Step | Hyp | Ref
| Expression |
1 | | 1pru 7546 |
. . . 4
⊢
(2nd ‘1P) = {𝑤 ∣
1Q <Q 𝑤} |
2 | 1 | abeq2i 2288 |
. . 3
⊢ (𝑤 ∈ (2nd
‘1P) ↔ 1Q
<Q 𝑤) |
3 | | prop 7465 |
. . . . . 6
⊢ (𝐴 ∈ P →
〈(1st ‘𝐴), (2nd ‘𝐴)〉 ∈
P) |
4 | | prmuloc2 7557 |
. . . . . 6
⊢
((〈(1st ‘𝐴), (2nd ‘𝐴)〉 ∈ P ∧
1Q <Q 𝑤) → ∃𝑣 ∈ (1st ‘𝐴)(𝑣 ·Q 𝑤) ∈ (2nd
‘𝐴)) |
5 | 3, 4 | sylan 283 |
. . . . 5
⊢ ((𝐴 ∈ P ∧
1Q <Q 𝑤) → ∃𝑣 ∈ (1st ‘𝐴)(𝑣 ·Q 𝑤) ∈ (2nd
‘𝐴)) |
6 | | prnminu 7479 |
. . . . . . . 8
⊢
((〈(1st ‘𝐴), (2nd ‘𝐴)〉 ∈ P ∧ (𝑣
·Q 𝑤) ∈ (2nd ‘𝐴)) → ∃𝑧 ∈ (2nd
‘𝐴)𝑧 <Q (𝑣
·Q 𝑤)) |
7 | 3, 6 | sylan 283 |
. . . . . . 7
⊢ ((𝐴 ∈ P ∧
(𝑣
·Q 𝑤) ∈ (2nd ‘𝐴)) → ∃𝑧 ∈ (2nd
‘𝐴)𝑧 <Q (𝑣
·Q 𝑤)) |
8 | 7 | ad2ant2rl 511 |
. . . . . 6
⊢ (((𝐴 ∈ P ∧
1Q <Q 𝑤) ∧ (𝑣 ∈ (1st ‘𝐴) ∧ (𝑣 ·Q 𝑤) ∈ (2nd
‘𝐴))) →
∃𝑧 ∈
(2nd ‘𝐴)𝑧 <Q (𝑣
·Q 𝑤)) |
9 | | simp3 999 |
. . . . . . . . . . 11
⊢ (((𝐴 ∈ P ∧
1Q <Q 𝑤) ∧ (𝑣 ∈ (1st ‘𝐴) ∧ (𝑣 ·Q 𝑤) ∈ (2nd
‘𝐴)) ∧ 𝑧 <Q
(𝑣
·Q 𝑤)) → 𝑧 <Q (𝑣
·Q 𝑤)) |
10 | | simp2l 1023 |
. . . . . . . . . . . 12
⊢ (((𝐴 ∈ P ∧
1Q <Q 𝑤) ∧ (𝑣 ∈ (1st ‘𝐴) ∧ (𝑣 ·Q 𝑤) ∈ (2nd
‘𝐴)) ∧ 𝑧 <Q
(𝑣
·Q 𝑤)) → 𝑣 ∈ (1st ‘𝐴)) |
11 | | elprnql 7471 |
. . . . . . . . . . . . . . . . . 18
⊢
((〈(1st ‘𝐴), (2nd ‘𝐴)〉 ∈ P ∧ 𝑣 ∈ (1st
‘𝐴)) → 𝑣 ∈
Q) |
12 | 3, 11 | sylan 283 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝐴 ∈ P ∧
𝑣 ∈ (1st
‘𝐴)) → 𝑣 ∈
Q) |
13 | 12 | ad2ant2r 509 |
. . . . . . . . . . . . . . . 16
⊢ (((𝐴 ∈ P ∧
1Q <Q 𝑤) ∧ (𝑣 ∈ (1st ‘𝐴) ∧ (𝑣 ·Q 𝑤) ∈ (2nd
‘𝐴))) → 𝑣 ∈
Q) |
14 | 13 | 3adant3 1017 |
. . . . . . . . . . . . . . 15
⊢ (((𝐴 ∈ P ∧
1Q <Q 𝑤) ∧ (𝑣 ∈ (1st ‘𝐴) ∧ (𝑣 ·Q 𝑤) ∈ (2nd
‘𝐴)) ∧ 𝑧 <Q
(𝑣
·Q 𝑤)) → 𝑣 ∈ Q) |
15 | | simp1r 1022 |
. . . . . . . . . . . . . . . 16
⊢ (((𝐴 ∈ P ∧
1Q <Q 𝑤) ∧ (𝑣 ∈ (1st ‘𝐴) ∧ (𝑣 ·Q 𝑤) ∈ (2nd
‘𝐴)) ∧ 𝑧 <Q
(𝑣
·Q 𝑤)) → 1Q
<Q 𝑤) |
16 | | ltrelnq 7355 |
. . . . . . . . . . . . . . . . . 18
⊢
<Q ⊆ (Q ×
Q) |
17 | 16 | brel 4675 |
. . . . . . . . . . . . . . . . 17
⊢
(1Q <Q 𝑤 →
(1Q ∈ Q ∧ 𝑤 ∈ Q)) |
18 | 17 | simprd 114 |
. . . . . . . . . . . . . . . 16
⊢
