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Theorem addcmpblnq0 6426
Description: Lemma showing compatibility of addition on non-negative fractions. (Contributed by Jim Kingdon, 23-Nov-2019.)
Assertion
Ref Expression
addcmpblnq0 ((((A 𝜔 B N) (𝐶 𝜔 𝐷 N)) ((𝐹 𝜔 𝐺 N) (𝑅 𝜔 𝑆 N))) → (((A ·𝑜 𝐷) = (B ·𝑜 𝐶) (𝐹 ·𝑜 𝑆) = (𝐺 ·𝑜 𝑅)) → ⟨((A ·𝑜 𝐺) +𝑜 (B ·𝑜 𝐹)), (B ·𝑜 𝐺)⟩ ~Q0 ⟨((𝐶 ·𝑜 𝑆) +𝑜 (𝐷 ·𝑜 𝑅)), (𝐷 ·𝑜 𝑆)⟩))

Proof of Theorem addcmpblnq0
Dummy variables x y z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nndi 6004 . . . . . . . 8 ((x 𝜔 y 𝜔 z 𝜔) → (x ·𝑜 (y +𝑜 z)) = ((x ·𝑜 y) +𝑜 (x ·𝑜 z)))
21adantl 262 . . . . . . 7 (((((A 𝜔 B N) (𝐶 𝜔 𝐷 N)) ((𝐹 𝜔 𝐺 N) (𝑅 𝜔 𝑆 N))) (x 𝜔 y 𝜔 z 𝜔)) → (x ·𝑜 (y +𝑜 z)) = ((x ·𝑜 y) +𝑜 (x ·𝑜 z)))
3 simplll 485 . . . . . . . 8 ((((A 𝜔 B N) (𝐶 𝜔 𝐷 N)) ((𝐹 𝜔 𝐺 N) (𝑅 𝜔 𝑆 N))) → A 𝜔)
4 simprlr 490 . . . . . . . . 9 ((((A 𝜔 B N) (𝐶 𝜔 𝐷 N)) ((𝐹 𝜔 𝐺 N) (𝑅 𝜔 𝑆 N))) → 𝐺 N)
5 pinn 6293 . . . . . . . . 9 (𝐺 N𝐺 𝜔)
64, 5syl 14 . . . . . . . 8 ((((A 𝜔 B N) (𝐶 𝜔 𝐷 N)) ((𝐹 𝜔 𝐺 N) (𝑅 𝜔 𝑆 N))) → 𝐺 𝜔)
7 nnmcl 5999 . . . . . . . 8 ((A 𝜔 𝐺 𝜔) → (A ·𝑜 𝐺) 𝜔)
83, 6, 7syl2anc 391 . . . . . . 7 ((((A 𝜔 B N) (𝐶 𝜔 𝐷 N)) ((𝐹 𝜔 𝐺 N) (𝑅 𝜔 𝑆 N))) → (A ·𝑜 𝐺) 𝜔)
9 simpllr 486 . . . . . . . . 9 ((((A 𝜔 B N) (𝐶 𝜔 𝐷 N)) ((𝐹 𝜔 𝐺 N) (𝑅 𝜔 𝑆 N))) → B N)
10 pinn 6293 . . . . . . . . 9 (B NB 𝜔)
119, 10syl 14 . . . . . . . 8 ((((A 𝜔 B N) (𝐶 𝜔 𝐷 N)) ((𝐹 𝜔 𝐺 N) (𝑅 𝜔 𝑆 N))) → B 𝜔)
