑖 ·s 𝑙) -s (𝑖 ·s 𝑘)) <s ((𝑗 ·s 𝑙) -s (𝑗 ·s 𝑘)))))) |
108 | 98, 107 | imbi12d 344 |
. . . . . . . 8
⊢ (𝑓 = 𝑙 → ((𝑥 = ((( bday
‘𝑔) +no ( bday ‘ℎ)) ∪ (((( bday
‘𝑖) +no ( bday ‘𝑘)) ∪ (( bday
‘𝑗) +no ( bday ‘𝑓))) ∪ ((( bday
‘𝑖) +no ( bday ‘𝑓)) ∪ (( bday
‘𝑗) +no ( bday ‘𝑘))))) → ((𝑔 ·s ℎ) ∈ No
∧ ((𝑖 <s 𝑗 ∧ 𝑘 <s 𝑓) → ((𝑖 ·s 𝑓) -s (𝑖 ·s 𝑘)) <s ((𝑗 ·s 𝑓) -s (𝑗 ·s 𝑘))))) ↔ (𝑥 = ((( bday
‘𝑔) +no ( bday ‘ℎ)) ∪ (((( bday
‘𝑖) +no ( bday ‘𝑘)) ∪ (( bday
‘𝑗) +no ( bday ‘𝑙))) ∪ ((( bday
‘𝑖) +no ( bday ‘𝑙)) ∪ (( bday
‘𝑗) +no ( bday ‘𝑘))))) → ((𝑔 ·s ℎ) ∈ No
∧ ((𝑖 <s 𝑗 ∧ 𝑘 <s 𝑙) → ((𝑖 ·s 𝑙) -s (𝑖 ·s 𝑘)) <s ((𝑗 ·s 𝑙) -s (𝑗 ·s 𝑘))))))) |
109 | 30, 38, 55, 72, 90, 108 | cbvral6vw 3242 |
. . . . . . 7
⊢
(∀𝑎 ∈
No ∀𝑏 ∈ No
∀𝑐 ∈ No ∀𝑑 ∈ No
∀𝑒 ∈ No ∀𝑓 ∈ No
(𝑥 = ((( bday ‘𝑎) +no ( bday
‘𝑏)) ∪
(((( bday ‘𝑐) +no ( bday
‘𝑒)) ∪
(( bday ‘𝑑) +no ( bday
‘𝑓))) ∪
((( bday ‘𝑐) +no ( bday
‘𝑓)) ∪
(( bday ‘𝑑) +no ( bday
‘𝑒))))) →
((𝑎 ·s
𝑏) ∈ No ∧ ((𝑐 <s 𝑑 ∧ 𝑒 <s 𝑓) → ((𝑐 ·s 𝑓) -s (𝑐 ·s 𝑒)) <s ((𝑑 ·s 𝑓) -s (𝑑 ·s 𝑒))))) ↔ ∀𝑔 ∈ No
∀ℎ ∈ No ∀𝑖 ∈ No
∀𝑗 ∈ No ∀𝑘 ∈ No
∀𝑙 ∈ No (𝑥 = ((( bday
‘𝑔) +no ( bday ‘ℎ)) ∪ (((( bday
‘𝑖) +no ( bday ‘𝑘)) ∪ (( bday
‘𝑗) +no ( bday ‘𝑙))) ∪ ((( bday
‘𝑖) +no ( bday ‘𝑙)) ∪ (( bday
‘𝑗) +no ( bday ‘𝑘))))) → ((𝑔 ·s ℎ) ∈ No
∧ ((𝑖 <s 𝑗 ∧ 𝑘 <s 𝑙) → ((𝑖 ·s 𝑙) -s (𝑖 ·s 𝑘)) <s ((𝑗 ·s 𝑙) -s (𝑗 ·s 𝑘)))))) |
110 | | eqeq1 2736 |
. . . . . . . . 9
⊢ (𝑥 = 𝑦 → (𝑥 = ((( bday
‘𝑎) +no ( bday ‘𝑏)) ∪ (((( bday
‘𝑐) +no ( bday ‘𝑒)) ∪ (( bday
‘𝑑) +no ( bday ‘𝑓))) ∪ ((( bday
‘𝑐) +no ( bday ‘𝑓)) ∪ (( bday
‘𝑑) +no ( bday ‘𝑒))))) ↔ 𝑦 = ((( bday
‘𝑎) +no ( bday ‘𝑏)) ∪ (((( bday
‘𝑐) +no ( bday ‘𝑒)) ∪ (( bday
‘𝑑) +no ( bday ‘𝑓))) ∪ ((( bday
‘𝑐) +no ( bday ‘𝑓)) ∪ (( bday
‘𝑑) +no ( bday ‘𝑒))))))) |
111 | 110 | imbi1d 341 |
. . . . . . . 8
⊢ (𝑥 = 𝑦 → ((𝑥 = ((( bday
‘𝑎) +no ( bday ‘𝑏)) ∪ (((( bday
‘𝑐) +no ( bday ‘𝑒)) ∪ (( bday
‘𝑑) +no ( bday ‘𝑓))) ∪ ((( bday
‘𝑐) +no ( bday ‘𝑓)) ∪ (( bday
‘𝑑) +no ( bday ‘𝑒))))) → ((𝑎 ·s 𝑏) ∈ No
∧ ((𝑐 <s 𝑑 ∧ 𝑒 <s 𝑓) → ((𝑐 ·s 𝑓) -s (𝑐 ·s 𝑒)) <s ((𝑑 ·s 𝑓) -s (𝑑 ·s 𝑒))))) ↔ (𝑦 = ((( bday
‘𝑎) +no ( bday ‘𝑏)) ∪ (((( bday
‘𝑐) +no ( bday ‘𝑒)) ∪ (( bday
‘𝑑) +no ( bday ‘𝑓))) ∪ ((( bday
‘𝑐) +no ( bday ‘𝑓)) ∪ (( bday
‘𝑑) +no ( bday ‘𝑒))))) → ((𝑎 ·s 𝑏) ∈ No
∧ ((𝑐 <s 𝑑 ∧ 𝑒 <s 𝑓) → ((𝑐 ·s 𝑓) -s (𝑐 ·s 𝑒)) <s ((𝑑 ·s 𝑓) -s (𝑑 ·s 𝑒))))))) |
112 | 111 | 6ralbidv 3223 |
. . . . . . 7
⊢ (𝑥 = 𝑦 → (∀𝑎 ∈ No
∀𝑏 ∈ No ∀𝑐 ∈ No
∀𝑑 ∈ No ∀𝑒 ∈ No
∀𝑓 ∈ No (𝑥 = ((( bday
‘𝑎) +no ( bday ‘𝑏)) ∪ (((( bday
‘𝑐) +no ( bday ‘𝑒)) ∪ (( bday
‘𝑑) +no ( bday ‘𝑓))) ∪ ((( bday
‘𝑐) +no ( bday ‘𝑓)) ∪ (( bday
‘𝑑) +no ( bday ‘𝑒))))) → ((𝑎 ·s 𝑏) ∈ No
∧ ((𝑐 <s 𝑑 ∧ 𝑒 <s 𝑓) → ((𝑐 ·s 𝑓) -s (𝑐 ·s 𝑒)) <s ((𝑑 ·s 𝑓) -s (𝑑 ·s 𝑒))))) ↔ ∀𝑎 ∈ No
∀𝑏 ∈ No ∀𝑐 ∈ No
∀𝑑 ∈ No ∀𝑒 ∈ No
∀𝑓 ∈ No (𝑦 = ((( bday
‘𝑎) +no ( bday ‘𝑏)) ∪ (((( bday
‘𝑐) +no ( bday ‘𝑒)) ∪ (( bday
‘𝑑) +no ( bday ‘𝑓))) ∪ ((( bday
‘𝑐) +no ( bday ‘𝑓)) ∪ (( bday
‘𝑑) +no ( bday ‘𝑒))))) → ((𝑎 ·s 𝑏) ∈ No
∧ ((𝑐 <s 𝑑 ∧ 𝑒 <s 𝑓) → ((𝑐 ·s 𝑓) -s (𝑐 ·s 𝑒)) <s ((𝑑 ·s 𝑓) -s (𝑑 ·s 𝑒))))))) |
113 | 109, 112 | bitr3id 284 |
. . . . . 6
⊢ (𝑥 = 𝑦 → (∀𝑔 ∈ No
∀ℎ ∈ No ∀𝑖 ∈ No
∀𝑗 ∈ No ∀𝑘 ∈ No
∀𝑙 ∈ No (𝑥 = ((( bday
‘𝑔) +no ( bday ‘ℎ)) ∪ (((( bday
‘𝑖) +no ( bday ‘𝑘)) ∪ (( bday
‘𝑗) +no ( bday ‘𝑙))) ∪ ((( bday
‘𝑖) +no ( bday ‘𝑙)) ∪ (( bday
‘𝑗) +no ( bday ‘𝑘))))) → ((𝑔 ·s ℎ) ∈ No
∧ ((𝑖 <s 𝑗 ∧ 𝑘 <s 𝑙) → ((𝑖 ·s 𝑙) -s (𝑖 ·s 𝑘)) <s ((𝑗 ·s 𝑙) -s (𝑗 ·s 𝑘))))) ↔ ∀𝑎 ∈ No
∀𝑏 ∈ No ∀𝑐 ∈ No
∀𝑑 ∈ No ∀𝑒 ∈ No
∀𝑓 ∈ No (𝑦 = ((( bday
‘𝑎) +no ( bday ‘𝑏)) ∪ (((( bday
‘𝑐) +no ( bday ‘𝑒)) ∪ (( bday
‘𝑑) +no ( bday ‘𝑓))) ∪ ((( bday
‘𝑐) +no ( bday ‘𝑓)) ∪ (( bday
‘𝑑) +no ( bday ‘𝑒))))) → ((𝑎 ·s 𝑏) ∈ No
∧ ((𝑐 <s 𝑑 ∧ 𝑒 <s 𝑓) → ((𝑐 ·s 𝑓) -s (𝑐 ·s 𝑒)) <s ((𝑑 ·s 𝑓) -s (𝑑 ·s 𝑒))))))) |
114 | | raleq 3322 |
. . . . . . . . . . . . . 14
⊢ (𝑥 = (((
bday ‘𝑔) +no
( bday ‘ℎ)) ∪ (((( bday
‘𝑖) +no ( bday ‘𝑘)) ∪ (( bday
‘𝑗) +no ( bday ‘𝑙))) ∪ ((( bday
‘𝑖) +no ( bday ‘𝑙)) ∪ (( bday
‘𝑗) +no ( bday ‘𝑘))))) → (∀𝑦 ∈ 𝑥 ∀𝑎 ∈ No
∀𝑏 ∈ No ∀𝑐 ∈ No
∀𝑑 ∈ No ∀𝑒 ∈ No
∀𝑓 ∈ No (𝑦 = ((( bday
‘𝑎) +no ( bday ‘𝑏)) ∪ (((( bday
‘𝑐) +no ( bday ‘𝑒)) ∪ (( bday
‘𝑑) +no ( bday ‘𝑓))) ∪ ((( bday
‘𝑐) +no ( bday ‘𝑓)) ∪ (( bday
‘𝑑) +no ( bday ‘𝑒))))) → ((𝑎 ·s 𝑏) ∈ No
∧ ((𝑐 <s 𝑑 ∧ 𝑒 <s 𝑓) → ((𝑐 ·s 𝑓) -s (𝑐 ·s 𝑒)) <s ((𝑑 ·s 𝑓) -s (𝑑 ·s 𝑒))))) ↔ ∀𝑦 ∈ ((( bday
‘𝑔) +no ( bday ‘ℎ)) ∪ (((( bday
‘𝑖) +no ( bday ‘𝑘)) ∪ (( bday
‘𝑗) +no ( bday ‘𝑙))) ∪ ((( bday
‘𝑖) +no ( bday ‘𝑙)) ∪ (( bday
‘𝑗) +no ( bday ‘𝑘)))))∀𝑎 ∈ No
∀𝑏 ∈ No ∀𝑐 ∈ No
∀𝑑 ∈ No ∀𝑒 ∈ No
∀𝑓 ∈ No (𝑦 = ((( bday
‘𝑎) +no ( bday ‘𝑏)) ∪ (((( bday
‘𝑐) +no ( bday ‘𝑒)) ∪ (( bday
‘𝑑) +no ( bday ‘𝑓))) ∪ ((( bday
‘𝑐) +no ( bday ‘𝑓)) ∪ (( bday
‘𝑑) +no ( bday ‘𝑒))))) → ((𝑎 ·s 𝑏) ∈ No
∧ ((𝑐 <s 𝑑 ∧ 𝑒 <s 𝑓) → ((𝑐 ·s 𝑓) -s (𝑐 ·s 𝑒)) <s ((𝑑 ·s 𝑓) -s (𝑑 ·s 𝑒))))))) |
115 | | ralrot3 3290 |
. . . . . . . . . . . . . . 15
⊢
(∀𝑎 ∈
No ∀𝑏 ∈ No
∀𝑦 ∈ ((( bday ‘𝑔) +no ( bday
‘ℎ)) ∪
(((( bday ‘𝑖) +no ( bday
‘𝑘)) ∪
(( bday ‘𝑗) +no ( bday
‘𝑙))) ∪
((( bday ‘𝑖) +no ( bday
‘𝑙)) ∪
(( bday ‘𝑗) +no ( bday
‘𝑘)))))∀𝑐 ∈ No
∀𝑑 ∈ No ∀𝑒 ∈ No
∀𝑓 ∈ No (𝑦 = ((( bday
‘𝑎) +no ( bday ‘𝑏)) ∪ (((( bday
‘𝑐) +no ( bday ‘𝑒)) ∪ (( bday
‘𝑑) +no ( bday ‘𝑓))) ∪ ((( bday
‘𝑐) +no ( bday ‘𝑓)) ∪ (( bday
‘𝑑) +no ( bday ‘𝑒))))) → ((𝑎 ·s 𝑏) ∈ No
∧ ((𝑐 <s 𝑑 ∧ 𝑒 <s 𝑓) → ((𝑐 ·s 𝑓) -s (𝑐 ·s 𝑒)) <s ((𝑑 ·s 𝑓) -s (𝑑 ·s 𝑒))))) ↔ ∀𝑦 ∈ ((( bday
‘𝑔) +no ( bday ‘ℎ)) ∪ (((( bday
‘𝑖) +no ( bday ‘𝑘)) ∪ (( bday
‘𝑗) +no ( bday ‘𝑙))) ∪ ((( bday
‘𝑖) +no ( bday ‘𝑙)) ∪ (( bday
‘𝑗) +no ( bday ‘𝑘)))))∀𝑎 ∈ No
∀𝑏 ∈ No ∀𝑐 ∈ No
∀𝑑 ∈ No ∀𝑒 ∈ No
∀𝑓 ∈ No (𝑦 = ((( bday
‘𝑎) +no ( bday ‘𝑏)) ∪ (((( bday
‘𝑐) +no ( bday ‘𝑒)) ∪ (( bday
‘𝑑) +no ( bday ‘𝑓))) ∪ ((( bday
‘𝑐) +no ( bday ‘𝑓)) ∪ (( bday
‘𝑑) +no ( bday ‘𝑒))))) → ((𝑎 ·s 𝑏) ∈ No
∧ ((𝑐 <s 𝑑 ∧ 𝑒 <s 𝑓) → ((𝑐 ·s 𝑓) -s (𝑐 ·s 𝑒)) <s ((𝑑 ·s 𝑓) -s (𝑑 ·s 𝑒)))))) |
116 | | ralrot3 3290 |
. . . . . . . . . . . . . . . . 17
⊢
(∀𝑐 ∈
No ∀𝑑 ∈ No
∀𝑦 ∈ ((( bday ‘𝑔) +no ( bday
‘ℎ)) ∪
(((( bday ‘𝑖) +no ( bday
‘𝑘)) ∪
(( bday ‘𝑗) +no ( bday
‘𝑙))) ∪
((( bday ‘𝑖) +no ( bday
‘𝑙)) ∪
(( bday ‘𝑗) +no ( bday
‘𝑘)))))∀𝑒 ∈ No
∀𝑓 ∈ No (𝑦 = ((( bday
‘𝑎) +no ( bday ‘𝑏)) ∪ (((( bday
‘𝑐) +no ( bday ‘𝑒)) ∪ (( bday
‘𝑑) +no ( bday ‘𝑓))) ∪ ((( bday
‘𝑐) +no ( bday ‘𝑓)) ∪ (( bday
‘𝑑) +no ( bday ‘𝑒))))) → ((𝑎 ·s 𝑏) ∈ No
∧ ((𝑐 <s 𝑑 ∧ 𝑒 <s 𝑓) → ((𝑐 ·s 𝑓) -s (𝑐 ·s 𝑒)) <s ((𝑑 ·s 𝑓) -s (𝑑 ·s 𝑒))))) ↔ ∀𝑦 ∈ ((( bday
‘𝑔) +no ( bday ‘ℎ)) ∪ (((( bday
‘𝑖) +no ( bday ‘𝑘)) ∪ (( bday
‘𝑗) +no ( bday ‘𝑙))) ∪ ((( bday
‘𝑖) +no ( bday ‘𝑙)) ∪ (( bday
‘𝑗) +no ( bday ‘𝑘)))))∀𝑐 ∈ No
∀𝑑 ∈ No ∀𝑒 ∈ No
∀𝑓 ∈ No (𝑦 = ((( bday
‘𝑎) +no ( bday ‘𝑏)) ∪ (((( bday
‘𝑐) +no ( bday ‘𝑒)) ∪ (( bday
‘𝑑) +no ( bday ‘𝑓))) ∪ ((( bday
‘𝑐) +no ( bday ‘𝑓)) ∪ (( bday
‘𝑑) +no ( bday ‘𝑒))))) → ((𝑎 ·s 𝑏) ∈ No
∧ ((𝑐 <s 𝑑 ∧ 𝑒 <s 𝑓) → ((𝑐 ·s 𝑓) -s (𝑐 ·s 𝑒)) <s ((𝑑 ·s 𝑓) -s (𝑑 ·s 𝑒)))))) |
117 | | ralrot3 3290 |
. . . . . . . . . . . . . . . . . . 19
⊢
(∀𝑒 ∈
No ∀𝑓 ∈ No
∀𝑦 ∈ ((( bday ‘𝑔) +no ( bday
‘ℎ)) ∪
(((( bday ‘𝑖) +no ( bday
‘𝑘)) ∪
(( bday ‘𝑗) +no ( bday
‘𝑙))) ∪
((( bday ‘𝑖) +no ( bday
‘𝑙)) ∪
(( bday ‘𝑗) +no ( bday
‘𝑘)))))(𝑦 = (((
bday ‘𝑎) +no
( bday ‘𝑏)) ∪ (((( bday
‘𝑐) +no ( bday ‘𝑒)) ∪ (( bday
‘𝑑) +no ( bday ‘𝑓))) ∪ ((( bday