(1Q <Q 𝑤 → 𝑤 ∈ Q) |
19 | 15, 18 | syl 14 |
. . . . . . . . . . . . . . 15
⊢ (((𝐴 ∈ P ∧
1Q <Q 𝑤) ∧ (𝑣 ∈ (1st ‘𝐴) ∧ (𝑣 ·Q 𝑤) ∈ (2nd
‘𝐴)) ∧ 𝑧 <Q
(𝑣
·Q 𝑤)) → 𝑤 ∈ Q) |
20 | | recclnq 7382 |
. . . . . . . . . . . . . . . 16
⊢ (𝑤 ∈ Q →
(*Q‘𝑤) ∈ Q) |
21 | 19, 20 | syl 14 |
. . . . . . . . . . . . . . 15
⊢ (((𝐴 ∈ P ∧
1Q <Q 𝑤) ∧ (𝑣 ∈ (1st ‘𝐴) ∧ (𝑣 ·Q 𝑤) ∈ (2nd
‘𝐴)) ∧ 𝑧 <Q
(𝑣
·Q 𝑤)) →
(*Q‘𝑤) ∈ Q) |
22 | | mulassnqg 7374 |
. . . . . . . . . . . . . . 15
⊢ ((𝑣 ∈ Q ∧
𝑤 ∈ Q
∧ (*Q‘𝑤) ∈ Q) → ((𝑣
·Q 𝑤) ·Q
(*Q‘𝑤)) = (𝑣 ·Q (𝑤
·Q (*Q‘𝑤)))) |
23 | 14, 19, 21, 22 | syl3anc 1238 |
. . . . . . . . . . . . . 14
⊢ (((𝐴 ∈ P ∧
1Q <Q 𝑤) ∧ (𝑣 ∈ (1st ‘𝐴) ∧ (𝑣 ·Q 𝑤) ∈ (2nd
‘𝐴)) ∧ 𝑧 <Q
(𝑣
·Q 𝑤)) → ((𝑣 ·Q 𝑤)
·Q (*Q‘𝑤)) = (𝑣 ·Q (𝑤
·Q (*Q‘𝑤)))) |
24 | | recidnq 7383 |
. . . . . . . . . . . . . . . 16
⊢ (𝑤 ∈ Q →
(𝑤
·Q (*Q‘𝑤)) =
1Q) |
25 | 19, 24 | syl 14 |
. . . . . . . . . . . . . . 15
⊢ (((𝐴 ∈ P ∧
1Q <Q 𝑤) ∧ (𝑣 ∈ (1st ‘𝐴) ∧ (𝑣 ·Q 𝑤) ∈ (2nd
‘𝐴)) ∧ 𝑧 <Q
(𝑣
·Q 𝑤)) → (𝑤 ·Q
(*Q‘𝑤)) =
1Q) |
26 | 25 | oveq2d 5885 |
. . . . . . . . . . . . . 14
⊢ (((𝐴 ∈ P ∧
1Q <Q 𝑤) ∧ (𝑣 ∈ (1st ‘𝐴) ∧ (𝑣 ·Q 𝑤) ∈ (2nd
‘𝐴)) ∧ 𝑧 <Q
(𝑣
·Q 𝑤)) → (𝑣 ·Q (𝑤
·Q (*Q‘𝑤))) = (𝑣 ·Q
1Q)) |
27 | | mulidnq 7379 |
. . . . . . . . . . . . . . 15
⊢ (𝑣 ∈ Q →
(𝑣
·Q 1Q) = 𝑣) |
28 | 14, 27 | syl 14 |
. . . . . . . . . . . . . 14
⊢ (((𝐴 ∈ P ∧
1Q <Q 𝑤) ∧ (𝑣 ∈ (1st ‘𝐴) ∧ (𝑣 ·Q 𝑤) ∈ (2nd
‘𝐴)) ∧ 𝑧 <Q
(𝑣
·Q 𝑤)) → (𝑣 ·Q
1Q) = 𝑣) |
29 | 23, 26, 28 | 3eqtrd 2214 |
. . . . . . . . . . . . 13
⊢ (((𝐴 ∈ P ∧
1Q <Q 𝑤) ∧ (𝑣 ∈ (1st ‘𝐴) ∧ (𝑣 ·Q 𝑤) ∈ (2nd
‘𝐴)) ∧ 𝑧 <Q
(𝑣
·Q 𝑤)) → ((𝑣 ·Q 𝑤)
·Q (*Q‘𝑤)) = 𝑣) |
30 | 29 | eleq1d 2246 |
. . . . . . . . . . . 12
⊢ (((𝐴 ∈ P ∧
1Q <Q 𝑤) ∧ (𝑣 ∈ (1st ‘𝐴) ∧ (𝑣 ·Q 𝑤) ∈ (2nd
‘𝐴)) ∧ 𝑧 <Q
(𝑣
·Q 𝑤)) → (((𝑣 ·Q 𝑤)
·Q (*Q‘𝑤)) ∈ (1st
‘𝐴) ↔ 𝑣 ∈ (1st
‘𝐴))) |
31 | 10, 30 | mpbird 167 |
. . . . . . . . . . 11
⊢ (((𝐴 ∈ P ∧
1Q <Q 𝑤) ∧ (𝑣 ∈ (1st ‘𝐴) ∧ (𝑣 ·Q 𝑤) ∈ (2nd
‘𝐴)) ∧ 𝑧 <Q
(𝑣
·Q 𝑤)) → ((𝑣 ·Q 𝑤)
·Q (*Q‘𝑤)) ∈ (1st
‘𝐴)) |
32 | | ltrnqi 7411 |
. . . . . . . . . . . . 13
⊢ (𝑧 <Q
