Step | Hyp | Ref
| Expression |
1 | | df-nqqs 7310 |
. 2
⊢
Q = ((N × N) /
~Q ) |
2 | | addpipqqs 7332 |
. 2
⊢ (((𝑧 ∈ N ∧
𝑤 ∈ N)
∧ (𝑣 ∈
N ∧ 𝑢
∈ N)) → ([〈𝑧, 𝑤〉] ~Q
+Q [〈𝑣, 𝑢〉] ~Q ) =
[〈((𝑧
·N 𝑢) +N (𝑤
·N 𝑣)), (𝑤 ·N 𝑢)〉]
~Q ) |
3 | | mulpipqqs 7335 |
. . 3
⊢ (((𝑥 ∈ N ∧
𝑦 ∈ N)
∧ (((𝑧
·N 𝑢) +N (𝑤
·N 𝑣)) ∈ N ∧ (𝑤
·N 𝑢) ∈ N)) →
([〈𝑥, 𝑦〉]
~Q ·Q [〈((𝑧
·N 𝑢) +N (𝑤
·N 𝑣)), (𝑤 ·N 𝑢)〉]
~Q ) = [〈(𝑥 ·N ((𝑧
·N 𝑢) +N (𝑤
·N 𝑣))), (𝑦 ·N (𝑤
·N 𝑢))〉] ~Q
) |
4 | | mulclpi 7290 |
. . . . . . 7
⊢ ((𝑥 ∈ N ∧
((𝑧
·N 𝑢) +N (𝑤
·N 𝑣)) ∈ N) → (𝑥
·N ((𝑧 ·N 𝑢) +N
(𝑤
·N 𝑣))) ∈ N) |
5 | | simpl 108 |
. . . . . . . 8
⊢ ((𝑦 ∈ N ∧
(𝑤
·N 𝑢) ∈ N) → 𝑦 ∈
N) |
6 | | mulclpi 7290 |
. . . . . . . 8
⊢ ((𝑦 ∈ N ∧
(𝑤
·N 𝑢) ∈ N) → (𝑦
·N (𝑤 ·N 𝑢)) ∈
N) |
7 | 5, 6 | jca 304 |
. . . . . . 7
⊢ ((𝑦 ∈ N ∧
(𝑤
·N 𝑢) ∈ N) → (𝑦 ∈ N ∧
(𝑦
·N (𝑤 ·N 𝑢)) ∈
N)) |
8 | 4, 7 | anim12i 336 |
. . . . . 6
⊢ (((𝑥 ∈ N ∧
((𝑧
·N 𝑢) +N (𝑤
·N 𝑣)) ∈ N) ∧ (𝑦 ∈ N ∧
(𝑤
·N 𝑢) ∈ N)) → ((𝑥
·N ((𝑧 ·N 𝑢) +N
(𝑤
·N 𝑣))) ∈ N ∧ (𝑦 ∈ N ∧
(𝑦
·N (𝑤 ·N 𝑢)) ∈
N))) |
9 | | an12 556 |
. . . . . . 7
⊢ (((𝑥
·N ((𝑧 ·N 𝑢) +N
(𝑤
·N 𝑣))) ∈ N ∧ (𝑦 ∈ N ∧
(𝑦
·N (𝑤 ·N 𝑢)) ∈ N))
↔ (𝑦 ∈
N ∧ ((𝑥
·N ((𝑧 ·N 𝑢) +N
(𝑤
·N 𝑣))) ∈ N ∧ (𝑦
·N (𝑤 ·N 𝑢)) ∈
N))) |
10 | | 3anass 977 |
. . . . . . 7
⊢ ((𝑦 ∈ N ∧
(𝑥
·N ((𝑧 ·N 𝑢) +N
(𝑤
·N 𝑣))) ∈ N ∧ (𝑦
·N (𝑤 ·N 𝑢)) ∈ N)
↔ (𝑦 ∈
N ∧ ((𝑥
·N ((𝑧 ·N 𝑢) +N
(𝑤
·N 𝑣))) ∈ N ∧ (𝑦
·N (𝑤 ·N 𝑢)) ∈
N))) |
11 | 9, 10 | bitr4i 186 |
. . . . . 6
⊢ (((𝑥
·N ((𝑧 ·N 𝑢) +N
(𝑤
·N 𝑣))) ∈ N ∧ (𝑦 ∈ N ∧
(𝑦
·N (𝑤 ·N 𝑢)) ∈ N))