‘𝑐) +no ( bday ‘𝑓)) ∪ (( bday
‘𝑑) +no ( bday ‘𝑒))))) → ((𝑎 ·s 𝑏) ∈ No
∧ ((𝑐 <s 𝑑 ∧ 𝑒 <s 𝑓) → ((𝑐 ·s 𝑓) -s (𝑐 ·s 𝑒)) <s ((𝑑 ·s 𝑓) -s (𝑑 ·s 𝑒))))) ↔ ∀𝑦 ∈ ((( bday
‘𝑔) +no ( bday ‘ℎ)) ∪ (((( bday
‘𝑖) +no ( bday ‘𝑘)) ∪ (( bday
‘𝑗) +no ( bday ‘𝑙))) ∪ ((( bday
‘𝑖) +no ( bday ‘𝑙)) ∪ (( bday
‘𝑗) +no ( bday ‘𝑘)))))∀𝑒 ∈ No
∀𝑓 ∈ No (𝑦 = ((( bday
‘𝑎) +no ( bday ‘𝑏)) ∪ (((( bday
‘𝑐) +no ( bday ‘𝑒)) ∪ (( bday
‘𝑑) +no ( bday ‘𝑓))) ∪ ((( bday
‘𝑐) +no ( bday ‘𝑓)) ∪ (( bday
‘𝑑) +no ( bday ‘𝑒))))) → ((𝑎 ·s 𝑏) ∈ No
∧ ((𝑐 <s 𝑑 ∧ 𝑒 <s 𝑓) → ((𝑐 ·s 𝑓) -s (𝑐 ·s 𝑒)) <s ((𝑑 ·s 𝑓) -s (𝑑 ·s 𝑒)))))) |
118 | | r19.23v 3182 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
(∀𝑦 ∈
((( bday ‘𝑔) +no ( bday
‘ℎ)) ∪
(((( bday ‘𝑖) +no ( bday
‘𝑘)) ∪
(( bday ‘𝑗) +no ( bday
‘𝑙))) ∪
((( bday ‘𝑖) +no ( bday
‘𝑙)) ∪
(( bday ‘𝑗) +no ( bday
‘𝑘)))))(𝑦 = (((
bday ‘𝑎) +no
( bday ‘𝑏)) ∪ (((( bday
‘𝑐) +no ( bday ‘𝑒)) ∪ (( bday
‘𝑑) +no ( bday ‘𝑓))) ∪ ((( bday
‘𝑐) +no ( bday ‘𝑓)) ∪ (( bday
‘𝑑) +no ( bday ‘𝑒))))) → ((𝑎 ·s 𝑏) ∈ No
∧ ((𝑐 <s 𝑑 ∧ 𝑒 <s 𝑓) → ((𝑐 ·s 𝑓) -s (𝑐 ·s 𝑒)) <s ((𝑑 ·s 𝑓) -s (𝑑 ·s 𝑒))))) ↔ (∃𝑦 ∈ ((( bday
‘𝑔) +no ( bday ‘ℎ)) ∪ (((( bday
‘𝑖) +no ( bday ‘𝑘)) ∪ (( bday
‘𝑗) +no ( bday ‘𝑙))) ∪ ((( bday
‘𝑖) +no ( bday ‘𝑙)) ∪ (( bday
‘𝑗) +no ( bday ‘𝑘)))))𝑦 = ((( bday
‘𝑎) +no ( bday ‘𝑏)) ∪ (((( bday
‘𝑐) +no ( bday ‘𝑒)) ∪ (( bday
‘𝑑) +no ( bday ‘𝑓))) ∪ ((( bday
‘𝑐) +no ( bday ‘𝑓)) ∪ (( bday
‘𝑑) +no ( bday ‘𝑒))))) → ((𝑎 ·s 𝑏) ∈ No
∧ ((𝑐 <s 𝑑 ∧ 𝑒 <s 𝑓) → ((𝑐 ·s 𝑓) -s (𝑐 ·s 𝑒)) <s ((𝑑 ·s 𝑓) -s (𝑑 ·s 𝑒)))))) |
119 | | risset 3230 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (((( bday ‘𝑎) +no ( bday
‘𝑏)) ∪
(((( bday ‘𝑐) +no ( bday
‘𝑒)) ∪
(( bday ‘𝑑) +no ( bday
‘𝑓))) ∪
((( bday ‘𝑐) +no ( bday
‘𝑓)) ∪
(( bday ‘𝑑) +no ( bday
‘𝑒))))) ∈
((( bday ‘𝑔) +no ( bday
‘ℎ)) ∪
(((( bday ‘𝑖) +no ( bday
‘𝑘)) ∪
(( bday ‘𝑗) +no ( bday
‘𝑙))) ∪
((( bday ‘𝑖) +no ( bday
‘𝑙)) ∪
(( bday ‘𝑗) +no ( bday
‘𝑘))))) ↔
∃𝑦 ∈ ((( bday ‘𝑔) +no ( bday
‘ℎ)) ∪
(((( bday ‘𝑖) +no ( bday
‘𝑘)) ∪
(( bday ‘𝑗) +no ( bday
‘𝑙))) ∪
((( bday ‘𝑖) +no ( bday
‘𝑙)) ∪
(( bday ‘𝑗) +no ( bday
‘𝑘)))))𝑦 = (((
bday ‘𝑎) +no
( bday ‘𝑏)) ∪ (((( bday
‘𝑐) +no ( bday ‘𝑒)) ∪ (( bday
‘𝑑) +no ( bday ‘𝑓))) ∪ ((( bday
‘𝑐) +no ( bday ‘𝑓)) ∪ (( bday
‘𝑑) +no ( bday ‘𝑒)))))) |
120 | 119 | imbi1i 349 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
((((( bday ‘𝑎) +no ( bday
‘𝑏)) ∪
(((( bday ‘𝑐) +no ( bday
‘𝑒)) ∪
(( bday ‘𝑑) +no ( bday
‘𝑓))) ∪
((( bday ‘𝑐) +no ( bday
‘𝑓)) ∪
(( bday ‘𝑑) +no ( bday
‘𝑒))))) ∈
((( bday ‘𝑔) +no ( bday
‘ℎ)) ∪
(((( bday ‘𝑖) +no ( bday
‘𝑘)) ∪
(( bday ‘𝑗) +no ( bday
‘𝑙))) ∪
((( bday ‘𝑖) +no ( bday
‘𝑙)) ∪
(( bday ‘𝑗) +no ( bday
‘𝑘))))) →
((𝑎 ·s
𝑏) ∈ No ∧ ((𝑐 <s 𝑑 ∧ 𝑒 <s 𝑓) → ((𝑐 ·s 𝑓) -s (𝑐 ·s 𝑒)) <s ((𝑑 ·s 𝑓) -s (𝑑 ·s 𝑒))))) ↔ (∃𝑦 ∈ ((( bday
‘𝑔) +no ( bday ‘ℎ)) ∪ (((( bday
‘𝑖) +no ( bday ‘𝑘)) ∪ (( bday
‘𝑗) +no ( bday ‘𝑙))) ∪ ((( bday
‘𝑖) +no ( bday ‘𝑙)) ∪ (( bday
‘𝑗) +no ( bday ‘𝑘)))))𝑦 = ((( bday
‘𝑎) +no ( bday ‘𝑏)) ∪ (((( bday
‘𝑐) +no ( bday ‘𝑒)) ∪ (( bday
‘𝑑) +no ( bday ‘𝑓))) ∪ ((( bday
‘𝑐) +no ( bday ‘𝑓)) ∪ (( bday
‘𝑑) +no ( bday ‘𝑒))))) → ((𝑎 ·s 𝑏) ∈ No
∧ ((𝑐 <s 𝑑 ∧ 𝑒 <s 𝑓) → ((𝑐 ·s 𝑓) -s (𝑐 ·s 𝑒)) <s ((𝑑 ·s 𝑓) -s (𝑑 ·s 𝑒)))))) |
121 | 118, 120 | bitr4i 277 |
. . . . . . . . . . . . . . . . . . . 20
⊢
(∀𝑦 ∈
((( bday ‘𝑔) +no ( bday
‘ℎ)) ∪
(((( bday ‘𝑖) +no ( bday
‘𝑘)) ∪
(( bday ‘𝑗) +no ( bday
‘𝑙))) ∪
((( bday ‘𝑖) +no ( bday
‘𝑙)) ∪
(( bday ‘𝑗) +no ( bday
‘𝑘)))))(𝑦 = (((
bday ‘𝑎) +no
( bday ‘𝑏)) ∪ (((( bday
‘𝑐) +no ( bday ‘𝑒)) ∪ (( bday
‘𝑑) +no ( bday ‘𝑓))) ∪ ((( bday
‘𝑐) +no ( bday ‘𝑓)) ∪ (( bday
‘𝑑) +no ( bday ‘𝑒))))) → ((𝑎 ·s 𝑏) ∈ No
∧ ((𝑐 <s 𝑑 ∧ 𝑒 <s 𝑓) → ((𝑐 ·s 𝑓) -s (𝑐 ·s 𝑒)) <s ((𝑑 ·s 𝑓) -s (𝑑 ·s 𝑒))))) ↔ (((( bday
‘𝑎) +no ( bday ‘𝑏)) ∪ (((( bday
‘𝑐) +no ( bday ‘𝑒)) ∪ (( bday
‘𝑑) +no ( bday ‘𝑓))) ∪ ((( bday
‘𝑐) +no ( bday ‘𝑓)) ∪ (( bday
‘𝑑) +no ( bday ‘𝑒))))) ∈ ((( bday
‘𝑔) +no ( bday ‘ℎ)) ∪ (((( bday
‘𝑖) +no ( bday ‘𝑘)) ∪ (( bday
‘𝑗) +no ( bday ‘𝑙))) ∪ ((( bday
‘𝑖) +no ( bday ‘𝑙)) ∪ (( bday
‘𝑗) +no ( bday ‘𝑘))))) → ((𝑎 ·s 𝑏) ∈ No
∧ ((𝑐 <s 𝑑 ∧ 𝑒 <s 𝑓) → ((𝑐 ·s 𝑓) -s (𝑐 ·s 𝑒)) <s ((𝑑 ·s 𝑓) -s (𝑑 ·s 𝑒)))))) |
122 | 121 | 2ralbii 3128 |
. . . . . . . . . . . . . . . . . . 19
⊢
(∀𝑒 ∈
No ∀𝑓 ∈ No
∀𝑦 ∈ ((( bday ‘𝑔) +no ( bday
‘ℎ)) ∪
(((( bday ‘𝑖) +no ( bday
‘𝑘)) ∪
(( bday ‘𝑗) +no ( bday
‘𝑙))) ∪
((( bday ‘𝑖) +no ( bday
‘𝑙)) ∪
(( bday ‘𝑗) +no ( bday
‘𝑘)))))(𝑦 = (((
bday ‘𝑎) +no
( bday ‘𝑏)) ∪ (((( bday
‘𝑐) +no ( bday ‘𝑒)) ∪ (( bday
‘𝑑) +no ( bday ‘𝑓))) ∪ ((( bday
‘𝑐) +no ( bday ‘𝑓)) ∪ (( bday
‘𝑑) +no ( bday ‘𝑒))))) → ((𝑎 ·s 𝑏) ∈ No
∧ ((𝑐 <s 𝑑 ∧ 𝑒 <s 𝑓) → ((𝑐 ·s 𝑓) -s (𝑐 ·s 𝑒)) <s ((𝑑 ·s 𝑓) -s (𝑑 ·s 𝑒))))) ↔ ∀𝑒 ∈ No
∀𝑓 ∈ No (((( bday ‘𝑎) +no (
bday ‘𝑏))
∪ (((( bday ‘𝑐) +no ( bday
‘𝑒)) ∪
(( bday ‘𝑑) +no ( bday
‘𝑓))) ∪
((( bday ‘𝑐) +no ( bday
‘𝑓)) ∪
(( bday ‘𝑑) +no ( bday
‘𝑒))))) ∈
((( bday ‘𝑔) +no ( bday
‘ℎ)) ∪
(((( bday ‘𝑖) +no ( bday
‘𝑘)) ∪
(( bday ‘𝑗) +no ( bday
‘𝑙))) ∪
((( bday ‘𝑖) +no ( bday
‘𝑙)) ∪
(( bday ‘𝑗) +no ( bday
‘𝑘))))) →
((𝑎 ·s
𝑏) ∈ No ∧ ((𝑐 <s 𝑑 ∧ 𝑒 <s 𝑓) → ((𝑐 ·s 𝑓) -s (𝑐 ·s 𝑒)) <s ((𝑑 ·s 𝑓) -s (𝑑 ·s 𝑒)))))) |
123 | 117, 122 | bitr3i 276 |
. . . . . . . . . . . . . . . . . 18
⊢
(∀𝑦 ∈
((( bday ‘𝑔) +no ( bday
‘ℎ)) ∪
(((( bday ‘𝑖) +no ( bday
‘𝑘)) ∪
(( bday ‘𝑗) +no ( bday
‘𝑙))) ∪
((( bday ‘𝑖) +no ( bday
‘𝑙)) ∪
(( bday ‘𝑗) +no ( bday
‘𝑘)))))∀𝑒 ∈ No
∀𝑓 ∈ No (𝑦 = ((( bday
‘𝑎) +no ( bday ‘𝑏)) ∪ (((( bday
‘𝑐) +no ( bday ‘𝑒)) ∪ (( bday
‘𝑑) +no ( bday ‘𝑓))) ∪ ((( bday
‘𝑐) +no ( bday ‘𝑓)) ∪ (( bday
‘𝑑) +no ( bday ‘𝑒))))) → ((𝑎 ·s 𝑏) ∈ No
∧ ((𝑐 <s 𝑑 ∧ 𝑒 <s 𝑓) → ((𝑐 ·s 𝑓) -s (𝑐 ·s 𝑒)) <s ((𝑑 ·s 𝑓) -s (𝑑 ·s 𝑒))))) ↔ ∀𝑒 ∈ No
∀𝑓 ∈ No (((( bday ‘𝑎) +no (
bday ‘𝑏))
∪ (((( bday ‘𝑐) +no ( bday
‘𝑒)) ∪
(( bday ‘𝑑) +no ( bday
‘𝑓))) ∪
((( bday ‘𝑐) +no ( bday
‘𝑓)) ∪
(( bday ‘𝑑) +no ( bday
‘𝑒))))) ∈
((( bday ‘𝑔) +no ( bday
‘ℎ)) ∪
(((( bday ‘𝑖) +no ( bday
‘𝑘)) ∪
(( bday ‘𝑗) +no ( bday
‘𝑙))) ∪
((( bday ‘𝑖) +no ( bday
‘𝑙)) ∪
(( bday ‘𝑗) +no ( bday
‘𝑘))))) →
((𝑎 ·s
𝑏) ∈ No ∧ ((𝑐 <s 𝑑 ∧ 𝑒 <s 𝑓) → ((𝑐 ·s 𝑓) -s (𝑐 ·s 𝑒)) <s ((𝑑 ·s 𝑓) -s (𝑑 ·s 𝑒)))))) |
124 | 123 | 2ralbii 3128 |
. . . . . . . . . . . . . . . . 17
⊢
(∀𝑐 ∈
No ∀𝑑 ∈ No
∀𝑦 ∈ ((( bday ‘𝑔) +no ( bday
‘ℎ)) ∪
(((( bday ‘𝑖) +no ( bday
‘𝑘)) ∪
(( bday ‘𝑗) +no ( bday
‘𝑙))) ∪
((( bday ‘𝑖) +no ( bday
‘𝑙)) ∪
(( bday ‘𝑗) +no ( bday
‘𝑘)))))∀𝑒 ∈ No
∀𝑓 ∈ No (𝑦 = ((( bday
‘𝑎) +no ( bday ‘𝑏)) ∪ (((( bday
‘𝑐) +no ( bday ‘𝑒)) ∪ (( bday
‘𝑑) +no ( bday ‘𝑓))) ∪ ((( bday
‘𝑐) +no ( bday ‘𝑓)) ∪ (( bday
‘𝑑) +no ( bday ‘𝑒))))) → ((𝑎 ·s 𝑏) ∈ No
∧ ((𝑐 <s 𝑑 ∧ 𝑒 <s 𝑓) → ((𝑐 ·s 𝑓) -s (𝑐 ·s 𝑒)) <s ((𝑑 ·s 𝑓) -s (𝑑 ·s 𝑒))))) ↔ ∀𝑐 ∈ No
∀𝑑 ∈ No ∀𝑒 ∈ No
∀𝑓 ∈ No (((( bday ‘𝑎) +no (
bday ‘𝑏))
∪ (((( bday ‘𝑐) +no ( bday
‘𝑒)) ∪
(( bday ‘𝑑) +no ( bday
‘𝑓))) ∪
((( bday ‘𝑐) +no ( bday
‘𝑓)) ∪
(( bday ‘𝑑) +no ( bday
‘𝑒))))) ∈
((( bday ‘𝑔) +no ( bday
‘ℎ)) ∪
(((( bday ‘𝑖) +no ( bday
‘𝑘)) ∪
(( bday ‘𝑗) +no ( bday
‘𝑙))) ∪
((( bday ‘𝑖) +no ( bday
‘𝑙)) ∪
(( bday ‘𝑗) +no ( bday
‘𝑘))))) →
((𝑎 ·s
𝑏) ∈ No ∧ ((𝑐 <s 𝑑 ∧ 𝑒 <s 𝑓) → ((𝑐 ·s 𝑓) -s (𝑐 ·s 𝑒)) <s ((𝑑 ·s 𝑓) -s (𝑑 ·s 𝑒)))))) |
125 | 116, 124 | bitr3i 276 |