(𝑣
·Q 𝑤) →
(*Q‘(𝑣 ·Q 𝑤))
<Q (*Q‘𝑧)) |
33 | | ltmnqg 7391 |
. . . . . . . . . . . . . . 15
⊢ ((𝑓 ∈ Q ∧
𝑔 ∈ Q
∧ ℎ ∈
Q) → (𝑓
<Q 𝑔 ↔ (ℎ ·Q 𝑓) <Q
(ℎ
·Q 𝑔))) |
34 | 33 | adantl 277 |
. . . . . . . . . . . . . 14
⊢ ((((𝐴 ∈ P ∧
1Q <Q 𝑤) ∧ (𝑣 ∈ (1st ‘𝐴) ∧ (𝑣 ·Q 𝑤) ∈ (2nd
‘𝐴)) ∧ 𝑧 <Q
(𝑣
·Q 𝑤)) ∧ (𝑓 ∈ Q ∧ 𝑔 ∈ Q ∧
ℎ ∈ Q))
→ (𝑓
<Q 𝑔 ↔ (ℎ ·Q 𝑓) <Q
(ℎ
·Q 𝑔))) |
35 | | mulclnq 7366 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑣 ∈ Q ∧
𝑤 ∈ Q)
→ (𝑣
·Q 𝑤) ∈ Q) |
36 | 14, 19, 35 | syl2anc 411 |
. . . . . . . . . . . . . . 15
⊢ (((𝐴 ∈ P ∧
1Q <Q 𝑤) ∧ (𝑣 ∈ (1st ‘𝐴) ∧ (𝑣 ·Q 𝑤) ∈ (2nd
‘𝐴)) ∧ 𝑧 <Q
(𝑣
·Q 𝑤)) → (𝑣 ·Q 𝑤) ∈
Q) |
37 | | recclnq 7382 |
. . . . . . . . . . . . . . 15
⊢ ((𝑣
·Q 𝑤) ∈ Q →
(*Q‘(𝑣 ·Q 𝑤)) ∈
Q) |
38 | 36, 37 | syl 14 |
. . . . . . . . . . . . . 14
⊢ (((𝐴 ∈ P ∧
1Q <Q 𝑤) ∧ (𝑣 ∈ (1st ‘𝐴) ∧ (𝑣 ·Q 𝑤) ∈ (2nd
‘𝐴)) ∧ 𝑧 <Q
(𝑣
·Q 𝑤)) →
(*Q‘(𝑣 ·Q 𝑤)) ∈
Q) |
39 | 16 | brel 4675 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑧 <Q
(𝑣
·Q 𝑤) → (𝑧 ∈ Q ∧ (𝑣
·Q 𝑤) ∈ Q)) |
40 | 39 | simpld 112 |
. . . . . . . . . . . . . . . 16
⊢ (𝑧 <Q
(𝑣
·Q 𝑤) → 𝑧 ∈ Q) |
41 | 9, 40 | syl 14 |
. . . . . . . . . . . . . . 15
⊢ (((𝐴 ∈ P ∧
1Q <Q 𝑤) ∧ (𝑣 ∈ (1st ‘𝐴) ∧ (𝑣 ·Q 𝑤) ∈ (2nd
‘𝐴)) ∧ 𝑧 <Q
(𝑣
·Q 𝑤)) → 𝑧 ∈ Q) |
42 | | recclnq 7382 |
. . . . . . . . . . . . . . 15
⊢ (𝑧 ∈ Q →
(*Q‘𝑧) ∈ Q) |
43 | 41, 42 | syl 14 |
. . . . . . . . . . . . . 14
⊢ (((𝐴 ∈ P ∧
1Q <Q 𝑤) ∧ (𝑣 ∈ (1st ‘𝐴) ∧ (𝑣 ·Q 𝑤) ∈ (2nd
‘𝐴)) ∧ 𝑧 <Q
(𝑣
·Q 𝑤)) →
(*Q‘𝑧) ∈ Q) |
44 | | mulcomnqg 7373 |
. . . . . . . . . . . . . . 15
⊢ ((𝑓 ∈ Q ∧
𝑔 ∈ Q)
→ (𝑓
·Q 𝑔) = (𝑔 ·Q 𝑓)) |
45 | 44 | adantl 277 |
. . . . . . . . . . . . . 14
⊢ ((((𝐴 ∈ P ∧
1Q <Q 𝑤) ∧ (𝑣 ∈ (1st ‘𝐴) ∧ (𝑣 ·Q 𝑤) ∈ (2nd
‘𝐴)) ∧ 𝑧 <Q
(𝑣
·Q 𝑤)) ∧ (𝑓 ∈ Q ∧ 𝑔 ∈ Q)) →
(𝑓
·Q 𝑔) = (𝑔 ·Q 𝑓)) |
46 | 34, 38, 43, 19, 45 | caovord2d 6038 |
. . . . . . . . . . . . 13
⊢ (((𝐴 ∈ P ∧
1Q <Q 𝑤) ∧ (𝑣 ∈ (1st ‘𝐴) ∧ (𝑣 ·Q 𝑤) ∈ (2nd
‘𝐴)) ∧ 𝑧 <Q
(𝑣
·Q 𝑤)) →
((*Q‘(𝑣 ·Q 𝑤))
<Q (*Q‘𝑧) ↔
((*Q‘(𝑣 ·Q 𝑤))
·Q 𝑤) <Q
((*Q‘𝑧) ·Q 𝑤))) |