↔ (𝑦 ∈
N ∧ (𝑥
·N ((𝑧 ·N 𝑢) +N
(𝑤
·N 𝑣))) ∈ N ∧ (𝑦
·N (𝑤 ·N 𝑢)) ∈
N)) |
12 | 8, 11 | sylib 121 |
. . . . 5
⊢ (((𝑥 ∈ N ∧
((𝑧
·N 𝑢) +N (𝑤
·N 𝑣)) ∈ N) ∧ (𝑦 ∈ N ∧
(𝑤
·N 𝑢) ∈ N)) → (𝑦 ∈ N ∧
(𝑥
·N ((𝑧 ·N 𝑢) +N
(𝑤
·N 𝑣))) ∈ N ∧ (𝑦
·N (𝑤 ·N 𝑢)) ∈
N)) |
13 | 12 | an4s 583 |
. . . 4
⊢ (((𝑥 ∈ N ∧
𝑦 ∈ N)
∧ (((𝑧
·N 𝑢) +N (𝑤
·N 𝑣)) ∈ N ∧ (𝑤
·N 𝑢) ∈ N)) → (𝑦 ∈ N ∧
(𝑥
·N ((𝑧 ·N 𝑢) +N
(𝑤
·N 𝑣))) ∈ N ∧ (𝑦
·N (𝑤 ·N 𝑢)) ∈
N)) |
14 | | mulcanenqec 7348 |
. . . 4
⊢ ((𝑦 ∈ N ∧
(𝑥
·N ((𝑧 ·N 𝑢) +N
(𝑤
·N 𝑣))) ∈ N ∧ (𝑦
·N (𝑤 ·N 𝑢)) ∈ N)
→ [〈(𝑦
·N (𝑥 ·N ((𝑧
·N 𝑢) +N (𝑤
·N 𝑣)))), (𝑦 ·N (𝑦
·N (𝑤 ·N 𝑢)))〉]
~Q = [〈(𝑥 ·N ((𝑧
·N 𝑢) +N (𝑤
·N 𝑣))), (𝑦 ·N (𝑤
·N 𝑢))〉] ~Q
) |
15 | 13, 14 | syl 14 |
. . 3
⊢ (((𝑥 ∈ N ∧
𝑦 ∈ N)
∧ (((𝑧
·N 𝑢) +N (𝑤
·N 𝑣)) ∈ N ∧ (𝑤
·N 𝑢) ∈ N)) →
[〈(𝑦
·N (𝑥 ·N ((𝑧
·N 𝑢) +N (𝑤
·N 𝑣)))), (𝑦 ·N (𝑦
·N (𝑤 ·N 𝑢)))〉]
~Q = [〈(𝑥 ·N ((𝑧
·N 𝑢) +N (𝑤
·N 𝑣))), (𝑦 ·N (𝑤
·N 𝑢))〉] ~Q
) |
16 | 3, 15 | eqtr4d 2206 |
. 2
⊢ (((𝑥 ∈ N ∧
𝑦 ∈ N)
∧ (((𝑧
·N 𝑢) +N (𝑤
·N 𝑣)) ∈ N ∧ (𝑤
·N 𝑢) ∈ N)) →
([〈𝑥, 𝑦〉]
~Q ·Q [〈((𝑧
·N 𝑢) +N (𝑤
·N 𝑣)), (𝑤 ·N 𝑢)〉]
~Q ) = [〈(𝑦 ·N (𝑥
·N ((𝑧 ·N 𝑢) +N
(𝑤
·N 𝑣)))), (𝑦 ·N (𝑦
·N (𝑤 ·N 𝑢)))〉]
~Q ) |
17 | | mulpipqqs 7335 |
. 2
⊢ (((𝑥 ∈ N ∧
𝑦 ∈ N)
∧ (𝑧 ∈
N ∧ 𝑤
∈ N)) → ([〈𝑥, 𝑦〉] ~Q
·Q [〈𝑧, 𝑤〉] ~Q ) =
[〈(𝑥
·N 𝑧), (𝑦 ·N 𝑤)〉]
~Q ) |
18 | | mulpipqqs 7335 |
. 2
⊢ (((𝑥 ∈ N ∧
𝑦 ∈ N)
∧ (𝑣 ∈
N ∧ 𝑢
∈ N)) → ([〈𝑥, 𝑦〉] ~Q
·Q [〈𝑣, 𝑢〉] ~Q ) =
[〈(𝑥
·N 𝑣), (𝑦 ·N 𝑢)〉]
~Q ) |
19 | | addpipqqs 7332 |
. 2
⊢ ((((𝑥
·N 𝑧) ∈ N ∧ (𝑦
·N 𝑤) ∈ N) ∧ ((𝑥
