Step | Hyp | Ref
| Expression |
1 | | df-nqqs 7299 |
. 2
⊢
Q = ((N × N) /
~Q ) |
2 | | addpipqqs 7321 |
. 2
⊢ (((𝑥 ∈ N ∧
𝑦 ∈ N)
∧ (𝑧 ∈
N ∧ 𝑤
∈ N)) → ([〈𝑥, 𝑦〉] ~Q
+Q [〈𝑧, 𝑤〉] ~Q ) =
[〈((𝑥
·N 𝑤) +N (𝑦
·N 𝑧)), (𝑦 ·N 𝑤)〉]
~Q ) |
3 | | addpipqqs 7321 |
. 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 ) |
4 | | addpipqqs 7321 |
. 2
⊢
(((((𝑥
·N 𝑤) +N (𝑦
·N 𝑧)) ∈ N ∧ (𝑦
·N 𝑤) ∈ N) ∧ (𝑣 ∈ N ∧
𝑢 ∈ N))
→ ([〈((𝑥
·N 𝑤) +N (𝑦
·N 𝑧)), (𝑦 ·N 𝑤)〉]
~Q +Q [〈𝑣, 𝑢〉] ~Q ) =
[〈((((𝑥
·N 𝑤) +N (𝑦
·N 𝑧)) ·N 𝑢) +N
((𝑦
·N 𝑤) ·N 𝑣)), ((𝑦 ·N 𝑤)
·N 𝑢)〉] ~Q
) |
5 | | addpipqqs 7321 |
. 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 ) = [〈((𝑥 ·N (𝑤
·N 𝑢)) +N (𝑦
·N ((𝑧 ·N 𝑢) +N
(𝑤
·N 𝑣)))), (𝑦 ·N (𝑤
·N 𝑢))〉] ~Q
) |
6 | | mulclpi 7279 |
. . . . 5
⊢ ((𝑥 ∈ N ∧
𝑤 ∈ N)
→ (𝑥
·N 𝑤) ∈ N) |
7 | 6 | ad2ant2rl 508 |
. . . 4
⊢ (((𝑥 ∈ N ∧
𝑦 ∈ N)
∧ (𝑧 ∈
N ∧ 𝑤
∈ N)) → (𝑥 ·N 𝑤) ∈
N) |
8 | | mulclpi 7279 |
. . . . 5
⊢ ((𝑦 ∈ N ∧
𝑧 ∈ N)
→ (𝑦
·N 𝑧) ∈ N) |
9 | 8 | ad2ant2lr 507 |
. . . 4
⊢ (((𝑥 ∈ N ∧
𝑦 ∈ N)
∧ (𝑧 ∈
N ∧ 𝑤
∈ N)) → (𝑦 ·N 𝑧) ∈
N) |
10 | | addclpi 7278 |
. . . 4
⊢ (((𝑥
·N 𝑤) ∈ N ∧ (𝑦
·N 𝑧) ∈ N) → ((𝑥
·N 𝑤) +N (𝑦
·N 𝑧)) ∈ N) |
11 | 7, 9, 10 | syl2anc 409 |
. . 3
⊢ (((𝑥 ∈ N ∧
𝑦 ∈ N)
∧ (𝑧 ∈
N ∧ 𝑤
∈ N)) → ((𝑥 ·N 𝑤) +N
(𝑦
·N 𝑧)) ∈ N) |
12 | | mulclpi 7279 |
. . . 4
⊢ ((𝑦 ∈ N ∧
𝑤 ∈ N)
→ (𝑦
·N 𝑤) ∈ N) |
13 | 12 | ad2ant2l 505 |
. . 3
⊢ (((𝑥 ∈ N ∧
𝑦 ∈ N)
∧ (𝑧 ∈
N ∧ 𝑤
∈ N)) → (𝑦 ·N 𝑤) ∈
N) |
14 | 11, 13 | jca 304 |
. 2
⊢ (((𝑥 ∈ N ∧
𝑦 ∈ N)
∧ (𝑧 ∈
N ∧ 𝑤
∈ N)) → (((𝑥 ·N 𝑤) +N
(𝑦
·N 𝑧)) ∈ N ∧ (𝑦