47 | 32, 46 | imbitrid 154 |
. . . . . . . . . . . 12
⊢ (((𝐴 ∈ P ∧
1Q <Q 𝑤) ∧ (𝑣 ∈ (1st ‘𝐴) ∧ (𝑣 ·Q 𝑤) ∈ (2nd
‘𝐴)) ∧ 𝑧 <Q
(𝑣
·Q 𝑤)) → (𝑧 <Q (𝑣
·Q 𝑤) →
((*Q‘(𝑣 ·Q 𝑤))
·Q 𝑤) <Q
((*Q‘𝑧) ·Q 𝑤))) |
48 | | mulcomnqg 7373 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (((𝑣
·Q 𝑤) ∈ Q ∧
(*Q‘(𝑣 ·Q 𝑤)) ∈ Q)
→ ((𝑣
·Q 𝑤) ·Q
(*Q‘(𝑣 ·Q 𝑤))) =
((*Q‘(𝑣 ·Q 𝑤))
·Q (𝑣 ·Q 𝑤))) |
49 | 37, 48 | mpdan 421 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑣
·Q 𝑤) ∈ Q → ((𝑣
·Q 𝑤) ·Q
(*Q‘(𝑣 ·Q 𝑤))) =
((*Q‘(𝑣 ·Q 𝑤))
·Q (𝑣 ·Q 𝑤))) |
50 | | recidnq 7383 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑣
·Q 𝑤) ∈ Q → ((𝑣
·Q 𝑤) ·Q
(*Q‘(𝑣 ·Q 𝑤))) =
1Q) |
51 | 49, 50 | eqtr3d 2212 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝑣
·Q 𝑤) ∈ Q →
((*Q‘(𝑣 ·Q 𝑤))
·Q (𝑣 ·Q 𝑤)) =
1Q) |
52 | 51, 24 | oveqan12d 5888 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝑣
·Q 𝑤) ∈ Q ∧ 𝑤 ∈ Q) →
(((*Q‘(𝑣 ·Q 𝑤))
·Q (𝑣 ·Q 𝑤))
·Q (𝑤 ·Q
(*Q‘𝑤))) = (1Q
·Q
1Q)) |
53 | 36, 19, 52 | syl2anc 411 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝐴 ∈ P ∧
1Q <Q 𝑤) ∧ (𝑣 ∈ (1st ‘𝐴) ∧ (𝑣 ·Q 𝑤) ∈ (2nd
‘𝐴)) ∧ 𝑧 <Q
(𝑣
·Q 𝑤)) →
(((*Q‘(𝑣 ·Q 𝑤))
·Q (𝑣 ·Q 𝑤))
·Q (𝑤 ·Q
(*Q‘𝑤))) = (1Q
·Q
1Q)) |
54 | | mulassnqg 7374 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝑓 ∈ Q ∧
𝑔 ∈ Q
∧ ℎ ∈
Q) → ((𝑓
·Q 𝑔) ·Q ℎ) = (𝑓 ·Q (𝑔
·Q ℎ))) |
55 | 54 | adantl 277 |
. . . . . . . . . . . . . . . . . 18
⊢ ((((𝐴 ∈ P ∧
1Q <Q 𝑤) ∧ (𝑣 ∈ (1st ‘𝐴) ∧ (𝑣 ·Q 𝑤) ∈ (2nd
‘𝐴)) ∧ 𝑧 <Q
(𝑣
·Q 𝑤)) ∧ (𝑓 ∈ Q ∧ 𝑔 ∈ Q ∧
ℎ ∈ Q))
→ ((𝑓
·Q 𝑔) ·Q ℎ) = (𝑓 ·Q (𝑔
·Q ℎ))) |
56 | | mulclnq 7366 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝑓 ∈ Q ∧
𝑔 ∈ Q)
→ (𝑓
·Q 𝑔) ∈ Q) |
57 | 56 | adantl 277 |
. . . . . . . . . . . . . . . . . 18
⊢ ((((𝐴 ∈ P ∧
1Q <Q 𝑤) ∧ (𝑣 ∈ (1st ‘𝐴) ∧ (𝑣 ·Q 𝑤) ∈ (2nd
‘𝐴)) ∧ 𝑧 <Q
(𝑣
·Q 𝑤)) ∧ (𝑓 ∈ Q ∧ 𝑔 ∈ Q)) →
(𝑓
·Q 𝑔) ∈ Q) |
58 | 38, 36, 19, 45, 55, 21, 57 | caov4d 6053 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝐴 ∈ P ∧
1Q <Q 𝑤) ∧ (𝑣 ∈ (1st ‘𝐴) ∧ (𝑣 ·Q 𝑤) ∈ (2nd