·N 𝑣) ∈ N ∧ (𝑦
·N 𝑢) ∈ N)) →
([〈(𝑥
·N 𝑧), (𝑦 ·N 𝑤)〉]
~Q +Q [〈(𝑥
·N 𝑣), (𝑦 ·N 𝑢)〉]
~Q ) = [〈(((𝑥 ·N 𝑧)
·N (𝑦 ·N 𝑢)) +N
((𝑦
·N 𝑤) ·N (𝑥
·N 𝑣))), ((𝑦 ·N 𝑤)
·N (𝑦 ·N 𝑢))〉]
~Q ) |
20 | | mulclpi 7290 |
. . . . 5
⊢ ((𝑧 ∈ N ∧
𝑢 ∈ N)
→ (𝑧
·N 𝑢) ∈ N) |
21 | | mulclpi 7290 |
. . . . 5
⊢ ((𝑤 ∈ N ∧
𝑣 ∈ N)
→ (𝑤
·N 𝑣) ∈ N) |
22 | | addclpi 7289 |
. . . . 5
⊢ (((𝑧
·N 𝑢) ∈ N ∧ (𝑤
·N 𝑣) ∈ N) → ((𝑧
·N 𝑢) +N (𝑤
·N 𝑣)) ∈ N) |
23 | 20, 21, 22 | syl2an 287 |
. . . 4
⊢ (((𝑧 ∈ N ∧
𝑢 ∈ N)
∧ (𝑤 ∈
N ∧ 𝑣
∈ N)) → ((𝑧 ·N 𝑢) +N
(𝑤
·N 𝑣)) ∈ N) |
24 | 23 | an42s 584 |
. . 3
⊢ (((𝑧 ∈ N ∧
𝑤 ∈ N)
∧ (𝑣 ∈
N ∧ 𝑢
∈ N)) → ((𝑧 ·N 𝑢) +N
(𝑤
·N 𝑣)) ∈ N) |
25 | | mulclpi 7290 |
. . . 4
⊢ ((𝑤 ∈ N ∧
𝑢 ∈ N)
→ (𝑤
·N 𝑢) ∈ N) |
26 | 25 | ad2ant2l 505 |
. . 3
⊢ (((𝑧 ∈ N ∧
𝑤 ∈ N)
∧ (𝑣 ∈
N ∧ 𝑢
∈ N)) → (𝑤 ·N 𝑢) ∈
N) |
27 | 24, 26 | jca 304 |
. 2
⊢ (((𝑧 ∈ N ∧
𝑤 ∈ N)
∧ (𝑣 ∈
N ∧ 𝑢
∈ N)) → (((𝑧 ·N 𝑢) +N
(𝑤
·N 𝑣)) ∈ N ∧ (𝑤
·N 𝑢) ∈ N)) |
28 | | mulclpi 7290 |
. . . 4
⊢ ((𝑥 ∈ N ∧
𝑧 ∈ N)
→ (𝑥
·N 𝑧) ∈ N) |
29 | | mulclpi 7290 |
. . . 4
⊢ ((𝑦 ∈ N ∧
𝑤 ∈ N)
→ (𝑦
·N 𝑤) ∈ N) |
30 | 28, 29 | anim12i 336 |
. . 3
⊢ (((𝑥 ∈ N ∧
𝑧 ∈ N)
∧ (𝑦 ∈
N ∧ 𝑤
∈ N)) → ((𝑥 ·N 𝑧) ∈ N ∧
(𝑦
·N 𝑤) ∈ N)) |
31 | 30 | an4s 583 |
. 2
⊢ (((𝑥 ∈ N ∧
𝑦 ∈ N)
∧ (𝑧 ∈
N ∧ 𝑤
∈ N)) → ((𝑥 ·N 𝑧) ∈ N ∧
(𝑦
·N 𝑤) ∈ N)) |
32 | | mulclpi 7290 |
. . . 4
⊢ ((𝑥 ∈ N ∧
𝑣 ∈ N)
→ (𝑥
·N 𝑣) ∈ N) |
33 | | mulclpi 7290 |
. . . 4
⊢ ((𝑦 ∈ N ∧
𝑢 ∈ N)
→ (𝑦
·N 𝑢) ∈ N) |
34 | 32, 33 | anim12i 336 |
. . 3
⊢ (((𝑥 ∈ N ∧
𝑣 ∈ N)
∧ (𝑦 ∈
N ∧ 𝑢