. . . . . . . . . . . . . . . 16
⊢
(∀𝑦 ∈
((( bday ‘𝑔) +no ( bday
‘ℎ)) ∪
(((( bday ‘𝑖) +no ( bday
‘𝑘)) ∪
(( bday ‘𝑗) +no ( bday
‘𝑙))) ∪
((( bday ‘𝑖) +no ( bday
‘𝑙)) ∪
(( bday ‘𝑗) +no ( bday
‘𝑘)))))∀𝑐 ∈ No
∀𝑑 ∈ No ∀𝑒 ∈ No
∀𝑓 ∈ No (𝑦 = ((( bday
‘𝑎) +no ( bday ‘𝑏)) ∪ (((( bday
‘𝑐) +no ( bday ‘𝑒)) ∪ (( bday
‘𝑑) +no ( bday ‘𝑓))) ∪ ((( bday
‘𝑐) +no ( bday ‘𝑓)) ∪ (( bday
‘𝑑) +no ( bday ‘𝑒))))) → ((𝑎 ·s 𝑏) ∈ No
∧ ((𝑐 <s 𝑑 ∧ 𝑒 <s 𝑓) → ((𝑐 ·s 𝑓) -s (𝑐 ·s 𝑒)) <s ((𝑑 ·s 𝑓) -s (𝑑 ·s 𝑒))))) ↔ ∀𝑐 ∈ No
∀𝑑 ∈ No ∀𝑒 ∈ No
∀𝑓 ∈ No (((( bday ‘𝑎) +no (
bday ‘𝑏))
∪ (((( bday ‘𝑐) +no ( bday
‘𝑒)) ∪
(( bday ‘𝑑) +no ( bday
‘𝑓))) ∪
((( bday ‘𝑐) +no ( bday
‘𝑓)) ∪
(( bday ‘𝑑) +no ( bday
‘𝑒))))) ∈
((( bday ‘𝑔) +no ( bday
‘ℎ)) ∪
(((( bday ‘𝑖) +no ( bday
‘𝑘)) ∪
(( bday ‘𝑗) +no ( bday
‘𝑙))) ∪
((( bday ‘𝑖) +no ( bday
‘𝑙)) ∪
(( bday ‘𝑗) +no ( bday
‘𝑘))))) →
((𝑎 ·s
𝑏) ∈ No ∧ ((𝑐 <s 𝑑 ∧ 𝑒 <s 𝑓) → ((𝑐 ·s 𝑓) -s (𝑐 ·s 𝑒)) <s ((𝑑 ·s 𝑓) -s (𝑑 ·s 𝑒)))))) |
126 | 125 | 2ralbii 3128 |
. . . . . . . . . . . . . . 15
⊢
(∀𝑎 ∈
No ∀𝑏 ∈ No
∀𝑦 ∈ ((( bday ‘𝑔) +no ( bday
‘ℎ)) ∪
(((( bday ‘𝑖) +no ( bday
‘𝑘)) ∪
(( bday ‘𝑗) +no ( bday
‘𝑙))) ∪
((( bday ‘𝑖) +no ( bday
‘𝑙)) ∪
(( bday ‘𝑗) +no ( bday
‘𝑘)))))∀𝑐 ∈ No
∀𝑑 ∈ No ∀𝑒 ∈ No
∀𝑓 ∈ No (𝑦 = ((( bday
‘𝑎) +no ( bday ‘𝑏)) ∪ (((( bday
‘𝑐) +no ( bday ‘𝑒)) ∪ (( bday
‘𝑑) +no ( bday ‘𝑓))) ∪ ((( bday
‘𝑐) +no ( bday ‘𝑓)) ∪ (( bday
‘𝑑) +no ( bday ‘𝑒))))) → ((𝑎 ·s 𝑏) ∈ No
∧ ((𝑐 <s 𝑑 ∧ 𝑒 <s 𝑓) → ((𝑐 ·s 𝑓) -s (𝑐 ·s 𝑒)) <s ((𝑑 ·s 𝑓) -s (𝑑 ·s 𝑒))))) ↔ ∀𝑎 ∈ No
∀𝑏 ∈ No ∀𝑐 ∈ No
∀𝑑 ∈ No ∀𝑒 ∈ No
∀𝑓 ∈ No (((( bday ‘𝑎) +no (
bday ‘𝑏))
∪ (((( bday ‘𝑐) +no ( bday
‘𝑒)) ∪
(( bday ‘𝑑) +no ( bday
‘𝑓))) ∪
((( bday ‘𝑐) +no ( bday
‘𝑓)) ∪
(( bday ‘𝑑) +no ( bday
‘𝑒))))) ∈
((( bday ‘𝑔) +no ( bday
‘ℎ)) ∪
(((( bday ‘𝑖) +no ( bday
‘𝑘)) ∪
(( bday ‘𝑗) +no ( bday
‘𝑙))) ∪
((( bday ‘𝑖) +no ( bday
‘𝑙)) ∪
(( bday ‘𝑗) +no ( bday
‘𝑘))))) →
((𝑎 ·s
𝑏) ∈ No ∧ ((𝑐 <s 𝑑 ∧ 𝑒 <s 𝑓) → ((𝑐 ·s 𝑓) -s (𝑐 ·s 𝑒)) <s ((𝑑 ·s 𝑓) -s (𝑑 ·s 𝑒)))))) |
127 | 115, 126 | bitr3i 276 |
. . . . . . . . . . . . . 14
⊢
(∀𝑦 ∈
((( bday ‘𝑔) +no ( bday
‘ℎ)) ∪
(((( bday ‘𝑖) +no ( bday
‘𝑘)) ∪
(( bday ‘𝑗) +no ( bday
‘𝑙))) ∪
((( bday ‘𝑖) +no ( bday
‘𝑙)) ∪
(( bday ‘𝑗) +no ( bday
‘𝑘)))))∀𝑎 ∈ No
∀𝑏 ∈ No ∀𝑐 ∈ No
∀𝑑 ∈ No ∀𝑒 ∈ No
∀𝑓 ∈ No (𝑦 = ((( bday
‘𝑎) +no ( bday ‘𝑏)) ∪ (((( bday
‘𝑐) +no ( bday ‘𝑒)) ∪ (( bday
‘𝑑) +no ( bday ‘𝑓))) ∪ ((( bday
‘𝑐) +no ( bday ‘𝑓)) ∪ (( bday
‘𝑑) +no ( bday ‘𝑒))))) → ((𝑎 ·s 𝑏) ∈ No
∧ ((𝑐 <s 𝑑 ∧ 𝑒 <s 𝑓) → ((𝑐 ·s 𝑓) -s (𝑐 ·s 𝑒)) <s ((𝑑 ·s 𝑓) -s (𝑑 ·s 𝑒))))) ↔ ∀𝑎 ∈ No
∀𝑏 ∈ No ∀𝑐 ∈ No
∀𝑑 ∈ No ∀𝑒 ∈ No
∀𝑓 ∈ No (((( bday ‘𝑎) +no (
bday ‘𝑏))
∪ (((( bday ‘𝑐) +no ( bday
‘𝑒)) ∪
(( bday ‘𝑑) +no ( bday
‘𝑓))) ∪
((( bday ‘𝑐) +no ( bday
‘𝑓)) ∪
(( bday ‘𝑑) +no ( bday
‘𝑒))))) ∈
((( bday ‘𝑔) +no ( bday
‘ℎ)) ∪
(((( bday ‘𝑖) +no ( bday
‘𝑘)) ∪
(( bday ‘𝑗) +no ( bday
‘𝑙))) ∪
((( bday ‘𝑖) +no ( bday
‘𝑙)) ∪
(( bday ‘𝑗) +no ( bday
‘𝑘))))) →
((𝑎 ·s
𝑏) ∈ No ∧ ((𝑐 <s 𝑑 ∧ 𝑒 <s 𝑓) → ((𝑐 ·s 𝑓) -s (𝑐 ·s 𝑒)) <s ((𝑑 ·s 𝑓) -s (𝑑 ·s 𝑒)))))) |
128 | 114, 127 | bitrdi 286 |
. . . . . . . . . . . . 13
⊢ (𝑥 = (((
bday ‘𝑔) +no
( bday ‘ℎ)) ∪ (((( bday
‘𝑖) +no ( bday ‘𝑘)) ∪ (( bday
‘𝑗) +no ( bday ‘𝑙))) ∪ ((( bday
‘𝑖) +no ( bday ‘𝑙)) ∪ (( bday
‘𝑗) +no ( bday ‘𝑘))))) → (∀𝑦 ∈ 𝑥 ∀𝑎 ∈ No
∀𝑏 ∈ No ∀𝑐 ∈ No
∀𝑑 ∈ No ∀𝑒 ∈ No
∀𝑓 ∈ No (𝑦 = ((( bday
‘𝑎) +no ( bday ‘𝑏)) ∪ (((( bday
‘𝑐) +no ( bday ‘𝑒)) ∪ (( bday
‘𝑑) +no ( bday ‘𝑓))) ∪ ((( bday
‘𝑐) +no ( bday ‘𝑓)) ∪ (( bday
‘𝑑) +no ( bday ‘𝑒))))) → ((𝑎 ·s 𝑏) ∈ No
∧ ((𝑐 <s 𝑑 ∧ 𝑒 <s 𝑓) → ((𝑐 ·s 𝑓) -s (𝑐 ·s 𝑒)) <s ((𝑑 ·s 𝑓) -s (𝑑 ·s 𝑒))))) ↔ ∀𝑎 ∈ No
∀𝑏 ∈ No ∀𝑐 ∈ No
∀𝑑 ∈ No ∀𝑒 ∈ No
∀𝑓 ∈ No (((( bday ‘𝑎) +no (
bday ‘𝑏))
∪ (((( bday ‘𝑐) +no ( bday
‘𝑒)) ∪
(( bday ‘𝑑) +no ( bday
‘𝑓))) ∪
((( bday ‘𝑐) +no ( bday
‘𝑓)) ∪
(( bday ‘𝑑) +no ( bday
‘𝑒))))) ∈
((( bday ‘𝑔) +no ( bday
‘ℎ)) ∪
(((( bday ‘𝑖) +no ( bday
‘𝑘)) ∪
(( bday ‘𝑗) +no ( bday
‘𝑙))) ∪
((( bday ‘𝑖) +no ( bday
‘𝑙)) ∪
(( bday ‘𝑗) +no ( bday
‘𝑘))))) →
((𝑎 ·s
𝑏) ∈ No ∧ ((𝑐 <s 𝑑 ∧ 𝑒 <s 𝑓) → ((𝑐 ·s 𝑓) -s (𝑐 ·s 𝑒)) <s ((𝑑 ·s 𝑓) -s (𝑑 ·s 𝑒))))))) |
129 | | simpl 483 |
. . . . . . . . . . . . . . . 16
⊢
((∀𝑎 ∈
No ∀𝑏 ∈ No
∀𝑐 ∈ No ∀𝑑 ∈ No
∀𝑒 ∈ No ∀𝑓 ∈ No
(((( bday ‘𝑎) +no ( bday
‘𝑏)) ∪
(((( bday ‘𝑐) +no ( bday
‘𝑒)) ∪
(( bday ‘𝑑) +no ( bday
‘𝑓))) ∪
((( bday ‘𝑐) +no ( bday
‘𝑓)) ∪
(( bday ‘𝑑) +no ( bday
‘𝑒))))) ∈
((( bday ‘𝑔) +no ( bday
‘ℎ)) ∪
(((( bday ‘𝑖) +no ( bday
‘𝑘)) ∪
(( bday ‘𝑗) +no ( bday
‘𝑙))) ∪
((( bday ‘𝑖) +no ( bday
‘𝑙)) ∪
(( bday ‘𝑗) +no ( bday
‘𝑘))))) →
((𝑎 ·s
𝑏) ∈ No ∧ ((𝑐 <s 𝑑 ∧ 𝑒 <s 𝑓) → ((𝑐 ·s 𝑓) -s (𝑐 ·s 𝑒)) <s ((𝑑 ·s 𝑓) -s (𝑑 ·s 𝑒))))) ∧ ((𝑔 ∈ No
∧ ℎ ∈ No ∧ 𝑖 ∈ No )
∧ (𝑗 ∈ No ∧ 𝑘 ∈ No
∧ 𝑙 ∈ No ))) → ∀𝑎 ∈ No
∀𝑏 ∈ No ∀𝑐 ∈ No
∀𝑑 ∈ No ∀𝑒 ∈ No
∀𝑓 ∈ No (((( bday ‘𝑎) +no (
bday ‘𝑏))
∪ (((( bday ‘𝑐) +no ( bday
‘𝑒)) ∪
(( bday ‘𝑑) +no ( bday
‘𝑓))) ∪
((( bday ‘𝑐) +no ( bday
‘𝑓)) ∪
(( bday ‘𝑑) +no ( bday
‘𝑒))))) ∈
((( bday ‘𝑔) +no ( bday
‘ℎ)) ∪
(((( bday ‘𝑖) +no ( bday
‘𝑘)) ∪
(( bday ‘𝑗) +no ( bday
‘𝑙))) ∪
((( bday ‘𝑖) +no ( bday
‘𝑙)) ∪
(( bday ‘𝑗) +no ( bday
‘𝑘))))) →
((𝑎 ·s
𝑏) ∈ No ∧ ((𝑐 <s 𝑑 ∧ 𝑒 <s 𝑓) → ((𝑐 ·s 𝑓) -s (𝑐 ·s 𝑒)) <s ((𝑑 ·s 𝑓) -s (𝑑 ·s 𝑒)))))) |
130 | | simprl1 1218 |
. . . . . . . . . . . . . . . 16
⊢
((∀𝑎 ∈
No ∀𝑏 ∈ No
∀𝑐 ∈ No ∀𝑑 ∈ No
∀𝑒 ∈ No ∀𝑓 ∈ No
(((( bday ‘𝑎) +no ( bday
‘𝑏)) ∪
(((( bday ‘𝑐) +no ( bday
‘𝑒)) ∪
(( bday ‘𝑑) +no ( bday
‘𝑓))) ∪
((( bday ‘𝑐) +no ( bday
‘𝑓)) ∪
(( bday ‘𝑑) +no ( bday
‘𝑒))))) ∈
((( bday ‘𝑔) +no ( bday
‘ℎ)) ∪
(((( bday ‘𝑖) +no ( bday
‘𝑘)) ∪
(( bday ‘𝑗) +no ( bday
‘𝑙))) ∪
((( bday ‘𝑖) +no ( bday
‘𝑙)) ∪
(( bday ‘𝑗) +no ( bday
‘𝑘))))) →
((𝑎 ·s
𝑏) ∈ No ∧ ((𝑐 <s 𝑑 ∧ 𝑒 <s 𝑓) → ((𝑐 ·s 𝑓) -s (𝑐 ·s 𝑒)) <s ((𝑑 ·s 𝑓) -s (𝑑 ·s 𝑒))))) ∧ ((𝑔 ∈ No
∧ ℎ ∈ No ∧ 𝑖 ∈ No )
∧ (𝑗 ∈ No ∧ 𝑘 ∈ No
∧ 𝑙 ∈ No ))) → 𝑔 ∈ No
) |
131 | | simprl2 1219 |
. . . . . . . . . . . . . . . 16
⊢
((∀𝑎 ∈
No ∀𝑏 ∈ No
∀𝑐 ∈ No ∀𝑑 ∈ No
∀𝑒 ∈ No ∀𝑓 ∈ No
(((( bday ‘𝑎) +no ( bday
‘𝑏)) ∪
(((( bday ‘𝑐) +no ( bday
‘𝑒)) ∪
(( bday ‘𝑑) +no ( bday
‘𝑓))) ∪
((( bday ‘𝑐) +no ( bday
‘𝑓)) ∪
(( bday ‘𝑑) +no ( bday
‘𝑒))))) ∈
((( bday ‘𝑔) +no ( bday
‘ℎ)) ∪
(((( bday ‘𝑖) +no ( bday
‘𝑘)) ∪
(( bday ‘𝑗) +no ( bday
‘𝑙))) ∪
((( bday ‘𝑖) +no ( bday
‘𝑙)) ∪
(( bday ‘𝑗) +no ( bday
‘𝑘))))) →
((𝑎 ·s
𝑏) ∈ No ∧ ((𝑐 <s 𝑑 ∧ 𝑒 <s 𝑓) → ((𝑐 ·s 𝑓) -s (𝑐 ·s 𝑒)) <s ((𝑑 ·s 𝑓) -s (𝑑 ·s 𝑒))))) ∧ ((𝑔 ∈ No
∧ ℎ ∈ No ∧ 𝑖 ∈ No )
∧ (𝑗 ∈ No ∧ 𝑘 ∈ No
∧ 𝑙 ∈ No ))) → ℎ ∈ No
) |
132 | 129, 130,
131 | mulsproplem11 27511 |
. . . . . . . . . . . . . . 15
⊢
((∀𝑎 ∈
No ∀𝑏 ∈ No
∀𝑐 ∈ No ∀𝑑 ∈ No
∀𝑒 ∈ No ∀𝑓 ∈ No
(((( bday ‘𝑎) +no ( bday
‘𝑏)) ∪
(((( bday ‘𝑐) +no ( bday
‘𝑒)) ∪
(( bday ‘𝑑) +no ( bday
‘𝑓))) ∪
((( bday ‘𝑐) +no ( bday
‘𝑓)) ∪
(( bday ‘𝑑) +no ( bday
‘𝑒))))) ∈
((( bday ‘𝑔) +no ( bday
‘ℎ)) ∪
(((( bday ‘𝑖) +no ( bday
‘𝑘)) ∪
(( bday ‘𝑗) +no ( bday
‘𝑙))) ∪
((( bday ‘𝑖) +no ( bday
‘𝑙)) ∪
(( bday ‘𝑗) +no ( bday
‘𝑘))))) →
((𝑎 ·s
𝑏) ∈ No ∧ ((𝑐 <s 𝑑 ∧ 𝑒 <s 𝑓) → ((𝑐 ·s 𝑓) -s (𝑐 ·s 𝑒)) <s ((𝑑 ·s 𝑓) -s (𝑑 ·s 𝑒))))) ∧ ((𝑔 ∈ No
∧ ℎ ∈ No ∧ 𝑖 ∈ No )
∧ (𝑗 ∈ No ∧ 𝑘 ∈ No
∧ 𝑙 ∈ No ))) → (𝑔 ·s ℎ) ∈ No
) |
133 | 129 | adantr 481 |
. . . . . . . . . . . . . . . . 17
⊢
(((∀𝑎 ∈
No ∀𝑏 ∈ No
∀𝑐 ∈ No ∀𝑑 ∈ No
∀𝑒 ∈ No ∀𝑓 ∈ No
(((( bday ‘𝑎) +no ( bday
