Step | Hyp | Ref
| Expression |
1 | | fvex 6902 |
. . . . 5
⊢ ( L
‘𝑋) ∈
V |
2 | 1 | abrexex 7946 |
. . . 4
⊢ {𝑝 ∣ ∃𝑙 ∈ ( L ‘𝑋)𝑝 = (𝑙 +s 𝑌)} ∈ V |
3 | 2 | a1i 11 |
. . 3
⊢ (𝜑 → {𝑝 ∣ ∃𝑙 ∈ ( L ‘𝑋)𝑝 = (𝑙 +s 𝑌)} ∈ V) |
4 | | fvex 6902 |
. . . . 5
⊢ ( L
‘𝑌) ∈
V |
5 | 4 | abrexex 7946 |
. . . 4
⊢ {𝑞 ∣ ∃𝑚 ∈ ( L ‘𝑌)𝑞 = (𝑋 +s 𝑚)} ∈ V |
6 | 5 | a1i 11 |
. . 3
⊢ (𝜑 → {𝑞 ∣ ∃𝑚 ∈ ( L ‘𝑌)𝑞 = (𝑋 +s 𝑚)} ∈ V) |
7 | 3, 6 | unexd 7738 |
. 2
⊢ (𝜑 → ({𝑝 ∣ ∃𝑙 ∈ ( L ‘𝑋)𝑝 = (𝑙 +s 𝑌)} ∪ {𝑞 ∣ ∃𝑚 ∈ ( L ‘𝑌)𝑞 = (𝑋 +s 𝑚)}) ∈ V) |
8 | | fvex 6902 |
. . . . 5
⊢ ( R
‘𝑋) ∈
V |
9 | 8 | abrexex 7946 |
. . . 4
⊢ {𝑤 ∣ ∃𝑟 ∈ ( R ‘𝑋)𝑤 = (𝑟 +s 𝑌)} ∈ V |
10 | 9 | a1i 11 |
. . 3
⊢ (𝜑 → {𝑤 ∣ ∃𝑟 ∈ ( R ‘𝑋)𝑤 = (𝑟 +s 𝑌)} ∈ V) |
11 | | fvex 6902 |
. . . . 5
⊢ ( R
‘𝑌) ∈
V |
12 | 11 | abrexex 7946 |
. . . 4
⊢ {𝑡 ∣ ∃𝑠 ∈ ( R ‘𝑌)𝑡 = (𝑋 +s 𝑠)} ∈ V |
13 | 12 | a1i 11 |
. . 3
⊢ (𝜑 → {𝑡 ∣ ∃𝑠 ∈ ( R ‘𝑌)𝑡 = (𝑋 +s 𝑠)} ∈ V) |
14 | 10, 13 | unexd 7738 |
. 2
⊢ (𝜑 → ({𝑤 ∣ ∃𝑟 ∈ ( R ‘𝑋)𝑤 = (𝑟 +s 𝑌)} ∪ {𝑡 ∣ ∃𝑠 ∈ ( R ‘𝑌)𝑡 = (𝑋 +s 𝑠)}) ∈ V) |
15 | | addsproplem.1 |
. . . . . . . . 9
⊢ (𝜑 → ∀𝑥 ∈ No
∀𝑦 ∈ No ∀𝑧 ∈ No
(((( bday ‘𝑥) +no ( bday
‘𝑦)) ∪
(( bday ‘𝑥) +no ( bday
‘𝑧))) ∈
((( bday ‘𝑋) +no ( bday
‘𝑌)) ∪
(( bday ‘𝑋) +no ( bday
‘𝑍))) →
((𝑥 +s 𝑦) ∈
No ∧ (𝑦 <s
𝑧 → (𝑦 +s 𝑥) <s (𝑧 +s 𝑥))))) |
16 | 15 | adantr 482 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑙 ∈ ( L ‘𝑋)) → ∀𝑥 ∈ No
∀𝑦 ∈ No ∀𝑧 ∈ No
(((( bday ‘𝑥) +no ( bday
‘𝑦)) ∪
(( bday ‘𝑥) +no ( bday
‘𝑧))) ∈
((( bday ‘𝑋) +no ( bday
‘𝑌)) ∪
(( bday ‘𝑋) +no ( bday
‘𝑍))) →
((𝑥 +s 𝑦) ∈
No ∧ (𝑦 <s
𝑧 → (𝑦 +s 𝑥) <s (𝑧 +s 𝑥))))) |
17 | | leftssno 27365 |
. . . . . . . . . 10
⊢ ( L
‘𝑋) ⊆ No |
18 | 17 | sseli 3978 |
. . . . . . . . 9
⊢ (𝑙 ∈ ( L ‘𝑋) → 𝑙 ∈ No
) |
19 | 18 | adantl 483 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑙 ∈ ( L ‘𝑋)) → 𝑙 ∈ No
) |
20 | | addsproplem2.3 |
. . . . . . . . 9
⊢ (𝜑 → 𝑌 ∈ No
) |
21 | 20 | adantr 482 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑙 ∈ ( L ‘𝑋)) → 𝑌 ∈ No
) |
22 | | 0sno 27317 |
. . . . . . . . 9
⊢
0s ∈ No |
23 | 22 | a1i 11 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑙 ∈ ( L ‘𝑋)) → 0s ∈ No ) |
24 | | bday0s 27319 |
. . . . . . . . . . . . 13
⊢ ( bday ‘ 0s ) = ∅ |
25 | 24 | oveq2i 7417 |
. . . . . . . . . . . 12
⊢ (( bday ‘𝑙) +no ( bday
‘ 0s )) = (( bday
‘𝑙) +no
∅) |
26 | | bdayelon 27268 |
. . . . . . . . . . . . 13
⊢ ( bday ‘𝑙) ∈ On |
27 | | naddrid 8679 |
. . . . . . . . . . . . 13
⊢ (( bday ‘𝑙) ∈ On → ((
bday ‘𝑙) +no
∅) = ( bday ‘𝑙)) |
28 | 26, 27 | ax-mp 5 |
. . . . . . . . . . . 12
⊢ (( bday ‘𝑙) +no ∅) = ( bday
‘𝑙) |
29 | 25, 28 | eqtri 2761 |
. . . . . . . . . . 11
⊢ (( bday ‘𝑙) +no ( bday
‘ 0s )) = ( bday
‘𝑙) |
30 | 29 | uneq2i 4160 |
. . . . . . . . . 10
⊢ ((( bday ‘𝑙) +no ( bday
‘𝑌)) ∪
(( bday ‘𝑙) +no ( bday
‘ 0s ))) = ((( bday
‘𝑙) +no ( bday ‘𝑌)) ∪ ( bday
‘𝑙)) |
31 | | bdayelon 27268 |
. . . . . . . . . . . 12
⊢ ( bday ‘𝑌) ∈ On |
32 | | naddword1 8687 |
. . . . . . . . . . . 12
⊢ ((( bday ‘𝑙) ∈ On ∧ (
bday ‘𝑌)
∈ On) → ( bday ‘𝑙) ⊆ (( bday
‘𝑙) +no ( bday ‘𝑌))) |
33 | 26, 31, 32 | mp2an 691 |
. . . . . . . . . . 11
⊢ ( bday ‘𝑙) ⊆ (( bday
‘𝑙) +no ( bday ‘𝑌)) |
34 | | ssequn2 4183 |
. . . . . . . . . . 11
⊢ (( bday ‘𝑙) ⊆ (( bday
‘𝑙) +no ( bday ‘𝑌)) ↔ ((( bday
‘𝑙) +no ( bday ‘𝑌)) ∪ ( bday
‘𝑙)) = (( bday ‘𝑙) +no ( bday
‘𝑌))) |
35 | 33, 34 | mpbi 229 |
. . . . . . . . . 10
⊢ ((( bday ‘𝑙) +no ( bday
‘𝑌)) ∪
( bday ‘𝑙)) = (( bday
‘𝑙) +no ( bday ‘𝑌)) |
36 | 30, 35 | eqtri 2761 |
. . . . . . . . 9
⊢ ((( bday ‘𝑙) +no ( bday
‘𝑌)) ∪
(( bday ‘𝑙) +no ( bday
‘ 0s ))) = (( bday
‘𝑙) +no ( bday ‘𝑌)) |
37 | | leftssold 27363 |
. . . . . . . . . . . . . 14
⊢ ( L
‘𝑋) ⊆ ( O
‘( bday ‘𝑋)) |
38 | 37 | sseli 3978 |
. . . . . . . . . . . . 13
⊢ (𝑙 ∈ ( L ‘𝑋) → 𝑙 ∈ ( O ‘(
bday ‘𝑋))) |
39 | | bdayelon 27268 |
. . . . . . . . . . . . . 14
⊢ ( bday ‘𝑋) ∈ On |
40 | | oldbday 27385 |
. . . . . . . . . . . . . 14
⊢ ((( bday ‘𝑋) ∈ On ∧ 𝑙 ∈ No )
→ (𝑙 ∈ ( O
‘( bday ‘𝑋)) ↔ ( bday
‘𝑙) ∈
( bday ‘𝑋))) |
41 | 39, 18, 40 | sylancr 588 |
. . . . . . . . . . . . 13
⊢ (𝑙 ∈ ( L ‘𝑋) → (𝑙 ∈ ( O ‘(
bday ‘𝑋))
↔ ( bday ‘𝑙) ∈ ( bday
‘𝑋))) |
42 | 38, 41 | mpbid 231 |
. . . . . . . . . . . 12
⊢ (𝑙 ∈ ( L ‘𝑋) → (
bday ‘𝑙)
∈ ( bday ‘𝑋)) |
43 | | naddel1 8683 |
. . . . . . . . . . . . 13
⊢ ((( bday ‘𝑙) ∈ On ∧ (
bday ‘𝑋)
∈ On ∧ ( bday ‘𝑌) ∈ On) → ((
bday ‘𝑙)
∈ ( bday ‘𝑋) ↔ (( bday
‘𝑙) +no ( bday ‘𝑌)) ∈ (( bday
‘𝑋) +no ( bday ‘𝑌)))) |
44 | 26, 39, 31, 43 | mp3an 1462 |
. . . . . . . . . . . 12
⊢ (( bday ‘𝑙) ∈ ( bday
‘𝑋) ↔
(( bday ‘𝑙) +no ( bday
‘𝑌)) ∈
(( bday ‘𝑋) +no ( bday
‘𝑌))) |
45 | 42, 44 | sylib 217 |
. . . . . . . . . . 11
⊢ (𝑙 ∈ ( L ‘𝑋) → (( bday ‘𝑙) +no ( bday
‘𝑌)) ∈
(( bday ‘𝑋) +no ( bday
‘𝑌))) |
46 | 45 | adantl 483 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑙 ∈ ( L ‘𝑋)) → (( bday
‘𝑙) +no ( bday ‘𝑌)) ∈ (( bday
‘𝑋) +no ( bday ‘𝑌))) |
47 | | elun1 4176 |
. . . . . . . . . 10
⊢ ((( bday ‘𝑙) +no ( bday
‘𝑌)) ∈
(( bday ‘𝑋) +no ( bday
‘𝑌)) →
(( bday ‘𝑙) +no ( bday
‘𝑌)) ∈
((( bday ‘𝑋) +no ( bday
‘𝑌)) ∪
(( bday ‘𝑋) +no ( bday
‘𝑍)))) |
48 | 46, 47 | syl 17 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑙 ∈ ( L ‘𝑋)) → (( bday
‘𝑙) +no ( bday ‘𝑌)) ∈ ((( bday
‘𝑋) +no ( bday ‘𝑌)) ∪ (( bday
‘𝑋) +no ( bday ‘𝑍)))) |
49 | 36, 48 | eqeltrid 2838 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑙 ∈ ( L ‘𝑋)) → ((( bday
‘𝑙) +no ( bday ‘𝑌)) ∪ (( bday
‘𝑙) +no ( bday ‘ 0s ))) ∈ ((( bday ‘𝑋) +no ( bday
‘𝑌)) ∪
(( bday ‘𝑋) +no ( bday
‘𝑍)))) |
50 | 16, 19, 21, 23, 49 | addsproplem1 27443 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑙 ∈ ( L ‘𝑋)) → ((𝑙 +s 𝑌) ∈ No
∧ (𝑌 <s
0s → (𝑌
+s 𝑙) <s (
0s +s 𝑙)))) |
51 | 50 | simpld 496 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑙 ∈ ( L ‘𝑋)) → (𝑙 +s 𝑌) ∈ No
) |
52 | | eleq1a 2829 |
. . . . . 6
⊢ ((𝑙 +s 𝑌) ∈ No
→ (𝑝 = (𝑙 +s 𝑌) → 𝑝 ∈ No
)) |
53 | 51, 52 | syl 17 |
. . . . 5
⊢ ((𝜑 ∧ 𝑙 ∈ ( L ‘𝑋)) → (𝑝 = (𝑙 +s 𝑌) → 𝑝 ∈ No
)) |
54 | 53 | rexlimdva 3156 |
. . . 4
⊢ (𝜑 → (∃𝑙 ∈ ( L ‘𝑋)𝑝 = (𝑙 +s 𝑌) → 𝑝 ∈ No
)) |
55 | 54 | abssdv 4065 |
. . 3
⊢ (𝜑 → {𝑝 ∣ ∃𝑙 ∈ ( L ‘𝑋)𝑝 = (𝑙 +s 𝑌)} ⊆ No
) |
56 | 15 | adantr 482 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑚 ∈ ( L ‘𝑌)) → ∀𝑥 ∈ No
∀𝑦 ∈ No ∀𝑧 ∈ No
(((( bday ‘𝑥) +no ( bday
‘𝑦)) ∪
(( bday ‘𝑥) +no ( bday
‘𝑧))) ∈
((( bday ‘𝑋) +no ( bday
‘𝑌)) ∪
(( bday ‘𝑋) +no ( bday
‘𝑍))) →
((𝑥 +s 𝑦) ∈
No ∧ (𝑦 <s
𝑧 → (𝑦 +s 𝑥) <s (𝑧 +s 𝑥))))) |
57 | | addsproplem2.2 |
. . . . . . . . 9
⊢ (𝜑 → 𝑋 ∈ No
) |
58 | 57 | adantr 482 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑚 ∈ ( L ‘𝑌)) → 𝑋 ∈ No
) |
59 | | leftssno 27365 |
. . . . . . . . . 10
⊢ ( L
‘𝑌) ⊆ No |
60 | 59 | sseli 3978 |
. . . . . . . . 9
⊢ (𝑚 ∈ ( L ‘𝑌) → 𝑚 ∈ No
) |
61 | 60 | adantl 483 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑚 ∈ ( L ‘𝑌)) → 𝑚 ∈ No
) |
62 | 22 | a1i 11 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑚 ∈ ( L ‘𝑌)) → 0s ∈ No ) |
63 | 24 | oveq2i 7417 |
. . . . . . . . . . . 12
⊢ (( bday ‘𝑋) +no ( bday
‘ 0s )) = (( bday
‘𝑋) +no
∅) |
64 | | naddrid 8679 |
. . . . . . . . . . . . 13
⊢ (( bday ‘𝑋) ∈ On → ((
bday ‘𝑋) +no
∅) = ( bday ‘𝑋)) |
65 | 39, 64 | ax-mp 5 |
. . . . . . . . . . . 12
⊢ (( bday ‘𝑋) +no ∅) = (
bday ‘𝑋) |
66 | 63, 65 | eqtri 2761 |
. . . . . . . . . . 11
⊢ (( bday ‘𝑋) +no ( bday
‘ 0s )) = ( bday
‘𝑋) |
67 | 66 | uneq2i 4160 |
. . . . . . . . . 10
⊢ ((( bday ‘𝑋) +no ( bday
‘𝑚)) ∪
(( bday ‘𝑋) +no ( bday
‘ 0s ))) = ((( bday
‘𝑋) +no ( bday ‘𝑚)) ∪ ( bday
‘𝑋)) |
68 | | bdayelon 27268 |
. . . . . . . . . . . 12
⊢ ( bday ‘𝑚) ∈ On |
69 | | naddword1 8687 |
. . . . . . . . . . . 12
⊢ ((( bday ‘𝑋) ∈ On ∧ (
bday ‘𝑚)
∈ On) → ( bday ‘𝑋) ⊆ (( bday
‘𝑋) +no ( bday ‘𝑚))) |
70 | 39, 68, 69 | mp2an 691 |
. . . . . . . . . . 11
⊢ ( bday ‘𝑋) ⊆ (( bday
‘𝑋) +no ( bday ‘𝑚)) |
71 | | ssequn2 4183 |
. . . . . . . . . . 11
⊢ (( bday ‘𝑋) ⊆ (( bday
‘𝑋) +no ( bday ‘𝑚)) ↔ ((( bday