∈ N)) → ((𝑥 ·N 𝑣) ∈ N ∧
(𝑦
·N 𝑢) ∈ N)) |
35 | 34 | an4s 583 |
. 2
⊢ (((𝑥 ∈ N ∧
𝑦 ∈ N)
∧ (𝑣 ∈
N ∧ 𝑢
∈ N)) → ((𝑥 ·N 𝑣) ∈ N ∧
(𝑦
·N 𝑢) ∈ N)) |
36 | | an42 582 |
. . . . 5
⊢ (((𝑧 ∈ N ∧
𝑢 ∈ N)
∧ (𝑤 ∈
N ∧ 𝑣
∈ N)) ↔ ((𝑧 ∈ N ∧ 𝑤 ∈ N) ∧
(𝑣 ∈ N
∧ 𝑢 ∈
N))) |
37 | 36 | anbi2i 454 |
. . . 4
⊢ (((𝑥 ∈ N ∧
𝑦 ∈ N)
∧ ((𝑧 ∈
N ∧ 𝑢
∈ N) ∧ (𝑤 ∈ N ∧ 𝑣 ∈ N)))
↔ ((𝑥 ∈
N ∧ 𝑦
∈ N) ∧ ((𝑧 ∈ N ∧ 𝑤 ∈ N) ∧
(𝑣 ∈ N
∧ 𝑢 ∈
N)))) |
38 | | 3anass 977 |
. . . 4
⊢ (((𝑥 ∈ N ∧
𝑦 ∈ N)
∧ (𝑧 ∈
N ∧ 𝑢
∈ N) ∧ (𝑤 ∈ N ∧ 𝑣 ∈ N)) ↔
((𝑥 ∈ N
∧ 𝑦 ∈
N) ∧ ((𝑧
∈ N ∧ 𝑢 ∈ N) ∧ (𝑤 ∈ N ∧
𝑣 ∈
N)))) |
39 | | 3anass 977 |
. . . 4
⊢ (((𝑥 ∈ N ∧
𝑦 ∈ N)
∧ (𝑧 ∈
N ∧ 𝑤
∈ N) ∧ (𝑣 ∈ N ∧ 𝑢 ∈ N)) ↔
((𝑥 ∈ N
∧ 𝑦 ∈
N) ∧ ((𝑧
∈ N ∧ 𝑤 ∈ N) ∧ (𝑣 ∈ N ∧
𝑢 ∈
N)))) |
40 | 37, 38, 39 | 3bitr4i 211 |
. . 3
⊢ (((𝑥 ∈ N ∧
𝑦 ∈ N)
∧ (𝑧 ∈
N ∧ 𝑢
∈ N) ∧ (𝑤 ∈ N ∧ 𝑣 ∈ N)) ↔
((𝑥 ∈ N
∧ 𝑦 ∈
N) ∧ (𝑧
∈ N ∧ 𝑤 ∈ N) ∧ (𝑣 ∈ N ∧
𝑢 ∈
N))) |
41 | | mulclpi 7290 |
. . . . . 6
⊢ ((𝑦 ∈ N ∧
𝑥 ∈ N)
→ (𝑦
·N 𝑥) ∈ N) |
42 | 41 | ancoms 266 |
. . . . 5
⊢ ((𝑥 ∈ N ∧
𝑦 ∈ N)
→ (𝑦
·N 𝑥) ∈ N) |
43 | | distrpig 7295 |
. . . . 5
⊢ (((𝑦
·N 𝑥) ∈ N ∧ (𝑧
·N 𝑢) ∈ N ∧ (𝑤
·N 𝑣) ∈ N) → ((𝑦
·N 𝑥) ·N ((𝑧
·N 𝑢) +N (𝑤
·N 𝑣))) = (((𝑦 ·N 𝑥)
·N (𝑧 ·N 𝑢)) +N
((𝑦
·N 𝑥) ·N (𝑤
·N 𝑣)))) |
44 | 42, 20, 21, 43 | syl3an 1275 |
. . . 4
⊢ (((𝑥 ∈ N ∧
𝑦 ∈ N)
∧ (𝑧 ∈
N ∧ 𝑢
∈ N) ∧ (𝑤 ∈ N ∧ 𝑣 ∈ N)) →
((𝑦
·N 𝑥) ·N ((𝑧
·N 𝑢) +N (𝑤
·N 𝑣))) = (((𝑦 ·N 𝑥)
·N (𝑧 ·N 𝑢)) +N
((𝑦
·N 𝑥) ·N (𝑤
·N 𝑣)))) |
45 | | simp1r 1017 |
. . . . 5
⊢ (((𝑥 ∈ N ∧
𝑦 ∈ N)
∧ (𝑧 ∈
N ∧ 𝑢
∈ N) ∧ (𝑤 ∈ N ∧ 𝑣 ∈ N)) →
𝑦 ∈