‘𝑏)) ∪
(((( bday ‘𝑐) +no ( bday
‘𝑒)) ∪
(( bday ‘𝑑) +no ( bday
‘𝑓))) ∪
((( bday ‘𝑐) +no ( bday
‘𝑓)) ∪
(( bday ‘𝑑) +no ( bday
‘𝑒))))) ∈
((( bday ‘𝑔) +no ( bday
‘ℎ)) ∪
(((( bday ‘𝑖) +no ( bday
‘𝑘)) ∪
(( bday ‘𝑗) +no ( bday
‘𝑙))) ∪
((( bday ‘𝑖) +no ( bday
‘𝑙)) ∪
(( bday ‘𝑗) +no ( bday
‘𝑘))))) →
((𝑎 ·s
𝑏) ∈ No ∧ ((𝑐 <s 𝑑 ∧ 𝑒 <s 𝑓) → ((𝑐 ·s 𝑓) -s (𝑐 ·s 𝑒)) <s ((𝑑 ·s 𝑓) -s (𝑑 ·s 𝑒))))) ∧ ((𝑔 ∈ No
∧ ℎ ∈ No ∧ 𝑖 ∈ No )
∧ (𝑗 ∈ No ∧ 𝑘 ∈ No
∧ 𝑙 ∈ No ))) ∧ (𝑖 <s 𝑗 ∧ 𝑘 <s 𝑙)) → ∀𝑎 ∈ No
∀𝑏 ∈ No ∀𝑐 ∈ No
∀𝑑 ∈ No ∀𝑒 ∈ No
∀𝑓 ∈ No (((( bday ‘𝑎) +no (
bday ‘𝑏))
∪ (((( bday ‘𝑐) +no ( bday
‘𝑒)) ∪
(( bday ‘𝑑) +no ( bday
‘𝑓))) ∪
((( bday ‘𝑐) +no ( bday
‘𝑓)) ∪
(( bday ‘𝑑) +no ( bday
‘𝑒))))) ∈
((( bday ‘𝑔) +no ( bday
‘ℎ)) ∪
(((( bday ‘𝑖) +no ( bday
‘𝑘)) ∪
(( bday ‘𝑗) +no ( bday
‘𝑙))) ∪
((( bday ‘𝑖) +no ( bday
‘𝑙)) ∪
(( bday ‘𝑗) +no ( bday
‘𝑘))))) →
((𝑎 ·s
𝑏) ∈ No ∧ ((𝑐 <s 𝑑 ∧ 𝑒 <s 𝑓) → ((𝑐 ·s 𝑓) -s (𝑐 ·s 𝑒)) <s ((𝑑 ·s 𝑓) -s (𝑑 ·s 𝑒)))))) |
134 | | simprl3 1220 |
. . . . . . . . . . . . . . . . . 18
⊢
((∀𝑎 ∈
No ∀𝑏 ∈ No
∀𝑐 ∈ No ∀𝑑 ∈ No
∀𝑒 ∈ No ∀𝑓 ∈ No
(((( bday ‘𝑎) +no ( bday
‘𝑏)) ∪
(((( bday ‘𝑐) +no ( bday
‘𝑒)) ∪
(( bday ‘𝑑) +no ( bday
‘𝑓))) ∪
((( bday ‘𝑐) +no ( bday
‘𝑓)) ∪
(( bday ‘𝑑) +no ( bday
‘𝑒))))) ∈
((( bday ‘𝑔) +no ( bday
‘ℎ)) ∪
(((( bday ‘𝑖) +no ( bday
‘𝑘)) ∪
(( bday ‘𝑗) +no ( bday
‘𝑙))) ∪
((( bday ‘𝑖) +no ( bday
‘𝑙)) ∪
(( bday ‘𝑗) +no ( bday
‘𝑘))))) →
((𝑎 ·s
𝑏) ∈ No ∧ ((𝑐 <s 𝑑 ∧ 𝑒 <s 𝑓) → ((𝑐 ·s 𝑓) -s (𝑐 ·s 𝑒)) <s ((𝑑 ·s 𝑓) -s (𝑑 ·s 𝑒))))) ∧ ((𝑔 ∈ No
∧ ℎ ∈ No ∧ 𝑖 ∈ No )
∧ (𝑗 ∈ No ∧ 𝑘 ∈ No
∧ 𝑙 ∈ No ))) → 𝑖 ∈ No
) |
135 | 134 | adantr 481 |
. . . . . . . . . . . . . . . . 17
⊢
(((∀𝑎 ∈
No ∀𝑏 ∈ No
∀𝑐 ∈ No ∀𝑑 ∈ No
∀𝑒 ∈ No ∀𝑓 ∈ No
(((( bday ‘𝑎) +no ( bday
‘𝑏)) ∪
(((( bday ‘𝑐) +no ( bday
‘𝑒)) ∪
(( bday ‘𝑑) +no ( bday
‘𝑓))) ∪
((( bday ‘𝑐) +no ( bday
‘𝑓)) ∪
(( bday ‘𝑑) +no ( bday
‘𝑒))))) ∈
((( bday ‘𝑔) +no ( bday
‘ℎ)) ∪
(((( bday ‘𝑖) +no ( bday
‘𝑘)) ∪
(( bday ‘𝑗) +no ( bday
‘𝑙))) ∪
((( bday ‘𝑖) +no ( bday
‘𝑙)) ∪
(( bday ‘𝑗) +no ( bday
‘𝑘))))) →
((𝑎 ·s
𝑏) ∈ No ∧ ((𝑐 <s 𝑑 ∧ 𝑒 <s 𝑓) → ((𝑐 ·s 𝑓) -s (𝑐 ·s 𝑒)) <s ((𝑑 ·s 𝑓) -s (𝑑 ·s 𝑒))))) ∧ ((𝑔 ∈ No
∧ ℎ ∈ No ∧ 𝑖 ∈ No )
∧ (𝑗 ∈ No ∧ 𝑘 ∈ No
∧ 𝑙 ∈ No ))) ∧ (𝑖 <s 𝑗 ∧ 𝑘 <s 𝑙)) → 𝑖 ∈ No
) |
136 | | simprr1 1221 |
. . . . . . . . . . . . . . . . . 18
⊢
((∀𝑎 ∈
No ∀𝑏 ∈ No
∀𝑐 ∈ No ∀𝑑 ∈ No
∀𝑒 ∈ No ∀𝑓 ∈ No
(((( bday ‘𝑎) +no ( bday
‘𝑏)) ∪
(((( bday ‘𝑐) +no ( bday
‘𝑒)) ∪
(( bday ‘𝑑) +no ( bday
‘𝑓))) ∪
((( bday ‘𝑐) +no ( bday
‘𝑓)) ∪
(( bday ‘𝑑) +no ( bday
‘𝑒))))) ∈
((( bday ‘𝑔) +no ( bday
‘ℎ)) ∪
(((( bday ‘𝑖) +no ( bday
‘𝑘)) ∪
(( bday ‘𝑗) +no ( bday
‘𝑙))) ∪
((( bday ‘𝑖) +no ( bday
‘𝑙)) ∪
(( bday ‘𝑗) +no ( bday
‘𝑘))))) →
((𝑎 ·s
𝑏) ∈ No ∧ ((𝑐 <s 𝑑 ∧ 𝑒 <s 𝑓) → ((𝑐 ·s 𝑓) -s (𝑐 ·s 𝑒)) <s ((𝑑 ·s 𝑓) -s (𝑑 ·s 𝑒))))) ∧ ((𝑔 ∈ No
∧ ℎ ∈ No ∧ 𝑖 ∈ No )
∧ (𝑗 ∈ No ∧ 𝑘 ∈ No
∧ 𝑙 ∈ No ))) → 𝑗 ∈ No
) |
137 | 136 | adantr 481 |
. . . . . . . . . . . . . . . . 17
⊢
(((∀𝑎 ∈
No ∀𝑏 ∈ No
∀𝑐 ∈ No ∀𝑑 ∈ No
∀𝑒 ∈ No ∀𝑓 ∈ No
(((( bday ‘𝑎) +no ( bday
‘𝑏)) ∪
(((( bday ‘𝑐) +no ( bday
‘𝑒)) ∪
(( bday ‘𝑑) +no ( bday
‘𝑓))) ∪
((( bday ‘𝑐) +no ( bday
‘𝑓)) ∪
(( bday ‘𝑑) +no ( bday
‘𝑒))))) ∈
((( bday ‘𝑔) +no ( bday
‘ℎ)) ∪
(((( bday ‘𝑖) +no ( bday
‘𝑘)) ∪
(( bday ‘𝑗) +no ( bday
‘𝑙))) ∪
((( bday ‘𝑖) +no ( bday
‘𝑙)) ∪
(( bday ‘𝑗) +no ( bday
‘𝑘))))) →
((𝑎 ·s
𝑏) ∈ No ∧ ((𝑐 <s 𝑑 ∧ 𝑒 <s 𝑓) → ((𝑐 ·s 𝑓) -s (𝑐 ·s 𝑒)) <s ((𝑑 ·s 𝑓) -s (𝑑 ·s 𝑒))))) ∧ ((𝑔 ∈ No
∧ ℎ ∈ No ∧ 𝑖 ∈ No )
∧ (𝑗 ∈ No ∧ 𝑘 ∈ No
∧ 𝑙 ∈ No ))) ∧ (𝑖 <s 𝑗 ∧ 𝑘 <s 𝑙)) → 𝑗 ∈ No
) |
138 | | simprr2 1222 |
. . . . . . . . . . . . . . . . . 18
⊢
((∀𝑎 ∈
No ∀𝑏 ∈ No
∀𝑐 ∈ No ∀𝑑 ∈ No
∀𝑒 ∈ No ∀𝑓 ∈ No
(((( bday ‘𝑎) +no ( bday
‘𝑏)) ∪
(((( bday ‘𝑐) +no ( bday
‘𝑒)) ∪
(( bday ‘𝑑) +no ( bday
‘𝑓))) ∪
((( bday ‘𝑐) +no ( bday
‘𝑓)) ∪
(( bday ‘𝑑) +no ( bday
‘𝑒))))) ∈
((( bday ‘𝑔) +no ( bday
‘ℎ)) ∪
(((( bday ‘𝑖) +no ( bday
‘𝑘)) ∪
(( bday ‘𝑗) +no ( bday
‘𝑙))) ∪
((( bday ‘𝑖) +no ( bday
‘𝑙)) ∪
(( bday ‘𝑗) +no ( bday
‘𝑘))))) →
((𝑎 ·s
𝑏) ∈ No ∧ ((𝑐 <s 𝑑 ∧ 𝑒 <s 𝑓) → ((𝑐 ·s 𝑓) -s (𝑐 ·s 𝑒)) <s ((𝑑 ·s 𝑓) -s (𝑑 ·s 𝑒))))) ∧ ((𝑔 ∈ No
∧ ℎ ∈ No ∧ 𝑖 ∈ No )
∧ (𝑗 ∈ No ∧ 𝑘 ∈ No
∧ 𝑙 ∈ No ))) → 𝑘 ∈ No
) |
139 | 138 | adantr 481 |
. . . . . . . . . . . . . . . . 17
⊢
(((∀𝑎 ∈
No ∀𝑏 ∈ No
∀𝑐 ∈ No ∀𝑑 ∈ No
∀𝑒 ∈ No ∀𝑓 ∈ No
(((( bday ‘𝑎) +no ( bday
‘𝑏)) ∪
(((( bday ‘𝑐) +no ( bday
‘𝑒)) ∪
(( bday ‘𝑑) +no ( bday
‘𝑓))) ∪
((( bday ‘𝑐) +no ( bday
‘𝑓)) ∪
(( bday ‘𝑑) +no ( bday
‘𝑒))))) ∈
((( bday ‘𝑔) +no ( bday
‘ℎ)) ∪
(((( bday ‘𝑖) +no ( bday
‘𝑘)) ∪
(( bday ‘𝑗) +no ( bday
‘𝑙))) ∪
((( bday ‘𝑖) +no ( bday
‘𝑙)) ∪
(( bday ‘𝑗) +no ( bday
‘𝑘))))) →
((𝑎 ·s
𝑏) ∈ No ∧ ((𝑐 <s 𝑑 ∧ 𝑒 <s 𝑓) → ((𝑐 ·s 𝑓) -s (𝑐 ·s 𝑒)) <s ((𝑑 ·s 𝑓) -s (𝑑 ·s 𝑒))))) ∧ ((𝑔 ∈ No
∧ ℎ ∈ No ∧ 𝑖 ∈ No )
∧ (𝑗 ∈ No ∧ 𝑘 ∈ No
∧ 𝑙 ∈ No ))) ∧ (𝑖 <s 𝑗 ∧ 𝑘 <s 𝑙)) → 𝑘 ∈ No
) |
140 | | simprr3 1223 |
. . . . . . . . . . . . . . . . . 18
⊢
((∀𝑎 ∈
No ∀𝑏 ∈ No
∀𝑐 ∈ No ∀𝑑 ∈ No
∀𝑒 ∈ No ∀𝑓 ∈ No
(((( bday ‘𝑎) +no ( bday
‘𝑏)) ∪
(((( bday ‘𝑐) +no ( bday
‘𝑒)) ∪
(( bday ‘𝑑) +no ( bday
‘𝑓))) ∪
((( bday ‘𝑐) +no ( bday
‘𝑓)) ∪
(( bday ‘𝑑) +no ( bday
‘𝑒))))) ∈
((( bday ‘𝑔) +no ( bday
‘ℎ)) ∪
(((( bday ‘𝑖) +no ( bday
‘𝑘)) ∪
(( bday ‘𝑗) +no ( bday
‘𝑙))) ∪
((( bday ‘𝑖) +no ( bday
‘𝑙)) ∪
(( bday ‘𝑗) +no ( bday
‘𝑘))))) →
((𝑎 ·s
𝑏) ∈ No ∧ ((𝑐 <s 𝑑 ∧ 𝑒 <s 𝑓) → ((𝑐 ·s 𝑓) -s (𝑐 ·s 𝑒)) <s ((𝑑 ·s 𝑓) -s (𝑑 ·s 𝑒))))) ∧ ((𝑔 ∈ No
∧ ℎ ∈ No ∧ 𝑖 ∈ No )
∧ (𝑗 ∈ No ∧ 𝑘 ∈ No
∧ 𝑙 ∈ No ))) → 𝑙 ∈ No
) |
141 | 140 | adantr 481 |
. . . . . . . . . . . . . . . . 17
⊢
(((∀𝑎 ∈
No ∀𝑏 ∈ No
∀𝑐 ∈ No ∀𝑑 ∈ No
∀𝑒 ∈ No ∀𝑓 ∈ No
(((( bday ‘𝑎) +no ( bday
‘𝑏)) ∪
(((( bday ‘𝑐) +no ( bday
‘𝑒)) ∪
(( bday ‘𝑑) +no ( bday
‘𝑓))) ∪
((( bday ‘𝑐) +no ( bday
‘𝑓)) ∪
(( bday ‘𝑑) +no ( bday
‘𝑒))))) ∈
((( bday ‘𝑔) +no ( bday
‘ℎ)) ∪
(((( bday ‘𝑖) +no ( bday
‘𝑘)) ∪
(( bday ‘𝑗) +no ( bday
‘𝑙))) ∪
((( bday ‘𝑖) +no ( bday
‘𝑙)) ∪
(( bday ‘𝑗) +no ( bday
‘𝑘))))) →
((𝑎 ·s
𝑏) ∈ No ∧ ((𝑐 <s 𝑑 ∧ 𝑒 <s 𝑓) → ((𝑐 ·s 𝑓) -s (𝑐 ·s 𝑒)) <s ((𝑑 ·s 𝑓) -s (𝑑 ·s 𝑒))))) ∧ ((𝑔 ∈ No
∧ ℎ ∈ No ∧ 𝑖 ∈ No )
∧ (𝑗 ∈ No ∧ 𝑘 ∈ No
∧ 𝑙 ∈ No ))) ∧ (𝑖 <s 𝑗 ∧ 𝑘 <s 𝑙)) → 𝑙 ∈ No
) |
142 | | simprl 769 |
. . . . . . . . . . . . . . . . 17
⊢
(((∀𝑎 ∈
No ∀𝑏 ∈ No
∀𝑐 ∈ No ∀𝑑 ∈ No
∀𝑒 ∈ No ∀𝑓 ∈ No
(((( bday ‘𝑎) +no ( bday
‘𝑏)) ∪
(((( bday ‘𝑐) +no ( bday
‘𝑒)) ∪
(( bday ‘𝑑) +no ( bday
‘𝑓))) ∪
((( bday ‘𝑐) +no ( bday
‘𝑓)) ∪
(( bday ‘𝑑) +no ( bday
‘𝑒))))) ∈
((( bday ‘𝑔) +no ( bday
‘ℎ)) ∪
(((( bday ‘𝑖) +no ( bday
‘𝑘)) ∪
(( bday ‘𝑗) +no ( bday
‘𝑙))) ∪
((( bday ‘𝑖) +no ( bday
‘𝑙)) ∪
(( bday ‘𝑗) +no ( bday
‘𝑘))))) →
((𝑎 ·s
𝑏) ∈ No ∧ ((𝑐 <s 𝑑 ∧ 𝑒 <s 𝑓) → ((𝑐 ·s 𝑓) -s (𝑐 ·s 𝑒)) <s ((𝑑 ·s 𝑓) -s (𝑑 ·s 𝑒))))) ∧ ((𝑔 ∈ No
∧ ℎ ∈ No ∧ 𝑖 ∈ No )
∧ (𝑗 ∈ No ∧ 𝑘 ∈ No
∧ 𝑙 ∈ No ))) ∧ (𝑖 <s 𝑗 ∧ 𝑘 <s 𝑙)) → 𝑖 <s 𝑗) |
143 | | simprr 771 |
. . . . . . . . . . . . . . . . 17
⊢
(((∀𝑎 ∈
No ∀𝑏 ∈ No
∀𝑐 ∈ No ∀𝑑 ∈ No
∀𝑒 ∈ No ∀𝑓 ∈ No
(((( bday ‘𝑎) +no ( bday
‘𝑏)) ∪
(((( bday ‘𝑐) +no ( bday
‘𝑒)) ∪
(( bday ‘𝑑) +no ( bday
‘𝑓))) ∪
((( bday ‘𝑐) +no ( bday
‘𝑓)) ∪
(( bday ‘𝑑) +no ( bday
‘𝑒))))) ∈
((( bday ‘𝑔) +no ( bday
‘ℎ)) ∪
(((( bday ‘𝑖) +no ( bday
‘𝑘)) ∪
(( bday ‘𝑗) +no ( bday
‘𝑙))) ∪
((( bday ‘𝑖) +no ( bday
‘𝑙)) ∪
(( bday ‘𝑗) +no ( bday
‘𝑘))))) →
((𝑎 ·s