‘𝑋) +no ( bday ‘𝑚)) ∪ ( bday
‘𝑋)) = (( bday ‘𝑋) +no ( bday
‘𝑚))) |
72 | 70, 71 | mpbi 229 |
. . . . . . . . . 10
⊢ ((( bday ‘𝑋) +no ( bday
‘𝑚)) ∪
( bday ‘𝑋)) = (( bday
‘𝑋) +no ( bday ‘𝑚)) |
73 | 67, 72 | eqtri 2761 |
. . . . . . . . 9
⊢ ((( bday ‘𝑋) +no ( bday
‘𝑚)) ∪
(( bday ‘𝑋) +no ( bday
‘ 0s ))) = (( bday
‘𝑋) +no ( bday ‘𝑚)) |
74 | | leftssold 27363 |
. . . . . . . . . . . . . 14
⊢ ( L
‘𝑌) ⊆ ( O
‘( bday ‘𝑌)) |
75 | 74 | sseli 3978 |
. . . . . . . . . . . . 13
⊢ (𝑚 ∈ ( L ‘𝑌) → 𝑚 ∈ ( O ‘(
bday ‘𝑌))) |
76 | | oldbday 27385 |
. . . . . . . . . . . . . 14
⊢ ((( bday ‘𝑌) ∈ On ∧ 𝑚 ∈ No )
→ (𝑚 ∈ ( O
‘( bday ‘𝑌)) ↔ ( bday
‘𝑚) ∈
( bday ‘𝑌))) |
77 | 31, 60, 76 | sylancr 588 |
. . . . . . . . . . . . 13
⊢ (𝑚 ∈ ( L ‘𝑌) → (𝑚 ∈ ( O ‘(
bday ‘𝑌))
↔ ( bday ‘𝑚) ∈ ( bday
‘𝑌))) |
78 | 75, 77 | mpbid 231 |
. . . . . . . . . . . 12
⊢ (𝑚 ∈ ( L ‘𝑌) → (
bday ‘𝑚)
∈ ( bday ‘𝑌)) |
79 | | naddel2 8684 |
. . . . . . . . . . . . 13
⊢ ((( bday ‘𝑚) ∈ On ∧ (
bday ‘𝑌)
∈ On ∧ ( bday ‘𝑋) ∈ On) → ((
bday ‘𝑚)
∈ ( bday ‘𝑌) ↔ (( bday
‘𝑋) +no ( bday ‘𝑚)) ∈ (( bday
‘𝑋) +no ( bday ‘𝑌)))) |
80 | 68, 31, 39, 79 | mp3an 1462 |
. . . . . . . . . . . 12
⊢ (( bday ‘𝑚) ∈ ( bday
‘𝑌) ↔
(( bday ‘𝑋) +no ( bday
‘𝑚)) ∈
(( bday ‘𝑋) +no ( bday
‘𝑌))) |
81 | 78, 80 | sylib 217 |
. . . . . . . . . . 11
⊢ (𝑚 ∈ ( L ‘𝑌) → (( bday ‘𝑋) +no ( bday
‘𝑚)) ∈
(( bday ‘𝑋) +no ( bday
‘𝑌))) |
82 | 81 | adantl 483 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑚 ∈ ( L ‘𝑌)) → (( bday
‘𝑋) +no ( bday ‘𝑚)) ∈ (( bday
‘𝑋) +no ( bday ‘𝑌))) |
83 | | elun1 4176 |
. . . . . . . . . 10
⊢ ((( bday ‘𝑋) +no ( bday
‘𝑚)) ∈
(( bday ‘𝑋) +no ( bday
‘𝑌)) →
(( bday ‘𝑋) +no ( bday
‘𝑚)) ∈
((( bday ‘𝑋) +no ( bday
‘𝑌)) ∪
(( bday ‘𝑋) +no ( bday
‘𝑍)))) |
84 | 82, 83 | syl 17 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑚 ∈ ( L ‘𝑌)) → (( bday
‘𝑋) +no ( bday ‘𝑚)) ∈ ((( bday
‘𝑋) +no ( bday ‘𝑌)) ∪ (( bday
‘𝑋) +no ( bday ‘𝑍)))) |
85 | 73, 84 | eqeltrid 2838 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑚 ∈ ( L ‘𝑌)) → ((( bday
‘𝑋) +no ( bday ‘𝑚)) ∪ (( bday
‘𝑋) +no ( bday ‘ 0s ))) ∈ ((( bday ‘𝑋) +no ( bday
‘𝑌)) ∪
(( bday ‘𝑋) +no ( bday
‘𝑍)))) |
86 | 56, 58, 61, 62, 85 | addsproplem1 27443 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑚 ∈ ( L ‘𝑌)) → ((𝑋 +s 𝑚) ∈ No
∧ (𝑚 <s
0s → (𝑚
+s 𝑋) <s (
0s +s 𝑋)))) |
87 | 86 | simpld 496 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑚 ∈ ( L ‘𝑌)) → (𝑋 +s 𝑚) ∈ No
) |
88 | | eleq1a 2829 |
. . . . . 6
⊢ ((𝑋 +s 𝑚) ∈
No → (𝑞 =
(𝑋 +s 𝑚) → 𝑞 ∈ No
)) |
89 | 87, 88 | syl 17 |
. . . . 5
⊢ ((𝜑 ∧ 𝑚 ∈ ( L ‘𝑌)) → (𝑞 = (𝑋 +s 𝑚) → 𝑞 ∈ No
)) |
90 | 89 | rexlimdva 3156 |
. . . 4
⊢ (𝜑 → (∃𝑚 ∈ ( L ‘𝑌)𝑞 = (𝑋 +s 𝑚) → 𝑞 ∈ No
)) |
91 | 90 | abssdv 4065 |
. . 3
⊢ (𝜑 → {𝑞 ∣ ∃𝑚 ∈ ( L ‘𝑌)𝑞 = (𝑋 +s 𝑚)} ⊆ No
) |
92 | 55, 91 | unssd 4186 |
. 2
⊢ (𝜑 → ({𝑝 ∣ ∃𝑙 ∈ ( L ‘𝑋)𝑝 = (𝑙 +s 𝑌)} ∪ {𝑞 ∣ ∃𝑚 ∈ ( L ‘𝑌)𝑞 = (𝑋 +s 𝑚)}) ⊆ No
) |
93 | 15 | adantr 482 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑟 ∈ ( R ‘𝑋)) → ∀𝑥 ∈ No
∀𝑦 ∈ No ∀𝑧 ∈ No
(((( bday ‘𝑥) +no ( bday
‘𝑦)) ∪
(( bday ‘𝑥) +no ( bday
‘𝑧))) ∈
((( bday ‘𝑋) +no ( bday
‘𝑌)) ∪
(( bday ‘𝑋) +no ( bday
‘𝑍))) →
((𝑥 +s 𝑦) ∈
No ∧ (𝑦 <s
𝑧 → (𝑦 +s 𝑥) <s (𝑧 +s 𝑥))))) |
94 | | rightssno 27366 |
. . . . . . . . . 10
⊢ ( R
‘𝑋) ⊆ No |
95 | 94 | sseli 3978 |
. . . . . . . . 9
⊢ (𝑟 ∈ ( R ‘𝑋) → 𝑟 ∈ No
) |
96 | 95 | adantl 483 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑟 ∈ ( R ‘𝑋)) → 𝑟 ∈ No
) |
97 | 20 | adantr 482 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑟 ∈ ( R ‘𝑋)) → 𝑌 ∈ No
) |
98 | 22 | a1i 11 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑟 ∈ ( R ‘𝑋)) → 0s ∈ No ) |
99 | 24 | oveq2i 7417 |
. . . . . . . . . . . 12
⊢ (( bday ‘𝑟) +no ( bday
‘ 0s )) = (( bday
‘𝑟) +no
∅) |
100 | | bdayelon 27268 |
. . . . . . . . . . . . 13
⊢ ( bday ‘𝑟) ∈ On |
101 | | naddrid 8679 |
. . . . . . . . . . . . 13
⊢ (( bday ‘𝑟) ∈ On → ((
bday ‘𝑟) +no
∅) = ( bday ‘𝑟)) |
102 | 100, 101 | ax-mp 5 |
. . . . . . . . . . . 12
⊢ (( bday ‘𝑟) +no ∅) = ( bday
‘𝑟) |
103 | 99, 102 | eqtri 2761 |
. . . . . . . . . . 11
⊢ (( bday ‘𝑟) +no ( bday
‘ 0s )) = ( bday
‘𝑟) |
104 | 103 | uneq2i 4160 |
. . . . . . . . . 10
⊢ ((( bday ‘𝑟) +no ( bday
‘𝑌)) ∪
(( bday ‘𝑟) +no ( bday
‘ 0s ))) = ((( bday
‘𝑟) +no ( bday ‘𝑌)) ∪ ( bday
‘𝑟)) |
105 | | naddword1 8687 |
. . . . . . . . . . . 12
⊢ ((( bday ‘𝑟) ∈ On ∧ (
bday ‘𝑌)
∈ On) → ( bday ‘𝑟) ⊆ (( bday
‘𝑟) +no ( bday ‘𝑌))) |
106 | 100, 31, 105 | mp2an 691 |
. . . . . . . . . . 11
⊢ ( bday ‘𝑟) ⊆ (( bday
‘𝑟) +no ( bday ‘𝑌)) |
107 | | ssequn2 4183 |
. . . . . . . . . . 11
⊢ (( bday ‘𝑟) ⊆ (( bday
‘𝑟) +no ( bday ‘𝑌)) ↔ ((( bday
‘𝑟) +no ( bday ‘𝑌)) ∪ ( bday
‘𝑟)) = (( bday ‘𝑟) +no ( bday
‘𝑌))) |
108 | 106, 107 | mpbi 229 |
. . . . . . . . . 10
⊢ ((( bday ‘𝑟) +no ( bday
‘𝑌)) ∪
( bday ‘𝑟)) = (( bday
‘𝑟) +no ( bday ‘𝑌)) |
109 | 104, 108 | eqtri 2761 |
. . . . . . . . 9
⊢ ((( bday ‘𝑟) +no ( bday
‘𝑌)) ∪
(( bday ‘𝑟) +no ( bday
‘ 0s ))) = (( bday
‘𝑟) +no ( bday ‘𝑌)) |
110 | | rightssold 27364 |
. . . . . . . . . . . . . 14
⊢ ( R
‘𝑋) ⊆ ( O
‘( bday ‘𝑋)) |
111 | 110 | sseli 3978 |
. . . . . . . . . . . . 13
⊢ (𝑟 ∈ ( R ‘𝑋) → 𝑟 ∈ ( O ‘(
bday ‘𝑋))) |
112 | | oldbday 27385 |
. . . . . . . . . . . . . 14
⊢ ((( bday ‘𝑋) ∈ On ∧ 𝑟 ∈ No )
→ (𝑟 ∈ ( O
‘( bday ‘𝑋)) ↔ ( bday
‘𝑟) ∈
( bday ‘𝑋))) |
113 | 39, 95, 112 | sylancr 588 |
. . . . . . . . . . . . 13
⊢ (𝑟 ∈ ( R ‘𝑋) → (𝑟 ∈ ( O ‘(
bday ‘𝑋))
↔ ( bday ‘𝑟) ∈ ( bday
‘𝑋))) |
114 | 111, 113 | mpbid 231 |
. . . . . . . . . . . 12
⊢ (𝑟 ∈ ( R ‘𝑋) → (
bday ‘𝑟)
∈ ( bday ‘𝑋)) |
115 | | naddel1 8683 |
. . . . . . . . . . . . 13
⊢ ((( bday ‘𝑟) ∈ On ∧ (
bday ‘𝑋)
∈ On ∧ ( bday ‘𝑌) ∈ On) → ((
bday ‘𝑟)
∈ ( bday ‘𝑋) ↔ (( bday
‘𝑟) +no ( bday ‘𝑌)) ∈ (( bday
‘𝑋) +no ( bday ‘𝑌)))) |
116 | 100, 39, 31, 115 | mp3an 1462 |
. . . . . . . . . . . 12
⊢ (( bday ‘𝑟) ∈ ( bday
‘𝑋) ↔
(( bday ‘𝑟) +no ( bday
‘𝑌)) ∈
(( bday ‘𝑋) +no ( bday
‘𝑌))) |
117 | 114, 116 | sylib 217 |
. . . . . . . . . . 11
⊢ (𝑟 ∈ ( R ‘𝑋) → (( bday ‘𝑟) +no ( bday
‘𝑌)) ∈
(( bday ‘𝑋) +no ( bday
‘𝑌))) |
118 | 117 | adantl 483 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑟 ∈ ( R ‘𝑋)) → (( bday
‘𝑟) +no ( bday ‘𝑌)) ∈ (( bday
‘𝑋) +no ( bday ‘𝑌))) |
119 | | elun1 4176 |
. . . . . . . . . 10
⊢ ((( bday ‘𝑟) +no ( bday
‘𝑌)) ∈
(( bday ‘𝑋) +no ( bday
‘𝑌)) →
(( bday ‘𝑟) +no ( bday
‘𝑌)) ∈
((( bday ‘𝑋) +no ( bday
‘𝑌)) ∪
(( bday ‘𝑋) +no ( bday
‘𝑍)))) |
120 | 118, 119 | syl 17 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑟 ∈ ( R ‘𝑋)) → (( bday
‘𝑟) +no ( bday ‘𝑌)) ∈ ((( bday
‘𝑋) +no ( bday ‘𝑌)) ∪ (( bday
‘𝑋) +no ( bday ‘𝑍)))) |
121 | 109, 120 | eqeltrid 2838 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑟 ∈ ( R ‘𝑋)) → ((( bday
‘𝑟) +no ( bday ‘𝑌)) ∪ (( bday
‘𝑟) +no ( bday ‘ 0s ))) ∈ ((( bday ‘𝑋) +no ( bday
‘𝑌)) ∪
(( bday ‘𝑋) +no ( bday
‘𝑍)))) |
122 | 93, 96, 97, 98, 121 | addsproplem1 27443 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑟 ∈ ( R ‘𝑋)) → ((𝑟 +s 𝑌) ∈ No
∧ (𝑌 <s
0s → (𝑌
+s 𝑟) <s (
0s +s 𝑟)))) |
123 | 122 | simpld 496 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑟 ∈ ( R ‘𝑋)) → (𝑟 +s 𝑌) ∈ No
) |
124 | | eleq1a 2829 |
. . . . . 6
⊢ ((𝑟 +s 𝑌) ∈ No
→ (𝑤 = (𝑟 +s 𝑌) → 𝑤 ∈ No
)) |
125 | 123, 124 | syl 17 |
. . . . 5
⊢ ((𝜑 ∧ 𝑟 ∈ ( R ‘𝑋)) → (𝑤 = (𝑟 +s 𝑌) → 𝑤 ∈ No
)) |
126 | 125 | rexlimdva 3156 |
. . . 4
⊢ (𝜑 → (∃𝑟 ∈ ( R ‘𝑋)𝑤 = (𝑟 +s 𝑌) → 𝑤 ∈ No
)) |
127 | 126 | abssdv 4065 |
. . 3
⊢ (𝜑 → {𝑤 ∣ ∃𝑟 ∈ ( R ‘𝑋)𝑤 = (𝑟 +s 𝑌)} ⊆ No
) |
128 | 15 | adantr 482 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑠 ∈ ( R ‘𝑌)) → ∀𝑥 ∈ No
∀𝑦 ∈ No ∀𝑧 ∈ No
(((( bday ‘𝑥) +no ( bday
‘𝑦)) ∪
(( bday ‘𝑥) +no ( bday
‘𝑧))) ∈
((( bday ‘𝑋) +no ( bday
‘𝑌)) ∪
(( bday ‘𝑋) +no ( bday
‘𝑍))) →
((𝑥 +s 𝑦) ∈
No ∧ (𝑦 <s
𝑧 → (𝑦 +s 𝑥) <s (𝑧 +s 𝑥))))) |
129 | 57 | adantr 482 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑠 ∈ ( R ‘𝑌)) → 𝑋 ∈ No
) |
130 | | rightssno 27366 |
. . . . . . . . . 10