N) |
46 | | simp1l 1016 |
. . . . 5
⊢ (((𝑥 ∈ N ∧
𝑦 ∈ N)
∧ (𝑧 ∈
N ∧ 𝑢
∈ N) ∧ (𝑤 ∈ N ∧ 𝑣 ∈ N)) →
𝑥 ∈
N) |
47 | 20 | 3ad2ant2 1014 |
. . . . . 6
⊢ (((𝑥 ∈ N ∧
𝑦 ∈ N)
∧ (𝑧 ∈
N ∧ 𝑢
∈ N) ∧ (𝑤 ∈ N ∧ 𝑣 ∈ N)) →
(𝑧
·N 𝑢) ∈ N) |
48 | 21 | 3ad2ant3 1015 |
. . . . . 6
⊢ (((𝑥 ∈ N ∧
𝑦 ∈ N)
∧ (𝑧 ∈
N ∧ 𝑢
∈ N) ∧ (𝑤 ∈ N ∧ 𝑣 ∈ N)) →
(𝑤
·N 𝑣) ∈ N) |
49 | 47, 48, 22 | syl2anc 409 |
. . . . 5
⊢ (((𝑥 ∈ N ∧
𝑦 ∈ N)
∧ (𝑧 ∈
N ∧ 𝑢
∈ N) ∧ (𝑤 ∈ N ∧ 𝑣 ∈ N)) →
((𝑧
·N 𝑢) +N (𝑤
·N 𝑣)) ∈ N) |
50 | | mulasspig 7294 |
. . . . 5
⊢ ((𝑦 ∈ N ∧
𝑥 ∈ N
∧ ((𝑧
·N 𝑢) +N (𝑤
·N 𝑣)) ∈ N) → ((𝑦
·N 𝑥) ·N ((𝑧
·N 𝑢) +N (𝑤
·N 𝑣))) = (𝑦 ·N (𝑥
·N ((𝑧 ·N 𝑢) +N
(𝑤
·N 𝑣))))) |
51 | 45, 46, 49, 50 | syl3anc 1233 |
. . . 4
⊢ (((𝑥 ∈ N ∧
𝑦 ∈ N)
∧ (𝑧 ∈
N ∧ 𝑢
∈ N) ∧ (𝑤 ∈ N ∧ 𝑣 ∈ N)) →
((𝑦
·N 𝑥) ·N ((𝑧
·N 𝑢) +N (𝑤
·N 𝑣))) = (𝑦 ·N (𝑥
·N ((𝑧 ·N 𝑢) +N
(𝑤
·N 𝑣))))) |
52 | | mulcompig 7293 |
. . . . . . . . 9
⊢ ((𝑥 ∈ N ∧
𝑦 ∈ N)
→ (𝑥
·N 𝑦) = (𝑦 ·N 𝑥)) |
53 | 52 | oveq1d 5868 |
. . . . . . . 8
⊢ ((𝑥 ∈ N ∧
𝑦 ∈ N)
→ ((𝑥
·N 𝑦) ·N (𝑧
·N 𝑢)) = ((𝑦 ·N 𝑥)
·N (𝑧 ·N 𝑢))) |
54 | 53 | adantr 274 |
. . . . . . 7
⊢ (((𝑥 ∈ N ∧
𝑦 ∈ N)
∧ (𝑧 ∈
N ∧ 𝑢
∈ N)) → ((𝑥 ·N 𝑦)
·N (𝑧 ·N 𝑢)) = ((𝑦 ·N 𝑥)
·N (𝑧 ·N 𝑢))) |
55 | | simpll 524 |
. . . . . . . 8
⊢ (((𝑥 ∈ N ∧
𝑦 ∈ N)
∧ (𝑧 ∈
N ∧ 𝑢
∈ N)) → 𝑥 ∈ N) |
56 | | simplr 525 |
. . . . . . . 8
⊢ (((𝑥 ∈ N ∧
𝑦 ∈ N)
∧ (𝑧 ∈
N ∧ 𝑢
∈ N)) → 𝑦 ∈ N) |
57 | | simprl 526 |
. . . . . . . 8
⊢ (((𝑥 ∈ N ∧
𝑦 ∈ N)
∧ (𝑧 ∈
N ∧ 𝑢
∈ N)) → 𝑧 ∈ N) |
58 | | mulcompig 7293 |
. . . . . . . . 9
⊢ ((𝑓 ∈ N ∧