𝑏) ∈ No ∧ ((𝑐 <s 𝑑 ∧ 𝑒 <s 𝑓) → ((𝑐 ·s 𝑓) -s (𝑐 ·s 𝑒)) <s ((𝑑 ·s 𝑓) -s (𝑑 ·s 𝑒))))) ∧ ((𝑔 ∈ No
∧ ℎ ∈ No ∧ 𝑖 ∈ No )
∧ (𝑗 ∈ No ∧ 𝑘 ∈ No
∧ 𝑙 ∈ No ))) ∧ (𝑖 <s 𝑗 ∧ 𝑘 <s 𝑙)) → 𝑘 <s 𝑙) |
144 | 133, 135,
137, 139, 141, 142, 143 | mulsproplem14 27514 |
. . . . . . . . . . . . . . . 16
⊢
(((∀𝑎 ∈
No ∀𝑏 ∈ No
∀𝑐 ∈ No ∀𝑑 ∈ No
∀𝑒 ∈ No ∀𝑓 ∈ No
(((( bday ‘𝑎) +no ( bday
‘𝑏)) ∪
(((( bday ‘𝑐) +no ( bday
‘𝑒)) ∪
(( bday ‘𝑑) +no ( bday
‘𝑓))) ∪
((( bday ‘𝑐) +no ( bday
‘𝑓)) ∪
(( bday ‘𝑑) +no ( bday
‘𝑒))))) ∈
((( bday ‘𝑔) +no ( bday
‘ℎ)) ∪
(((( bday ‘𝑖) +no ( bday
‘𝑘)) ∪
(( bday ‘𝑗) +no ( bday
‘𝑙))) ∪
((( bday ‘𝑖) +no ( bday
‘𝑙)) ∪
(( bday ‘𝑗) +no ( bday
‘𝑘))))) →
((𝑎 ·s
𝑏) ∈ No ∧ ((𝑐 <s 𝑑 ∧ 𝑒 <s 𝑓) → ((𝑐 ·s 𝑓) -s (𝑐 ·s 𝑒)) <s ((𝑑 ·s 𝑓) -s (𝑑 ·s 𝑒))))) ∧ ((𝑔 ∈ No
∧ ℎ ∈ No ∧ 𝑖 ∈ No )
∧ (𝑗 ∈ No ∧ 𝑘 ∈ No
∧ 𝑙 ∈ No ))) ∧ (𝑖 <s 𝑗 ∧ 𝑘 <s 𝑙)) → ((𝑖 ·s 𝑙) -s (𝑖 ·s 𝑘)) <s ((𝑗 ·s 𝑙) -s (𝑗 ·s 𝑘))) |
145 | 144 | ex 413 |
. . . . . . . . . . . . . . 15
⊢
((∀𝑎 ∈
No ∀𝑏 ∈ No
∀𝑐 ∈ No ∀𝑑 ∈ No
∀𝑒 ∈ No ∀𝑓 ∈ No
(((( bday ‘𝑎) +no ( bday
‘𝑏)) ∪
(((( bday ‘𝑐) +no ( bday
‘𝑒)) ∪
(( bday ‘𝑑) +no ( bday
‘𝑓))) ∪
((( bday ‘𝑐) +no ( bday
‘𝑓)) ∪
(( bday ‘𝑑) +no ( bday
‘𝑒))))) ∈
((( bday ‘𝑔) +no ( bday
‘ℎ)) ∪
(((( bday ‘𝑖) +no ( bday
‘𝑘)) ∪
(( bday ‘𝑗) +no ( bday
‘𝑙))) ∪
((( bday ‘𝑖) +no ( bday
‘𝑙)) ∪
(( bday ‘𝑗) +no ( bday
‘𝑘))))) →
((𝑎 ·s
𝑏) ∈ No ∧ ((𝑐 <s 𝑑 ∧ 𝑒 <s 𝑓) → ((𝑐 ·s 𝑓) -s (𝑐 ·s 𝑒)) <s ((𝑑 ·s 𝑓) -s (𝑑 ·s 𝑒))))) ∧ ((𝑔 ∈ No
∧ ℎ ∈ No ∧ 𝑖 ∈ No )
∧ (𝑗 ∈ No ∧ 𝑘 ∈ No
∧ 𝑙 ∈ No ))) → ((𝑖 <s 𝑗 ∧ 𝑘 <s 𝑙) → ((𝑖 ·s 𝑙) -s (𝑖 ·s 𝑘)) <s ((𝑗 ·s 𝑙) -s (𝑗 ·s 𝑘)))) |
146 | 132, 145 | jca 512 |
. . . . . . . . . . . . . 14
⊢
((∀𝑎 ∈
No ∀𝑏 ∈ No
∀𝑐 ∈ No ∀𝑑 ∈ No
∀𝑒 ∈ No ∀𝑓 ∈ No
(((( bday ‘𝑎) +no ( bday
‘𝑏)) ∪
(((( bday ‘𝑐) +no ( bday
‘𝑒)) ∪
(( bday ‘𝑑) +no ( bday
‘𝑓))) ∪
((( bday ‘𝑐) +no ( bday
‘𝑓)) ∪
(( bday ‘𝑑) +no ( bday
‘𝑒))))) ∈
((( bday ‘𝑔) +no ( bday
‘ℎ)) ∪
(((( bday ‘𝑖) +no ( bday
‘𝑘)) ∪
(( bday ‘𝑗) +no ( bday
‘𝑙))) ∪
((( bday ‘𝑖) +no ( bday
‘𝑙)) ∪
(( bday ‘𝑗) +no ( bday
‘𝑘))))) →
((𝑎 ·s
𝑏) ∈ No ∧ ((𝑐 <s 𝑑 ∧ 𝑒 <s 𝑓) → ((𝑐 ·s 𝑓) -s (𝑐 ·s 𝑒)) <s ((𝑑 ·s 𝑓) -s (𝑑 ·s 𝑒))))) ∧ ((𝑔 ∈ No
∧ ℎ ∈ No ∧ 𝑖 ∈ No )
∧ (𝑗 ∈ No ∧ 𝑘 ∈ No
∧ 𝑙 ∈ No ))) → ((𝑔 ·s ℎ) ∈ No
∧ ((𝑖 <s 𝑗 ∧ 𝑘 <s 𝑙) → ((𝑖 ·s 𝑙) -s (𝑖 ·s 𝑘)) <s ((𝑗 ·s 𝑙) -s (𝑗 ·s 𝑘))))) |
147 | 146 | ex 413 |
. . . . . . . . . . . . 13
⊢
(∀𝑎 ∈
No ∀𝑏 ∈ No
∀𝑐 ∈ No ∀𝑑 ∈ No
∀𝑒 ∈ No ∀𝑓 ∈ No
(((( bday ‘𝑎) +no ( bday
‘𝑏)) ∪
(((( bday ‘𝑐) +no ( bday
‘𝑒)) ∪
(( bday ‘𝑑) +no ( bday
‘𝑓))) ∪
((( bday ‘𝑐) +no ( bday
‘𝑓)) ∪
(( bday ‘𝑑) +no ( bday
‘𝑒))))) ∈
((( bday ‘𝑔) +no ( bday
‘ℎ)) ∪
(((( bday ‘𝑖) +no ( bday
‘𝑘)) ∪
(( bday ‘𝑗) +no ( bday
‘𝑙))) ∪
((( bday ‘𝑖) +no ( bday
‘𝑙)) ∪
(( bday ‘𝑗) +no ( bday
‘𝑘))))) →
((𝑎 ·s
𝑏) ∈ No ∧ ((𝑐 <s 𝑑 ∧ 𝑒 <s 𝑓) → ((𝑐 ·s 𝑓) -s (𝑐 ·s 𝑒)) <s ((𝑑 ·s 𝑓) -s (𝑑 ·s 𝑒))))) → (((𝑔 ∈ No
∧ ℎ ∈ No ∧ 𝑖 ∈ No )
∧ (𝑗 ∈ No ∧ 𝑘 ∈ No
∧ 𝑙 ∈ No )) → ((𝑔 ·s ℎ) ∈ No
∧ ((𝑖 <s 𝑗 ∧ 𝑘 <s 𝑙) → ((𝑖 ·s 𝑙) -s (𝑖 ·s 𝑘)) <s ((𝑗 ·s 𝑙) -s (𝑗 ·s 𝑘)))))) |
148 | 128, 147 | syl6bi 252 |
. . . . . . . . . . . 12
⊢ (𝑥 = (((
bday ‘𝑔) +no
( bday ‘ℎ)) ∪ (((( bday
‘𝑖) +no ( bday ‘𝑘)) ∪ (( bday
‘𝑗) +no ( bday ‘𝑙))) ∪ ((( bday
‘𝑖) +no ( bday ‘𝑙)) ∪ (( bday
‘𝑗) +no ( bday ‘𝑘))))) → (∀𝑦 ∈ 𝑥 ∀𝑎 ∈ No
∀𝑏 ∈ No ∀𝑐 ∈ No
∀𝑑 ∈ No ∀𝑒 ∈ No
∀𝑓 ∈ No (𝑦 = ((( bday
‘𝑎) +no ( bday ‘𝑏)) ∪ (((( bday
‘𝑐) +no ( bday ‘𝑒)) ∪ (( bday
‘𝑑) +no ( bday ‘𝑓))) ∪ ((( bday
‘𝑐) +no ( bday ‘𝑓)) ∪ (( bday
‘𝑑) +no ( bday ‘𝑒))))) → ((𝑎 ·s 𝑏) ∈ No
∧ ((𝑐 <s 𝑑 ∧ 𝑒 <s 𝑓) → ((𝑐 ·s 𝑓) -s (𝑐 ·s 𝑒)) <s ((𝑑 ·s 𝑓) -s (𝑑 ·s 𝑒))))) → (((𝑔 ∈ No
∧ ℎ ∈ No ∧ 𝑖 ∈ No )
∧ (𝑗 ∈ No ∧ 𝑘 ∈ No
∧ 𝑙 ∈ No )) → ((𝑔 ·s ℎ) ∈ No
∧ ((𝑖 <s 𝑗 ∧ 𝑘 <s 𝑙) → ((𝑖 ·s 𝑙) -s (𝑖 ·s 𝑘)) <s ((𝑗 ·s 𝑙) -s (𝑗 ·s 𝑘))))))) |
149 | 148 | impd 411 |
. . . . . . . . . . 11
⊢ (𝑥 = (((
bday ‘𝑔) +no
( bday ‘ℎ)) ∪ (((( bday
‘𝑖) +no ( bday ‘𝑘)) ∪ (( bday
‘𝑗) +no ( bday ‘𝑙))) ∪ ((( bday
‘𝑖) +no ( bday ‘𝑙)) ∪ (( bday
‘𝑗) +no ( bday ‘𝑘))))) → ((∀𝑦 ∈ 𝑥 ∀𝑎 ∈ No
∀𝑏 ∈ No ∀𝑐 ∈ No
∀𝑑 ∈ No ∀𝑒 ∈ No
∀𝑓 ∈ No (𝑦 = ((( bday
‘𝑎) +no ( bday ‘𝑏)) ∪ (((( bday
‘𝑐) +no ( bday ‘𝑒)) ∪ (( bday
‘𝑑) +no ( bday ‘𝑓))) ∪ ((( bday
‘𝑐) +no ( bday ‘𝑓)) ∪ (( bday
‘𝑑) +no ( bday ‘𝑒))))) → ((𝑎 ·s 𝑏) ∈ No
∧ ((𝑐 <s 𝑑 ∧ 𝑒 <s 𝑓) → ((𝑐 ·s 𝑓) -s (𝑐 ·s 𝑒)) <s ((𝑑 ·s 𝑓) -s (𝑑 ·s 𝑒))))) ∧ ((𝑔 ∈ No
∧ ℎ ∈ No ∧ 𝑖 ∈ No )
∧ (𝑗 ∈ No ∧ 𝑘 ∈ No
∧ 𝑙 ∈ No ))) → ((𝑔 ·s ℎ) ∈ No
∧ ((𝑖 <s 𝑗 ∧ 𝑘 <s 𝑙) → ((𝑖 ·s 𝑙) -s (𝑖 ·s 𝑘)) <s ((𝑗 ·s 𝑙) -s (𝑗 ·s 𝑘)))))) |
150 | 149 | com12 32 |
. . . . . . . . . 10
⊢
((∀𝑦 ∈
𝑥 ∀𝑎 ∈
No ∀𝑏 ∈
No ∀𝑐 ∈ No
∀𝑑 ∈ No ∀𝑒 ∈ No
∀𝑓 ∈ No (𝑦 = ((( bday
‘𝑎) +no ( bday ‘𝑏)) ∪ (((( bday
‘𝑐) +no ( bday ‘𝑒)) ∪ (( bday
‘𝑑) +no ( bday ‘𝑓))) ∪ ((( bday
‘𝑐) +no ( bday ‘𝑓)) ∪ (( bday
‘𝑑) +no ( bday ‘𝑒))))) → ((𝑎 ·s 𝑏) ∈ No
∧ ((𝑐 <s 𝑑 ∧ 𝑒 <s 𝑓) → ((𝑐 ·s 𝑓) -s (𝑐 ·s 𝑒)) <s ((𝑑 ·s 𝑓) -s (𝑑 ·s 𝑒))))) ∧ ((𝑔 ∈ No
∧ ℎ ∈ No ∧ 𝑖 ∈ No )
∧ (𝑗 ∈ No ∧ 𝑘 ∈ No
∧ 𝑙 ∈ No ))) → (𝑥 = ((( bday
‘𝑔) +no ( bday ‘ℎ)) ∪ (((( bday
‘𝑖) +no ( bday ‘𝑘)) ∪ (( bday
‘𝑗) +no ( bday ‘𝑙))) ∪ ((( bday
‘𝑖) +no ( bday ‘𝑙)) ∪ (( bday
‘𝑗) +no ( bday ‘𝑘))))) → ((𝑔 ·s ℎ) ∈ No
∧ ((𝑖 <s 𝑗 ∧ 𝑘 <s 𝑙) → ((𝑖 ·s 𝑙) -s (𝑖 ·s 𝑘)) <s ((𝑗 ·s 𝑙) -s (𝑗 ·s 𝑘)))))) |
151 | 150 | anassrs 468 |
. . . . . . . . 9
⊢
(((∀𝑦 ∈
𝑥 ∀𝑎 ∈
No ∀𝑏 ∈
No ∀𝑐 ∈ No
∀𝑑 ∈ No ∀𝑒 ∈ No
∀𝑓 ∈ No (𝑦 = ((( bday
‘𝑎) +no ( bday ‘𝑏)) ∪ (((( bday
‘𝑐) +no ( bday ‘𝑒)) ∪ (( bday
‘𝑑) +no ( bday ‘𝑓))) ∪ ((( bday
‘𝑐) +no ( bday ‘𝑓)) ∪ (( bday
‘𝑑) +no ( bday ‘𝑒))))) → ((𝑎 ·s 𝑏) ∈ No
∧ ((𝑐 <s 𝑑 ∧ 𝑒 <s 𝑓) → ((𝑐 ·s 𝑓) -s (𝑐 ·s 𝑒)) <s ((𝑑 ·s 𝑓) -s (𝑑 ·s 𝑒))))) ∧ (𝑔 ∈ No
∧ ℎ ∈ No ∧ 𝑖 ∈ No ))
∧ (𝑗 ∈ No ∧ 𝑘 ∈ No
∧ 𝑙 ∈ No )) → (𝑥 = ((( bday
‘𝑔) +no ( bday ‘ℎ)) ∪ (((( bday
‘𝑖) +no ( bday ‘𝑘)) ∪ (( bday
‘𝑗) +no ( bday ‘𝑙))) ∪ ((( bday
‘𝑖) +no ( bday ‘𝑙)) ∪ (( bday
‘𝑗) +no ( bday ‘𝑘))))) → ((𝑔 ·s ℎ) ∈ No
∧ ((𝑖 <s 𝑗 ∧ 𝑘 <s 𝑙) → ((𝑖 ·s 𝑙) -s (𝑖 ·s 𝑘)) <s ((𝑗 ·s 𝑙) -s (𝑗 ·s 𝑘)))))) |
152 | 151 | ralrimivvva 3203 |
. . . . . . . 8
⊢
((∀𝑦 ∈
𝑥 ∀𝑎 ∈
No ∀𝑏 ∈
No ∀𝑐 ∈ No
∀𝑑 ∈ No ∀𝑒 ∈ No
∀𝑓 ∈ No (𝑦 = ((( bday
‘𝑎) +no ( bday ‘𝑏)) ∪ (((( bday
‘𝑐) +no ( bday ‘𝑒)) ∪ (( bday
‘𝑑) +no ( bday ‘𝑓))) ∪ ((( bday
‘𝑐) +no ( bday ‘𝑓)) ∪ (( bday
‘𝑑) +no ( bday ‘𝑒))))) → ((𝑎 ·s 𝑏) ∈ No
∧ ((𝑐 <s 𝑑 ∧ 𝑒 <s 𝑓) → ((𝑐 ·s 𝑓) -s (𝑐 ·s 𝑒)) <s ((𝑑 ·s 𝑓) -s (𝑑 ·s 𝑒))))) ∧ (𝑔 ∈ No
∧ ℎ ∈ No ∧ 𝑖 ∈ No ))
→ ∀𝑗 ∈
No ∀𝑘 ∈ No
∀𝑙 ∈ No (𝑥 = ((( bday
‘𝑔) +no ( bday ‘ℎ)) ∪ (((( bday
‘𝑖) +no ( bday ‘𝑘)) ∪ (( bday
‘𝑗) +no ( bday ‘𝑙))) ∪ ((( bday
‘𝑖) +no ( bday ‘𝑙)) ∪ (( bday
‘𝑗) +no ( bday ‘𝑘))))) → ((𝑔 ·s ℎ) ∈ No
∧ ((𝑖 <s 𝑗 ∧ 𝑘 <s 𝑙) → ((𝑖 ·s 𝑙) -s (𝑖 ·s 𝑘)) <s ((𝑗 ·s 𝑙) -s (𝑗 ·s 𝑘)))))) |
153 | 152 | ralrimivvva 3203 |
. . . . . . 7
⊢
(∀𝑦 ∈
𝑥 ∀𝑎 ∈
No ∀𝑏 ∈
No ∀𝑐 ∈ No
∀𝑑 ∈ No ∀𝑒 ∈ No
∀𝑓 ∈ No (𝑦 = ((( bday
‘𝑎) +no ( bday ‘𝑏)) ∪ (((( bday
‘𝑐) +no ( bday ‘𝑒)) ∪ (( bday
‘𝑑) +no ( bday ‘𝑓))) ∪ ((( bday
‘𝑐) +no ( bday ‘𝑓)) ∪ (( bday
‘𝑑) +no ( bday ‘𝑒))))) → ((𝑎 ·s 𝑏) ∈ No
∧ ((𝑐 <s 𝑑 ∧ 𝑒 <s 𝑓) → ((𝑐 ·s 𝑓) -s (𝑐 ·s 𝑒)) <s ((𝑑 ·s 𝑓) -s (𝑑 ·s 𝑒))))) → ∀𝑔 ∈ No
∀ℎ ∈ No ∀𝑖 ∈ No
∀𝑗 ∈ No ∀𝑘 ∈ No
∀𝑙 ∈ No (𝑥 = ((( bday
‘𝑔) +no ( bday ‘ℎ)) ∪ (((( bday
‘𝑖) +no ( bday ‘𝑘)) ∪ (( bday
‘𝑗) +no ( bday ‘𝑙))) ∪ ((( bday
‘𝑖) +no ( bday ‘𝑙)) ∪ (( bday
‘𝑗) +no ( bday ‘𝑘))))) → ((𝑔 ·s ℎ) ∈ No