⊢ ( R
‘𝑌) ⊆ No |
131 | 130 | sseli 3978 |
. . . . . . . . 9
⊢ (𝑠 ∈ ( R ‘𝑌) → 𝑠 ∈ No
) |
132 | 131 | adantl 483 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑠 ∈ ( R ‘𝑌)) → 𝑠 ∈ No
) |
133 | 22 | a1i 11 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑠 ∈ ( R ‘𝑌)) → 0s ∈ No ) |
134 | 66 | uneq2i 4160 |
. . . . . . . . . 10
⊢ ((( bday ‘𝑋) +no ( bday
‘𝑠)) ∪
(( bday ‘𝑋) +no ( bday
‘ 0s ))) = ((( bday
‘𝑋) +no ( bday ‘𝑠)) ∪ ( bday
‘𝑋)) |
135 | | bdayelon 27268 |
. . . . . . . . . . . 12
⊢ ( bday ‘𝑠) ∈ On |
136 | | naddword1 8687 |
. . . . . . . . . . . 12
⊢ ((( bday ‘𝑋) ∈ On ∧ (
bday ‘𝑠)
∈ On) → ( bday ‘𝑋) ⊆ (( bday
‘𝑋) +no ( bday ‘𝑠))) |
137 | 39, 135, 136 | mp2an 691 |
. . . . . . . . . . 11
⊢ ( bday ‘𝑋) ⊆ (( bday
‘𝑋) +no ( bday ‘𝑠)) |
138 | | ssequn2 4183 |
. . . . . . . . . . 11
⊢ (( bday ‘𝑋) ⊆ (( bday
‘𝑋) +no ( bday ‘𝑠)) ↔ ((( bday
‘𝑋) +no ( bday ‘𝑠)) ∪ ( bday
‘𝑋)) = (( bday ‘𝑋) +no ( bday
‘𝑠))) |
139 | 137, 138 | mpbi 229 |
. . . . . . . . . 10
⊢ ((( bday ‘𝑋) +no ( bday
‘𝑠)) ∪
( bday ‘𝑋)) = (( bday
‘𝑋) +no ( bday ‘𝑠)) |
140 | 134, 139 | eqtri 2761 |
. . . . . . . . 9
⊢ ((( bday ‘𝑋) +no ( bday
‘𝑠)) ∪
(( bday ‘𝑋) +no ( bday
‘ 0s ))) = (( bday
‘𝑋) +no ( bday ‘𝑠)) |
141 | | rightssold 27364 |
. . . . . . . . . . . . . 14
⊢ ( R
‘𝑌) ⊆ ( O
‘( bday ‘𝑌)) |
142 | 141 | sseli 3978 |
. . . . . . . . . . . . 13
⊢ (𝑠 ∈ ( R ‘𝑌) → 𝑠 ∈ ( O ‘(
bday ‘𝑌))) |
143 | | oldbday 27385 |
. . . . . . . . . . . . . 14
⊢ ((( bday ‘𝑌) ∈ On ∧ 𝑠 ∈ No )
→ (𝑠 ∈ ( O
‘( bday ‘𝑌)) ↔ ( bday
‘𝑠) ∈
( bday ‘𝑌))) |
144 | 31, 131, 143 | sylancr 588 |
. . . . . . . . . . . . 13
⊢ (𝑠 ∈ ( R ‘𝑌) → (𝑠 ∈ ( O ‘(
bday ‘𝑌))
↔ ( bday ‘𝑠) ∈ ( bday
‘𝑌))) |
145 | 142, 144 | mpbid 231 |
. . . . . . . . . . . 12
⊢ (𝑠 ∈ ( R ‘𝑌) → (
bday ‘𝑠)
∈ ( bday ‘𝑌)) |
146 | | naddel2 8684 |
. . . . . . . . . . . . 13
⊢ ((( bday ‘𝑠) ∈ On ∧ (
bday ‘𝑌)
∈ On ∧ ( bday ‘𝑋) ∈ On) → ((
bday ‘𝑠)
∈ ( bday ‘𝑌) ↔ (( bday
‘𝑋) +no ( bday ‘𝑠)) ∈ (( bday
‘𝑋) +no ( bday ‘𝑌)))) |
147 | 135, 31, 39, 146 | mp3an 1462 |
. . . . . . . . . . . 12
⊢ (( bday ‘𝑠) ∈ ( bday
‘𝑌) ↔
(( bday ‘𝑋) +no ( bday
‘𝑠)) ∈
(( bday ‘𝑋) +no ( bday
‘𝑌))) |
148 | 145, 147 | sylib 217 |
. . . . . . . . . . 11
⊢ (𝑠 ∈ ( R ‘𝑌) → (( bday ‘𝑋) +no ( bday
‘𝑠)) ∈
(( bday ‘𝑋) +no ( bday
‘𝑌))) |
149 | 148 | adantl 483 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑠 ∈ ( R ‘𝑌)) → (( bday
‘𝑋) +no ( bday ‘𝑠)) ∈ (( bday
‘𝑋) +no ( bday ‘𝑌))) |
150 | | elun1 4176 |
. . . . . . . . . 10
⊢ ((( bday ‘𝑋) +no ( bday
‘𝑠)) ∈
(( bday ‘𝑋) +no ( bday
‘𝑌)) →
(( bday ‘𝑋) +no ( bday
‘𝑠)) ∈
((( bday ‘𝑋) +no ( bday
‘𝑌)) ∪
(( bday ‘𝑋) +no ( bday
‘𝑍)))) |
151 | 149, 150 | syl 17 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑠 ∈ ( R ‘𝑌)) → (( bday
‘𝑋) +no ( bday ‘𝑠)) ∈ ((( bday
‘𝑋) +no ( bday ‘𝑌)) ∪ (( bday
‘𝑋) +no ( bday ‘𝑍)))) |
152 | 140, 151 | eqeltrid 2838 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑠 ∈ ( R ‘𝑌)) → ((( bday
‘𝑋) +no ( bday ‘𝑠)) ∪ (( bday
‘𝑋) +no ( bday ‘ 0s ))) ∈ ((( bday ‘𝑋) +no ( bday
‘𝑌)) ∪
(( bday ‘𝑋) +no ( bday
‘𝑍)))) |
153 | 128, 129,
132, 133, 152 | addsproplem1 27443 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑠 ∈ ( R ‘𝑌)) → ((𝑋 +s 𝑠) ∈ No
∧ (𝑠 <s
0s → (𝑠
+s 𝑋) <s (
0s +s 𝑋)))) |
154 | 153 | simpld 496 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑠 ∈ ( R ‘𝑌)) → (𝑋 +s 𝑠) ∈ No
) |
155 | | eleq1a 2829 |
. . . . . 6
⊢ ((𝑋 +s 𝑠) ∈
No → (𝑡 =
(𝑋 +s 𝑠) → 𝑡 ∈ No
)) |
156 | 154, 155 | syl 17 |
. . . . 5
⊢ ((𝜑 ∧ 𝑠 ∈ ( R ‘𝑌)) → (𝑡 = (𝑋 +s 𝑠) → 𝑡 ∈ No
)) |
157 | 156 | rexlimdva 3156 |
. . . 4
⊢ (𝜑 → (∃𝑠 ∈ ( R ‘𝑌)𝑡 = (𝑋 +s 𝑠) → 𝑡 ∈ No
)) |
158 | 157 | abssdv 4065 |
. . 3
⊢ (𝜑 → {𝑡 ∣ ∃𝑠 ∈ ( R ‘𝑌)𝑡 = (𝑋 +s 𝑠)} ⊆ No
) |
159 | 127, 158 | unssd 4186 |
. 2
⊢ (𝜑 → ({𝑤 ∣ ∃𝑟 ∈ ( R ‘𝑋)𝑤 = (𝑟 +s 𝑌)} ∪ {𝑡 ∣ ∃𝑠 ∈ ( R ‘𝑌)𝑡 = (𝑋 +s 𝑠)}) ⊆ No
) |
160 | | elun 4148 |
. . . . . . 7
⊢ (𝑎 ∈ ({𝑝 ∣ ∃𝑙 ∈ ( L ‘𝑋)𝑝 = (𝑙 +s 𝑌)} ∪ {𝑞 ∣ ∃𝑚 ∈ ( L ‘𝑌)𝑞 = (𝑋 +s 𝑚)}) ↔ (𝑎 ∈ {𝑝 ∣ ∃𝑙 ∈ ( L ‘𝑋)𝑝 = (𝑙 +s 𝑌)} ∨ 𝑎 ∈ {𝑞 ∣ ∃𝑚 ∈ ( L ‘𝑌)𝑞 = (𝑋 +s 𝑚)})) |
161 | | vex 3479 |
. . . . . . . . 9
⊢ 𝑎 ∈ V |
162 | | eqeq1 2737 |
. . . . . . . . . 10
⊢ (𝑝 = 𝑎 → (𝑝 = (𝑙 +s 𝑌) ↔ 𝑎 = (𝑙 +s 𝑌))) |
163 | 162 | rexbidv 3179 |
. . . . . . . . 9
⊢ (𝑝 = 𝑎 → (∃𝑙 ∈ ( L ‘𝑋)𝑝 = (𝑙 +s 𝑌) ↔ ∃𝑙 ∈ ( L ‘𝑋)𝑎 = (𝑙 +s 𝑌))) |
164 | 161, 163 | elab 3668 |
. . . . . . . 8
⊢ (𝑎 ∈ {𝑝 ∣ ∃𝑙 ∈ ( L ‘𝑋)𝑝 = (𝑙 +s 𝑌)} ↔ ∃𝑙 ∈ ( L ‘𝑋)𝑎 = (𝑙 +s 𝑌)) |
165 | | eqeq1 2737 |
. . . . . . . . . 10
⊢ (𝑞 = 𝑎 → (𝑞 = (𝑋 +s 𝑚) ↔ 𝑎 = (𝑋 +s 𝑚))) |
166 | 165 | rexbidv 3179 |
. . . . . . . . 9
⊢ (𝑞 = 𝑎 → (∃𝑚 ∈ ( L ‘𝑌)𝑞 = (𝑋 +s 𝑚) ↔ ∃𝑚 ∈ ( L ‘𝑌)𝑎 = (𝑋 +s 𝑚))) |
167 | 161, 166 | elab 3668 |
. . . . . . . 8
⊢ (𝑎 ∈ {𝑞 ∣ ∃𝑚 ∈ ( L ‘𝑌)𝑞 = (𝑋 +s 𝑚)} ↔ ∃𝑚 ∈ ( L ‘𝑌)𝑎 = (𝑋 +s 𝑚)) |
168 | 164, 167 | orbi12i 914 |
. . . . . . 7
⊢ ((𝑎 ∈ {𝑝 ∣ ∃𝑙 ∈ ( L ‘𝑋)𝑝 = (𝑙 +s 𝑌)} ∨ 𝑎 ∈ {𝑞 ∣ ∃𝑚 ∈ ( L ‘𝑌)𝑞 = (𝑋 +s 𝑚)}) ↔ (∃𝑙 ∈ ( L ‘𝑋)𝑎 = (𝑙 +s 𝑌) ∨ ∃𝑚 ∈ ( L ‘𝑌)𝑎 = (𝑋 +s 𝑚))) |
169 | 160, 168 | bitri 275 |
. . . . . 6
⊢ (𝑎 ∈ ({𝑝 ∣ ∃𝑙 ∈ ( L ‘𝑋)𝑝 = (𝑙 +s 𝑌)} ∪ {𝑞 ∣ ∃𝑚 ∈ ( L ‘𝑌)𝑞 = (𝑋 +s 𝑚)}) ↔ (∃𝑙 ∈ ( L ‘𝑋)𝑎 = (𝑙 +s 𝑌) ∨ ∃𝑚 ∈ ( L ‘𝑌)𝑎 = (𝑋 +s 𝑚))) |
170 | | elun 4148 |
. . . . . . 7
⊢ (𝑏 ∈ ({𝑤 ∣ ∃𝑟 ∈ ( R ‘𝑋)𝑤 = (𝑟 +s 𝑌)} ∪ {𝑡 ∣ ∃𝑠 ∈ ( R ‘𝑌)𝑡 = (𝑋 +s 𝑠)}) ↔ (𝑏 ∈ {𝑤 ∣ ∃𝑟 ∈ ( R ‘𝑋)𝑤 = (𝑟 +s 𝑌)} ∨ 𝑏 ∈ {𝑡 ∣ ∃𝑠 ∈ ( R ‘𝑌)𝑡 = (𝑋 +s 𝑠)})) |
171 | | vex 3479 |
. . . . . . . . 9
⊢ 𝑏 ∈ V |
172 | | eqeq1 2737 |
. . . . . . . . . 10
⊢ (𝑤 = 𝑏 → (𝑤 = (𝑟 +s 𝑌) ↔ 𝑏 = (𝑟 +s 𝑌))) |
173 | 172 | rexbidv 3179 |
. . . . . . . . 9
⊢ (𝑤 = 𝑏 → (∃𝑟 ∈ ( R ‘𝑋)𝑤 = (𝑟 +s 𝑌) ↔ ∃𝑟 ∈ ( R ‘𝑋)𝑏 = (𝑟 +s 𝑌))) |
174 | 171, 173 | elab 3668 |
. . . . . . . 8
⊢ (𝑏 ∈ {𝑤 ∣ ∃𝑟 ∈ ( R ‘𝑋)𝑤 = (𝑟 +s 𝑌)} ↔ ∃𝑟 ∈ ( R ‘𝑋)𝑏 = (𝑟 +s 𝑌)) |
175 | | eqeq1 2737 |
. . . . . . . . . 10
⊢ (𝑡 = 𝑏 → (𝑡 = (𝑋 +s 𝑠) ↔ 𝑏 = (𝑋 +s 𝑠))) |
176 | 175 | rexbidv 3179 |
. . . . . . . . 9
⊢ (𝑡 = 𝑏 → (∃𝑠 ∈ ( R ‘𝑌)𝑡 = (𝑋 +s 𝑠) ↔ ∃𝑠 ∈ ( R ‘𝑌)𝑏 = (𝑋 +s 𝑠))) |
177 | 171, 176 | elab 3668 |
. . . . . . . 8
⊢ (𝑏 ∈ {𝑡 ∣ ∃𝑠 ∈ ( R ‘𝑌)𝑡 = (𝑋 +s 𝑠)} ↔ ∃𝑠 ∈ ( R ‘𝑌)𝑏 = (𝑋 +s 𝑠)) |
178 | 174, 177 | orbi12i 914 |
. . . . . . 7
⊢ ((𝑏 ∈ {𝑤 ∣ ∃𝑟 ∈ ( R ‘𝑋)𝑤 = (𝑟 +s 𝑌)} ∨ 𝑏 ∈ {𝑡 ∣ ∃𝑠 ∈ ( R ‘𝑌)𝑡 = (𝑋 +s 𝑠)}) ↔ (∃𝑟 ∈ ( R ‘𝑋)𝑏 = (𝑟 +s 𝑌) ∨ ∃𝑠 ∈ ( R ‘𝑌)𝑏 = (𝑋 +s 𝑠))) |
179 | 170, 178 | bitri 275 |
. . . . . 6
⊢ (𝑏 ∈ ({𝑤 ∣ ∃𝑟 ∈ ( R ‘𝑋)𝑤 = (𝑟 +s 𝑌)} ∪ {𝑡 ∣ ∃𝑠 ∈ ( R ‘𝑌)𝑡 = (𝑋 +s 𝑠)}) ↔ (∃𝑟 ∈ ( R ‘𝑋)𝑏 = (𝑟 +s 𝑌) ∨ ∃𝑠 ∈ ( R ‘𝑌)𝑏 = (𝑋 +s 𝑠))) |
180 | 169, 179 | anbi12i 628 |
. . . . 5
⊢ ((𝑎 ∈ ({𝑝 ∣ ∃𝑙 ∈ ( L ‘𝑋)𝑝 = (𝑙 +s 𝑌)} ∪ {𝑞 ∣ ∃𝑚 ∈ ( L ‘𝑌)𝑞 = (𝑋 +s 𝑚)}) ∧ 𝑏 ∈ ({𝑤 ∣ ∃𝑟 ∈ ( R ‘𝑋)𝑤 = (𝑟 +s 𝑌)} ∪ {𝑡 ∣ ∃𝑠 ∈ ( R ‘𝑌)𝑡 = (𝑋 +s 𝑠)})) ↔ ((∃𝑙 ∈ ( L ‘𝑋)𝑎 = (𝑙 +s 𝑌) ∨ ∃𝑚 ∈ ( L ‘𝑌)𝑎 = (𝑋 +s 𝑚)) ∧ (∃𝑟 ∈ ( R ‘𝑋)𝑏 = (𝑟 +s 𝑌) ∨ ∃𝑠 ∈ ( R ‘𝑌)𝑏 = (𝑋 +s 𝑠)))) |
181 | | anddi 1010 |
. . . . 5
⊢
(((∃𝑙 ∈ (
L ‘𝑋)𝑎 = (𝑙 +s 𝑌) ∨ ∃𝑚 ∈ ( L ‘𝑌)𝑎 = (𝑋 +s 𝑚)) ∧ (∃𝑟 ∈ ( R ‘𝑋)𝑏 = (𝑟 +s 𝑌) ∨ ∃𝑠 ∈ ( R ‘𝑌)𝑏 = (𝑋 +s 𝑠))) ↔ (((∃𝑙 ∈ ( L ‘𝑋)𝑎 = (𝑙 +s 𝑌) ∧ ∃𝑟 ∈ ( R ‘𝑋)𝑏 = (𝑟 +s 𝑌)) ∨ (∃𝑙 ∈ ( L ‘𝑋)𝑎 = (𝑙 +s 𝑌) ∧ ∃𝑠 ∈ ( R ‘𝑌)𝑏 = (𝑋 +s 𝑠))) ∨ ((∃𝑚 ∈ ( L ‘𝑌)𝑎 = (𝑋 +s 𝑚) ∧ ∃𝑟 ∈ ( R ‘𝑋)𝑏 = (𝑟 +s 𝑌)) ∨ (∃𝑚 ∈ ( L ‘𝑌)𝑎 = (𝑋 +s 𝑚) ∧ ∃𝑠 ∈ ( R ‘𝑌)𝑏 = (𝑋 +s 𝑠))))) |
182 | 180, 181 | bitri 275 |
. . . 4
⊢ ((𝑎 ∈ ({𝑝 ∣ ∃𝑙 ∈ ( L ‘𝑋)𝑝 = (𝑙 +s 𝑌)} ∪ {𝑞 ∣ ∃𝑚 ∈ ( L ‘𝑌)𝑞 = (𝑋 +s 𝑚)}) ∧ 𝑏 ∈ ({𝑤 ∣ ∃𝑟 ∈ ( R ‘𝑋)𝑤 = (𝑟 +s 𝑌)} ∪ {𝑡 ∣ ∃𝑠 ∈ ( R ‘𝑌)𝑡 = (𝑋 +s 𝑠)})) ↔ (((∃𝑙 ∈ ( L ‘𝑋)𝑎 = (𝑙 +s 𝑌) ∧ ∃𝑟 ∈ ( R ‘𝑋)𝑏 = (𝑟 +s 𝑌)) ∨ (∃𝑙 ∈ ( L ‘𝑋)𝑎 = (𝑙 +s 𝑌) ∧ ∃𝑠 ∈ ( R ‘𝑌)𝑏 = (𝑋 +s 𝑠))) ∨ ((∃𝑚 ∈ ( L ‘𝑌)𝑎 = (𝑋 +s 𝑚) ∧ ∃𝑟 ∈ ( R ‘𝑋)𝑏 = (𝑟 +s 𝑌)) ∨ (∃𝑚 ∈ ( L ‘𝑌)𝑎 = (𝑋 +s 𝑚) ∧ ∃𝑠 ∈ ( R ‘𝑌)𝑏 = (𝑋 +s 𝑠))))) |
183 | | reeanv 3227 |
. . . . . . 7
⊢
(∃𝑙 ∈ ( L
‘𝑋)∃𝑟 ∈ ( R ‘𝑋)(𝑎 = (𝑙 +s 𝑌) ∧ 𝑏 = (𝑟 +s 𝑌)) ↔ (∃𝑙 ∈ ( L ‘𝑋)𝑎 = (𝑙 +s 𝑌) ∧ ∃𝑟 ∈ ( R ‘𝑋)𝑏 = (𝑟 +s 𝑌))) |
184 | | lltropt 27357 |
. . . . . . . . . . . 12
⊢ ( L
‘𝑋) <<s ( R
‘𝑋) |
185 | 184 | a1i 11 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ (𝑙 ∈ ( L ‘𝑋) ∧ 𝑟 ∈ ( R ‘𝑋))) → ( L ‘𝑋) <<s ( R ‘𝑋)) |
186 | | simprl 770 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ (𝑙 ∈ ( L ‘𝑋) ∧ 𝑟 ∈ ( R ‘𝑋))) → 𝑙 ∈ ( L ‘𝑋)) |