𝑔 ∈ N)
→ (𝑓
·N 𝑔) = (𝑔 ·N 𝑓)) |
59 | 58 | adantl 275 |
. . . . . . . 8
⊢ ((((𝑥 ∈ N ∧
𝑦 ∈ N)
∧ (𝑧 ∈
N ∧ 𝑢
∈ N)) ∧ (𝑓 ∈ N ∧ 𝑔 ∈ N)) →
(𝑓
·N 𝑔) = (𝑔 ·N 𝑓)) |
60 | | mulasspig 7294 |
. . . . . . . . 9
⊢ ((𝑓 ∈ N ∧
𝑔 ∈ N
∧ ℎ ∈
N) → ((𝑓
·N 𝑔) ·N ℎ) = (𝑓 ·N (𝑔
·N ℎ))) |
61 | 60 | adantl 275 |
. . . . . . . 8
⊢ ((((𝑥 ∈ N ∧
𝑦 ∈ N)
∧ (𝑧 ∈
N ∧ 𝑢
∈ N)) ∧ (𝑓 ∈ N ∧ 𝑔 ∈ N ∧
ℎ ∈ N))
→ ((𝑓
·N 𝑔) ·N ℎ) = (𝑓 ·N (𝑔
·N ℎ))) |
62 | | simprr 527 |
. . . . . . . 8
⊢ (((𝑥 ∈ N ∧
𝑦 ∈ N)
∧ (𝑧 ∈
N ∧ 𝑢
∈ N)) → 𝑢 ∈ N) |
63 | | mulclpi 7290 |
. . . . . . . . 9
⊢ ((𝑓 ∈ N ∧
𝑔 ∈ N)
→ (𝑓
·N 𝑔) ∈ N) |
64 | 63 | adantl 275 |
. . . . . . . 8
⊢ ((((𝑥 ∈ N ∧
𝑦 ∈ N)
∧ (𝑧 ∈
N ∧ 𝑢
∈ N)) ∧ (𝑓 ∈ N ∧ 𝑔 ∈ N)) →
(𝑓
·N 𝑔) ∈ N) |
65 | 55, 56, 57, 59, 61, 62, 64 | caov4d 6037 |
. . . . . . 7
⊢ (((𝑥 ∈ N ∧
𝑦 ∈ N)
∧ (𝑧 ∈
N ∧ 𝑢
∈ N)) → ((𝑥 ·N 𝑦)
·N (𝑧 ·N 𝑢)) = ((𝑥 ·N 𝑧)
·N (𝑦 ·N 𝑢))) |
66 | 54, 65 | eqtr3d 2205 |
. . . . . 6
⊢ (((𝑥 ∈ N ∧
𝑦 ∈ N)
∧ (𝑧 ∈
N ∧ 𝑢
∈ N)) → ((𝑦 ·N 𝑥)
·N (𝑧 ·N 𝑢)) = ((𝑥 ·N 𝑧)
·N (𝑦 ·N 𝑢))) |
67 | 66 | 3adant3 1012 |
. . . . 5
⊢ (((𝑥 ∈ N ∧
𝑦 ∈ N)
∧ (𝑧 ∈
N ∧ 𝑢
∈ N) ∧ (𝑤 ∈ N ∧ 𝑣 ∈ N)) →
((𝑦
·N 𝑥) ·N (𝑧
·N 𝑢)) = ((𝑥 ·N 𝑧)
·N (𝑦 ·N 𝑢))) |
68 | | simplr 525 |
. . . . . . 7
⊢ (((𝑥 ∈ N ∧
𝑦 ∈ N)
∧ (𝑤 ∈
N ∧ 𝑣
∈ N)) → 𝑦 ∈ N) |
69 | | simpll 524 |
. . . . . . 7
⊢ (((𝑥 ∈ N ∧
𝑦 ∈ N)
∧ (𝑤 ∈
N ∧ 𝑣
∈ N)) → 𝑥 ∈ N) |
70 | | simprl 526 |
. . . . . . 7
⊢ (((𝑥 ∈ N ∧
𝑦 ∈ N)
∧ (𝑤 ∈
N ∧ 𝑣
∈ N)) → 𝑤 ∈ N) |
71 | 58 | adantl 275 |
. . . . . . 7
⊢ ((((𝑥 ∈ N ∧
𝑦 ∈ N)
∧ (𝑤 ∈