·N 𝑤) ∈ N)) |
15 | | mulclpi 7279 |
. . . . 5
⊢ ((𝑧 ∈ N ∧
𝑢 ∈ N)
→ (𝑧
·N 𝑢) ∈ N) |
16 | 15 | ad2ant2rl 508 |
. . . 4
⊢ (((𝑧 ∈ N ∧
𝑤 ∈ N)
∧ (𝑣 ∈
N ∧ 𝑢
∈ N)) → (𝑧 ·N 𝑢) ∈
N) |
17 | | mulclpi 7279 |
. . . . 5
⊢ ((𝑤 ∈ N ∧
𝑣 ∈ N)
→ (𝑤
·N 𝑣) ∈ N) |
18 | 17 | ad2ant2lr 507 |
. . . 4
⊢ (((𝑧 ∈ N ∧
𝑤 ∈ N)
∧ (𝑣 ∈
N ∧ 𝑢
∈ N)) → (𝑤 ·N 𝑣) ∈
N) |
19 | | addclpi 7278 |
. . . 4
⊢ (((𝑧
·N 𝑢) ∈ N ∧ (𝑤
·N 𝑣) ∈ N) → ((𝑧
·N 𝑢) +N (𝑤
·N 𝑣)) ∈ N) |
20 | 16, 18, 19 | syl2anc 409 |
. . 3
⊢ (((𝑧 ∈ N ∧
𝑤 ∈ N)
∧ (𝑣 ∈
N ∧ 𝑢
∈ N)) → ((𝑧 ·N 𝑢) +N
(𝑤
·N 𝑣)) ∈ N) |
21 | | mulclpi 7279 |
. . . 4
⊢ ((𝑤 ∈ N ∧
𝑢 ∈ N)
→ (𝑤
·N 𝑢) ∈ N) |
22 | 21 | ad2ant2l 505 |
. . 3
⊢ (((𝑧 ∈ N ∧
𝑤 ∈ N)
∧ (𝑣 ∈
N ∧ 𝑢
∈ N)) → (𝑤 ·N 𝑢) ∈
N) |
23 | 20, 22 | jca 304 |
. 2
⊢ (((𝑧 ∈ N ∧
𝑤 ∈ N)
∧ (𝑣 ∈
N ∧ 𝑢
∈ N)) → (((𝑧 ·N 𝑢) +N
(𝑤
·N 𝑣)) ∈ N ∧ (𝑤
·N 𝑢) ∈ N)) |
24 | | simp1l 1016 |
. . . . 5
⊢ (((𝑥 ∈ N ∧
𝑦 ∈ N)
∧ (𝑧 ∈
N ∧ 𝑤
∈ N) ∧ (𝑣 ∈ N ∧ 𝑢 ∈ N)) →
𝑥 ∈
N) |
25 | | simp2r 1019 |
. . . . . 6
⊢ (((𝑥 ∈ N ∧
𝑦 ∈ N)
∧ (𝑧 ∈
N ∧ 𝑤
∈ N) ∧ (𝑣 ∈ N ∧ 𝑢 ∈ N)) →
𝑤 ∈
N) |
26 | | simp3r 1021 |
. . . . . 6
⊢ (((𝑥 ∈ N ∧
𝑦 ∈ N)
∧ (𝑧 ∈
N ∧ 𝑤
∈ N) ∧ (𝑣 ∈ N ∧ 𝑢 ∈ N)) →
𝑢 ∈
N) |
27 | 25, 26, 21 | syl2anc 409 |
. . . . 5
⊢ (((𝑥 ∈ N ∧
𝑦 ∈ N)
∧ (𝑧 ∈
N ∧ 𝑤
∈ N) ∧ (𝑣 ∈ N ∧ 𝑢 ∈ N)) →
(𝑤
·N 𝑢) ∈ N) |
28 | | mulclpi 7279 |
. . . . 5
⊢ ((𝑥 ∈ N ∧
(𝑤
·N 𝑢) ∈ N) → (𝑥
·N (𝑤 ·N 𝑢)) ∈
N) |
29 | 24, 27, 28 | syl2anc 409 |
. . . 4
⊢ (((𝑥 ∈ N ∧
𝑦 ∈ N)
∧ (𝑧 ∈
N ∧ 𝑤
∈ N) ∧ (𝑣 ∈ N ∧ 𝑢 ∈ N)) →
(𝑥
·N (𝑤 ·N 𝑢)) ∈
N) |
30 | | simp1r 1017 |
. . . . 5