‘𝐴)) ∧ 𝑧 <Q
(𝑣
·Q 𝑤)) →
(((*Q‘(𝑣 ·Q 𝑤))
·Q (𝑣 ·Q 𝑤))
·Q (𝑤 ·Q
(*Q‘𝑤))) =
(((*Q‘(𝑣 ·Q 𝑤))
·Q 𝑤) ·Q ((𝑣
·Q 𝑤) ·Q
(*Q‘𝑤)))) |
59 | 53, 58 | eqtr3d 2212 |
. . . . . . . . . . . . . . . 16
⊢ (((𝐴 ∈ P ∧
1Q <Q 𝑤) ∧ (𝑣 ∈ (1st ‘𝐴) ∧ (𝑣 ·Q 𝑤) ∈ (2nd
‘𝐴)) ∧ 𝑧 <Q
(𝑣
·Q 𝑤)) → (1Q
·Q 1Q) =
(((*Q‘(𝑣 ·Q 𝑤))
·Q 𝑤) ·Q ((𝑣
·Q 𝑤) ·Q
(*Q‘𝑤)))) |
60 | | 1nq 7356 |
. . . . . . . . . . . . . . . . 17
⊢
1Q ∈ Q |
61 | | mulidnq 7379 |
. . . . . . . . . . . . . . . . 17
⊢
(1Q ∈ Q →
(1Q ·Q
1Q) = 1Q) |
62 | 60, 61 | ax-mp 5 |
. . . . . . . . . . . . . . . 16
⊢
(1Q ·Q
1Q) = 1Q |
63 | 59, 62 | eqtr3di 2225 |
. . . . . . . . . . . . . . 15
⊢ (((𝐴 ∈ P ∧
1Q <Q 𝑤) ∧ (𝑣 ∈ (1st ‘𝐴) ∧ (𝑣 ·Q 𝑤) ∈ (2nd
‘𝐴)) ∧ 𝑧 <Q
(𝑣
·Q 𝑤)) →
(((*Q‘(𝑣 ·Q 𝑤))
·Q 𝑤) ·Q ((𝑣
·Q 𝑤) ·Q
(*Q‘𝑤))) =
1Q) |
64 | 57, 38, 19 | caovcld 6022 |
. . . . . . . . . . . . . . . 16
⊢ (((𝐴 ∈ P ∧
1Q <Q 𝑤) ∧ (𝑣 ∈ (1st ‘𝐴) ∧ (𝑣 ·Q 𝑤) ∈ (2nd
‘𝐴)) ∧ 𝑧 <Q
(𝑣
·Q 𝑤)) →
((*Q‘(𝑣 ·Q 𝑤))
·Q 𝑤) ∈ Q) |
65 | 57, 36, 21 | caovcld 6022 |
. . . . . . . . . . . . . . . 16
⊢ (((𝐴 ∈ P ∧
1Q <Q 𝑤) ∧ (𝑣 ∈ (1st ‘𝐴) ∧ (𝑣 ·Q 𝑤) ∈ (2nd
‘𝐴)) ∧ 𝑧 <Q
(𝑣
·Q 𝑤)) → ((𝑣 ·Q 𝑤)
·Q (*Q‘𝑤)) ∈
Q) |
66 | | recmulnqg 7381 |
. . . . . . . . . . . . . . . 16
⊢
((((*Q‘(𝑣 ·Q 𝑤))
·Q 𝑤) ∈ Q ∧ ((𝑣
·Q 𝑤) ·Q
(*Q‘𝑤)) ∈ Q) →
((*Q‘((*Q‘(𝑣
·Q 𝑤)) ·Q 𝑤)) = ((𝑣 ·Q 𝑤)
·Q (*Q‘𝑤)) ↔
(((*Q‘(𝑣 ·Q 𝑤))
·Q 𝑤) ·Q ((𝑣
·Q 𝑤) ·Q
(*Q‘𝑤))) =
1Q)) |
67 | 64, 65, 66 | syl2anc 411 |
. . . . . . . . . . . . . . 15
⊢ (((𝐴 ∈ P ∧
1Q <Q 𝑤) ∧ (𝑣 ∈ (1st ‘𝐴) ∧ (𝑣 ·Q 𝑤) ∈ (2nd
‘𝐴)) ∧ 𝑧 <Q
(𝑣
·Q 𝑤)) →
((*Q‘((*Q‘(𝑣
·Q 𝑤)) ·Q 𝑤)) = ((𝑣 ·Q 𝑤)
·Q (*Q‘𝑤)) ↔
(((*Q‘(𝑣 ·Q 𝑤))
·Q 𝑤) ·Q ((𝑣
·Q 𝑤) ·Q
(*Q‘𝑤))) =
1Q)) |
68 | 63, 67 | mpbird 167 |
. . . . . . . . . . . . . 14
⊢ (((𝐴 ∈ P ∧
1Q <Q 𝑤) ∧ (𝑣 ∈ (1st ‘𝐴) ∧ (𝑣 ·Q 𝑤) ∈ (2nd
‘𝐴)) ∧ 𝑧 <Q
(𝑣
·Q 𝑤)) →
(*Q‘((*Q‘(𝑣
·Q 𝑤)) ·Q 𝑤)) = ((𝑣 ·Q 𝑤)
·Q (*Q‘𝑤))) |
69 | 68 | eleq1d 2246 |
. . . . . . . . . . . . 13
⊢ (((𝐴 ∈ P ∧
1Q <Q 𝑤) ∧ (𝑣 ∈ (1st ‘𝐴) ∧ (𝑣 ·Q 𝑤) ∈ (2nd
‘𝐴)) ∧ 𝑧 <Q
(𝑣
·Q 𝑤)) →
((*Q‘((*Q‘(𝑣
·Q 𝑤)) ·Q 𝑤)) ∈ (1st