12 simprll 489 . . . . . . . 8 ((((A 𝜔 B N) (𝐶 𝜔 𝐷 N)) ((𝐹 𝜔 𝐺 N) (𝑅 𝜔 𝑆 N))) → 𝐹 𝜔)
13 nnmcl 5999 . . . . . . . 8 ((B 𝜔 𝐹 𝜔) → (B ·𝑜 𝐹) 𝜔)
1411, 12, 13syl2anc 391 . . . . . . 7 ((((A 𝜔 B N) (𝐶 𝜔 𝐷 N)) ((𝐹 𝜔 𝐺 N) (𝑅 𝜔 𝑆 N))) → (B ·𝑜 𝐹) 𝜔)
15 simplrr 488 . . . . . . . . 9 ((((A 𝜔 B N) (𝐶 𝜔 𝐷 N)) ((𝐹 𝜔 𝐺 N) (𝑅 𝜔 𝑆 N))) → 𝐷 N)
16 pinn 6293 . . . . . . . . 9 (𝐷 N𝐷 𝜔)
1715, 16syl 14 . . . . . . . 8 ((((A 𝜔 B N) (𝐶 𝜔 𝐷 N)) ((𝐹 𝜔 𝐺 N) (𝑅 𝜔 𝑆 N))) → 𝐷 𝜔)
18 simprrr 492 . . . . . . . . 9 ((((A 𝜔 B N) (𝐶 𝜔 𝐷 N)) ((𝐹 𝜔 𝐺 N) (𝑅 𝜔 𝑆 N))) → 𝑆 N)
19 pinn 6293 . . . . . . . . 9 (𝑆 N𝑆 𝜔)
2018, 19syl 14 . . . . . . . 8 ((((A 𝜔 B N) (𝐶 𝜔 𝐷 N)) ((𝐹 𝜔 𝐺 N) (𝑅 𝜔 𝑆 N))) → 𝑆 𝜔)
21 nnmcl 5999 . . . . . . . 8 ((𝐷 𝜔 𝑆 𝜔) → (𝐷 ·𝑜 𝑆) 𝜔)
2217, 20, 21syl2anc 391 . . . . . . 7 ((((A 𝜔 B N) (𝐶 𝜔 𝐷 N)) ((𝐹 𝜔 𝐺 N) (𝑅 𝜔 𝑆 N))) → (𝐷 ·𝑜 𝑆) 𝜔)
23 nnacl 5998 . . . . . . . 8 ((x 𝜔 y 𝜔) → (x +𝑜 y) 𝜔)
2423adantl 262 . . . . . . 7 (((((A 𝜔 B N) (𝐶 𝜔 𝐷 N)) ((𝐹 𝜔 𝐺 N) (𝑅 𝜔 𝑆 N))) (x 𝜔 y 𝜔)) → (x +𝑜 y) 𝜔)
25 nnmcom 6007 . . . . . . . 8 ((x 𝜔 y 𝜔) → (x ·𝑜 y) = (y ·𝑜 x))
2625adantl 262 . . . . . . 7 (((((A 𝜔 B N) (𝐶 𝜔 𝐷 N)) ((𝐹 𝜔 𝐺 N) (𝑅 𝜔 𝑆 N))) (x 𝜔 y 𝜔)) → (x ·𝑜 y) = (y ·𝑜 x))
272, 8, 14, 22, 24, 26caovdir2d 5619 . . . . . 6 ((((A 𝜔 B N) (𝐶 𝜔 𝐷 N)) ((𝐹 𝜔 𝐺 N) (𝑅 𝜔 𝑆 N))) → (((A ·𝑜 𝐺) +𝑜 (B ·𝑜 𝐹)) ·𝑜 (𝐷 ·𝑜 𝑆)) = (((A ·𝑜 𝐺) ·𝑜 (𝐷 ·𝑜 𝑆)) +𝑜 ((B ·𝑜 𝐹) ·𝑜 (𝐷 ·𝑜 𝑆))))
28 nnmass 6005 . . . . . . . . 9 ((x 𝜔 y 𝜔 z 𝜔) → ((x ·𝑜 y) ·𝑜 z) = (x ·𝑜 (y ·𝑜 z)))