⊢ (((𝑥 ∈ N ∧
𝑦 ∈ N)
∧ (𝑧 ∈
N ∧ 𝑤
∈ N) ∧ (𝑣 ∈ N ∧ 𝑢 ∈ N)) →
𝑦 ∈
N) |
31 | | simp2l 1018 |
. . . . . 6
⊢ (((𝑥 ∈ N ∧
𝑦 ∈ N)
∧ (𝑧 ∈
N ∧ 𝑤
∈ N) ∧ (𝑣 ∈ N ∧ 𝑢 ∈ N)) →
𝑧 ∈
N) |
32 | 31, 26, 15 | syl2anc 409 |
. . . . 5
⊢ (((𝑥 ∈ N ∧
𝑦 ∈ N)
∧ (𝑧 ∈
N ∧ 𝑤
∈ N) ∧ (𝑣 ∈ N ∧ 𝑢 ∈ N)) →
(𝑧
·N 𝑢) ∈ N) |
33 | | mulclpi 7279 |
. . . . 5
⊢ ((𝑦 ∈ N ∧
(𝑧
·N 𝑢) ∈ N) → (𝑦
·N (𝑧 ·N 𝑢)) ∈
N) |
34 | 30, 32, 33 | syl2anc 409 |
. . . 4
⊢ (((𝑥 ∈ N ∧
𝑦 ∈ N)
∧ (𝑧 ∈
N ∧ 𝑤
∈ N) ∧ (𝑣 ∈ N ∧ 𝑢 ∈ N)) →
(𝑦
·N (𝑧 ·N 𝑢)) ∈
N) |
35 | | simp3l 1020 |
. . . . . 6
⊢ (((𝑥 ∈ N ∧
𝑦 ∈ N)
∧ (𝑧 ∈
N ∧ 𝑤
∈ N) ∧ (𝑣 ∈ N ∧ 𝑢 ∈ N)) →
𝑣 ∈
N) |
36 | 25, 35, 17 | syl2anc 409 |
. . . . 5
⊢ (((𝑥 ∈ N ∧
𝑦 ∈ N)
∧ (𝑧 ∈
N ∧ 𝑤
∈ N) ∧ (𝑣 ∈ N ∧ 𝑢 ∈ N)) →
(𝑤
·N 𝑣) ∈ N) |
37 | | mulclpi 7279 |
. . . . 5
⊢ ((𝑦 ∈ N ∧
(𝑤
·N 𝑣) ∈ N) → (𝑦
·N (𝑤 ·N 𝑣)) ∈
N) |
38 | 30, 36, 37 | syl2anc 409 |
. . . 4
⊢ (((𝑥 ∈ N ∧
𝑦 ∈ N)
∧ (𝑧 ∈
N ∧ 𝑤
∈ N) ∧ (𝑣 ∈ N ∧ 𝑢 ∈ N)) →
(𝑦
·N (𝑤 ·N 𝑣)) ∈
N) |
39 | | addasspig 7281 |
. . . 4
⊢ (((𝑥
·N (𝑤 ·N 𝑢)) ∈ N ∧
(𝑦
·N (𝑧 ·N 𝑢)) ∈ N ∧
(𝑦
·N (𝑤 ·N 𝑣)) ∈ N)
→ (((𝑥
·N (𝑤 ·N 𝑢)) +N
(𝑦
·N (𝑧 ·N 𝑢))) +N
(𝑦
·N (𝑤 ·N 𝑣))) = ((𝑥 ·N (𝑤
·N 𝑢)) +N ((𝑦
·N (𝑧 ·N 𝑢)) +N
(𝑦
·N (𝑤 ·N 𝑣))))) |
40 | 29, 34, 38, 39 | syl3anc 1233 |
. . 3
⊢ (((𝑥 ∈ N ∧
𝑦 ∈ N)
∧ (𝑧 ∈
N ∧ 𝑤
∈ N) ∧ (𝑣 ∈ N ∧ 𝑢 ∈ N)) →
(((𝑥
·N (𝑤 ·N 𝑢)) +N
(𝑦
·N (𝑧 ·N 𝑢))) +N
(𝑦
·N (𝑤 ·N 𝑣))) = ((𝑥 ·N (𝑤
·N 𝑢)) +N ((𝑦
·N (𝑧 ·N 𝑢)) +N
(𝑦
·N (𝑤 ·N 𝑣))))) |
41 | | mulcompig 7282 |
. . . . . 6
⊢ ((𝑓 ∈ N ∧
𝑔 ∈ N)
→ (𝑓
·N 𝑔) = (𝑔 ·N 𝑓)) |
42 | 41 | adantl 275 |
. . . . 5