‘𝐴) ↔ ((𝑣
·Q 𝑤) ·Q
(*Q‘𝑤)) ∈ (1st ‘𝐴))) |
70 | 69 | biimprd 158 |
. . . . . . . . . . . 12
⊢ (((𝐴 ∈ P ∧
1Q <Q 𝑤) ∧ (𝑣 ∈ (1st ‘𝐴) ∧ (𝑣 ·Q 𝑤) ∈ (2nd
‘𝐴)) ∧ 𝑧 <Q
(𝑣
·Q 𝑤)) → (((𝑣 ·Q 𝑤)
·Q (*Q‘𝑤)) ∈ (1st
‘𝐴) →
(*Q‘((*Q‘(𝑣
·Q 𝑤)) ·Q 𝑤)) ∈ (1st
‘𝐴))) |
71 | | breq1 4003 |
. . . . . . . . . . . . . . . 16
⊢ (𝑦 =
((*Q‘(𝑣 ·Q 𝑤))
·Q 𝑤) → (𝑦 <Q
((*Q‘𝑧) ·Q 𝑤) ↔
((*Q‘(𝑣 ·Q 𝑤))
·Q 𝑤) <Q
((*Q‘𝑧) ·Q 𝑤))) |
72 | | fveq2 5511 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑦 =
((*Q‘(𝑣 ·Q 𝑤))
·Q 𝑤) →
(*Q‘𝑦) =
(*Q‘((*Q‘(𝑣
·Q 𝑤)) ·Q 𝑤))) |
73 | 72 | eleq1d 2246 |
. . . . . . . . . . . . . . . 16
⊢ (𝑦 =
((*Q‘(𝑣 ·Q 𝑤))
·Q 𝑤) →
((*Q‘𝑦) ∈ (1st ‘𝐴) ↔
(*Q‘((*Q‘(𝑣
·Q 𝑤)) ·Q 𝑤)) ∈ (1st
‘𝐴))) |
74 | 71, 73 | anbi12d 473 |
. . . . . . . . . . . . . . 15
⊢ (𝑦 =
((*Q‘(𝑣 ·Q 𝑤))
·Q 𝑤) → ((𝑦 <Q
((*Q‘𝑧) ·Q 𝑤) ∧
(*Q‘𝑦) ∈ (1st ‘𝐴)) ↔
(((*Q‘(𝑣 ·Q 𝑤))
·Q 𝑤) <Q
((*Q‘𝑧) ·Q 𝑤) ∧
(*Q‘((*Q‘(𝑣
·Q 𝑤)) ·Q 𝑤)) ∈ (1st
‘𝐴)))) |
75 | 74 | spcegv 2825 |
. . . . . . . . . . . . . 14
⊢
(((*Q‘(𝑣 ·Q 𝑤))
·Q 𝑤) ∈ Q →
((((*Q‘(𝑣 ·Q 𝑤))
·Q 𝑤) <Q
((*Q‘𝑧) ·Q 𝑤) ∧
(*Q‘((*Q‘(𝑣
·Q 𝑤)) ·Q 𝑤)) ∈ (1st
‘𝐴)) →
∃𝑦(𝑦 <Q
((*Q‘𝑧) ·Q 𝑤) ∧
(*Q‘𝑦) ∈ (1st ‘𝐴)))) |
76 | 64, 75 | syl 14 |
. . . . . . . . . . . . 13
⊢ (((𝐴 ∈ P ∧
1Q <Q 𝑤) ∧ (𝑣 ∈ (1st ‘𝐴) ∧ (𝑣 ·Q 𝑤) ∈ (2nd
‘𝐴)) ∧ 𝑧 <Q
(𝑣
·Q 𝑤)) →
((((*Q‘(𝑣 ·Q 𝑤))
·Q 𝑤) <Q
((*Q‘𝑧) ·Q 𝑤) ∧
(*Q‘((*Q‘(𝑣
·Q 𝑤)) ·Q 𝑤)) ∈ (1st
‘𝐴)) →
∃𝑦(𝑦 <Q
((*Q‘𝑧) ·Q 𝑤) ∧
(*Q‘𝑦) ∈ (1st ‘𝐴)))) |
77 | | recexpr.1 |
. . . . . . . . . . . . . 14
⊢ 𝐵 = 〈{𝑥 ∣ ∃𝑦(𝑥 <Q 𝑦 ∧
(*Q‘𝑦) ∈ (2nd ‘𝐴))}, {𝑥 ∣ ∃𝑦(𝑦 <Q 𝑥 ∧
(*Q‘𝑦) ∈ (1st ‘𝐴))}〉 |
78 | 77 | recexprlemelu 7613 |
. . . . . . . . . . . . 13
⊢
(((*Q‘𝑧) ·Q 𝑤) ∈ (2nd
‘𝐵) ↔
∃𝑦(𝑦 <Q
((*Q‘𝑧) ·Q 𝑤) ∧
(*Q‘𝑦) ∈ (1st ‘𝐴))) |
79 | 76, 78 | syl6ibr 162 |
. . . . . . . . . . . 12
⊢ (((𝐴 ∈ P ∧
1Q <Q 𝑤) ∧ (𝑣 ∈ (1st ‘𝐴) ∧ (𝑣 ·Q 𝑤) ∈ (2nd
‘𝐴)) ∧ 𝑧 <Q
(𝑣
·Q 𝑤)) →
((((*Q‘(𝑣 ·Q 𝑤))
·Q 𝑤) <Q
((*Q‘𝑧) ·Q 𝑤) ∧
(*Q‘((*Q‘(𝑣
·Q 𝑤)) ·Q 𝑤)) ∈ (1st
‘𝐴)) →
((*Q‘𝑧) ·Q 𝑤) ∈ (2nd
‘𝐵))) |
80 | 47, 70, 79 | syl2and 295 |