∧ ((𝑖 <s 𝑗 ∧ 𝑘 <s 𝑙) → ((𝑖 ·s 𝑙) -s (𝑖 ·s 𝑘)) <s ((𝑗 ·s 𝑙) -s (𝑗 ·s 𝑘)))))) |
154 | 153 | a1i 11 |
. . . . . 6
⊢ (𝑥 ∈ On → (∀𝑦 ∈ 𝑥 ∀𝑎 ∈ No
∀𝑏 ∈ No ∀𝑐 ∈ No
∀𝑑 ∈ No ∀𝑒 ∈ No
∀𝑓 ∈ No (𝑦 = ((( bday
‘𝑎) +no ( bday ‘𝑏)) ∪ (((( bday
‘𝑐) +no ( bday ‘𝑒)) ∪ (( bday
‘𝑑) +no ( bday ‘𝑓))) ∪ ((( bday
‘𝑐) +no ( bday ‘𝑓)) ∪ (( bday
‘𝑑) +no ( bday ‘𝑒))))) → ((𝑎 ·s 𝑏) ∈ No
∧ ((𝑐 <s 𝑑 ∧ 𝑒 <s 𝑓) → ((𝑐 ·s 𝑓) -s (𝑐 ·s 𝑒)) <s ((𝑑 ·s 𝑓) -s (𝑑 ·s 𝑒))))) → ∀𝑔 ∈ No
∀ℎ ∈ No ∀𝑖 ∈ No
∀𝑗 ∈ No ∀𝑘 ∈ No
∀𝑙 ∈ No (𝑥 = ((( bday
‘𝑔) +no ( bday ‘ℎ)) ∪ (((( bday
‘𝑖) +no ( bday ‘𝑘)) ∪ (( bday
‘𝑗) +no ( bday ‘𝑙))) ∪ ((( bday
‘𝑖) +no ( bday ‘𝑙)) ∪ (( bday
‘𝑗) +no ( bday ‘𝑘))))) → ((𝑔 ·s ℎ) ∈ No
∧ ((𝑖 <s 𝑗 ∧ 𝑘 <s 𝑙) → ((𝑖 ·s 𝑙) -s (𝑖 ·s 𝑘)) <s ((𝑗 ·s 𝑙) -s (𝑗 ·s 𝑘))))))) |
155 | 113, 154 | tfis2 7830 |
. . . . 5
⊢ (𝑥 ∈ On → ∀𝑔 ∈
No ∀ℎ ∈
No ∀𝑖 ∈ No
∀𝑗 ∈ No ∀𝑘 ∈ No
∀𝑙 ∈ No (𝑥 = ((( bday
‘𝑔) +no ( bday ‘ℎ)) ∪ (((( bday
‘𝑖) +no ( bday ‘𝑘)) ∪ (( bday
‘𝑗) +no ( bday ‘𝑙))) ∪ ((( bday
‘𝑖) +no ( bday ‘𝑙)) ∪ (( bday
‘𝑗) +no ( bday ‘𝑘))))) → ((𝑔 ·s ℎ) ∈ No
∧ ((𝑖 <s 𝑗 ∧ 𝑘 <s 𝑙) → ((𝑖 ·s 𝑙) -s (𝑖 ·s 𝑘)) <s ((𝑗 ·s 𝑙) -s (𝑗 ·s 𝑘)))))) |
156 | | fveq2 6879 |
. . . . . . . . . 10
⊢ (𝑔 = 𝐴 → ( bday
‘𝑔) = ( bday ‘𝐴)) |
157 | 156 | oveq1d 7409 |
. . . . . . . . 9
⊢ (𝑔 = 𝐴 → (( bday
‘𝑔) +no ( bday ‘ℎ)) = (( bday
‘𝐴) +no ( bday ‘ℎ))) |
158 | 157 | uneq1d 4159 |
. . . . . . . 8
⊢ (𝑔 = 𝐴 → ((( bday
‘𝑔) +no ( bday ‘ℎ)) ∪ (((( bday
‘𝑖) +no ( bday ‘𝑘)) ∪ (( bday
‘𝑗) +no ( bday ‘𝑙))) ∪ ((( bday
‘𝑖) +no ( bday ‘𝑙)) ∪ (( bday
‘𝑗) +no ( bday ‘𝑘))))) = ((( bday
‘𝐴) +no ( bday ‘ℎ)) ∪ (((( bday
‘𝑖) +no ( bday ‘𝑘)) ∪ (( bday
‘𝑗) +no ( bday ‘𝑙))) ∪ ((( bday
‘𝑖) +no ( bday ‘𝑙)) ∪ (( bday
‘𝑗) +no ( bday ‘𝑘)))))) |
159 | 158 | eqeq2d 2743 |
. . . . . . 7
⊢ (𝑔 = 𝐴 → (𝑥 = ((( bday
‘𝑔) +no ( bday ‘ℎ)) ∪ (((( bday
‘𝑖) +no ( bday ‘𝑘)) ∪ (( bday
‘𝑗) +no ( bday ‘𝑙))) ∪ ((( bday
‘𝑖) +no ( bday ‘𝑙)) ∪ (( bday
‘𝑗) +no ( bday ‘𝑘))))) ↔ 𝑥 = ((( bday
‘𝐴) +no ( bday ‘ℎ)) ∪ (((( bday
‘𝑖) +no ( bday ‘𝑘)) ∪ (( bday
‘𝑗) +no ( bday ‘𝑙))) ∪ ((( bday
‘𝑖) +no ( bday ‘𝑙)) ∪ (( bday
‘𝑗) +no ( bday ‘𝑘))))))) |
160 | | oveq1 7401 |
. . . . . . . . 9
⊢ (𝑔 = 𝐴 → (𝑔 ·s ℎ) = (𝐴 ·s ℎ)) |
161 | 160 | eleq1d 2818 |
. . . . . . . 8
⊢ (𝑔 = 𝐴 → ((𝑔 ·s ℎ) ∈ No
↔ (𝐴
·s ℎ)
∈ No )) |
162 | 161 | anbi1d 630 |
. . . . . . 7
⊢ (𝑔 = 𝐴 → (((𝑔 ·s ℎ) ∈ No
∧ ((𝑖 <s 𝑗 ∧ 𝑘 <s 𝑙) → ((𝑖 ·s 𝑙) -s (𝑖 ·s 𝑘)) <s ((𝑗 ·s 𝑙) -s (𝑗 ·s 𝑘)))) ↔ ((𝐴 ·s ℎ) ∈ No
∧ ((𝑖 <s 𝑗 ∧ 𝑘 <s 𝑙) → ((𝑖 ·s 𝑙) -s (𝑖 ·s 𝑘)) <s ((𝑗 ·s 𝑙) -s (𝑗 ·s 𝑘)))))) |
163 | 159, 162 | imbi12d 344 |
. . . . . 6
⊢ (𝑔 = 𝐴 → ((𝑥 = ((( bday
‘𝑔) +no ( bday ‘ℎ)) ∪ (((( bday
‘𝑖) +no ( bday ‘𝑘)) ∪ (( bday
‘𝑗) +no ( bday ‘𝑙))) ∪ ((( bday
‘𝑖) +no ( bday ‘𝑙)) ∪ (( bday
‘𝑗) +no ( bday ‘𝑘))))) → ((𝑔 ·s ℎ) ∈ No
∧ ((𝑖 <s 𝑗 ∧ 𝑘 <s 𝑙) → ((𝑖 ·s 𝑙) -s (𝑖 ·s 𝑘)) <s ((𝑗 ·s 𝑙) -s (𝑗 ·s 𝑘))))) ↔ (𝑥 = ((( bday
‘𝐴) +no ( bday ‘ℎ)) ∪ (((( bday
‘𝑖) +no ( bday ‘𝑘)) ∪ (( bday
‘𝑗) +no ( bday ‘𝑙))) ∪ ((( bday
‘𝑖) +no ( bday ‘𝑙)) ∪ (( bday
‘𝑗) +no ( bday ‘𝑘))))) → ((𝐴 ·s ℎ) ∈ No
∧ ((𝑖 <s 𝑗 ∧ 𝑘 <s 𝑙) → ((𝑖 ·s 𝑙) -s (𝑖 ·s 𝑘)) <s ((𝑗 ·s 𝑙) -s (𝑗 ·s 𝑘))))))) |
164 | | fveq2 6879 |
. . . . . . . . . 10
⊢ (ℎ = 𝐵 → ( bday
‘ℎ) = ( bday ‘𝐵)) |
165 | 164 | oveq2d 7410 |
. . . . . . . . 9
⊢ (ℎ = 𝐵 → (( bday
‘𝐴) +no ( bday ‘ℎ)) = (( bday
‘𝐴) +no ( bday ‘𝐵))) |
166 | 165 | uneq1d 4159 |
. . . . . . . 8
⊢ (ℎ = 𝐵 → ((( bday
‘𝐴) +no ( bday ‘ℎ)) ∪ (((( bday
‘𝑖) +no ( bday ‘𝑘)) ∪ (( bday
‘𝑗) +no ( bday ‘𝑙))) ∪ ((( bday
‘𝑖) +no ( bday ‘𝑙)) ∪ (( bday
‘𝑗) +no ( bday ‘𝑘))))) = ((( bday
‘𝐴) +no ( bday ‘𝐵)) ∪ (((( bday
‘𝑖) +no ( bday ‘𝑘)) ∪ (( bday
‘𝑗) +no ( bday ‘𝑙))) ∪ ((( bday
‘𝑖) +no ( bday ‘𝑙)) ∪ (( bday
‘𝑗) +no ( bday ‘𝑘)))))) |
167 | 166 | eqeq2d 2743 |
. . . . . . 7
⊢ (ℎ = 𝐵 → (𝑥 = ((( bday
‘𝐴) +no ( bday ‘ℎ)) ∪ (((( bday
‘𝑖) +no ( bday ‘𝑘)) ∪ (( bday
‘𝑗) +no ( bday ‘𝑙))) ∪ ((( bday
‘𝑖) +no ( bday ‘𝑙)) ∪ (( bday
‘𝑗) +no ( bday ‘𝑘))))) ↔ 𝑥 = ((( bday
‘𝐴) +no ( bday ‘𝐵)) ∪ (((( bday
‘𝑖) +no ( bday ‘𝑘)) ∪ (( bday
‘𝑗) +no ( bday ‘𝑙))) ∪ ((( bday
‘𝑖) +no ( bday ‘𝑙)) ∪ (( bday
‘𝑗) +no ( bday ‘𝑘))))))) |
168 | | oveq2 7402 |
. . . . . . . . 9
⊢ (ℎ = 𝐵 → (𝐴 ·s ℎ) = (𝐴 ·s 𝐵)) |
169 | 168 | eleq1d 2818 |
. . . . . . . 8
⊢ (ℎ = 𝐵 → ((𝐴 ·s ℎ) ∈ No
↔ (𝐴
·s 𝐵)
∈ No )) |
170 | 169 | anbi1d 630 |
. . . . . . 7
⊢ (ℎ = 𝐵 → (((𝐴 ·s ℎ) ∈ No
∧ ((𝑖 <s 𝑗 ∧ 𝑘 <s 𝑙) → ((𝑖 ·s 𝑙) -s (𝑖 ·s 𝑘)) <s ((𝑗 ·s 𝑙) -s (𝑗 ·s 𝑘)))) ↔ ((𝐴 ·s 𝐵) ∈ No
∧ ((𝑖 <s 𝑗 ∧ 𝑘 <s 𝑙) → ((𝑖 ·s 𝑙) -s (𝑖 ·s 𝑘)) <s ((𝑗 ·s 𝑙) -s (𝑗 ·s 𝑘)))))) |
171 | 167, 170 | imbi12d 344 |
. . . . . 6
⊢ (ℎ = 𝐵 → ((𝑥 = ((( bday
‘𝐴) +no ( bday ‘ℎ)) ∪ (((( bday
‘𝑖) +no ( bday ‘𝑘)) ∪ (( bday
‘𝑗) +no ( bday ‘𝑙))) ∪ ((( bday
‘𝑖) +no ( bday ‘𝑙)) ∪ (( bday
‘𝑗) +no ( bday ‘𝑘))))) → ((𝐴 ·s ℎ) ∈ No
∧ ((𝑖 <s 𝑗 ∧ 𝑘 <s 𝑙) → ((𝑖 ·s 𝑙) -s (𝑖 ·s 𝑘)) <s ((𝑗 ·s 𝑙) -s (𝑗 ·s 𝑘))))) ↔ (𝑥 = ((( bday
‘𝐴) +no ( bday ‘𝐵)) ∪ (((( bday
‘𝑖) +no ( bday ‘𝑘)) ∪ (( bday
‘𝑗) +no ( bday ‘𝑙))) ∪ ((( bday
‘𝑖) +no ( bday ‘𝑙)) ∪ (( bday
‘𝑗) +no ( bday ‘𝑘))))) → ((𝐴 ·s 𝐵) ∈ No
∧ ((𝑖 <s 𝑗 ∧ 𝑘 <s 𝑙) → ((𝑖 ·s 𝑙) -s (𝑖 ·s 𝑘)) <s ((𝑗 ·s 𝑙) -s (𝑗 ·s 𝑘))))))) |
172 | | fveq2 6879 |
. . . . . . . . . . . 12
⊢ (𝑖 = 𝐶 → ( bday
‘𝑖) = ( bday ‘𝐶)) |
173 | 172 | oveq1d 7409 |
. . . . . . . . . . 11
⊢ (𝑖 = 𝐶 → (( bday
‘𝑖) +no ( bday ‘𝑘)) = (( bday
‘𝐶) +no ( bday ‘𝑘))) |
174 | 173 | uneq1d 4159 |
. . . . . . . . . 10
⊢ (𝑖 = 𝐶 → ((( bday
‘𝑖) +no ( bday ‘𝑘)) ∪ (( bday
‘𝑗) +no ( bday ‘𝑙))) = ((( bday
‘𝐶) +no ( bday ‘𝑘)) ∪ (( bday
‘𝑗) +no ( bday ‘𝑙)))) |
175 | 172 | oveq1d 7409 |
. . . . . . . . . . 11
⊢ (𝑖 = 𝐶 → (( bday
‘𝑖) +no ( bday ‘𝑙)) = (( bday
‘𝐶) +no ( bday ‘𝑙))) |
176 | 175 | uneq1d 4159 |
. . . . . . . . . 10
⊢ (𝑖 = 𝐶 → ((( bday
‘𝑖) +no ( bday ‘𝑙)) ∪ (( bday
‘𝑗) +no ( bday ‘𝑘))) = ((( bday
‘𝐶) +no ( bday ‘𝑙)) ∪ (( bday
‘𝑗) +no ( bday ‘𝑘)))) |
177 | 174, 176 | uneq12d 4161 |
. . . . . . . . 9
⊢ (𝑖 = 𝐶 → (((( bday
‘𝑖) +no ( bday ‘𝑘)) ∪ (( bday
‘𝑗) +no ( bday ‘𝑙))) ∪ ((( bday
‘𝑖) +no ( bday ‘𝑙)) ∪ (( bday
‘𝑗) +no ( bday ‘𝑘)))) = (((( bday
‘𝐶) +no ( bday ‘𝑘)) ∪ (( bday
‘𝑗) +no ( bday ‘𝑙))) ∪ ((( bday
‘𝐶) +no ( bday ‘𝑙)) ∪ (( bday
‘𝑗) +no ( bday ‘𝑘))))) |
178 | 177 | uneq2d 4160 |
. . . . . . . 8
⊢ (𝑖 = 𝐶 → ((( bday
‘𝐴) +no ( bday ‘𝐵)) ∪ (((( bday
‘𝑖) +no ( bday ‘𝑘)) ∪ (( bday
‘𝑗) +no ( bday ‘𝑙))) ∪ ((( bday
‘𝑖) +no ( bday ‘𝑙)) ∪ (( bday
‘𝑗) +no ( bday ‘𝑘))))) = ((( bday
‘𝐴) +no ( bday ‘𝐵)) ∪ (((( bday
‘𝐶) +no ( bday ‘𝑘)) ∪ (( bday
‘𝑗) +no ( bday ‘𝑙))) ∪ ((( bday
‘𝐶) +no ( bday ‘𝑙)) ∪ (( bday
‘𝑗) +no ( bday ‘𝑘)))))) |
179 | 178 | eqeq2d 2743 |
. . . . . . 7
⊢ (𝑖 = 𝐶 → (𝑥 = ((( bday
‘𝐴) +no ( bday ‘𝐵)) ∪ (((( bday
‘𝑖) +no ( bday ‘𝑘)) ∪ (( bday
‘𝑗) +no ( bday ‘𝑙))) ∪ ((( bday
‘𝑖) +no ( bday ‘𝑙)) ∪ (( bday
‘𝑗) +no ( bday ‘𝑘))))) ↔ 𝑥 = ((( bday
‘𝐴) +no ( bday ‘𝐵)) ∪ (((( bday
‘𝐶) +no ( bday ‘𝑘)) ∪ (( bday
‘𝑗) +no ( bday ‘𝑙))) ∪ ((( bday
‘𝐶) +no ( bday ‘𝑙)) ∪ (( bday
‘𝑗) +no ( bday ‘𝑘))))))) |
180 | | breq1 5145 |
. . . . . . . . . 10
⊢ (𝑖 = 𝐶 → (𝑖 <s 𝑗 ↔ 𝐶 <s 𝑗)) |
181 | 180 | anbi1d 630 |
. . . . . . . . 9
⊢ (𝑖 = 𝐶 → ((𝑖 <s 𝑗 ∧ 𝑘 <s 𝑙) ↔ (𝐶 <s 𝑗 ∧ 𝑘 <s 𝑙))) |
182 | | oveq1 7401 |
. . . . . . . . . . 11