187 | | simprr 772 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ (𝑙 ∈ ( L ‘𝑋) ∧ 𝑟 ∈ ( R ‘𝑋))) → 𝑟 ∈ ( R ‘𝑋)) |
188 | 185, 186,
187 | ssltsepcd 27285 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑙 ∈ ( L ‘𝑋) ∧ 𝑟 ∈ ( R ‘𝑋))) → 𝑙 <s 𝑟) |
189 | 15 | adantr 482 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ (𝑙 ∈ ( L ‘𝑋) ∧ 𝑟 ∈ ( R ‘𝑋))) → ∀𝑥 ∈ No
∀𝑦 ∈ No ∀𝑧 ∈ No
(((( bday ‘𝑥) +no ( bday
‘𝑦)) ∪
(( bday ‘𝑥) +no ( bday
‘𝑧))) ∈
((( bday ‘𝑋) +no ( bday
‘𝑌)) ∪
(( bday ‘𝑋) +no ( bday
‘𝑍))) →
((𝑥 +s 𝑦) ∈
No ∧ (𝑦 <s
𝑧 → (𝑦 +s 𝑥) <s (𝑧 +s 𝑥))))) |
190 | 20 | adantr 482 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ (𝑙 ∈ ( L ‘𝑋) ∧ 𝑟 ∈ ( R ‘𝑋))) → 𝑌 ∈ No
) |
191 | 18 | ad2antrl 727 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ (𝑙 ∈ ( L ‘𝑋) ∧ 𝑟 ∈ ( R ‘𝑋))) → 𝑙 ∈ No
) |
192 | 95 | ad2antll 728 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ (𝑙 ∈ ( L ‘𝑋) ∧ 𝑟 ∈ ( R ‘𝑋))) → 𝑟 ∈ No
) |
193 | | naddcom 8678 |
. . . . . . . . . . . . . . . 16
⊢ ((( bday ‘𝑌) ∈ On ∧ (
bday ‘𝑙)
∈ On) → (( bday ‘𝑌) +no ( bday
‘𝑙)) = (( bday ‘𝑙) +no ( bday
‘𝑌))) |
194 | 31, 26, 193 | mp2an 691 |
. . . . . . . . . . . . . . 15
⊢ (( bday ‘𝑌) +no ( bday
‘𝑙)) = (( bday ‘𝑙) +no ( bday
‘𝑌)) |
195 | 45 | ad2antrl 727 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ (𝑙 ∈ ( L ‘𝑋) ∧ 𝑟 ∈ ( R ‘𝑋))) → (( bday
‘𝑙) +no ( bday ‘𝑌)) ∈ (( bday
‘𝑋) +no ( bday ‘𝑌))) |
196 | 194, 195 | eqeltrid 2838 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ (𝑙 ∈ ( L ‘𝑋) ∧ 𝑟 ∈ ( R ‘𝑋))) → (( bday
‘𝑌) +no ( bday ‘𝑙)) ∈ (( bday
‘𝑋) +no ( bday ‘𝑌))) |
197 | | naddcom 8678 |
. . . . . . . . . . . . . . . 16
⊢ ((( bday ‘𝑌) ∈ On ∧ (
bday ‘𝑟)
∈ On) → (( bday ‘𝑌) +no ( bday
‘𝑟)) = (( bday ‘𝑟) +no ( bday
‘𝑌))) |
198 | 31, 100, 197 | mp2an 691 |
. . . . . . . . . . . . . . 15
⊢ (( bday ‘𝑌) +no ( bday
‘𝑟)) = (( bday ‘𝑟) +no ( bday
‘𝑌)) |
199 | 117 | ad2antll 728 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ (𝑙 ∈ ( L ‘𝑋) ∧ 𝑟 ∈ ( R ‘𝑋))) → (( bday
‘𝑟) +no ( bday ‘𝑌)) ∈ (( bday
‘𝑋) +no ( bday ‘𝑌))) |
200 | 198, 199 | eqeltrid 2838 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ (𝑙 ∈ ( L ‘𝑋) ∧ 𝑟 ∈ ( R ‘𝑋))) → (( bday
‘𝑌) +no ( bday ‘𝑟)) ∈ (( bday
‘𝑋) +no ( bday ‘𝑌))) |
201 | | naddcl 8673 |
. . . . . . . . . . . . . . . 16
⊢ ((( bday ‘𝑌) ∈ On ∧ (
bday ‘𝑙)
∈ On) → (( bday ‘𝑌) +no ( bday
‘𝑙)) ∈
On) |
202 | 31, 26, 201 | mp2an 691 |
. . . . . . . . . . . . . . 15
⊢ (( bday ‘𝑌) +no ( bday
‘𝑙)) ∈
On |
203 | | naddcl 8673 |
. . . . . . . . . . . . . . . 16
⊢ ((( bday ‘𝑌) ∈ On ∧ (
bday ‘𝑟)
∈ On) → (( bday ‘𝑌) +no ( bday
‘𝑟)) ∈
On) |
204 | 31, 100, 203 | mp2an 691 |
. . . . . . . . . . . . . . 15
⊢ (( bday ‘𝑌) +no ( bday
‘𝑟)) ∈
On |
205 | | naddcl 8673 |
. . . . . . . . . . . . . . . 16
⊢ ((( bday ‘𝑋) ∈ On ∧ (
bday ‘𝑌)
∈ On) → (( bday ‘𝑋) +no ( bday
‘𝑌)) ∈
On) |
206 | 39, 31, 205 | mp2an 691 |
. . . . . . . . . . . . . . 15
⊢ (( bday ‘𝑋) +no ( bday
‘𝑌)) ∈
On |
207 | | onunel 6467 |
. . . . . . . . . . . . . . 15
⊢ (((( bday ‘𝑌) +no ( bday
‘𝑙)) ∈ On
∧ (( bday ‘𝑌) +no ( bday
‘𝑟)) ∈ On
∧ (( bday ‘𝑋) +no ( bday
‘𝑌)) ∈
On) → (((( bday ‘𝑌) +no ( bday
‘𝑙)) ∪
(( bday ‘𝑌) +no ( bday
‘𝑟))) ∈
(( bday ‘𝑋) +no ( bday
‘𝑌)) ↔
((( bday ‘𝑌) +no ( bday
‘𝑙)) ∈
(( bday ‘𝑋) +no ( bday
‘𝑌)) ∧
(( bday ‘𝑌) +no ( bday
‘𝑟)) ∈
(( bday ‘𝑋) +no ( bday
‘𝑌))))) |
208 | 202, 204,
206, 207 | mp3an 1462 |
. . . . . . . . . . . . . 14
⊢ (((( bday ‘𝑌) +no ( bday
‘𝑙)) ∪
(( bday ‘𝑌) +no ( bday
‘𝑟))) ∈
(( bday ‘𝑋) +no ( bday
‘𝑌)) ↔
((( bday ‘𝑌) +no ( bday
‘𝑙)) ∈
(( bday ‘𝑋) +no ( bday
‘𝑌)) ∧
(( bday ‘𝑌) +no ( bday
‘𝑟)) ∈
(( bday ‘𝑋) +no ( bday
‘𝑌)))) |
209 | 196, 200,
208 | sylanbrc 584 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ (𝑙 ∈ ( L ‘𝑋) ∧ 𝑟 ∈ ( R ‘𝑋))) → ((( bday
‘𝑌) +no ( bday ‘𝑙)) ∪ (( bday
‘𝑌) +no ( bday ‘𝑟))) ∈ (( bday
‘𝑋) +no ( bday ‘𝑌))) |
210 | | elun1 4176 |
. . . . . . . . . . . . 13
⊢ (((( bday ‘𝑌) +no ( bday
‘𝑙)) ∪
(( bday ‘𝑌) +no ( bday
‘𝑟))) ∈
(( bday ‘𝑋) +no ( bday
‘𝑌)) →
((( bday ‘𝑌) +no ( bday
‘𝑙)) ∪
(( bday ‘𝑌) +no ( bday
‘𝑟))) ∈
((( bday ‘𝑋) +no ( bday
‘𝑌)) ∪
(( bday ‘𝑋) +no ( bday
‘𝑍)))) |
211 | 209, 210 | syl 17 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ (𝑙 ∈ ( L ‘𝑋) ∧ 𝑟 ∈ ( R ‘𝑋))) → ((( bday
‘𝑌) +no ( bday ‘𝑙)) ∪ (( bday
‘𝑌) +no ( bday ‘𝑟))) ∈ ((( bday
‘𝑋) +no ( bday ‘𝑌)) ∪ (( bday
‘𝑋) +no ( bday ‘𝑍)))) |
212 | 189, 190,
191, 192, 211 | addsproplem1 27443 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ (𝑙 ∈ ( L ‘𝑋) ∧ 𝑟 ∈ ( R ‘𝑋))) → ((𝑌 +s 𝑙) ∈ No
∧ (𝑙 <s 𝑟 → (𝑙 +s 𝑌) <s (𝑟 +s 𝑌)))) |
213 | 212 | simprd 497 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑙 ∈ ( L ‘𝑋) ∧ 𝑟 ∈ ( R ‘𝑋))) → (𝑙 <s 𝑟 → (𝑙 +s 𝑌) <s (𝑟 +s 𝑌))) |
214 | 188, 213 | mpd 15 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝑙 ∈ ( L ‘𝑋) ∧ 𝑟 ∈ ( R ‘𝑋))) → (𝑙 +s 𝑌) <s (𝑟 +s 𝑌)) |
215 | | breq12 5153 |
. . . . . . . . 9
⊢ ((𝑎 = (𝑙 +s 𝑌) ∧ 𝑏 = (𝑟 +s 𝑌)) → (𝑎 <s 𝑏 ↔ (𝑙 +s 𝑌) <s (𝑟 +s 𝑌))) |
216 | 214, 215 | syl5ibrcom 246 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑙 ∈ ( L ‘𝑋) ∧ 𝑟 ∈ ( R ‘𝑋))) → ((𝑎 = (𝑙 +s 𝑌) ∧ 𝑏 = (𝑟 +s 𝑌)) → 𝑎 <s 𝑏)) |
217 | 216 | rexlimdvva 3212 |
. . . . . . 7
⊢ (𝜑 → (∃𝑙 ∈ ( L ‘𝑋)∃𝑟 ∈ ( R ‘𝑋)(𝑎 = (𝑙 +s 𝑌) ∧ 𝑏 = (𝑟 +s 𝑌)) → 𝑎 <s 𝑏)) |
218 | 183, 217 | biimtrrid 242 |
. . . . . 6
⊢ (𝜑 → ((∃𝑙 ∈ ( L ‘𝑋)𝑎 = (𝑙 +s 𝑌) ∧ ∃𝑟 ∈ ( R ‘𝑋)𝑏 = (𝑟 +s 𝑌)) → 𝑎 <s 𝑏)) |
219 | | reeanv 3227 |
. . . . . . 7
⊢
(∃𝑙 ∈ ( L
‘𝑋)∃𝑠 ∈ ( R ‘𝑌)(𝑎 = (𝑙 +s 𝑌) ∧ 𝑏 = (𝑋 +s 𝑠)) ↔ (∃𝑙 ∈ ( L ‘𝑋)𝑎 = (𝑙 +s 𝑌) ∧ ∃𝑠 ∈ ( R ‘𝑌)𝑏 = (𝑋 +s 𝑠))) |
220 | 51 | adantrr 716 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑙 ∈ ( L ‘𝑋) ∧ 𝑠 ∈ ( R ‘𝑌))) → (𝑙 +s 𝑌) ∈ No
) |
221 | 15 | adantr 482 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ (𝑙 ∈ ( L ‘𝑋) ∧ 𝑠 ∈ ( R ‘𝑌))) → ∀𝑥 ∈ No
∀𝑦 ∈ No ∀𝑧 ∈ No
(((( bday ‘𝑥) +no ( bday
‘𝑦)) ∪
(( bday ‘𝑥) +no ( bday
‘𝑧))) ∈
((( bday ‘𝑋) +no ( bday
‘𝑌)) ∪
(( bday ‘𝑋) +no ( bday
‘𝑍))) →
((𝑥 +s 𝑦) ∈
No ∧ (𝑦 <s
𝑧 → (𝑦 +s 𝑥) <s (𝑧 +s 𝑥))))) |
222 | 18 | ad2antrl 727 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ (𝑙 ∈ ( L ‘𝑋) ∧ 𝑠 ∈ ( R ‘𝑌))) → 𝑙 ∈ No
) |
223 | 131 | ad2antll 728 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ (𝑙 ∈ ( L ‘𝑋) ∧ 𝑠 ∈ ( R ‘𝑌))) → 𝑠 ∈ No
) |
224 | 22 | a1i 11 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ (𝑙 ∈ ( L ‘𝑋) ∧ 𝑠 ∈ ( R ‘𝑌))) → 0s ∈ No ) |
225 | 29 | uneq2i 4160 |
. . . . . . . . . . . . . 14
⊢ ((( bday ‘𝑙) +no ( bday
‘𝑠)) ∪
(( bday ‘𝑙) +no ( bday
‘ 0s ))) = ((( bday
‘𝑙) +no ( bday ‘𝑠)) ∪ ( bday
‘𝑙)) |
226 | | naddword1 8687 |
. . . . . . . . . . . . . . . 16
⊢ ((( bday ‘𝑙) ∈ On ∧ (
bday ‘𝑠)
∈ On) → ( bday ‘𝑙) ⊆ (( bday
‘𝑙) +no ( bday ‘𝑠))) |
227 | 26, 135, 226 | mp2an 691 |
. . . . . . . . . . . . . . 15
⊢ ( bday ‘𝑙) ⊆ (( bday
‘𝑙) +no ( bday ‘𝑠)) |
228 | | ssequn2 4183 |
. . . . . . . . . . . . . . 15
⊢ (( bday ‘𝑙) ⊆ (( bday
‘𝑙) +no ( bday ‘𝑠)) ↔ ((( bday
‘𝑙) +no ( bday ‘𝑠)) ∪ ( bday
‘𝑙)) = (( bday ‘𝑙) +no ( bday
‘𝑠))) |
229 | 227, 228 | mpbi 229 |
. . . . . . . . . . . . . 14
⊢ ((( bday ‘𝑙) +no ( bday
‘𝑠)) ∪
( bday ‘𝑙)) = (( bday
‘𝑙) +no ( bday ‘𝑠)) |
230 | 225, 229 | eqtri 2761 |
. . . . . . . . . . . . 13
⊢ ((( bday ‘𝑙) +no ( bday
‘𝑠)) ∪
(( bday ‘𝑙) +no ( bday
‘ 0s ))) = (( bday
‘𝑙) +no ( bday ‘𝑠)) |
231 | | naddel1 8683 |
. . . . . . . . . . . . . . . . . 18
⊢ ((( bday ‘𝑙) ∈ On ∧ (
bday ‘𝑋)
∈ On ∧ ( bday ‘𝑠) ∈ On) → ((
bday ‘𝑙)
∈ ( bday ‘𝑋) ↔ (( bday
‘𝑙) +no ( bday ‘𝑠)) ∈ (( bday
‘𝑋) +no ( bday ‘𝑠)))) |
232 | 26, 39, 135, 231 | mp3an 1462 |
. . . . . . . . . . . . . . . . 17
⊢ (( bday ‘𝑙) ∈ ( bday
‘𝑋) ↔
(( bday ‘𝑙) +no ( bday
‘𝑠)) ∈
(( bday ‘𝑋) +no ( bday
‘𝑠))) |
233 | 42, 232 | sylib 217 |
. . . . . . . . . . . . . . . 16
⊢ (𝑙 ∈ ( L ‘𝑋) → (( bday ‘𝑙) +no ( bday
‘𝑠)) ∈
(( bday ‘𝑋) +no ( bday
‘𝑠))) |
234 | 233 | ad2antrl 727 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ (𝑙 ∈ ( L ‘𝑋) ∧ 𝑠 ∈ ( R ‘𝑌))) → (( bday