⊢ ((((𝑥 ∈ N ∧
𝑦 ∈ N)
∧ (𝑧 ∈
N ∧ 𝑤
∈ N) ∧ (𝑣 ∈ N ∧ 𝑢 ∈ N)) ∧
(𝑓 ∈ N
∧ 𝑔 ∈
N)) → (𝑓
·N 𝑔) = (𝑔 ·N 𝑓)) |
43 | | distrpig 7284 |
. . . . . . . 8
⊢ ((ℎ ∈ N ∧
𝑓 ∈ N
∧ 𝑔 ∈
N) → (ℎ
·N (𝑓 +N 𝑔)) = ((ℎ ·N 𝑓) +N
(ℎ
·N 𝑔))) |
44 | 43 | 3coml 1205 |
. . . . . . 7
⊢ ((𝑓 ∈ N ∧
𝑔 ∈ N
∧ ℎ ∈
N) → (ℎ
·N (𝑓 +N 𝑔)) = ((ℎ ·N 𝑓) +N
(ℎ
·N 𝑔))) |
45 | | addclpi 7278 |
. . . . . . . . . 10
⊢ ((𝑓 ∈ N ∧
𝑔 ∈ N)
→ (𝑓
+N 𝑔) ∈ N) |
46 | | mulcompig 7282 |
. . . . . . . . . 10
⊢ ((ℎ ∈ N ∧
(𝑓
+N 𝑔) ∈ N) → (ℎ
·N (𝑓 +N 𝑔)) = ((𝑓 +N 𝑔)
·N ℎ)) |
47 | 45, 46 | sylan2 284 |
. . . . . . . . 9
⊢ ((ℎ ∈ N ∧
(𝑓 ∈ N
∧ 𝑔 ∈
N)) → (ℎ
·N (𝑓 +N 𝑔)) = ((𝑓 +N 𝑔)
·N ℎ)) |
48 | 47 | ancoms 266 |
. . . . . . . 8
⊢ (((𝑓 ∈ N ∧
𝑔 ∈ N)
∧ ℎ ∈
N) → (ℎ
·N (𝑓 +N 𝑔)) = ((𝑓 +N 𝑔)
·N ℎ)) |
49 | 48 | 3impa 1189 |
. . . . . . 7
⊢ ((𝑓 ∈ N ∧
𝑔 ∈ N
∧ ℎ ∈
N) → (ℎ
·N (𝑓 +N 𝑔)) = ((𝑓 +N 𝑔)
·N ℎ)) |
50 | | mulcompig 7282 |
. . . . . . . . . 10
⊢ ((ℎ ∈ N ∧
𝑓 ∈ N)
→ (ℎ
·N 𝑓) = (𝑓 ·N ℎ)) |
51 | 50 | ancoms 266 |
. . . . . . . . 9
⊢ ((𝑓 ∈ N ∧
ℎ ∈ N)
→ (ℎ
·N 𝑓) = (𝑓 ·N ℎ)) |
52 | 51 | 3adant2 1011 |
. . . . . . . 8
⊢ ((𝑓 ∈ N ∧
𝑔 ∈ N
∧ ℎ ∈
N) → (ℎ
·N 𝑓) = (𝑓 ·N ℎ)) |
53 | | mulcompig 7282 |
. . . . . . . . . 10
⊢ ((ℎ ∈ N ∧
𝑔 ∈ N)
→ (ℎ
·N 𝑔) = (𝑔 ·N ℎ)) |
54 | 53 | ancoms 266 |
. . . . . . . . 9
⊢ ((𝑔 ∈ N ∧
ℎ ∈ N)
→ (ℎ
·N 𝑔) = (𝑔 ·N ℎ)) |
55 | 54 | 3adant1 1010 |
. . . . . . . 8
⊢ ((𝑓 ∈ N ∧
𝑔 ∈ N
∧ ℎ ∈
N) → (ℎ
·N 𝑔) = (𝑔 ·N ℎ)) |
56 | 52, 55 | oveq12d 5869 |
. . . . . . 7
⊢ ((𝑓 ∈ N ∧
𝑔 ∈ N
∧ ℎ ∈
N) → ((ℎ
·N 𝑓) +N (ℎ
·N 𝑔)) = ((𝑓 ·N ℎ) +N