N ∧ 𝑣
∈ N)) ∧ (𝑓 ∈ N ∧ 𝑔 ∈ N)) →
(𝑓
·N 𝑔) = (𝑔 ·N 𝑓)) |
72 | 60 | adantl 275 |
. . . . . . 7
⊢ ((((𝑥 ∈ N ∧
𝑦 ∈ N)
∧ (𝑤 ∈
N ∧ 𝑣
∈ N)) ∧ (𝑓 ∈ N ∧ 𝑔 ∈ N ∧
ℎ ∈ N))
→ ((𝑓
·N 𝑔) ·N ℎ) = (𝑓 ·N (𝑔
·N ℎ))) |
73 | | simprr 527 |
. . . . . . 7
⊢ (((𝑥 ∈ N ∧
𝑦 ∈ N)
∧ (𝑤 ∈
N ∧ 𝑣
∈ N)) → 𝑣 ∈ N) |
74 | 63 | adantl 275 |
. . . . . . 7
⊢ ((((𝑥 ∈ N ∧
𝑦 ∈ N)
∧ (𝑤 ∈
N ∧ 𝑣
∈ N)) ∧ (𝑓 ∈ N ∧ 𝑔 ∈ N)) →
(𝑓
·N 𝑔) ∈ N) |
75 | 68, 69, 70, 71, 72, 73, 74 | caov4d 6037 |
. . . . . 6
⊢ (((𝑥 ∈ N ∧
𝑦 ∈ N)
∧ (𝑤 ∈
N ∧ 𝑣
∈ N)) → ((𝑦 ·N 𝑥)
·N (𝑤 ·N 𝑣)) = ((𝑦 ·N 𝑤)
·N (𝑥 ·N 𝑣))) |
76 | 75 | 3adant2 1011 |
. . . . 5
⊢ (((𝑥 ∈ N ∧
𝑦 ∈ N)
∧ (𝑧 ∈
N ∧ 𝑢
∈ N) ∧ (𝑤 ∈ N ∧ 𝑣 ∈ N)) →
((𝑦
·N 𝑥) ·N (𝑤
·N 𝑣)) = ((𝑦 ·N 𝑤)
·N (𝑥 ·N 𝑣))) |
77 | 67, 76 | oveq12d 5871 |
. . . 4
⊢ (((𝑥 ∈ N ∧
𝑦 ∈ N)
∧ (𝑧 ∈
N ∧ 𝑢
∈ N) ∧ (𝑤 ∈ N ∧ 𝑣 ∈ N)) →
(((𝑦
·N 𝑥) ·N (𝑧
·N 𝑢)) +N ((𝑦
·N 𝑥) ·N (𝑤
·N 𝑣))) = (((𝑥 ·N 𝑧)
·N (𝑦 ·N 𝑢)) +N
((𝑦
·N 𝑤) ·N (𝑥
·N 𝑣)))) |
78 | 44, 51, 77 | 3eqtr3d 2211 |
. . 3
⊢ (((𝑥 ∈ N ∧
𝑦 ∈ N)
∧ (𝑧 ∈
N ∧ 𝑢
∈ N) ∧ (𝑤 ∈ N ∧ 𝑣 ∈ N)) →
(𝑦
·N (𝑥 ·N ((𝑧
·N 𝑢) +N (𝑤
·N 𝑣)))) = (((𝑥 ·N 𝑧)
·N (𝑦 ·N 𝑢)) +N
((𝑦
·N 𝑤) ·N (𝑥
·N 𝑣)))) |
79 | 40, 78 | sylbir 134 |
. 2
⊢ (((𝑥 ∈ N ∧
𝑦 ∈ N)
∧ (𝑧 ∈
N ∧ 𝑤
∈ N) ∧ (𝑣 ∈ N ∧ 𝑢 ∈ N)) →
(𝑦
·N (𝑥 ·N ((𝑧
·N 𝑢) +N (𝑤
·N 𝑣)))) = (((𝑥 ·N 𝑧)
·N (𝑦 ·N 𝑢)) +N
((𝑦
·N 𝑤) ·N (𝑥
·N 𝑣)))) |
80 | 70 | 3adant2 1011 |
. . . . . 6
⊢ (((𝑥 ∈ N ∧
𝑦 ∈ N)
∧ (𝑧 ∈
N ∧ 𝑢
∈ N) ∧ (𝑤 ∈ N ∧ 𝑣 ∈ N)) →
𝑤 ∈
N) |
81 | 62 | 3adant3 1012 |
. . . . . 6
⊢ (((𝑥 ∈ N ∧
𝑦 ∈ N)
∧ (𝑧 ∈