‘𝑙) +no ( bday ‘𝑠)) ∈ (( bday
‘𝑋) +no ( bday ‘𝑠))) |
235 | 148 | ad2antll 728 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ (𝑙 ∈ ( L ‘𝑋) ∧ 𝑠 ∈ ( R ‘𝑌))) → (( bday
‘𝑋) +no ( bday ‘𝑠)) ∈ (( bday
‘𝑋) +no ( bday ‘𝑌))) |
236 | | ontr1 6408 |
. . . . . . . . . . . . . . . 16
⊢ ((( bday ‘𝑋) +no ( bday
‘𝑌)) ∈ On
→ (((( bday ‘𝑙) +no ( bday
‘𝑠)) ∈
(( bday ‘𝑋) +no ( bday
‘𝑠)) ∧
(( bday ‘𝑋) +no ( bday
‘𝑠)) ∈
(( bday ‘𝑋) +no ( bday
‘𝑌))) →
(( bday ‘𝑙) +no ( bday
‘𝑠)) ∈
(( bday ‘𝑋) +no ( bday
‘𝑌)))) |
237 | 206, 236 | ax-mp 5 |
. . . . . . . . . . . . . . 15
⊢ (((( bday ‘𝑙) +no ( bday
‘𝑠)) ∈
(( bday ‘𝑋) +no ( bday
‘𝑠)) ∧
(( bday ‘𝑋) +no ( bday
‘𝑠)) ∈
(( bday ‘𝑋) +no ( bday
‘𝑌))) →
(( bday ‘𝑙) +no ( bday
‘𝑠)) ∈
(( bday ‘𝑋) +no ( bday
‘𝑌))) |
238 | 234, 235,
237 | syl2anc 585 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ (𝑙 ∈ ( L ‘𝑋) ∧ 𝑠 ∈ ( R ‘𝑌))) → (( bday
‘𝑙) +no ( bday ‘𝑠)) ∈ (( bday
‘𝑋) +no ( bday ‘𝑌))) |
239 | | elun1 4176 |
. . . . . . . . . . . . . 14
⊢ ((( bday ‘𝑙) +no ( bday
‘𝑠)) ∈
(( bday ‘𝑋) +no ( bday
‘𝑌)) →
(( bday ‘𝑙) +no ( bday
‘𝑠)) ∈
((( bday ‘𝑋) +no ( bday
‘𝑌)) ∪
(( bday ‘𝑋) +no ( bday
‘𝑍)))) |
240 | 238, 239 | syl 17 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ (𝑙 ∈ ( L ‘𝑋) ∧ 𝑠 ∈ ( R ‘𝑌))) → (( bday
‘𝑙) +no ( bday ‘𝑠)) ∈ ((( bday
‘𝑋) +no ( bday ‘𝑌)) ∪ (( bday
‘𝑋) +no ( bday ‘𝑍)))) |
241 | 230, 240 | eqeltrid 2838 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ (𝑙 ∈ ( L ‘𝑋) ∧ 𝑠 ∈ ( R ‘𝑌))) → ((( bday
‘𝑙) +no ( bday ‘𝑠)) ∪ (( bday
‘𝑙) +no ( bday ‘ 0s ))) ∈ ((( bday ‘𝑋) +no ( bday
‘𝑌)) ∪
(( bday ‘𝑋) +no ( bday
‘𝑍)))) |
242 | 221, 222,
223, 224, 241 | addsproplem1 27443 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ (𝑙 ∈ ( L ‘𝑋) ∧ 𝑠 ∈ ( R ‘𝑌))) → ((𝑙 +s 𝑠) ∈ No
∧ (𝑠 <s
0s → (𝑠
+s 𝑙) <s (
0s +s 𝑙)))) |
243 | 242 | simpld 496 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑙 ∈ ( L ‘𝑋) ∧ 𝑠 ∈ ( R ‘𝑌))) → (𝑙 +s 𝑠) ∈ No
) |
244 | 154 | adantrl 715 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑙 ∈ ( L ‘𝑋) ∧ 𝑠 ∈ ( R ‘𝑌))) → (𝑋 +s 𝑠) ∈ No
) |
245 | | rightval 27349 |
. . . . . . . . . . . . . . 15
⊢ ( R
‘𝑌) = {𝑠 ∈ ( O ‘( bday ‘𝑌)) ∣ 𝑌 <s 𝑠} |
246 | 245 | reqabi 3455 |
. . . . . . . . . . . . . 14
⊢ (𝑠 ∈ ( R ‘𝑌) ↔ (𝑠 ∈ ( O ‘(
bday ‘𝑌))
∧ 𝑌 <s 𝑠)) |
247 | 246 | simprbi 498 |
. . . . . . . . . . . . 13
⊢ (𝑠 ∈ ( R ‘𝑌) → 𝑌 <s 𝑠) |
248 | 247 | ad2antll 728 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ (𝑙 ∈ ( L ‘𝑋) ∧ 𝑠 ∈ ( R ‘𝑌))) → 𝑌 <s 𝑠) |
249 | 20 | adantr 482 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ (𝑙 ∈ ( L ‘𝑋) ∧ 𝑠 ∈ ( R ‘𝑌))) → 𝑌 ∈ No
) |
250 | 45 | ad2antrl 727 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ (𝑙 ∈ ( L ‘𝑋) ∧ 𝑠 ∈ ( R ‘𝑌))) → (( bday
‘𝑙) +no ( bday ‘𝑌)) ∈ (( bday
‘𝑋) +no ( bday ‘𝑌))) |
251 | | naddcl 8673 |
. . . . . . . . . . . . . . . . . 18
⊢ ((( bday ‘𝑙) ∈ On ∧ (
bday ‘𝑌)
∈ On) → (( bday ‘𝑙) +no (
bday ‘𝑌))
∈ On) |
252 | 26, 31, 251 | mp2an 691 |
. . . . . . . . . . . . . . . . 17
⊢ (( bday ‘𝑙) +no ( bday
‘𝑌)) ∈
On |
253 | | naddcl 8673 |
. . . . . . . . . . . . . . . . . 18
⊢ ((( bday ‘𝑙) ∈ On ∧ (
bday ‘𝑠)
∈ On) → (( bday ‘𝑙) +no (
bday ‘𝑠))
∈ On) |
254 | 26, 135, 253 | mp2an 691 |
. . . . . . . . . . . . . . . . 17
⊢ (( bday ‘𝑙) +no ( bday
‘𝑠)) ∈
On |
255 | | onunel 6467 |
. . . . . . . . . . . . . . . . 17
⊢ (((( bday ‘𝑙) +no ( bday
‘𝑌)) ∈ On
∧ (( bday ‘𝑙) +no ( bday
‘𝑠)) ∈ On
∧ (( bday ‘𝑋) +no ( bday
‘𝑌)) ∈
On) → (((( bday ‘𝑙) +no ( bday
‘𝑌)) ∪
(( bday ‘𝑙) +no ( bday
‘𝑠))) ∈
(( bday ‘𝑋) +no ( bday
‘𝑌)) ↔
((( bday ‘𝑙) +no ( bday
‘𝑌)) ∈
(( bday ‘𝑋) +no ( bday
‘𝑌)) ∧
(( bday ‘𝑙) +no ( bday
‘𝑠)) ∈
(( bday ‘𝑋) +no ( bday
‘𝑌))))) |
256 | 252, 254,
206, 255 | mp3an 1462 |
. . . . . . . . . . . . . . . 16
⊢ (((( bday ‘𝑙) +no ( bday
‘𝑌)) ∪
(( bday ‘𝑙) +no ( bday
‘𝑠))) ∈
(( bday ‘𝑋) +no ( bday
‘𝑌)) ↔
((( bday ‘𝑙) +no ( bday
‘𝑌)) ∈
(( bday ‘𝑋) +no ( bday
‘𝑌)) ∧
(( bday ‘𝑙) +no ( bday
‘𝑠)) ∈
(( bday ‘𝑋) +no ( bday
‘𝑌)))) |
257 | 250, 238,
256 | sylanbrc 584 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ (𝑙 ∈ ( L ‘𝑋) ∧ 𝑠 ∈ ( R ‘𝑌))) → ((( bday
‘𝑙) +no ( bday ‘𝑌)) ∪ (( bday
‘𝑙) +no ( bday ‘𝑠))) ∈ (( bday
‘𝑋) +no ( bday ‘𝑌))) |
258 | | elun1 4176 |
. . . . . . . . . . . . . . 15
⊢ (((( bday ‘𝑙) +no ( bday
‘𝑌)) ∪
(( bday ‘𝑙) +no ( bday
‘𝑠))) ∈
(( bday ‘𝑋) +no ( bday
‘𝑌)) →
((( bday ‘𝑙) +no ( bday
‘𝑌)) ∪
(( bday ‘𝑙) +no ( bday
‘𝑠))) ∈
((( bday ‘𝑋) +no ( bday
‘𝑌)) ∪
(( bday ‘𝑋) +no ( bday
‘𝑍)))) |
259 | 257, 258 | syl 17 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ (𝑙 ∈ ( L ‘𝑋) ∧ 𝑠 ∈ ( R ‘𝑌))) → ((( bday
‘𝑙) +no ( bday ‘𝑌)) ∪ (( bday
‘𝑙) +no ( bday ‘𝑠))) ∈ ((( bday
‘𝑋) +no ( bday ‘𝑌)) ∪ (( bday
‘𝑋) +no ( bday ‘𝑍)))) |
260 | 221, 222,
249, 223, 259 | addsproplem1 27443 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ (𝑙 ∈ ( L ‘𝑋) ∧ 𝑠 ∈ ( R ‘𝑌))) → ((𝑙 +s 𝑌) ∈ No
∧ (𝑌 <s 𝑠 → (𝑌 +s 𝑙) <s (𝑠 +s 𝑙)))) |
261 | 260 | simprd 497 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ (𝑙 ∈ ( L ‘𝑋) ∧ 𝑠 ∈ ( R ‘𝑌))) → (𝑌 <s 𝑠 → (𝑌 +s 𝑙) <s (𝑠 +s 𝑙))) |
262 | 248, 261 | mpd 15 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ (𝑙 ∈ ( L ‘𝑋) ∧ 𝑠 ∈ ( R ‘𝑌))) → (𝑌 +s 𝑙) <s (𝑠 +s 𝑙)) |
263 | 222, 249 | addscomd 27441 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ (𝑙 ∈ ( L ‘𝑋) ∧ 𝑠 ∈ ( R ‘𝑌))) → (𝑙 +s 𝑌) = (𝑌 +s 𝑙)) |
264 | 222, 223 | addscomd 27441 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ (𝑙 ∈ ( L ‘𝑋) ∧ 𝑠 ∈ ( R ‘𝑌))) → (𝑙 +s 𝑠) = (𝑠 +s 𝑙)) |
265 | 262, 263,
264 | 3brtr4d 5180 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑙 ∈ ( L ‘𝑋) ∧ 𝑠 ∈ ( R ‘𝑌))) → (𝑙 +s 𝑌) <s (𝑙 +s 𝑠)) |
266 | | leftval 27348 |
. . . . . . . . . . . . . 14
⊢ ( L
‘𝑋) = {𝑙 ∈ ( O ‘( bday ‘𝑋)) ∣ 𝑙 <s 𝑋} |
267 | 266 | reqabi 3455 |
. . . . . . . . . . . . 13
⊢ (𝑙 ∈ ( L ‘𝑋) ↔ (𝑙 ∈ ( O ‘(
bday ‘𝑋))
∧ 𝑙 <s 𝑋)) |
268 | 267 | simprbi 498 |
. . . . . . . . . . . 12
⊢ (𝑙 ∈ ( L ‘𝑋) → 𝑙 <s 𝑋) |
269 | 268 | ad2antrl 727 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ (𝑙 ∈ ( L ‘𝑋) ∧ 𝑠 ∈ ( R ‘𝑌))) → 𝑙 <s 𝑋) |
270 | 57 | adantr 482 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ (𝑙 ∈ ( L ‘𝑋) ∧ 𝑠 ∈ ( R ‘𝑌))) → 𝑋 ∈ No
) |
271 | | naddcom 8678 |
. . . . . . . . . . . . . . . . 17
⊢ ((( bday ‘𝑠) ∈ On ∧ (
bday ‘𝑙)
∈ On) → (( bday ‘𝑠) +no (
bday ‘𝑙)) =
(( bday ‘𝑙) +no ( bday
‘𝑠))) |
272 | 135, 26, 271 | mp2an 691 |
. . . . . . . . . . . . . . . 16
⊢ (( bday ‘𝑠) +no ( bday
‘𝑙)) = (( bday ‘𝑙) +no ( bday
‘𝑠)) |
273 | 272, 238 | eqeltrid 2838 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ (𝑙 ∈ ( L ‘𝑋) ∧ 𝑠 ∈ ( R ‘𝑌))) → (( bday
‘𝑠) +no ( bday ‘𝑙)) ∈ (( bday
‘𝑋) +no ( bday ‘𝑌))) |
274 | | naddcom 8678 |
. . . . . . . . . . . . . . . . 17
⊢ ((( bday ‘𝑠) ∈ On ∧ (
bday ‘𝑋)
∈ On) → (( bday ‘𝑠) +no (
bday ‘𝑋)) =
(( bday ‘𝑋) +no ( bday
‘𝑠))) |
275 | 135, 39, 274 | mp2an 691 |
. . . . . . . . . . . . . . . 16
⊢ (( bday ‘𝑠) +no ( bday
‘𝑋)) = (( bday ‘𝑋) +no ( bday
‘𝑠)) |
276 | 275, 235 | eqeltrid 2838 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ (𝑙 ∈ ( L ‘𝑋) ∧ 𝑠 ∈ ( R ‘𝑌))) → (( bday
‘𝑠) +no ( bday ‘𝑋)) ∈ (( bday
‘𝑋) +no ( bday ‘𝑌))) |
277 | | naddcl 8673 |
. . . . . . . . . . . . . . . . 17
⊢ ((( bday ‘𝑠) ∈ On ∧ (
bday ‘𝑙)
∈ On) → (( bday ‘𝑠) +no (
bday ‘𝑙))
∈ On) |
278 | 135, 26, 277 | mp2an 691 |
. . . . . . . . . . . . . . . 16
⊢ (( bday ‘𝑠) +no ( bday
‘𝑙)) ∈
On |
279 | | naddcl 8673 |
. . . . . . . . . . . . . . . . 17
⊢ ((( bday ‘𝑠) ∈ On ∧ (
bday ‘𝑋)
∈ On) → (( bday ‘𝑠) +no (
bday ‘𝑋))
∈ On) |
280 | 135, 39, 279 | mp2an 691 |
. . . . . . . . . . . . . . . 16
⊢ (( bday ‘𝑠) +no ( bday
‘𝑋)) ∈
On |
281 | | onunel 6467 |
. . . . . . . . . . . . . . . 16
⊢ (((( bday ‘𝑠) +no ( bday
‘𝑙)) ∈ On
∧ (( bday ‘𝑠) +no ( bday
‘𝑋)) ∈ On
∧ (( bday ‘𝑋) +no ( bday
‘𝑌)) ∈
On) → (((( bday ‘𝑠) +no ( bday
‘𝑙)) ∪
(( bday ‘𝑠) +no ( bday
‘𝑋))) ∈
(( bday ‘𝑋) +no ( bday