⊢ (𝑖 = 𝐶 → (𝑖 ·s 𝑙) = (𝐶 ·s 𝑙)) |
183 | | oveq1 7401 |
. . . . . . . . . . 11
⊢ (𝑖 = 𝐶 → (𝑖 ·s 𝑘) = (𝐶 ·s 𝑘)) |
184 | 182, 183 | oveq12d 7412 |
. . . . . . . . . 10
⊢ (𝑖 = 𝐶 → ((𝑖 ·s 𝑙) -s (𝑖 ·s 𝑘)) = ((𝐶 ·s 𝑙) -s (𝐶 ·s 𝑘))) |
185 | 184 | breq1d 5152 |
. . . . . . . . 9
⊢ (𝑖 = 𝐶 → (((𝑖 ·s 𝑙) -s (𝑖 ·s 𝑘)) <s ((𝑗 ·s 𝑙) -s (𝑗 ·s 𝑘)) ↔ ((𝐶 ·s 𝑙) -s (𝐶 ·s 𝑘)) <s ((𝑗 ·s 𝑙) -s (𝑗 ·s 𝑘)))) |
186 | 181, 185 | imbi12d 344 |
. . . . . . . 8
⊢ (𝑖 = 𝐶 → (((𝑖 <s 𝑗 ∧ 𝑘 <s 𝑙) → ((𝑖 ·s 𝑙) -s (𝑖 ·s 𝑘)) <s ((𝑗 ·s 𝑙) -s (𝑗 ·s 𝑘))) ↔ ((𝐶 <s 𝑗 ∧ 𝑘 <s 𝑙) → ((𝐶 ·s 𝑙) -s (𝐶 ·s 𝑘)) <s ((𝑗 ·s 𝑙) -s (𝑗 ·s 𝑘))))) |
187 | 186 | anbi2d 629 |
. . . . . . 7
⊢ (𝑖 = 𝐶 → (((𝐴 ·s 𝐵) ∈ No
∧ ((𝑖 <s 𝑗 ∧ 𝑘 <s 𝑙) → ((𝑖 ·s 𝑙) -s (𝑖 ·s 𝑘)) <s ((𝑗 ·s 𝑙) -s (𝑗 ·s 𝑘)))) ↔ ((𝐴 ·s 𝐵) ∈ No
∧ ((𝐶 <s 𝑗 ∧ 𝑘 <s 𝑙) → ((𝐶 ·s 𝑙) -s (𝐶 ·s 𝑘)) <s ((𝑗 ·s 𝑙) -s (𝑗 ·s 𝑘)))))) |
188 | 179, 187 | imbi12d 344 |
. . . . . 6
⊢ (𝑖 = 𝐶 → ((𝑥 = ((( bday
‘𝐴) +no ( bday ‘𝐵)) ∪ (((( bday
‘𝑖) +no ( bday ‘𝑘)) ∪ (( bday
‘𝑗) +no ( bday ‘𝑙))) ∪ ((( bday
‘𝑖) +no ( bday ‘𝑙)) ∪ (( bday
‘𝑗) +no ( bday ‘𝑘))))) → ((𝐴 ·s 𝐵) ∈ No
∧ ((𝑖 <s 𝑗 ∧ 𝑘 <s 𝑙) → ((𝑖 ·s 𝑙) -s (𝑖 ·s 𝑘)) <s ((𝑗 ·s 𝑙) -s (𝑗 ·s 𝑘))))) ↔ (𝑥 = ((( bday
‘𝐴) +no ( bday ‘𝐵)) ∪ (((( bday
‘𝐶) +no ( bday ‘𝑘)) ∪ (( bday
‘𝑗) +no ( bday ‘𝑙))) ∪ ((( bday
‘𝐶) +no ( bday ‘𝑙)) ∪ (( bday
‘𝑗) +no ( bday ‘𝑘))))) → ((𝐴 ·s 𝐵) ∈ No
∧ ((𝐶 <s 𝑗 ∧ 𝑘 <s 𝑙) → ((𝐶 ·s 𝑙) -s (𝐶 ·s 𝑘)) <s ((𝑗 ·s 𝑙) -s (𝑗 ·s 𝑘))))))) |
189 | | fveq2 6879 |
. . . . . . . . . . . 12
⊢ (𝑗 = 𝐷 → ( bday
‘𝑗) = ( bday ‘𝐷)) |
190 | 189 | oveq1d 7409 |
. . . . . . . . . . 11
⊢ (𝑗 = 𝐷 → (( bday
‘𝑗) +no ( bday ‘𝑙)) = (( bday
‘𝐷) +no ( bday ‘𝑙))) |
191 | 190 | uneq2d 4160 |
. . . . . . . . . 10
⊢ (𝑗 = 𝐷 → ((( bday
‘𝐶) +no ( bday ‘𝑘)) ∪ (( bday
‘𝑗) +no ( bday ‘𝑙))) = ((( bday
‘𝐶) +no ( bday ‘𝑘)) ∪ (( bday
‘𝐷) +no ( bday ‘𝑙)))) |
192 | 189 | oveq1d 7409 |
. . . . . . . . . . 11
⊢ (𝑗 = 𝐷 → (( bday
‘𝑗) +no ( bday ‘𝑘)) = (( bday
‘𝐷) +no ( bday ‘𝑘))) |
193 | 192 | uneq2d 4160 |
. . . . . . . . . 10
⊢ (𝑗 = 𝐷 → ((( bday
‘𝐶) +no ( bday ‘𝑙)) ∪ (( bday
‘𝑗) +no ( bday ‘𝑘))) = ((( bday
‘𝐶) +no ( bday ‘𝑙)) ∪ (( bday
‘𝐷) +no ( bday ‘𝑘)))) |
194 | 191, 193 | uneq12d 4161 |
. . . . . . . . 9
⊢ (𝑗 = 𝐷 → (((( bday
‘𝐶) +no ( bday ‘𝑘)) ∪ (( bday
‘𝑗) +no ( bday ‘𝑙))) ∪ ((( bday
‘𝐶) +no ( bday ‘𝑙)) ∪ (( bday
‘𝑗) +no ( bday ‘𝑘)))) = (((( bday
‘𝐶) +no ( bday ‘𝑘)) ∪ (( bday
‘𝐷) +no ( bday ‘𝑙))) ∪ ((( bday
‘𝐶) +no ( bday ‘𝑙)) ∪ (( bday
‘𝐷) +no ( bday ‘𝑘))))) |
195 | 194 | uneq2d 4160 |
. . . . . . . 8
⊢ (𝑗 = 𝐷 → ((( bday
‘𝐴) +no ( bday ‘𝐵)) ∪ (((( bday
‘𝐶) +no ( bday ‘𝑘)) ∪ (( bday
‘𝑗) +no ( bday ‘𝑙))) ∪ ((( bday
‘𝐶) +no ( bday ‘𝑙)) ∪ (( bday
‘𝑗) +no ( bday ‘𝑘))))) = ((( bday
‘𝐴) +no ( bday ‘𝐵)) ∪ (((( bday
‘𝐶) +no ( bday ‘𝑘)) ∪ (( bday
‘𝐷) +no ( bday ‘𝑙))) ∪ ((( bday
‘𝐶) +no ( bday ‘𝑙)) ∪ (( bday
‘𝐷) +no ( bday ‘𝑘)))))) |
196 | 195 | eqeq2d 2743 |
. . . . . . 7
⊢ (𝑗 = 𝐷 → (𝑥 = ((( bday
‘𝐴) +no ( bday ‘𝐵)) ∪ (((( bday
‘𝐶) +no ( bday ‘𝑘)) ∪ (( bday
‘𝑗) +no ( bday ‘𝑙))) ∪ ((( bday
‘𝐶) +no ( bday ‘𝑙)) ∪ (( bday
‘𝑗) +no ( bday ‘𝑘))))) ↔ 𝑥 = ((( bday
‘𝐴) +no ( bday ‘𝐵)) ∪ (((( bday
‘𝐶) +no ( bday ‘𝑘)) ∪ (( bday
‘𝐷) +no ( bday ‘𝑙))) ∪ ((( bday
‘𝐶) +no ( bday ‘𝑙)) ∪ (( bday
‘𝐷) +no ( bday ‘𝑘))))))) |
197 | | breq2 5146 |
. . . . . . . . . 10
⊢ (𝑗 = 𝐷 → (𝐶 <s 𝑗 ↔ 𝐶 <s 𝐷)) |
198 | 197 | anbi1d 630 |
. . . . . . . . 9
⊢ (𝑗 = 𝐷 → ((𝐶 <s 𝑗 ∧ 𝑘 <s 𝑙) ↔ (𝐶 <s 𝐷 ∧ 𝑘 <s 𝑙))) |
199 | | oveq1 7401 |
. . . . . . . . . . 11
⊢ (𝑗 = 𝐷 → (𝑗 ·s 𝑙) = (𝐷 ·s 𝑙)) |
200 | | oveq1 7401 |
. . . . . . . . . . 11
⊢ (𝑗 = 𝐷 → (𝑗 ·s 𝑘) = (𝐷 ·s 𝑘)) |
201 | 199, 200 | oveq12d 7412 |
. . . . . . . . . 10
⊢ (𝑗 = 𝐷 → ((𝑗 ·s 𝑙) -s (𝑗 ·s 𝑘)) = ((𝐷 ·s 𝑙) -s (𝐷 ·s 𝑘))) |
202 | 201 | breq2d 5154 |
. . . . . . . . 9
⊢ (𝑗 = 𝐷 → (((𝐶 ·s 𝑙) -s (𝐶 ·s 𝑘)) <s ((𝑗 ·s 𝑙) -s (𝑗 ·s 𝑘)) ↔ ((𝐶 ·s 𝑙) -s (𝐶 ·s 𝑘)) <s ((𝐷 ·s 𝑙) -s (𝐷 ·s 𝑘)))) |
203 | 198, 202 | imbi12d 344 |
. . . . . . . 8
⊢ (𝑗 = 𝐷 → (((𝐶 <s 𝑗 ∧ 𝑘 <s 𝑙) → ((𝐶 ·s 𝑙) -s (𝐶 ·s 𝑘)) <s ((𝑗 ·s 𝑙) -s (𝑗 ·s 𝑘))) ↔ ((𝐶 <s 𝐷 ∧ 𝑘 <s 𝑙) → ((𝐶 ·s 𝑙) -s (𝐶 ·s 𝑘)) <s ((𝐷 ·s 𝑙) -s (𝐷 ·s 𝑘))))) |
204 | 203 | anbi2d 629 |
. . . . . . 7
⊢ (𝑗 = 𝐷 → (((𝐴 ·s 𝐵) ∈ No
∧ ((𝐶 <s 𝑗 ∧ 𝑘 <s 𝑙) → ((𝐶 ·s 𝑙) -s (𝐶 ·s 𝑘)) <s ((𝑗 ·s 𝑙) -s (𝑗 ·s 𝑘)))) ↔ ((𝐴 ·s 𝐵) ∈ No
∧ ((𝐶 <s 𝐷 ∧ 𝑘 <s 𝑙) → ((𝐶 ·s 𝑙) -s (𝐶 ·s 𝑘)) <s ((𝐷 ·s 𝑙) -s (𝐷 ·s 𝑘)))))) |
205 | 196, 204 | imbi12d 344 |
. . . . . 6
⊢ (𝑗 = 𝐷 → ((𝑥 = ((( bday
‘𝐴) +no ( bday ‘𝐵)) ∪ (((( bday
‘𝐶) +no ( bday ‘𝑘)) ∪ (( bday
‘𝑗) +no ( bday ‘𝑙))) ∪ ((( bday
‘𝐶) +no ( bday ‘𝑙)) ∪ (( bday
‘𝑗) +no ( bday ‘𝑘))))) → ((𝐴 ·s 𝐵) ∈ No
∧ ((𝐶 <s 𝑗 ∧ 𝑘 <s 𝑙) → ((𝐶 ·s 𝑙) -s (𝐶 ·s 𝑘)) <s ((𝑗 ·s 𝑙) -s (𝑗 ·s 𝑘))))) ↔ (𝑥 = ((( bday
‘𝐴) +no ( bday ‘𝐵)) ∪ (((( bday
‘𝐶) +no ( bday ‘𝑘)) ∪ (( bday
‘𝐷) +no ( bday ‘𝑙))) ∪ ((( bday
‘𝐶) +no ( bday ‘𝑙)) ∪ (( bday
‘𝐷) +no ( bday ‘𝑘))))) → ((𝐴 ·s 𝐵) ∈ No
∧ ((𝐶 <s 𝐷 ∧ 𝑘 <s 𝑙) → ((𝐶 ·s 𝑙) -s (𝐶 ·s 𝑘)) <s ((𝐷 ·s 𝑙) -s (𝐷 ·s 𝑘))))))) |
206 | | fveq2 6879 |
. . . . . . . . . . . 12
⊢ (𝑘 = 𝐸 → ( bday
‘𝑘) = ( bday ‘𝐸)) |
207 | 206 | oveq2d 7410 |
. . . . . . . . . . 11
⊢ (𝑘 = 𝐸 → (( bday
‘𝐶) +no ( bday ‘𝑘)) = (( bday
‘𝐶) +no ( bday ‘𝐸))) |
208 | 207 | uneq1d 4159 |
. . . . . . . . . 10
⊢ (𝑘 = 𝐸 → ((( bday
‘𝐶) +no ( bday ‘𝑘)) ∪ (( bday
‘𝐷) +no ( bday ‘𝑙))) = ((( bday
‘𝐶) +no ( bday ‘𝐸)) ∪ (( bday
‘𝐷) +no ( bday ‘𝑙)))) |
209 | 206 | oveq2d 7410 |
. . . . . . . . . . 11
⊢ (𝑘 = 𝐸 → (( bday
‘𝐷) +no ( bday ‘𝑘)) = (( bday
‘𝐷) +no ( bday ‘𝐸))) |
210 | 209 | uneq2d 4160 |
. . . . . . . . . 10
⊢ (𝑘 = 𝐸 → ((( bday
‘𝐶) +no ( bday ‘𝑙)) ∪ (( bday
‘𝐷) +no ( bday ‘𝑘))) = ((( bday
‘𝐶) +no ( bday ‘𝑙)) ∪ (( bday
‘𝐷) +no ( bday ‘𝐸)))) |
211 | 208, 210 | uneq12d 4161 |
. . . . . . . . 9
⊢ (𝑘 = 𝐸 → (((( bday
‘𝐶) +no ( bday ‘𝑘)) ∪ (( bday
‘𝐷) +no ( bday ‘𝑙))) ∪ ((( bday
‘𝐶) +no ( bday ‘𝑙)) ∪ (( bday
‘𝐷) +no ( bday ‘𝑘)))) = (((( bday
‘𝐶) +no ( bday ‘𝐸)) ∪ (( bday
‘𝐷) +no ( bday ‘𝑙))) ∪ ((( bday
‘𝐶) +no ( bday ‘𝑙)) ∪ (( bday
‘𝐷) +no ( bday ‘𝐸))))) |
212 | 211 | uneq2d 4160 |
. . . . . . . 8
⊢ (𝑘 = 𝐸 → ((( bday
‘𝐴) +no ( bday ‘𝐵)) ∪ (((( bday
‘𝐶) +no ( bday ‘𝑘)) ∪ (( bday
‘𝐷) +no ( bday ‘𝑙))) ∪ ((( bday
‘𝐶) +no ( bday ‘𝑙)) ∪ (( bday
‘𝐷) +no ( bday ‘𝑘))))) = ((( bday
‘𝐴) +no ( bday ‘𝐵)) ∪ (((( bday
‘𝐶) +no ( bday ‘𝐸)) ∪ (( bday
‘𝐷) +no ( bday ‘𝑙))) ∪ ((( bday
‘𝐶) +no ( bday ‘𝑙)) ∪ (( bday
‘𝐷) +no ( bday ‘𝐸)))))) |
213 | 212 | eqeq2d 2743 |
. . . . . . 7
⊢ (𝑘 = 𝐸 → (𝑥 = ((( bday
‘𝐴) +no ( bday ‘𝐵)) ∪ (((( bday
‘𝐶) +no ( bday ‘𝑘)) ∪ (( bday
‘𝐷) +no ( bday ‘𝑙))) ∪ ((( bday
‘𝐶) +no ( bday ‘𝑙)) ∪ (( bday
‘𝐷) +no ( bday ‘𝑘))))) ↔ 𝑥 = ((( bday
‘𝐴) +no ( bday ‘𝐵)) ∪ (((( bday
‘𝐶) +no ( bday ‘𝐸)) ∪ (( bday
‘𝐷) +no ( bday ‘𝑙))) ∪ ((( bday
‘𝐶) +no ( bday ‘𝑙)) ∪ (( bday
‘𝐷) +no ( bday ‘𝐸))))))) |
214 | | breq1 5145 |
. . . . . . . . . 10
⊢ (𝑘 = 𝐸 → (𝑘 <s 𝑙 ↔ 𝐸 <s 𝑙)) |
215 | 214 | anbi2d 629 |
. . . . . . . . 9
⊢ (𝑘 = 𝐸 → ((𝐶 <s 𝐷 ∧ 𝑘 <s 𝑙) ↔ (𝐶 <s 𝐷 ∧ 𝐸 <s 𝑙))) |
216 | | oveq2 7402 |
. . . . . . . . . . 11
⊢ (𝑘 = 𝐸 → (𝐶 ·s 𝑘) = (𝐶 ·s 𝐸)) |
217 | 216 | oveq2d 7410 |
. . . . . . . . . 10
⊢ (𝑘 = 𝐸 → ((𝐶 ·s 𝑙) -s (𝐶 ·s 𝑘)) = ((𝐶 ·s 𝑙) -s (𝐶 ·s 𝐸))) |