(𝑔
·N ℎ))) |
57 | 44, 49, 56 | 3eqtr3d 2211 |
. . . . . 6
⊢ ((𝑓 ∈ N ∧
𝑔 ∈ N
∧ ℎ ∈
N) → ((𝑓
+N 𝑔) ·N ℎ) = ((𝑓 ·N ℎ) +N
(𝑔
·N ℎ))) |
58 | 57 | adantl 275 |
. . . . 5
⊢ ((((𝑥 ∈ N ∧
𝑦 ∈ N)
∧ (𝑧 ∈
N ∧ 𝑤
∈ N) ∧ (𝑣 ∈ N ∧ 𝑢 ∈ N)) ∧
(𝑓 ∈ N
∧ 𝑔 ∈
N ∧ ℎ
∈ N)) → ((𝑓 +N 𝑔)
·N ℎ) = ((𝑓 ·N ℎ) +N
(𝑔
·N ℎ))) |
59 | | mulasspig 7283 |
. . . . . 6
⊢ ((𝑓 ∈ N ∧
𝑔 ∈ N
∧ ℎ ∈
N) → ((𝑓
·N 𝑔) ·N ℎ) = (𝑓 ·N (𝑔
·N ℎ))) |
60 | 59 | adantl 275 |
. . . . 5
⊢ ((((𝑥 ∈ N ∧
𝑦 ∈ N)
∧ (𝑧 ∈
N ∧ 𝑤
∈ N) ∧ (𝑣 ∈ N ∧ 𝑢 ∈ N)) ∧
(𝑓 ∈ N
∧ 𝑔 ∈
N ∧ ℎ
∈ N)) → ((𝑓 ·N 𝑔)
·N ℎ) = (𝑓 ·N (𝑔
·N ℎ))) |
61 | | mulclpi 7279 |
. . . . . 6
⊢ ((𝑓 ∈ N ∧
𝑔 ∈ N)
→ (𝑓
·N 𝑔) ∈ N) |
62 | 61 | adantl 275 |
. . . . 5
⊢ ((((𝑥 ∈ N ∧
𝑦 ∈ N)
∧ (𝑧 ∈
N ∧ 𝑤
∈ N) ∧ (𝑣 ∈ N ∧ 𝑢 ∈ N)) ∧
(𝑓 ∈ N
∧ 𝑔 ∈
N)) → (𝑓
·N 𝑔) ∈ N) |
63 | 42, 58, 60, 62, 24, 30, 25, 31, 26 | caovdilemd 6042 |
. . . 4
⊢ (((𝑥 ∈ N ∧
𝑦 ∈ N)
∧ (𝑧 ∈
N ∧ 𝑤
∈ N) ∧ (𝑣 ∈ N ∧ 𝑢 ∈ N)) →
(((𝑥
·N 𝑤) +N (𝑦
·N 𝑧)) ·N 𝑢) = ((𝑥 ·N (𝑤
·N 𝑢)) +N (𝑦
·N (𝑧 ·N 𝑢)))) |
64 | | mulasspig 7283 |
. . . . . . 7
⊢ ((𝑦 ∈ N ∧
𝑤 ∈ N
∧ 𝑣 ∈
N) → ((𝑦
·N 𝑤) ·N 𝑣) = (𝑦 ·N (𝑤
·N 𝑣))) |
65 | 64 | 3adant1l 1225 |
. . . . . 6
⊢ (((𝑥 ∈ N ∧
𝑦 ∈ N)
∧ 𝑤 ∈
N ∧ 𝑣
∈ N) → ((𝑦 ·N 𝑤)
·N 𝑣) = (𝑦 ·N (𝑤
·N 𝑣))) |
66 | 65 | 3adant2l 1227 |
. . . . 5
⊢ (((𝑥 ∈ N ∧
𝑦 ∈ N)
∧ (𝑧 ∈
N ∧ 𝑤
∈ N) ∧ 𝑣 ∈ N) → ((𝑦
·N 𝑤) ·N 𝑣) = (𝑦 ·N (𝑤
·N 𝑣))) |
67 | 66 | 3adant3r 1230 |
. . . 4
⊢ (((𝑥 ∈ N ∧
𝑦 ∈ N)
∧ (𝑧 ∈
N ∧ 𝑤
∈ N) ∧ (𝑣 ∈ N ∧ 𝑢 ∈ N)) →
((𝑦
·N 𝑤) ·N 𝑣) = (𝑦 ·N (𝑤
·N 𝑣))) |
68 | 63, 67 | oveq12d 5869 |
. . 3