‘𝑌)) ↔
((( bday ‘𝑠) +no ( bday
‘𝑙)) ∈
(( bday ‘𝑋) +no ( bday
‘𝑌)) ∧
(( bday ‘𝑠) +no ( bday
‘𝑋)) ∈
(( bday ‘𝑋) +no ( bday
‘𝑌))))) |
282 | 278, 280,
206, 281 | mp3an 1462 |
. . . . . . . . . . . . . . 15
⊢ (((( bday ‘𝑠) +no ( bday
‘𝑙)) ∪
(( bday ‘𝑠) +no ( bday
‘𝑋))) ∈
(( bday ‘𝑋) +no ( bday
‘𝑌)) ↔
((( bday ‘𝑠) +no ( bday
‘𝑙)) ∈
(( bday ‘𝑋) +no ( bday
‘𝑌)) ∧
(( bday ‘𝑠) +no ( bday
‘𝑋)) ∈
(( bday ‘𝑋) +no ( bday
‘𝑌)))) |
283 | 273, 276,
282 | sylanbrc 584 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ (𝑙 ∈ ( L ‘𝑋) ∧ 𝑠 ∈ ( R ‘𝑌))) → ((( bday
‘𝑠) +no ( bday ‘𝑙)) ∪ (( bday
‘𝑠) +no ( bday ‘𝑋))) ∈ (( bday
‘𝑋) +no ( bday ‘𝑌))) |
284 | | elun1 4176 |
. . . . . . . . . . . . . 14
⊢ (((( bday ‘𝑠) +no ( bday
‘𝑙)) ∪
(( bday ‘𝑠) +no ( bday
‘𝑋))) ∈
(( bday ‘𝑋) +no ( bday
‘𝑌)) →
((( bday ‘𝑠) +no ( bday
‘𝑙)) ∪
(( bday ‘𝑠) +no ( bday
‘𝑋))) ∈
((( bday ‘𝑋) +no ( bday
‘𝑌)) ∪
(( bday ‘𝑋) +no ( bday
‘𝑍)))) |
285 | 283, 284 | syl 17 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ (𝑙 ∈ ( L ‘𝑋) ∧ 𝑠 ∈ ( R ‘𝑌))) → ((( bday
‘𝑠) +no ( bday ‘𝑙)) ∪ (( bday
‘𝑠) +no ( bday ‘𝑋))) ∈ ((( bday
‘𝑋) +no ( bday ‘𝑌)) ∪ (( bday
‘𝑋) +no ( bday ‘𝑍)))) |
286 | 221, 223,
222, 270, 285 | addsproplem1 27443 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ (𝑙 ∈ ( L ‘𝑋) ∧ 𝑠 ∈ ( R ‘𝑌))) → ((𝑠 +s 𝑙) ∈ No
∧ (𝑙 <s 𝑋 → (𝑙 +s 𝑠) <s (𝑋 +s 𝑠)))) |
287 | 286 | simprd 497 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ (𝑙 ∈ ( L ‘𝑋) ∧ 𝑠 ∈ ( R ‘𝑌))) → (𝑙 <s 𝑋 → (𝑙 +s 𝑠) <s (𝑋 +s 𝑠))) |
288 | 269, 287 | mpd 15 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑙 ∈ ( L ‘𝑋) ∧ 𝑠 ∈ ( R ‘𝑌))) → (𝑙 +s 𝑠) <s (𝑋 +s 𝑠)) |
289 | 220, 243,
244, 265, 288 | slttrd 27252 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝑙 ∈ ( L ‘𝑋) ∧ 𝑠 ∈ ( R ‘𝑌))) → (𝑙 +s 𝑌) <s (𝑋 +s 𝑠)) |
290 | | breq12 5153 |
. . . . . . . . 9
⊢ ((𝑎 = (𝑙 +s 𝑌) ∧ 𝑏 = (𝑋 +s 𝑠)) → (𝑎 <s 𝑏 ↔ (𝑙 +s 𝑌) <s (𝑋 +s 𝑠))) |
291 | 289, 290 | syl5ibrcom 246 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑙 ∈ ( L ‘𝑋) ∧ 𝑠 ∈ ( R ‘𝑌))) → ((𝑎 = (𝑙 +s 𝑌) ∧ 𝑏 = (𝑋 +s 𝑠)) → 𝑎 <s 𝑏)) |
292 | 291 | rexlimdvva 3212 |
. . . . . . 7
⊢ (𝜑 → (∃𝑙 ∈ ( L ‘𝑋)∃𝑠 ∈ ( R ‘𝑌)(𝑎 = (𝑙 +s 𝑌) ∧ 𝑏 = (𝑋 +s 𝑠)) → 𝑎 <s 𝑏)) |
293 | 219, 292 | biimtrrid 242 |
. . . . . 6
⊢ (𝜑 → ((∃𝑙 ∈ ( L ‘𝑋)𝑎 = (𝑙 +s 𝑌) ∧ ∃𝑠 ∈ ( R ‘𝑌)𝑏 = (𝑋 +s 𝑠)) → 𝑎 <s 𝑏)) |
294 | 218, 293 | jaod 858 |
. . . . 5
⊢ (𝜑 → (((∃𝑙 ∈ ( L ‘𝑋)𝑎 = (𝑙 +s 𝑌) ∧ ∃𝑟 ∈ ( R ‘𝑋)𝑏 = (𝑟 +s 𝑌)) ∨ (∃𝑙 ∈ ( L ‘𝑋)𝑎 = (𝑙 +s 𝑌) ∧ ∃𝑠 ∈ ( R ‘𝑌)𝑏 = (𝑋 +s 𝑠))) → 𝑎 <s 𝑏)) |
295 | | reeanv 3227 |
. . . . . . 7
⊢
(∃𝑚 ∈ ( L
‘𝑌)∃𝑟 ∈ ( R ‘𝑋)(𝑎 = (𝑋 +s 𝑚) ∧ 𝑏 = (𝑟 +s 𝑌)) ↔ (∃𝑚 ∈ ( L ‘𝑌)𝑎 = (𝑋 +s 𝑚) ∧ ∃𝑟 ∈ ( R ‘𝑋)𝑏 = (𝑟 +s 𝑌))) |
296 | 15 | adantr 482 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ (𝑚 ∈ ( L ‘𝑌) ∧ 𝑟 ∈ ( R ‘𝑋))) → ∀𝑥 ∈ No
∀𝑦 ∈ No ∀𝑧 ∈ No
(((( bday ‘𝑥) +no ( bday
‘𝑦)) ∪
(( bday ‘𝑥) +no ( bday
‘𝑧))) ∈
((( bday ‘𝑋) +no ( bday
‘𝑌)) ∪
(( bday ‘𝑋) +no ( bday
‘𝑍))) →
((𝑥 +s 𝑦) ∈
No ∧ (𝑦 <s
𝑧 → (𝑦 +s 𝑥) <s (𝑧 +s 𝑥))))) |
297 | 57 | adantr 482 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ (𝑚 ∈ ( L ‘𝑌) ∧ 𝑟 ∈ ( R ‘𝑋))) → 𝑋 ∈ No
) |
298 | 60 | ad2antrl 727 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ (𝑚 ∈ ( L ‘𝑌) ∧ 𝑟 ∈ ( R ‘𝑋))) → 𝑚 ∈ No
) |
299 | 22 | a1i 11 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ (𝑚 ∈ ( L ‘𝑌) ∧ 𝑟 ∈ ( R ‘𝑋))) → 0s ∈ No ) |
300 | 81 | ad2antrl 727 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ (𝑚 ∈ ( L ‘𝑌) ∧ 𝑟 ∈ ( R ‘𝑋))) → (( bday
‘𝑋) +no ( bday ‘𝑚)) ∈ (( bday
‘𝑋) +no ( bday ‘𝑌))) |
301 | 300, 83 | syl 17 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ (𝑚 ∈ ( L ‘𝑌) ∧ 𝑟 ∈ ( R ‘𝑋))) → (( bday
‘𝑋) +no ( bday ‘𝑚)) ∈ ((( bday
‘𝑋) +no ( bday ‘𝑌)) ∪ (( bday
‘𝑋) +no ( bday ‘𝑍)))) |
302 | 73, 301 | eqeltrid 2838 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ (𝑚 ∈ ( L ‘𝑌) ∧ 𝑟 ∈ ( R ‘𝑋))) → ((( bday
‘𝑋) +no ( bday ‘𝑚)) ∪ (( bday
‘𝑋) +no ( bday ‘ 0s ))) ∈ ((( bday ‘𝑋) +no ( bday
‘𝑌)) ∪
(( bday ‘𝑋) +no ( bday
‘𝑍)))) |
303 | 296, 297,
298, 299, 302 | addsproplem1 27443 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ (𝑚 ∈ ( L ‘𝑌) ∧ 𝑟 ∈ ( R ‘𝑋))) → ((𝑋 +s 𝑚) ∈ No
∧ (𝑚 <s
0s → (𝑚
+s 𝑋) <s (
0s +s 𝑋)))) |
304 | 303 | simpld 496 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑚 ∈ ( L ‘𝑌) ∧ 𝑟 ∈ ( R ‘𝑋))) → (𝑋 +s 𝑚) ∈ No
) |
305 | 95 | ad2antll 728 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ (𝑚 ∈ ( L ‘𝑌) ∧ 𝑟 ∈ ( R ‘𝑋))) → 𝑟 ∈ No
) |
306 | 103 | uneq2i 4160 |
. . . . . . . . . . . . . 14
⊢ ((( bday ‘𝑟) +no ( bday
‘𝑚)) ∪
(( bday ‘𝑟) +no ( bday
‘ 0s ))) = ((( bday
‘𝑟) +no ( bday ‘𝑚)) ∪ ( bday
‘𝑟)) |
307 | | naddword1 8687 |
. . . . . . . . . . . . . . . 16
⊢ ((( bday ‘𝑟) ∈ On ∧ (
bday ‘𝑚)
∈ On) → ( bday ‘𝑟) ⊆ (( bday
‘𝑟) +no ( bday ‘𝑚))) |
308 | 100, 68, 307 | mp2an 691 |
. . . . . . . . . . . . . . 15
⊢ ( bday ‘𝑟) ⊆ (( bday
‘𝑟) +no ( bday ‘𝑚)) |
309 | | ssequn2 4183 |
. . . . . . . . . . . . . . 15
⊢ (( bday ‘𝑟) ⊆ (( bday
‘𝑟) +no ( bday ‘𝑚)) ↔ ((( bday
‘𝑟) +no ( bday ‘𝑚)) ∪ ( bday
‘𝑟)) = (( bday ‘𝑟) +no ( bday
‘𝑚))) |
310 | 308, 309 | mpbi 229 |
. . . . . . . . . . . . . 14
⊢ ((( bday ‘𝑟) +no ( bday
‘𝑚)) ∪
( bday ‘𝑟)) = (( bday
‘𝑟) +no ( bday ‘𝑚)) |
311 | 306, 310 | eqtri 2761 |
. . . . . . . . . . . . 13
⊢ ((( bday ‘𝑟) +no ( bday
‘𝑚)) ∪
(( bday ‘𝑟) +no ( bday
‘ 0s ))) = (( bday
‘𝑟) +no ( bday ‘𝑚)) |
312 | | naddel1 8683 |
. . . . . . . . . . . . . . . . . 18
⊢ ((( bday ‘𝑟) ∈ On ∧ (
bday ‘𝑋)
∈ On ∧ ( bday ‘𝑚) ∈ On) → ((
bday ‘𝑟)
∈ ( bday ‘𝑋) ↔ (( bday
‘𝑟) +no ( bday ‘𝑚)) ∈ (( bday
‘𝑋) +no ( bday ‘𝑚)))) |
313 | 100, 39, 68, 312 | mp3an 1462 |
. . . . . . . . . . . . . . . . 17
⊢ (( bday ‘𝑟) ∈ ( bday
‘𝑋) ↔
(( bday ‘𝑟) +no ( bday
‘𝑚)) ∈
(( bday ‘𝑋) +no ( bday
‘𝑚))) |
314 | 114, 313 | sylib 217 |
. . . . . . . . . . . . . . . 16
⊢ (𝑟 ∈ ( R ‘𝑋) → (( bday ‘𝑟) +no ( bday
‘𝑚)) ∈
(( bday ‘𝑋) +no ( bday
‘𝑚))) |
315 | 314 | ad2antll 728 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ (𝑚 ∈ ( L ‘𝑌) ∧ 𝑟 ∈ ( R ‘𝑋))) → (( bday
‘𝑟) +no ( bday ‘𝑚)) ∈ (( bday
‘𝑋) +no ( bday ‘𝑚))) |
316 | | ontr1 6408 |
. . . . . . . . . . . . . . . 16
⊢ ((( bday ‘𝑋) +no ( bday
‘𝑌)) ∈ On
→ (((( bday ‘𝑟) +no ( bday
‘𝑚)) ∈
(( bday ‘𝑋) +no ( bday
‘𝑚)) ∧
(( bday ‘𝑋) +no ( bday
‘𝑚)) ∈
(( bday ‘𝑋) +no ( bday
‘𝑌))) →
(( bday ‘𝑟) +no ( bday
‘𝑚)) ∈
(( bday ‘𝑋) +no ( bday
‘𝑌)))) |
317 | 206, 316 | ax-mp 5 |
. . . . . . . . . . . . . . 15
⊢ (((( bday ‘𝑟) +no ( bday
‘𝑚)) ∈
(( bday ‘𝑋) +no ( bday
‘𝑚)) ∧
(( bday ‘𝑋) +no ( bday
‘𝑚)) ∈
(( bday ‘𝑋) +no ( bday
‘𝑌))) →
(( bday ‘𝑟) +no ( bday
‘𝑚)) ∈
(( bday ‘𝑋) +no ( bday
‘𝑌))) |
318 | 315, 300,
317 | syl2anc 585 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ (𝑚 ∈ ( L ‘𝑌) ∧ 𝑟 ∈ ( R ‘𝑋))) → (( bday
‘𝑟) +no ( bday ‘𝑚)) ∈ (( bday
‘𝑋) +no ( bday ‘𝑌))) |
319 | | elun1 4176 |
. . . . . . . . . . . . . 14
⊢ ((( bday ‘𝑟) +no ( bday
‘𝑚)) ∈
(( bday ‘𝑋) +no ( bday
‘𝑌)) →
(( bday ‘𝑟) +no ( bday
‘𝑚)) ∈
((( bday ‘𝑋) +no ( bday
‘𝑌)) ∪
(( bday ‘𝑋) +no ( bday
‘𝑍)))) |
320 | 318, 319 | syl 17 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ (𝑚 ∈ ( L ‘𝑌) ∧ 𝑟 ∈ ( R ‘𝑋))) → (( bday
‘𝑟) +no ( bday ‘𝑚)) ∈ ((( bday
‘𝑋) +no ( bday ‘𝑌)) ∪ (( bday
‘𝑋) +no ( bday ‘𝑍)))) |
321 | 311, 320 | eqeltrid 2838 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ (𝑚 ∈ ( L ‘𝑌) ∧ 𝑟 ∈ ( R ‘𝑋))) → ((( bday
‘𝑟) +no ( bday ‘𝑚)) ∪ (( bday
‘𝑟) +no ( bday ‘ 0s ))) ∈ ((( bday ‘𝑋) +no ( bday
‘𝑌)) ∪
(( bday ‘𝑋) +no ( bday
‘𝑍)))) |
322 | 296, 305,
298, 299, 321 | addsproplem1 27443 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ (𝑚 ∈ ( L ‘𝑌) ∧ 𝑟 ∈ ( R ‘𝑋))) → ((𝑟 +s 𝑚) ∈ No
∧ (𝑚 <s
0s → (𝑚
+s 𝑟) <s (
0s +s 𝑟)))) |
323 | 322 | simpld 496 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑚 ∈ ( L ‘𝑌) ∧ 𝑟 ∈ ( R ‘𝑋))) → (𝑟 +s 𝑚) ∈ No
) |
324 | 20 | adantr 482 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ (𝑚 ∈ ( L ‘𝑌) ∧ 𝑟 ∈ ( R ‘𝑋))) → 𝑌 ∈ No
) |