. . . . . . . . . . 11
⊢ (((𝐴 ∈ P ∧
1Q <Q 𝑤) ∧ (𝑣 ∈ (1st ‘𝐴) ∧ (𝑣 ·Q 𝑤) ∈ (2nd
‘𝐴)) ∧ 𝑧 <Q
(𝑣
·Q 𝑤)) → ((𝑧 <Q (𝑣
·Q 𝑤) ∧ ((𝑣 ·Q 𝑤)
·Q (*Q‘𝑤)) ∈ (1st
‘𝐴)) →
((*Q‘𝑧) ·Q 𝑤) ∈ (2nd
‘𝐵))) |
81 | 9, 31, 80 | mp2and 433 |
. . . . . . . . . 10
⊢ (((𝐴 ∈ P ∧
1Q <Q 𝑤) ∧ (𝑣 ∈ (1st ‘𝐴) ∧ (𝑣 ·Q 𝑤) ∈ (2nd
‘𝐴)) ∧ 𝑧 <Q
(𝑣
·Q 𝑤)) →
((*Q‘𝑧) ·Q 𝑤) ∈ (2nd
‘𝐵)) |
82 | | mulidnq 7379 |
. . . . . . . . . . . . . 14
⊢ (𝑤 ∈ Q →
(𝑤
·Q 1Q) = 𝑤) |
83 | | mulcomnqg 7373 |
. . . . . . . . . . . . . . 15
⊢ ((𝑤 ∈ Q ∧
1Q ∈ Q) → (𝑤 ·Q
1Q) = (1Q
·Q 𝑤)) |
84 | 60, 83 | mpan2 425 |
. . . . . . . . . . . . . 14
⊢ (𝑤 ∈ Q →
(𝑤
·Q 1Q) =
(1Q ·Q 𝑤)) |
85 | 82, 84 | eqtr3d 2212 |
. . . . . . . . . . . . 13
⊢ (𝑤 ∈ Q →
𝑤 =
(1Q ·Q 𝑤)) |
86 | 85 | adantl 277 |
. . . . . . . . . . . 12
⊢ ((𝑧 ∈ Q ∧
𝑤 ∈ Q)
→ 𝑤 =
(1Q ·Q 𝑤)) |
87 | | recidnq 7383 |
. . . . . . . . . . . . . 14
⊢ (𝑧 ∈ Q →
(𝑧
·Q (*Q‘𝑧)) =
1Q) |
88 | 87 | oveq1d 5884 |
. . . . . . . . . . . . 13
⊢ (𝑧 ∈ Q →
((𝑧
·Q (*Q‘𝑧))
·Q 𝑤) = (1Q
·Q 𝑤)) |
89 | 88 | adantr 276 |
. . . . . . . . . . . 12
⊢ ((𝑧 ∈ Q ∧
𝑤 ∈ Q)
→ ((𝑧
·Q (*Q‘𝑧))
·Q 𝑤) = (1Q
·Q 𝑤)) |
90 | | mulassnqg 7374 |
. . . . . . . . . . . . . 14
⊢ ((𝑧 ∈ Q ∧
(*Q‘𝑧) ∈ Q ∧ 𝑤 ∈ Q) →
((𝑧
·Q (*Q‘𝑧))
·Q 𝑤) = (𝑧 ·Q
((*Q‘𝑧) ·Q 𝑤))) |
91 | 42, 90 | syl3an2 1272 |
. . . . . . . . . . . . 13
⊢ ((𝑧 ∈ Q ∧
𝑧 ∈ Q
∧ 𝑤 ∈
Q) → ((𝑧
·Q (*Q‘𝑧))
·Q 𝑤) = (𝑧 ·Q
((*Q‘𝑧) ·Q 𝑤))) |
92 | 91 | 3anidm12 1295 |
. . . . . . . . . . . 12
⊢ ((𝑧 ∈ Q ∧
𝑤 ∈ Q)
→ ((𝑧
·Q (*Q‘𝑧))
·Q 𝑤) = (𝑧 ·Q
((*Q‘𝑧) ·Q 𝑤))) |
93 | 86, 89, 92 | 3eqtr2d 2216 |
. . . . . . . . . . 11
⊢ ((𝑧 ∈ Q ∧
𝑤 ∈ Q)
→ 𝑤 = (𝑧
·Q ((*Q‘𝑧)
·Q 𝑤))) |
94 | 41, 19, 93 | syl2anc 411 |
. . . . . . . . . 10
⊢ (((𝐴 ∈ P ∧
1Q <Q 𝑤) ∧ (𝑣 ∈ (1st ‘𝐴) ∧ (𝑣 ·Q 𝑤) ∈ (2nd
‘𝐴)) ∧ 𝑧 <Q
(𝑣
·Q 𝑤)) → 𝑤 = (𝑧 ·Q
((*Q‘𝑧) ·Q 𝑤))) |
95 | | oveq2 5877 |
. . . . . . . . . . . 12
⊢ (𝑥 =
((*Q‘𝑧) ·Q 𝑤) → (𝑧 ·Q 𝑥) = (𝑧 ·Q
((*Q‘𝑧) ·Q 𝑤))) |
96 | 95 | eqeq2d 2189 |
. . . . . . . . . . 11
⊢ (𝑥 =
((*Q‘𝑧) ·Q 𝑤) → (𝑤 = (𝑧 ·Q 𝑥) ↔ 𝑤 = (𝑧 ·Q
((*Q‘𝑧) ·Q 𝑤)))) |