N ∧ 𝑢
∈ N) ∧ (𝑤 ∈ N ∧ 𝑣 ∈ N)) →
𝑢 ∈
N) |
82 | 80, 81, 25 | syl2anc 409 |
. . . . 5
⊢ (((𝑥 ∈ N ∧
𝑦 ∈ N)
∧ (𝑧 ∈
N ∧ 𝑢
∈ N) ∧ (𝑤 ∈ N ∧ 𝑣 ∈ N)) →
(𝑤
·N 𝑢) ∈ N) |
83 | | mulasspig 7294 |
. . . . 5
⊢ ((𝑦 ∈ N ∧
𝑦 ∈ N
∧ (𝑤
·N 𝑢) ∈ N) → ((𝑦
·N 𝑦) ·N (𝑤
·N 𝑢)) = (𝑦 ·N (𝑦
·N (𝑤 ·N 𝑢)))) |
84 | 45, 45, 82, 83 | syl3anc 1233 |
. . . 4
⊢ (((𝑥 ∈ N ∧
𝑦 ∈ N)
∧ (𝑧 ∈
N ∧ 𝑢
∈ N) ∧ (𝑤 ∈ N ∧ 𝑣 ∈ N)) →
((𝑦
·N 𝑦) ·N (𝑤
·N 𝑢)) = (𝑦 ·N (𝑦
·N (𝑤 ·N 𝑢)))) |
85 | 58 | adantl 275 |
. . . . 5
⊢ ((((𝑥 ∈ N ∧
𝑦 ∈ N)
∧ (𝑧 ∈
N ∧ 𝑢
∈ N) ∧ (𝑤 ∈ N ∧ 𝑣 ∈ N)) ∧
(𝑓 ∈ N
∧ 𝑔 ∈
N)) → (𝑓
·N 𝑔) = (𝑔 ·N 𝑓)) |
86 | 60 | adantl 275 |
. . . . 5
⊢ ((((𝑥 ∈ N ∧
𝑦 ∈ N)
∧ (𝑧 ∈
N ∧ 𝑢
∈ N) ∧ (𝑤 ∈ N ∧ 𝑣 ∈ N)) ∧
(𝑓 ∈ N
∧ 𝑔 ∈
N ∧ ℎ
∈ N)) → ((𝑓 ·N 𝑔)
·N ℎ) = (𝑓 ·N (𝑔
·N ℎ))) |
87 | 63 | adantl 275 |
. . . . 5
⊢ ((((𝑥 ∈ N ∧
𝑦 ∈ N)
∧ (𝑧 ∈
N ∧ 𝑢
∈ N) ∧ (𝑤 ∈ N ∧ 𝑣 ∈ N)) ∧
(𝑓 ∈ N
∧ 𝑔 ∈
N)) → (𝑓
·N 𝑔) ∈ N) |
88 | 45, 45, 80, 85, 86, 81, 87 | caov4d 6037 |
. . . 4
⊢ (((𝑥 ∈ N ∧
𝑦 ∈ N)
∧ (𝑧 ∈
N ∧ 𝑢
∈ N) ∧ (𝑤 ∈ N ∧ 𝑣 ∈ N)) →
((𝑦
·N 𝑦) ·N (𝑤
·N 𝑢)) = ((𝑦 ·N 𝑤)
·N (𝑦 ·N 𝑢))) |
89 | 84, 88 | eqtr3d 2205 |
. . 3
⊢ (((𝑥 ∈ N ∧
𝑦 ∈ N)
∧ (𝑧 ∈
N ∧ 𝑢
∈ N) ∧ (𝑤 ∈ N ∧ 𝑣 ∈ N)) →
(𝑦
·N (𝑦 ·N (𝑤
·N 𝑢))) = ((𝑦 ·N 𝑤)
·N (𝑦 ·N 𝑢))) |
90 | 40, 89 | sylbir 134 |
. 2
⊢ (((𝑥 ∈ N ∧
𝑦 ∈ N)
∧ (𝑧 ∈
N ∧ 𝑤
∈ N) ∧ (𝑣 ∈ N ∧ 𝑢 ∈ N)) →
(𝑦
·N (𝑦 ·N (𝑤
·N 𝑢))) = ((𝑦 ·N 𝑤)
·N (𝑦 ·N 𝑢))) |
91 | 1, 2, 16, 17, 18, 19, 27, 31, 35, 79, 90 | ecovidi 6625 |
1
⊢ ((𝐴 ∈ Q ∧
𝐵 ∈ Q
∧ 𝐶 ∈
Q) → (𝐴
·Q (𝐵 +Q 𝐶)) = ((𝐴 ·Q 𝐵) +Q
(𝐴
·Q 𝐶))) |