325 | 117 | ad2antll 728 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ (𝑚 ∈ ( L ‘𝑌) ∧ 𝑟 ∈ ( R ‘𝑋))) → (( bday
‘𝑟) +no ( bday ‘𝑌)) ∈ (( bday
‘𝑋) +no ( bday ‘𝑌))) |
326 | 325, 119 | syl 17 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ (𝑚 ∈ ( L ‘𝑌) ∧ 𝑟 ∈ ( R ‘𝑋))) → (( bday
‘𝑟) +no ( bday ‘𝑌)) ∈ ((( bday
‘𝑋) +no ( bday ‘𝑌)) ∪ (( bday
‘𝑋) +no ( bday ‘𝑍)))) |
327 | 109, 326 | eqeltrid 2838 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ (𝑚 ∈ ( L ‘𝑌) ∧ 𝑟 ∈ ( R ‘𝑋))) → ((( bday
‘𝑟) +no ( bday ‘𝑌)) ∪ (( bday
‘𝑟) +no ( bday ‘ 0s ))) ∈ ((( bday ‘𝑋) +no ( bday
‘𝑌)) ∪
(( bday ‘𝑋) +no ( bday
‘𝑍)))) |
328 | 296, 305,
324, 299, 327 | addsproplem1 27443 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ (𝑚 ∈ ( L ‘𝑌) ∧ 𝑟 ∈ ( R ‘𝑋))) → ((𝑟 +s 𝑌) ∈ No
∧ (𝑌 <s
0s → (𝑌
+s 𝑟) <s (
0s +s 𝑟)))) |
329 | 328 | simpld 496 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑚 ∈ ( L ‘𝑌) ∧ 𝑟 ∈ ( R ‘𝑋))) → (𝑟 +s 𝑌) ∈ No
) |
330 | | rightval 27349 |
. . . . . . . . . . . . . . . 16
⊢ ( R
‘𝑋) = {𝑟 ∈ ( O ‘( bday ‘𝑋)) ∣ 𝑋 <s 𝑟} |
331 | 330 | eleq2i 2826 |
. . . . . . . . . . . . . . 15
⊢ (𝑟 ∈ ( R ‘𝑋) ↔ 𝑟 ∈ {𝑟 ∈ ( O ‘(
bday ‘𝑋))
∣ 𝑋 <s 𝑟}) |
332 | 331 | biimpi 215 |
. . . . . . . . . . . . . 14
⊢ (𝑟 ∈ ( R ‘𝑋) → 𝑟 ∈ {𝑟 ∈ ( O ‘(
bday ‘𝑋))
∣ 𝑋 <s 𝑟}) |
333 | 332 | ad2antll 728 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ (𝑚 ∈ ( L ‘𝑌) ∧ 𝑟 ∈ ( R ‘𝑋))) → 𝑟 ∈ {𝑟 ∈ ( O ‘(
bday ‘𝑋))
∣ 𝑋 <s 𝑟}) |
334 | | rabid 3453 |
. . . . . . . . . . . . 13
⊢ (𝑟 ∈ {𝑟 ∈ ( O ‘(
bday ‘𝑋))
∣ 𝑋 <s 𝑟} ↔ (𝑟 ∈ ( O ‘(
bday ‘𝑋))
∧ 𝑋 <s 𝑟)) |
335 | 333, 334 | sylib 217 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ (𝑚 ∈ ( L ‘𝑌) ∧ 𝑟 ∈ ( R ‘𝑋))) → (𝑟 ∈ ( O ‘(
bday ‘𝑋))
∧ 𝑋 <s 𝑟)) |
336 | 335 | simprd 497 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ (𝑚 ∈ ( L ‘𝑌) ∧ 𝑟 ∈ ( R ‘𝑋))) → 𝑋 <s 𝑟) |
337 | | naddcom 8678 |
. . . . . . . . . . . . . . . . 17
⊢ ((( bday ‘𝑚) ∈ On ∧ (
bday ‘𝑋)
∈ On) → (( bday ‘𝑚) +no (
bday ‘𝑋)) =
(( bday ‘𝑋) +no ( bday
‘𝑚))) |
338 | 68, 39, 337 | mp2an 691 |
. . . . . . . . . . . . . . . 16
⊢ (( bday ‘𝑚) +no ( bday
‘𝑋)) = (( bday ‘𝑋) +no ( bday
‘𝑚)) |
339 | 338, 300 | eqeltrid 2838 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ (𝑚 ∈ ( L ‘𝑌) ∧ 𝑟 ∈ ( R ‘𝑋))) → (( bday
‘𝑚) +no ( bday ‘𝑋)) ∈ (( bday
‘𝑋) +no ( bday ‘𝑌))) |
340 | | naddcom 8678 |
. . . . . . . . . . . . . . . . 17
⊢ ((( bday ‘𝑚) ∈ On ∧ (
bday ‘𝑟)
∈ On) → (( bday ‘𝑚) +no (
bday ‘𝑟)) =
(( bday ‘𝑟) +no ( bday
‘𝑚))) |
341 | 68, 100, 340 | mp2an 691 |
. . . . . . . . . . . . . . . 16
⊢ (( bday ‘𝑚) +no ( bday
‘𝑟)) = (( bday ‘𝑟) +no ( bday
‘𝑚)) |
342 | 341, 318 | eqeltrid 2838 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ (𝑚 ∈ ( L ‘𝑌) ∧ 𝑟 ∈ ( R ‘𝑋))) → (( bday
‘𝑚) +no ( bday ‘𝑟)) ∈ (( bday
‘𝑋) +no ( bday ‘𝑌))) |
343 | | naddcl 8673 |
. . . . . . . . . . . . . . . . 17
⊢ ((( bday ‘𝑚) ∈ On ∧ (
bday ‘𝑋)
∈ On) → (( bday ‘𝑚) +no (
bday ‘𝑋))
∈ On) |
344 | 68, 39, 343 | mp2an 691 |
. . . . . . . . . . . . . . . 16
⊢ (( bday ‘𝑚) +no ( bday
‘𝑋)) ∈
On |
345 | | naddcl 8673 |
. . . . . . . . . . . . . . . . 17
⊢ ((( bday ‘𝑚) ∈ On ∧ (
bday ‘𝑟)
∈ On) → (( bday ‘𝑚) +no (
bday ‘𝑟))
∈ On) |
346 | 68, 100, 345 | mp2an 691 |
. . . . . . . . . . . . . . . 16
⊢ (( bday ‘𝑚) +no ( bday
‘𝑟)) ∈
On |
347 | | onunel 6467 |
. . . . . . . . . . . . . . . 16
⊢ (((( bday ‘𝑚) +no ( bday
‘𝑋)) ∈ On
∧ (( bday ‘𝑚) +no ( bday
‘𝑟)) ∈ On
∧ (( bday ‘𝑋) +no ( bday
‘𝑌)) ∈
On) → (((( bday ‘𝑚) +no ( bday
‘𝑋)) ∪
(( bday ‘𝑚) +no ( bday
‘𝑟))) ∈
(( bday ‘𝑋) +no ( bday
‘𝑌)) ↔
((( bday ‘𝑚) +no ( bday
‘𝑋)) ∈
(( bday ‘𝑋) +no ( bday
‘𝑌)) ∧
(( bday ‘𝑚) +no ( bday
‘𝑟)) ∈
(( bday ‘𝑋) +no ( bday
‘𝑌))))) |
348 | 344, 346,
206, 347 | mp3an 1462 |
. . . . . . . . . . . . . . 15
⊢ (((( bday ‘𝑚) +no ( bday
‘𝑋)) ∪
(( bday ‘𝑚) +no ( bday
‘𝑟))) ∈
(( bday ‘𝑋) +no ( bday
‘𝑌)) ↔
((( bday ‘𝑚) +no ( bday
‘𝑋)) ∈
(( bday ‘𝑋) +no ( bday
‘𝑌)) ∧
(( bday ‘𝑚) +no ( bday
‘𝑟)) ∈
(( bday ‘𝑋) +no ( bday
‘𝑌)))) |
349 | 339, 342,
348 | sylanbrc 584 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ (𝑚 ∈ ( L ‘𝑌) ∧ 𝑟 ∈ ( R ‘𝑋))) → ((( bday
‘𝑚) +no ( bday ‘𝑋)) ∪ (( bday
‘𝑚) +no ( bday ‘𝑟))) ∈ (( bday
‘𝑋) +no ( bday ‘𝑌))) |
350 | | elun1 4176 |
. . . . . . . . . . . . . 14
⊢ (((( bday ‘𝑚) +no ( bday
‘𝑋)) ∪
(( bday ‘𝑚) +no ( bday
‘𝑟))) ∈
(( bday ‘𝑋) +no ( bday
‘𝑌)) →
((( bday ‘𝑚) +no ( bday
‘𝑋)) ∪
(( bday ‘𝑚) +no ( bday
‘𝑟))) ∈
((( bday ‘𝑋) +no ( bday
‘𝑌)) ∪
(( bday ‘𝑋) +no ( bday
‘𝑍)))) |
351 | 349, 350 | syl 17 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ (𝑚 ∈ ( L ‘𝑌) ∧ 𝑟 ∈ ( R ‘𝑋))) → ((( bday
‘𝑚) +no ( bday ‘𝑋)) ∪ (( bday
‘𝑚) +no ( bday ‘𝑟))) ∈ ((( bday
‘𝑋) +no ( bday ‘𝑌)) ∪ (( bday
‘𝑋) +no ( bday ‘𝑍)))) |
352 | 296, 298,
297, 305, 351 | addsproplem1 27443 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ (𝑚 ∈ ( L ‘𝑌) ∧ 𝑟 ∈ ( R ‘𝑋))) → ((𝑚 +s 𝑋) ∈ No
∧ (𝑋 <s 𝑟 → (𝑋 +s 𝑚) <s (𝑟 +s 𝑚)))) |
353 | 352 | simprd 497 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ (𝑚 ∈ ( L ‘𝑌) ∧ 𝑟 ∈ ( R ‘𝑋))) → (𝑋 <s 𝑟 → (𝑋 +s 𝑚) <s (𝑟 +s 𝑚))) |
354 | 336, 353 | mpd 15 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑚 ∈ ( L ‘𝑌) ∧ 𝑟 ∈ ( R ‘𝑋))) → (𝑋 +s 𝑚) <s (𝑟 +s 𝑚)) |
355 | | leftval 27348 |
. . . . . . . . . . . . . . . . 17
⊢ ( L
‘𝑌) = {𝑚 ∈ ( O ‘( bday ‘𝑌)) ∣ 𝑚 <s 𝑌} |
356 | 355 | eleq2i 2826 |
. . . . . . . . . . . . . . . 16
⊢ (𝑚 ∈ ( L ‘𝑌) ↔ 𝑚 ∈ {𝑚 ∈ ( O ‘(
bday ‘𝑌))
∣ 𝑚 <s 𝑌}) |
357 | 356 | biimpi 215 |
. . . . . . . . . . . . . . 15
⊢ (𝑚 ∈ ( L ‘𝑌) → 𝑚 ∈ {𝑚 ∈ ( O ‘(
bday ‘𝑌))
∣ 𝑚 <s 𝑌}) |
358 | 357 | ad2antrl 727 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ (𝑚 ∈ ( L ‘𝑌) ∧ 𝑟 ∈ ( R ‘𝑋))) → 𝑚 ∈ {𝑚 ∈ ( O ‘(
bday ‘𝑌))
∣ 𝑚 <s 𝑌}) |
359 | | rabid 3453 |
. . . . . . . . . . . . . 14
⊢ (𝑚 ∈ {𝑚 ∈ ( O ‘(
bday ‘𝑌))
∣ 𝑚 <s 𝑌} ↔ (𝑚 ∈ ( O ‘(
bday ‘𝑌))
∧ 𝑚 <s 𝑌)) |
360 | 358, 359 | sylib 217 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ (𝑚 ∈ ( L ‘𝑌) ∧ 𝑟 ∈ ( R ‘𝑋))) → (𝑚 ∈ ( O ‘(
bday ‘𝑌))
∧ 𝑚 <s 𝑌)) |
361 | 360 | simprd 497 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ (𝑚 ∈ ( L ‘𝑌) ∧ 𝑟 ∈ ( R ‘𝑋))) → 𝑚 <s 𝑌) |
362 | | naddcl 8673 |
. . . . . . . . . . . . . . . . . 18
⊢ ((( bday ‘𝑟) ∈ On ∧ (
bday ‘𝑚)
∈ On) → (( bday ‘𝑟) +no (
bday ‘𝑚))
∈ On) |
363 | 100, 68, 362 | mp2an 691 |
. . . . . . . . . . . . . . . . 17
⊢ (( bday ‘𝑟) +no ( bday
‘𝑚)) ∈
On |
364 | | naddcl 8673 |
. . . . . . . . . . . . . . . . . 18
⊢ ((( bday ‘𝑟) ∈ On ∧ (
bday ‘𝑌)
∈ On) → (( bday ‘𝑟) +no (
bday ‘𝑌))
∈ On) |
365 | 100, 31, 364 | mp2an 691 |
. . . . . . . . . . . . . . . . 17
⊢ (( bday ‘𝑟) +no ( bday
‘𝑌)) ∈
On |
366 | | onunel 6467 |
. . . . . . . . . . . . . . . . 17
⊢ (((( bday ‘𝑟) +no ( bday
‘𝑚)) ∈ On
∧ (( bday ‘𝑟) +no ( bday
‘𝑌)) ∈ On
∧ (( bday ‘𝑋) +no ( bday
‘𝑌)) ∈
On) → (((( bday ‘𝑟) +no ( bday
‘𝑚)) ∪
(( bday ‘𝑟) +no ( bday
‘𝑌))) ∈
(( bday ‘𝑋) +no ( bday
‘𝑌)) ↔
((( bday ‘𝑟) +no ( bday
‘𝑚)) ∈
(( bday ‘𝑋) +no ( bday
‘𝑌)) ∧
(( bday ‘𝑟) +no ( bday
‘𝑌)) ∈
(( bday ‘𝑋) +no ( bday
‘𝑌))))) |
367 | 363, 365,
206, 366 | mp3an 1462 |
. . . . . . . . . . . . . . . 16
⊢ (((( bday ‘𝑟) +no ( bday
‘𝑚)) ∪
(( bday ‘𝑟) +no ( bday
‘𝑌))) ∈
(( bday ‘𝑋) +no ( bday
‘𝑌)) ↔
((( bday ‘𝑟) +no ( bday
‘𝑚)) ∈
(( bday ‘𝑋) +no ( bday
‘𝑌)) ∧
(( bday ‘𝑟) +no ( bday
‘𝑌)) ∈
(( bday ‘𝑋) +no ( bday
‘𝑌)))) |
368 | 318, 325,
367 | sylanbrc 584 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ (𝑚 ∈ ( L ‘𝑌) ∧ 𝑟 ∈ ( R ‘𝑋))) → ((( bday
‘𝑟) +no ( bday ‘𝑚)) ∪ (( bday
‘𝑟) +no ( bday ‘𝑌))) ∈ (( bday
‘𝑋) +no ( bday ‘𝑌))) |
369 | | elun1 4176 |
. . . . . . . . . . . . . . 15
⊢ (((( bday ‘𝑟) +no ( bday
‘𝑚)) ∪
(( bday ‘𝑟) +no ( bday
‘𝑌))) ∈
(( bday ‘𝑋) +no ( bday
‘𝑌)) →
((( bday ‘𝑟) +no ( bday
‘𝑚)) ∪
(( bday ‘𝑟) +no ( bday
‘𝑌))) ∈
((( bday ‘𝑋) +no ( bday
‘𝑌)) ∪
(( bday ‘𝑋) +no ( bday
‘𝑍)))) |
370 | 368, 369 | syl 17 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ (𝑚 ∈ ( L ‘𝑌) ∧ 𝑟 ∈ ( R ‘𝑋))) → ((( bday
‘𝑟) +no ( bday ‘𝑚)) ∪ (( bday