⊢ (((𝑥 ∈ N ∧
𝑦 ∈ N)
∧ (𝑧 ∈
N ∧ 𝑤
∈ N) ∧ (𝑣 ∈ N ∧ 𝑢 ∈ N)) →
((((𝑥
·N 𝑤) +N (𝑦
·N 𝑧)) ·N 𝑢) +N
((𝑦
·N 𝑤) ·N 𝑣)) = (((𝑥 ·N (𝑤
·N 𝑢)) +N (𝑦
·N (𝑧 ·N 𝑢))) +N
(𝑦
·N (𝑤 ·N 𝑣)))) |
69 | | distrpig 7284 |
. . . . 5
⊢ ((𝑦 ∈ N ∧
(𝑧
·N 𝑢) ∈ N ∧ (𝑤
·N 𝑣) ∈ N) → (𝑦
·N ((𝑧 ·N 𝑢) +N
(𝑤
·N 𝑣))) = ((𝑦 ·N (𝑧
·N 𝑢)) +N (𝑦
·N (𝑤 ·N 𝑣)))) |
70 | 30, 32, 36, 69 | syl3anc 1233 |
. . . 4
⊢ (((𝑥 ∈ N ∧
𝑦 ∈ N)
∧ (𝑧 ∈
N ∧ 𝑤
∈ N) ∧ (𝑣 ∈ N ∧ 𝑢 ∈ N)) →
(𝑦
·N ((𝑧 ·N 𝑢) +N
(𝑤
·N 𝑣))) = ((𝑦 ·N (𝑧
·N 𝑢)) +N (𝑦
·N (𝑤 ·N 𝑣)))) |
71 | 70 | oveq2d 5867 |
. . 3
⊢ (((𝑥 ∈ N ∧
𝑦 ∈ N)
∧ (𝑧 ∈
N ∧ 𝑤
∈ N) ∧ (𝑣 ∈ N ∧ 𝑢 ∈ N)) →
((𝑥
·N (𝑤 ·N 𝑢)) +N
(𝑦
·N ((𝑧 ·N 𝑢) +N
(𝑤
·N 𝑣)))) = ((𝑥 ·N (𝑤
·N 𝑢)) +N ((𝑦
·N (𝑧 ·N 𝑢)) +N
(𝑦
·N (𝑤 ·N 𝑣))))) |
72 | 40, 68, 71 | 3eqtr4d 2213 |
. 2
⊢ (((𝑥 ∈ N ∧
𝑦 ∈ N)
∧ (𝑧 ∈
N ∧ 𝑤
∈ N) ∧ (𝑣 ∈ N ∧ 𝑢 ∈ N)) →
((((𝑥
·N 𝑤) +N (𝑦
·N 𝑧)) ·N 𝑢) +N
((𝑦
·N 𝑤) ·N 𝑣)) = ((𝑥 ·N (𝑤
·N 𝑢)) +N (𝑦
·N ((𝑧 ·N 𝑢) +N
(𝑤
·N 𝑣))))) |
73 | | mulasspig 7283 |
. . . . 5
⊢ ((𝑦 ∈ N ∧
𝑤 ∈ N
∧ 𝑢 ∈
N) → ((𝑦
·N 𝑤) ·N 𝑢) = (𝑦 ·N (𝑤
·N 𝑢))) |
74 | 73 | 3adant1l 1225 |
. . . 4
⊢ (((𝑥 ∈ N ∧
𝑦 ∈ N)
∧ 𝑤 ∈
N ∧ 𝑢
∈ N) → ((𝑦 ·N 𝑤)
·N 𝑢) = (𝑦 ·N (𝑤
·N 𝑢))) |
75 | 74 | 3adant2l 1227 |
. . 3
⊢ (((𝑥 ∈ N ∧
𝑦 ∈ N)
∧ (𝑧 ∈
N ∧ 𝑤
∈ N) ∧ 𝑢 ∈ N) → ((𝑦
·N 𝑤) ·N 𝑢) = (𝑦 ·N (𝑤
·N 𝑢))) |
76 | 75 | 3adant3l 1229 |
. 2
⊢ (((𝑥 ∈ N ∧
𝑦 ∈ N)
∧ (𝑧 ∈
N ∧ 𝑤
∈ N) ∧ (𝑣 ∈ N ∧ 𝑢 ∈ N)) →
((𝑦
·N 𝑤) ·N 𝑢) = (𝑦 ·N (𝑤
·N 𝑢))) |
77 | 1, 2, 3, 4, 5, 14,
23, 72, 76 | ecoviass 6620 |
1
⊢ ((𝐴 ∈ Q ∧
𝐵 ∈ Q
∧ 𝐶 ∈
Q) → ((𝐴
+Q 𝐵) +Q 𝐶) = (𝐴 +Q (𝐵 +Q
𝐶))) |