218 | | oveq2 7402 |
. . . . . . . . . . 11
⊢ (𝑘 = 𝐸 → (𝐷 ·s 𝑘) = (𝐷 ·s 𝐸)) |
219 | 218 | oveq2d 7410 |
. . . . . . . . . 10
⊢ (𝑘 = 𝐸 → ((𝐷 ·s 𝑙) -s (𝐷 ·s 𝑘)) = ((𝐷 ·s 𝑙) -s (𝐷 ·s 𝐸))) |
220 | 217, 219 | breq12d 5155 |
. . . . . . . . 9
⊢ (𝑘 = 𝐸 → (((𝐶 ·s 𝑙) -s (𝐶 ·s 𝑘)) <s ((𝐷 ·s 𝑙) -s (𝐷 ·s 𝑘)) ↔ ((𝐶 ·s 𝑙) -s (𝐶 ·s 𝐸)) <s ((𝐷 ·s 𝑙) -s (𝐷 ·s 𝐸)))) |
221 | 215, 220 | imbi12d 344 |
. . . . . . . 8
⊢ (𝑘 = 𝐸 → (((𝐶 <s 𝐷 ∧ 𝑘 <s 𝑙) → ((𝐶 ·s 𝑙) -s (𝐶 ·s 𝑘)) <s ((𝐷 ·s 𝑙) -s (𝐷 ·s 𝑘))) ↔ ((𝐶 <s 𝐷 ∧ 𝐸 <s 𝑙) → ((𝐶 ·s 𝑙) -s (𝐶 ·s 𝐸)) <s ((𝐷 ·s 𝑙) -s (𝐷 ·s 𝐸))))) |
222 | 221 | anbi2d 629 |
. . . . . . 7
⊢ (𝑘 = 𝐸 → (((𝐴 ·s 𝐵) ∈ No
∧ ((𝐶 <s 𝐷 ∧ 𝑘 <s 𝑙) → ((𝐶 ·s 𝑙) -s (𝐶 ·s 𝑘)) <s ((𝐷 ·s 𝑙) -s (𝐷 ·s 𝑘)))) ↔ ((𝐴 ·s 𝐵) ∈ No
∧ ((𝐶 <s 𝐷 ∧ 𝐸 <s 𝑙) → ((𝐶 ·s 𝑙) -s (𝐶 ·s 𝐸)) <s ((𝐷 ·s 𝑙) -s (𝐷 ·s 𝐸)))))) |
223 | 213, 222 | imbi12d 344 |
. . . . . 6
⊢ (𝑘 = 𝐸 → ((𝑥 = ((( bday
‘𝐴) +no ( bday ‘𝐵)) ∪ (((( bday
‘𝐶) +no ( bday ‘𝑘)) ∪ (( bday
‘𝐷) +no ( bday ‘𝑙))) ∪ ((( bday
‘𝐶) +no ( bday ‘𝑙)) ∪ (( bday
‘𝐷) +no ( bday ‘𝑘))))) → ((𝐴 ·s 𝐵) ∈ No
∧ ((𝐶 <s 𝐷 ∧ 𝑘 <s 𝑙) → ((𝐶 ·s 𝑙) -s (𝐶 ·s 𝑘)) <s ((𝐷 ·s 𝑙) -s (𝐷 ·s 𝑘))))) ↔ (𝑥 = ((( bday
‘𝐴) +no ( bday ‘𝐵)) ∪ (((( bday
‘𝐶) +no ( bday ‘𝐸)) ∪ (( bday
‘𝐷) +no ( bday ‘𝑙))) ∪ ((( bday
‘𝐶) +no ( bday ‘𝑙)) ∪ (( bday
‘𝐷) +no ( bday ‘𝐸))))) → ((𝐴 ·s 𝐵) ∈ No
∧ ((𝐶 <s 𝐷 ∧ 𝐸 <s 𝑙) → ((𝐶 ·s 𝑙) -s (𝐶 ·s 𝐸)) <s ((𝐷 ·s 𝑙) -s (𝐷 ·s 𝐸))))))) |
224 | | fveq2 6879 |
. . . . . . . . . . . 12
⊢ (𝑙 = 𝐹 → ( bday
‘𝑙) = ( bday ‘𝐹)) |
225 | 224 | oveq2d 7410 |
. . . . . . . . . . 11
⊢ (𝑙 = 𝐹 → (( bday
‘𝐷) +no ( bday ‘𝑙)) = (( bday
‘𝐷) +no ( bday ‘𝐹))) |
226 | 225 | uneq2d 4160 |
. . . . . . . . . 10
⊢ (𝑙 = 𝐹 → ((( bday
‘𝐶) +no ( bday ‘𝐸)) ∪ (( bday
‘𝐷) +no ( bday ‘𝑙))) = ((( bday
‘𝐶) +no ( bday ‘𝐸)) ∪ (( bday
‘𝐷) +no ( bday ‘𝐹)))) |
227 | 224 | oveq2d 7410 |
. . . . . . . . . . 11
⊢ (𝑙 = 𝐹 → (( bday
‘𝐶) +no ( bday ‘𝑙)) = (( bday
‘𝐶) +no ( bday ‘𝐹))) |
228 | 227 | uneq1d 4159 |
. . . . . . . . . 10
⊢ (𝑙 = 𝐹 → ((( bday
‘𝐶) +no ( bday ‘𝑙)) ∪ (( bday
‘𝐷) +no ( bday ‘𝐸))) = ((( bday
‘𝐶) +no ( bday ‘𝐹)) ∪ (( bday
‘𝐷) +no ( bday ‘𝐸)))) |
229 | 226, 228 | uneq12d 4161 |
. . . . . . . . 9
⊢ (𝑙 = 𝐹 → (((( bday
‘𝐶) +no ( bday ‘𝐸)) ∪ (( bday
‘𝐷) +no ( bday ‘𝑙))) ∪ ((( bday
‘𝐶) +no ( bday ‘𝑙)) ∪ (( bday
‘𝐷) +no ( bday ‘𝐸)))) = (((( bday
‘𝐶) +no ( bday ‘𝐸)) ∪ (( bday
‘𝐷) +no ( bday ‘𝐹))) ∪ ((( bday
‘𝐶) +no ( bday ‘𝐹)) ∪ (( bday
‘𝐷) +no ( bday ‘𝐸))))) |
230 | 229 | uneq2d 4160 |
. . . . . . . 8
⊢ (𝑙 = 𝐹 → ((( bday
‘𝐴) +no ( bday ‘𝐵)) ∪ (((( bday
‘𝐶) +no ( bday ‘𝐸)) ∪ (( bday
‘𝐷) +no ( bday ‘𝑙))) ∪ ((( bday
‘𝐶) +no ( bday ‘𝑙)) ∪ (( bday
‘𝐷) +no ( bday ‘𝐸))))) = ((( bday
‘𝐴) +no ( bday ‘𝐵)) ∪ (((( bday
‘𝐶) +no ( bday ‘𝐸)) ∪ (( bday
‘𝐷) +no ( bday ‘𝐹))) ∪ ((( bday
‘𝐶) +no ( bday ‘𝐹)) ∪ (( bday
‘𝐷) +no ( bday ‘𝐸)))))) |
231 | 230 | eqeq2d 2743 |
. . . . . . 7
⊢ (𝑙 = 𝐹 → (𝑥 = ((( bday
‘𝐴) +no ( bday ‘𝐵)) ∪ (((( bday
‘𝐶) +no ( bday ‘𝐸)) ∪ (( bday
‘𝐷) +no ( bday ‘𝑙))) ∪ ((( bday
‘𝐶) +no ( bday ‘𝑙)) ∪ (( bday
‘𝐷) +no ( bday ‘𝐸))))) ↔ 𝑥 = ((( bday
‘𝐴) +no ( bday ‘𝐵)) ∪ (((( bday
‘𝐶) +no ( bday ‘𝐸)) ∪ (( bday
‘𝐷) +no ( bday ‘𝐹))) ∪ ((( bday
‘𝐶) +no ( bday ‘𝐹)) ∪ (( bday
‘𝐷) +no ( bday ‘𝐸))))))) |
232 | | breq2 5146 |
. . . . . . . . . 10
⊢ (𝑙 = 𝐹 → (𝐸 <s 𝑙 ↔ 𝐸 <s 𝐹)) |
233 | 232 | anbi2d 629 |
. . . . . . . . 9
⊢ (𝑙 = 𝐹 → ((𝐶 <s 𝐷 ∧ 𝐸 <s 𝑙) ↔ (𝐶 <s 𝐷 ∧ 𝐸 <s 𝐹))) |
234 | | oveq2 7402 |
. . . . . . . . . . 11
⊢ (𝑙 = 𝐹 → (𝐶 ·s 𝑙) = (𝐶 ·s 𝐹)) |
235 | 234 | oveq1d 7409 |
. . . . . . . . . 10
⊢ (𝑙 = 𝐹 → ((𝐶 ·s 𝑙) -s (𝐶 ·s 𝐸)) = ((𝐶 ·s 𝐹) -s (𝐶 ·s 𝐸))) |
236 | | oveq2 7402 |
. . . . . . . . . . 11
⊢ (𝑙 = 𝐹 → (𝐷 ·s 𝑙) = (𝐷 ·s 𝐹)) |
237 | 236 | oveq1d 7409 |
. . . . . . . . . 10
⊢ (𝑙 = 𝐹 → ((𝐷 ·s 𝑙) -s (𝐷 ·s 𝐸)) = ((𝐷 ·s 𝐹) -s (𝐷 ·s 𝐸))) |
238 | 235, 237 | breq12d 5155 |
. . . . . . . . 9
⊢ (𝑙 = 𝐹 → (((𝐶 ·s 𝑙) -s (𝐶 ·s 𝐸)) <s ((𝐷 ·s 𝑙) -s (𝐷 ·s 𝐸)) ↔ ((𝐶 ·s 𝐹) -s (𝐶 ·s 𝐸)) <s ((𝐷 ·s 𝐹) -s (𝐷 ·s 𝐸)))) |
239 | 233, 238 | imbi12d 344 |
. . . . . . . 8
⊢ (𝑙 = 𝐹 → (((𝐶 <s 𝐷 ∧ 𝐸 <s 𝑙) → ((𝐶 ·s 𝑙) -s (𝐶 ·s 𝐸)) <s ((𝐷 ·s 𝑙) -s (𝐷 ·s 𝐸))) ↔ ((𝐶 <s 𝐷 ∧ 𝐸 <s 𝐹) → ((𝐶 ·s 𝐹) -s (𝐶 ·s 𝐸)) <s ((𝐷 ·s 𝐹) -s (𝐷 ·s 𝐸))))) |
240 | 239 | anbi2d 629 |
. . . . . . 7
⊢ (𝑙 = 𝐹 → (((𝐴 ·s 𝐵) ∈ No
∧ ((𝐶 <s 𝐷 ∧ 𝐸 <s 𝑙) → ((𝐶 ·s 𝑙) -s (𝐶 ·s 𝐸)) <s ((𝐷 ·s 𝑙) -s (𝐷 ·s 𝐸)))) ↔ ((𝐴 ·s 𝐵) ∈ No
∧ ((𝐶 <s 𝐷 ∧ 𝐸 <s 𝐹) → ((𝐶 ·s 𝐹) -s (𝐶 ·s 𝐸)) <s ((𝐷 ·s 𝐹) -s (𝐷 ·s 𝐸)))))) |
241 | 231, 240 | imbi12d 344 |
. . . . . 6
⊢ (𝑙 = 𝐹 → ((𝑥 = ((( bday
‘𝐴) +no ( bday ‘𝐵)) ∪ (((( bday
‘𝐶) +no ( bday ‘𝐸)) ∪ (( bday
‘𝐷) +no ( bday ‘𝑙))) ∪ ((( bday
‘𝐶) +no ( bday ‘𝑙)) ∪ (( bday
‘𝐷) +no ( bday ‘𝐸))))) → ((𝐴 ·s 𝐵) ∈ No
∧ ((𝐶 <s 𝐷 ∧ 𝐸 <s 𝑙) → ((𝐶 ·s 𝑙) -s (𝐶 ·s 𝐸)) <s ((𝐷 ·s 𝑙) -s (𝐷 ·s 𝐸))))) ↔ (𝑥 = ((( bday
‘𝐴) +no ( bday ‘𝐵)) ∪ (((( bday
‘𝐶) +no ( bday ‘𝐸)) ∪ (( bday
‘𝐷) +no ( bday ‘𝐹))) ∪ ((( bday
‘𝐶) +no ( bday ‘𝐹)) ∪ (( bday
‘𝐷) +no ( bday ‘𝐸))))) → ((𝐴 ·s 𝐵) ∈ No
∧ ((𝐶 <s 𝐷 ∧ 𝐸 <s 𝐹) → ((𝐶 ·s 𝐹) -s (𝐶 ·s 𝐸)) <s ((𝐷 ·s 𝐹) -s (𝐷 ·s 𝐸))))))) |
242 | 163, 171,
188, 205, 223, 241 | rspc6v 3628 |
. . . . 5
⊢ (((𝐴 ∈
No ∧ 𝐵 ∈
No ) ∧ (𝐶 ∈ No
∧ 𝐷 ∈ No ) ∧ (𝐸 ∈ No
∧ 𝐹 ∈ No )) → (∀𝑔 ∈ No
∀ℎ ∈ No ∀𝑖 ∈ No
∀𝑗 ∈ No ∀𝑘 ∈ No
∀𝑙 ∈ No (𝑥 = ((( bday
‘𝑔) +no ( bday ‘ℎ)) ∪ (((( bday
‘𝑖) +no ( bday ‘𝑘)) ∪ (( bday
‘𝑗) +no ( bday ‘𝑙))) ∪ ((( bday
‘𝑖) +no ( bday ‘𝑙)) ∪ (( bday
‘𝑗) +no ( bday ‘𝑘))))) → ((𝑔 ·s ℎ) ∈ No
∧ ((𝑖 <s 𝑗 ∧ 𝑘 <s 𝑙) → ((𝑖 ·s 𝑙) -s (𝑖 ·s 𝑘)) <s ((𝑗 ·s 𝑙) -s (𝑗 ·s 𝑘))))) → (𝑥 = ((( bday
‘𝐴) +no ( bday ‘𝐵)) ∪ (((( bday
‘𝐶) +no ( bday ‘𝐸)) ∪ (( bday
‘𝐷) +no ( bday ‘𝐹))) ∪ ((( bday
‘𝐶) +no ( bday ‘𝐹)) ∪ (( bday
‘𝐷) +no ( bday ‘𝐸))))) → ((𝐴 ·s 𝐵) ∈ No
∧ ((𝐶 <s 𝐷 ∧ 𝐸 <s 𝐹) → ((𝐶 ·s 𝐹) -s (𝐶 ·s 𝐸)) <s ((𝐷 ·s 𝐹) -s (𝐷 ·s 𝐸))))))) |
243 | 155, 242 | syl5com 31 |
. . . 4
⊢ (𝑥 ∈ On → (((𝐴 ∈
No ∧ 𝐵 ∈
No ) ∧ (𝐶 ∈ No
∧ 𝐷 ∈ No ) ∧ (𝐸 ∈ No
∧ 𝐹 ∈ No )) → (𝑥 = ((( bday
‘𝐴) +no ( bday ‘𝐵)) ∪ (((( bday
‘𝐶) +no ( bday ‘𝐸)) ∪ (( bday
‘𝐷) +no ( bday ‘𝐹))) ∪ ((( bday
‘𝐶) +no ( bday ‘𝐹)) ∪ (( bday
‘𝐷) +no ( bday ‘𝐸))))) → ((𝐴 ·s 𝐵) ∈ No
∧ ((𝐶 <s 𝐷 ∧ 𝐸 <s 𝐹) → ((𝐶 ·s 𝐹) -s (𝐶 ·s 𝐸)) <s ((𝐷 ·s 𝐹) -s (𝐷 ·s 𝐸))))))) |
244 | 243 | com23 86 |
. . 3
⊢ (𝑥 ∈ On → (𝑥 = (((
bday ‘𝐴) +no
( bday ‘𝐵)) ∪ (((( bday
‘𝐶) +no ( bday ‘𝐸)) ∪ (( bday
‘𝐷) +no ( bday ‘𝐹))) ∪ ((( bday
‘𝐶) +no ( bday ‘𝐹)) ∪ (( bday
‘𝐷) +no ( bday ‘𝐸))))) → (((𝐴 ∈ No
∧ 𝐵 ∈ No ) ∧ (𝐶 ∈ No
∧ 𝐷 ∈ No ) ∧ (𝐸 ∈ No
∧ 𝐹 ∈ No )) → ((𝐴 ·s 𝐵) ∈ No
∧ ((𝐶 <s 𝐷 ∧ 𝐸 <s 𝐹) → ((𝐶 ·s 𝐹) -s (𝐶 ·s 𝐸)) <s ((𝐷 ·s 𝐹) -s (𝐷 ·s 𝐸))))))) |
245 | 244 | rexlimiv 3148 |
. 2
⊢
(∃𝑥 ∈ On
𝑥 = ((( bday ‘𝐴) +no ( bday
‘𝐵)) ∪
(((( bday ‘𝐶) +no ( bday
‘𝐸)) ∪
(( bday ‘𝐷) +no ( bday
‘𝐹))) ∪
((( bday ‘𝐶) +no ( bday
‘𝐹)) ∪
(( bday ‘𝐷) +no ( bday
‘𝐸))))) →
(((𝐴 ∈ No ∧ 𝐵 ∈ No )
∧ (𝐶 ∈ No ∧ 𝐷 ∈ No )
∧ (𝐸 ∈ No ∧ 𝐹 ∈ No ))
→ ((𝐴
·s 𝐵)
∈ No ∧ ((𝐶 <s 𝐷 ∧ 𝐸 <s 𝐹) → ((𝐶 ·s 𝐹) -s (𝐶 ·s 𝐸)) <s ((𝐷 ·s 𝐹) -s (𝐷 ·s 𝐸)))))) |
246 | 22, 245 | ax-mp 5 |
1
⊢ (((𝐴 ∈
No ∧ 𝐵 ∈
No ) ∧ (𝐶 ∈ No
∧ 𝐷 ∈ No ) ∧ (𝐸 ∈ No
∧ 𝐹 ∈ No )) → ((𝐴 ·s 𝐵) ∈ No
∧ ((𝐶 <s 𝐷 ∧ 𝐸 <s 𝐹) → ((𝐶 ·s 𝐹) -s (𝐶 ·s 𝐸)) <s ((𝐷 ·s 𝐹) -s (𝐷 ·s 𝐸))))) |