97 | 96 | rspcev 2841 |
. . . . . . . . . 10
⊢
((((*Q‘𝑧) ·Q 𝑤) ∈ (2nd
‘𝐵) ∧ 𝑤 = (𝑧 ·Q
((*Q‘𝑧) ·Q 𝑤))) → ∃𝑥 ∈ (2nd
‘𝐵)𝑤 = (𝑧 ·Q 𝑥)) |
98 | 81, 94, 97 | syl2anc 411 |
. . . . . . . . 9
⊢ (((𝐴 ∈ P ∧
1Q <Q 𝑤) ∧ (𝑣 ∈ (1st ‘𝐴) ∧ (𝑣 ·Q 𝑤) ∈ (2nd
‘𝐴)) ∧ 𝑧 <Q
(𝑣
·Q 𝑤)) → ∃𝑥 ∈ (2nd ‘𝐵)𝑤 = (𝑧 ·Q 𝑥)) |
99 | 98 | 3expia 1205 |
. . . . . . . 8
⊢ (((𝐴 ∈ P ∧
1Q <Q 𝑤) ∧ (𝑣 ∈ (1st ‘𝐴) ∧ (𝑣 ·Q 𝑤) ∈ (2nd
‘𝐴))) → (𝑧 <Q
(𝑣
·Q 𝑤) → ∃𝑥 ∈ (2nd ‘𝐵)𝑤 = (𝑧 ·Q 𝑥))) |
100 | 99 | reximdv 2578 |
. . . . . . 7
⊢ (((𝐴 ∈ P ∧
1Q <Q 𝑤) ∧ (𝑣 ∈ (1st ‘𝐴) ∧ (𝑣 ·Q 𝑤) ∈ (2nd
‘𝐴))) →
(∃𝑧 ∈
(2nd ‘𝐴)𝑧 <Q (𝑣
·Q 𝑤) → ∃𝑧 ∈ (2nd ‘𝐴)∃𝑥 ∈ (2nd ‘𝐵)𝑤 = (𝑧 ·Q 𝑥))) |
101 | 77 | recexprlempr 7622 |
. . . . . . . . 9
⊢ (𝐴 ∈ P →
𝐵 ∈
P) |
102 | | df-imp 7459 |
. . . . . . . . . 10
⊢
·P = (𝑦 ∈ P, 𝑤 ∈ P ↦ 〈{𝑢 ∈ Q ∣
∃𝑓 ∈
Q ∃𝑔
∈ Q (𝑓
∈ (1st ‘𝑦) ∧ 𝑔 ∈ (1st ‘𝑤) ∧ 𝑢 = (𝑓 ·Q 𝑔))}, {𝑢 ∈ Q ∣ ∃𝑓 ∈ Q
∃𝑔 ∈
Q (𝑓 ∈
(2nd ‘𝑦)
∧ 𝑔 ∈
(2nd ‘𝑤)
∧ 𝑢 = (𝑓
·Q 𝑔))}〉) |
103 | 102, 56 | genpelvu 7503 |
. . . . . . . . 9
⊢ ((𝐴 ∈ P ∧
𝐵 ∈ P)
→ (𝑤 ∈
(2nd ‘(𝐴
·P 𝐵)) ↔ ∃𝑧 ∈ (2nd ‘𝐴)∃𝑥 ∈ (2nd ‘𝐵)𝑤 = (𝑧 ·Q 𝑥))) |
104 | 101, 103 | mpdan 421 |
. . . . . . . 8
⊢ (𝐴 ∈ P →
(𝑤 ∈ (2nd
‘(𝐴
·P 𝐵)) ↔ ∃𝑧 ∈ (2nd ‘𝐴)∃𝑥 ∈ (2nd ‘𝐵)𝑤 = (𝑧 ·Q 𝑥))) |
105 | 104 | ad2antrr 488 |
. . . . . . 7
⊢ (((𝐴 ∈ P ∧
1Q <Q 𝑤) ∧ (𝑣 ∈ (1st ‘𝐴) ∧ (𝑣 ·Q 𝑤) ∈ (2nd
‘𝐴))) → (𝑤 ∈ (2nd
‘(𝐴
·P 𝐵)) ↔ ∃𝑧 ∈ (2nd ‘𝐴)∃𝑥 ∈ (2nd ‘𝐵)𝑤 = (𝑧 ·Q 𝑥))) |
106 | 100, 105 | sylibrd 169 |
. . . . . 6
⊢ (((𝐴 ∈ P ∧
1Q <Q 𝑤) ∧ (𝑣 ∈ (1st ‘𝐴) ∧ (𝑣 ·Q 𝑤) ∈ (2nd
‘𝐴))) →
(∃𝑧 ∈
(2nd ‘𝐴)𝑧 <Q (𝑣
·Q 𝑤) → 𝑤 ∈ (2nd ‘(𝐴
·P 𝐵)))) |
107 | 8, 106 | mpd 13 |
. . . . 5
⊢ (((𝐴 ∈ P ∧
1Q <Q 𝑤) ∧ (𝑣 ∈ (1st ‘𝐴) ∧ (𝑣 ·Q 𝑤) ∈ (2nd
‘𝐴))) → 𝑤 ∈ (2nd
‘(𝐴
·P 𝐵))) |
108 | 5, 107 | rexlimddv 2599 |
. . . 4
⊢ ((𝐴 ∈ P ∧
1Q <Q 𝑤) → 𝑤 ∈ (2nd ‘(𝐴
·P 𝐵))) |
109 | 108 | ex 115 |
. . 3
⊢ (𝐴 ∈ P →
(1Q <Q 𝑤 → 𝑤 ∈ (2nd ‘(𝐴
·P 𝐵)))) |
110 | 2, 109 | biimtrid 152 |
. 2
⊢ (𝐴 ∈ P →
(𝑤 ∈ (2nd
‘1P) → 𝑤 ∈ (2nd ‘(𝐴
·P 𝐵)))) |
111 | 110 | ssrdv 3161 |
1
⊢ (𝐴 ∈ P →
(2nd ‘1P) ⊆ (2nd
‘(𝐴
·P 𝐵))) |