‘𝑟) +no ( bday ‘𝑌))) ∈ ((( bday
‘𝑋) +no ( bday ‘𝑌)) ∪ (( bday
‘𝑋) +no ( bday ‘𝑍)))) |
371 | 296, 305,
298, 324, 370 | addsproplem1 27443 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ (𝑚 ∈ ( L ‘𝑌) ∧ 𝑟 ∈ ( R ‘𝑋))) → ((𝑟 +s 𝑚) ∈ No
∧ (𝑚 <s 𝑌 → (𝑚 +s 𝑟) <s (𝑌 +s 𝑟)))) |
372 | 371 | simprd 497 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ (𝑚 ∈ ( L ‘𝑌) ∧ 𝑟 ∈ ( R ‘𝑋))) → (𝑚 <s 𝑌 → (𝑚 +s 𝑟) <s (𝑌 +s 𝑟))) |
373 | 361, 372 | mpd 15 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ (𝑚 ∈ ( L ‘𝑌) ∧ 𝑟 ∈ ( R ‘𝑋))) → (𝑚 +s 𝑟) <s (𝑌 +s 𝑟)) |
374 | 305, 298 | addscomd 27441 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ (𝑚 ∈ ( L ‘𝑌) ∧ 𝑟 ∈ ( R ‘𝑋))) → (𝑟 +s 𝑚) = (𝑚 +s 𝑟)) |
375 | 305, 324 | addscomd 27441 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ (𝑚 ∈ ( L ‘𝑌) ∧ 𝑟 ∈ ( R ‘𝑋))) → (𝑟 +s 𝑌) = (𝑌 +s 𝑟)) |
376 | 373, 374,
375 | 3brtr4d 5180 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑚 ∈ ( L ‘𝑌) ∧ 𝑟 ∈ ( R ‘𝑋))) → (𝑟 +s 𝑚) <s (𝑟 +s 𝑌)) |
377 | 304, 323,
329, 354, 376 | slttrd 27252 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝑚 ∈ ( L ‘𝑌) ∧ 𝑟 ∈ ( R ‘𝑋))) → (𝑋 +s 𝑚) <s (𝑟 +s 𝑌)) |
378 | | breq12 5153 |
. . . . . . . . 9
⊢ ((𝑎 = (𝑋 +s 𝑚) ∧ 𝑏 = (𝑟 +s 𝑌)) → (𝑎 <s 𝑏 ↔ (𝑋 +s 𝑚) <s (𝑟 +s 𝑌))) |
379 | 377, 378 | syl5ibrcom 246 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑚 ∈ ( L ‘𝑌) ∧ 𝑟 ∈ ( R ‘𝑋))) → ((𝑎 = (𝑋 +s 𝑚) ∧ 𝑏 = (𝑟 +s 𝑌)) → 𝑎 <s 𝑏)) |
380 | 379 | rexlimdvva 3212 |
. . . . . . 7
⊢ (𝜑 → (∃𝑚 ∈ ( L ‘𝑌)∃𝑟 ∈ ( R ‘𝑋)(𝑎 = (𝑋 +s 𝑚) ∧ 𝑏 = (𝑟 +s 𝑌)) → 𝑎 <s 𝑏)) |
381 | 295, 380 | biimtrrid 242 |
. . . . . 6
⊢ (𝜑 → ((∃𝑚 ∈ ( L ‘𝑌)𝑎 = (𝑋 +s 𝑚) ∧ ∃𝑟 ∈ ( R ‘𝑋)𝑏 = (𝑟 +s 𝑌)) → 𝑎 <s 𝑏)) |
382 | | reeanv 3227 |
. . . . . . 7
⊢
(∃𝑚 ∈ ( L
‘𝑌)∃𝑠 ∈ ( R ‘𝑌)(𝑎 = (𝑋 +s 𝑚) ∧ 𝑏 = (𝑋 +s 𝑠)) ↔ (∃𝑚 ∈ ( L ‘𝑌)𝑎 = (𝑋 +s 𝑚) ∧ ∃𝑠 ∈ ( R ‘𝑌)𝑏 = (𝑋 +s 𝑠))) |
383 | | lltropt 27357 |
. . . . . . . . . . . . 13
⊢ ( L
‘𝑌) <<s ( R
‘𝑌) |
384 | 383 | a1i 11 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ (𝑚 ∈ ( L ‘𝑌) ∧ 𝑠 ∈ ( R ‘𝑌))) → ( L ‘𝑌) <<s ( R ‘𝑌)) |
385 | | simprl 770 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ (𝑚 ∈ ( L ‘𝑌) ∧ 𝑠 ∈ ( R ‘𝑌))) → 𝑚 ∈ ( L ‘𝑌)) |
386 | | simprr 772 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ (𝑚 ∈ ( L ‘𝑌) ∧ 𝑠 ∈ ( R ‘𝑌))) → 𝑠 ∈ ( R ‘𝑌)) |
387 | 384, 385,
386 | ssltsepcd 27285 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ (𝑚 ∈ ( L ‘𝑌) ∧ 𝑠 ∈ ( R ‘𝑌))) → 𝑚 <s 𝑠) |
388 | 15 | adantr 482 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ (𝑚 ∈ ( L ‘𝑌) ∧ 𝑠 ∈ ( R ‘𝑌))) → ∀𝑥 ∈ No
∀𝑦 ∈ No ∀𝑧 ∈ No
(((( bday ‘𝑥) +no ( bday
‘𝑦)) ∪
(( bday ‘𝑥) +no ( bday
‘𝑧))) ∈
((( bday ‘𝑋) +no ( bday
‘𝑌)) ∪
(( bday ‘𝑋) +no ( bday
‘𝑍))) →
((𝑥 +s 𝑦) ∈
No ∧ (𝑦 <s
𝑧 → (𝑦 +s 𝑥) <s (𝑧 +s 𝑥))))) |
389 | 57 | adantr 482 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ (𝑚 ∈ ( L ‘𝑌) ∧ 𝑠 ∈ ( R ‘𝑌))) → 𝑋 ∈ No
) |
390 | 60 | ad2antrl 727 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ (𝑚 ∈ ( L ‘𝑌) ∧ 𝑠 ∈ ( R ‘𝑌))) → 𝑚 ∈ No
) |
391 | 131 | ad2antll 728 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ (𝑚 ∈ ( L ‘𝑌) ∧ 𝑠 ∈ ( R ‘𝑌))) → 𝑠 ∈ No
) |
392 | 81 | ad2antrl 727 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ (𝑚 ∈ ( L ‘𝑌) ∧ 𝑠 ∈ ( R ‘𝑌))) → (( bday
‘𝑋) +no ( bday ‘𝑚)) ∈ (( bday
‘𝑋) +no ( bday ‘𝑌))) |
393 | 148 | ad2antll 728 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ (𝑚 ∈ ( L ‘𝑌) ∧ 𝑠 ∈ ( R ‘𝑌))) → (( bday
‘𝑋) +no ( bday ‘𝑠)) ∈ (( bday
‘𝑋) +no ( bday ‘𝑌))) |
394 | | naddcl 8673 |
. . . . . . . . . . . . . . . . 17
⊢ ((( bday ‘𝑋) ∈ On ∧ (
bday ‘𝑚)
∈ On) → (( bday ‘𝑋) +no ( bday
‘𝑚)) ∈
On) |
395 | 39, 68, 394 | mp2an 691 |
. . . . . . . . . . . . . . . 16
⊢ (( bday ‘𝑋) +no ( bday
‘𝑚)) ∈
On |
396 | | naddcl 8673 |
. . . . . . . . . . . . . . . . 17
⊢ ((( bday ‘𝑋) ∈ On ∧ (
bday ‘𝑠)
∈ On) → (( bday ‘𝑋) +no ( bday
‘𝑠)) ∈
On) |
397 | 39, 135, 396 | mp2an 691 |
. . . . . . . . . . . . . . . 16
⊢ (( bday ‘𝑋) +no ( bday
‘𝑠)) ∈
On |
398 | | onunel 6467 |
. . . . . . . . . . . . . . . 16
⊢ (((( bday ‘𝑋) +no ( bday
‘𝑚)) ∈ On
∧ (( bday ‘𝑋) +no ( bday
‘𝑠)) ∈ On
∧ (( bday ‘𝑋) +no ( bday
‘𝑌)) ∈
On) → (((( bday ‘𝑋) +no ( bday
‘𝑚)) ∪
(( bday ‘𝑋) +no ( bday
‘𝑠))) ∈
(( bday ‘𝑋) +no ( bday
‘𝑌)) ↔
((( bday ‘𝑋) +no ( bday
‘𝑚)) ∈
(( bday ‘𝑋) +no ( bday
‘𝑌)) ∧
(( bday ‘𝑋) +no ( bday
‘𝑠)) ∈
(( bday ‘𝑋) +no ( bday
‘𝑌))))) |
399 | 395, 397,
206, 398 | mp3an 1462 |
. . . . . . . . . . . . . . 15
⊢ (((( bday ‘𝑋) +no ( bday
‘𝑚)) ∪
(( bday ‘𝑋) +no ( bday
‘𝑠))) ∈
(( bday ‘𝑋) +no ( bday
‘𝑌)) ↔
((( bday ‘𝑋) +no ( bday
‘𝑚)) ∈
(( bday ‘𝑋) +no ( bday
‘𝑌)) ∧
(( bday ‘𝑋) +no ( bday
‘𝑠)) ∈
(( bday ‘𝑋) +no ( bday
‘𝑌)))) |
400 | 392, 393,
399 | sylanbrc 584 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ (𝑚 ∈ ( L ‘𝑌) ∧ 𝑠 ∈ ( R ‘𝑌))) → ((( bday
‘𝑋) +no ( bday ‘𝑚)) ∪ (( bday
‘𝑋) +no ( bday ‘𝑠))) ∈ (( bday
‘𝑋) +no ( bday ‘𝑌))) |
401 | | elun1 4176 |
. . . . . . . . . . . . . 14
⊢ (((( bday ‘𝑋) +no ( bday
‘𝑚)) ∪
(( bday ‘𝑋) +no ( bday
‘𝑠))) ∈
(( bday ‘𝑋) +no ( bday
‘𝑌)) →
((( bday ‘𝑋) +no ( bday
‘𝑚)) ∪
(( bday ‘𝑋) +no ( bday
‘𝑠))) ∈
((( bday ‘𝑋) +no ( bday
‘𝑌)) ∪
(( bday ‘𝑋) +no ( bday
‘𝑍)))) |
402 | 400, 401 | syl 17 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ (𝑚 ∈ ( L ‘𝑌) ∧ 𝑠 ∈ ( R ‘𝑌))) → ((( bday
‘𝑋) +no ( bday ‘𝑚)) ∪ (( bday
‘𝑋) +no ( bday ‘𝑠))) ∈ ((( bday
‘𝑋) +no ( bday ‘𝑌)) ∪ (( bday
‘𝑋) +no ( bday ‘𝑍)))) |
403 | 388, 389,
390, 391, 402 | addsproplem1 27443 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ (𝑚 ∈ ( L ‘𝑌) ∧ 𝑠 ∈ ( R ‘𝑌))) → ((𝑋 +s 𝑚) ∈ No
∧ (𝑚 <s 𝑠 → (𝑚 +s 𝑋) <s (𝑠 +s 𝑋)))) |
404 | 403 | simprd 497 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ (𝑚 ∈ ( L ‘𝑌) ∧ 𝑠 ∈ ( R ‘𝑌))) → (𝑚 <s 𝑠 → (𝑚 +s 𝑋) <s (𝑠 +s 𝑋))) |
405 | 387, 404 | mpd 15 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑚 ∈ ( L ‘𝑌) ∧ 𝑠 ∈ ( R ‘𝑌))) → (𝑚 +s 𝑋) <s (𝑠 +s 𝑋)) |
406 | 389, 390 | addscomd 27441 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑚 ∈ ( L ‘𝑌) ∧ 𝑠 ∈ ( R ‘𝑌))) → (𝑋 +s 𝑚) = (𝑚 +s 𝑋)) |
407 | 389, 391 | addscomd 27441 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑚 ∈ ( L ‘𝑌) ∧ 𝑠 ∈ ( R ‘𝑌))) → (𝑋 +s 𝑠) = (𝑠 +s 𝑋)) |
408 | 405, 406,
407 | 3brtr4d 5180 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝑚 ∈ ( L ‘𝑌) ∧ 𝑠 ∈ ( R ‘𝑌))) → (𝑋 +s 𝑚) <s (𝑋 +s 𝑠)) |
409 | | breq12 5153 |
. . . . . . . . 9
⊢ ((𝑎 = (𝑋 +s 𝑚) ∧ 𝑏 = (𝑋 +s 𝑠)) → (𝑎 <s 𝑏 ↔ (𝑋 +s 𝑚) <s (𝑋 +s 𝑠))) |
410 | 408, 409 | syl5ibrcom 246 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑚 ∈ ( L ‘𝑌) ∧ 𝑠 ∈ ( R ‘𝑌))) → ((𝑎 = (𝑋 +s 𝑚) ∧ 𝑏 = (𝑋 +s 𝑠)) → 𝑎 <s 𝑏)) |
411 | 410 | rexlimdvva 3212 |
. . . . . . 7
⊢ (𝜑 → (∃𝑚 ∈ ( L ‘𝑌)∃𝑠 ∈ ( R ‘𝑌)(𝑎 = (𝑋 +s 𝑚) ∧ 𝑏 = (𝑋 +s 𝑠)) → 𝑎 <s 𝑏)) |
412 | 382, 411 | biimtrrid 242 |
. . . . . 6
⊢ (𝜑 → ((∃𝑚 ∈ ( L ‘𝑌)𝑎 = (𝑋 +s 𝑚) ∧ ∃𝑠 ∈ ( R ‘𝑌)𝑏 = (𝑋 +s 𝑠)) → 𝑎 <s 𝑏)) |
413 | 381, 412 | jaod 858 |
. . . . 5
⊢ (𝜑 → (((∃𝑚 ∈ ( L ‘𝑌)𝑎 = (𝑋 +s 𝑚) ∧ ∃𝑟 ∈ ( R ‘𝑋)𝑏 = (𝑟 +s 𝑌)) ∨ (∃𝑚 ∈ ( L ‘𝑌)𝑎 = (𝑋 +s 𝑚) ∧ ∃𝑠 ∈ ( R ‘𝑌)𝑏 = (𝑋 +s 𝑠))) → 𝑎 <s 𝑏)) |
414 | 294, 413 | jaod 858 |
. . . 4
⊢ (𝜑 → ((((∃𝑙 ∈ ( L ‘𝑋)𝑎 = (𝑙 +s 𝑌) ∧ ∃𝑟 ∈ ( R ‘𝑋)𝑏 = (𝑟 +s 𝑌)) ∨ (∃𝑙 ∈ ( L ‘𝑋)𝑎 = (𝑙 +s 𝑌) ∧ ∃𝑠 ∈ ( R ‘𝑌)𝑏 = (𝑋 +s 𝑠))) ∨ ((∃𝑚 ∈ ( L ‘𝑌)𝑎 = (𝑋 +s 𝑚) ∧ ∃𝑟 ∈ ( R ‘𝑋)𝑏 = (𝑟 +s 𝑌)) ∨ (∃𝑚 ∈ ( L ‘𝑌)𝑎 = (𝑋 +s 𝑚) ∧ ∃𝑠 ∈ ( R ‘𝑌)𝑏 = (𝑋 +s 𝑠)))) → 𝑎 <s 𝑏)) |
415 | 182, 414 | biimtrid 241 |
. . 3
⊢ (𝜑 → ((𝑎 ∈ ({𝑝 ∣ ∃𝑙 ∈ ( L ‘𝑋)𝑝 = (𝑙 +s 𝑌)} ∪ {𝑞 ∣ ∃𝑚 ∈ ( L ‘𝑌)𝑞 = (𝑋 +s 𝑚)}) ∧ 𝑏 ∈ ({𝑤 ∣ ∃𝑟 ∈ ( R ‘𝑋)𝑤 = (𝑟 +s 𝑌)} ∪ {𝑡 ∣ ∃𝑠 ∈ ( R ‘𝑌)𝑡 = (𝑋 +s 𝑠)})) → 𝑎 <s 𝑏)) |
416 | 415 | 3impib 1117 |
. 2
⊢ ((𝜑 ∧ 𝑎 ∈ ({𝑝 ∣ ∃𝑙 ∈ ( L ‘𝑋)𝑝 = (𝑙 +s 𝑌)} ∪ {𝑞 ∣ ∃𝑚 ∈ ( L ‘𝑌)𝑞 = (𝑋 +s 𝑚)}) ∧ 𝑏 ∈ ({𝑤 ∣ ∃𝑟 ∈ ( R ‘𝑋)𝑤 = (𝑟 +s 𝑌)} ∪ {𝑡 ∣ ∃𝑠 ∈ ( R ‘𝑌)𝑡 = (𝑋 +s 𝑠)})) → 𝑎 <s 𝑏) |
417 | 7, 14, 92, 159, 416 | ssltd 27283 |
1
⊢ (𝜑 → ({𝑝 ∣ ∃𝑙 ∈ ( L ‘𝑋)𝑝 = (𝑙 +s 𝑌)} ∪ {𝑞 ∣ ∃𝑚 ∈ ( L ‘𝑌)𝑞 = (𝑋 +s 𝑚)}) <<s ({𝑤 ∣ ∃𝑟 ∈ ( R ‘𝑋)𝑤 = (𝑟 +s 𝑌)} ∪ {𝑡 ∣ ∃𝑠 ∈ ( R ‘𝑌)𝑡 = (𝑋 +s 𝑠)})) |