2928adantl 262 . . . . . . . 8 (((((A 𝜔 B N) (𝐶 𝜔 𝐷 N)) ((𝐹 𝜔 𝐺 N) (𝑅 𝜔 𝑆 N))) (x 𝜔 y 𝜔 z 𝜔)) → ((x ·𝑜 y) ·𝑜 z) = (x ·𝑜 (y ·𝑜 z)))
30 nnmcl 5999 . . . . . . . . 9 ((x 𝜔 y 𝜔) → (x ·𝑜 y) 𝜔)
3130adantl 262 . . . . . . . 8 (((((A 𝜔 B N) (𝐶 𝜔 𝐷 N)) ((𝐹 𝜔 𝐺 N) (𝑅 𝜔 𝑆 N))) (x 𝜔 y 𝜔)) → (x ·𝑜 y) 𝜔)
323, 6, 17, 26, 29, 20, 31caov4d 5627 . . . . . . 7 ((((A 𝜔 B N) (𝐶 𝜔 𝐷 N)) ((𝐹 𝜔 𝐺 N) (𝑅 𝜔 𝑆 N))) → ((A ·𝑜 𝐺) ·𝑜 (𝐷 ·𝑜 𝑆)) = ((A ·𝑜 𝐷) ·𝑜 (𝐺 ·𝑜 𝑆)))
3311, 12, 17, 26, 29, 20, 31caov4d 5627 . . . . . . 7 ((((A 𝜔 B N) (𝐶 𝜔 𝐷 N)) ((𝐹 𝜔 𝐺 N) (𝑅 𝜔 𝑆 N))) → ((B ·𝑜 𝐹) ·𝑜 (𝐷 ·𝑜 𝑆)) = ((B ·𝑜 𝐷) ·𝑜 (𝐹 ·𝑜 𝑆)))
3432, 33oveq12d 5473 . . . . . 6 ((((A 𝜔 B N) (𝐶 𝜔 𝐷 N)) ((𝐹 𝜔 𝐺 N) (𝑅 𝜔 𝑆 N))) → (((A ·𝑜 𝐺) ·𝑜 (𝐷 ·𝑜 𝑆)) +𝑜 ((B ·𝑜 𝐹) ·𝑜 (𝐷 ·𝑜 𝑆))) = (((A ·𝑜 𝐷) ·𝑜 (𝐺 ·𝑜 𝑆)) +𝑜 ((B ·𝑜 𝐷) ·𝑜 (𝐹 ·𝑜 𝑆))))
3527, 34eqtrd 2069 . . . . 5 ((((A 𝜔 B N) (𝐶 𝜔 𝐷 N)) ((𝐹 𝜔 𝐺 N) (𝑅 𝜔 𝑆 N))) → (((A ·𝑜 𝐺) +𝑜 (B ·𝑜 𝐹)) ·𝑜 (𝐷 ·𝑜 𝑆)) = (((A ·𝑜 𝐷) ·𝑜 (𝐺 ·𝑜 𝑆)) +𝑜 ((B ·𝑜 𝐷) ·𝑜 (𝐹 ·𝑜 𝑆))))
36 oveq1 5462 . . . . . 6 ((A ·𝑜 𝐷) = (B ·𝑜 𝐶) → ((A ·𝑜 𝐷) ·𝑜 (𝐺 ·𝑜 𝑆)) = ((B ·𝑜 𝐶) ·𝑜 (𝐺 ·𝑜 𝑆)))
37 oveq2 5463 . . . . . 6 ((𝐹 ·𝑜 𝑆) = (𝐺 ·𝑜 𝑅) → ((B ·𝑜 𝐷) ·𝑜 (𝐹 ·𝑜 𝑆)) = ((B ·𝑜 𝐷) ·𝑜 (𝐺 ·𝑜 𝑅)))
3836, 37oveqan12d 5474 . . . . 5 (((A ·𝑜 𝐷) = (B ·𝑜 𝐶) (𝐹 ·𝑜 𝑆) = (𝐺 ·𝑜 𝑅)) → (((A ·𝑜 𝐷) ·𝑜 (𝐺 ·𝑜 𝑆)) +𝑜 ((B ·𝑜 𝐷) ·𝑜 (𝐹 ·𝑜 𝑆))) = (((B ·𝑜 𝐶) ·𝑜 (𝐺 ·𝑜 𝑆)) +𝑜 ((B ·𝑜 𝐷) ·𝑜 (𝐺 ·𝑜 𝑅))))
3935, 38sylan9eq 2089 . . . 4 (((((A 𝜔 B N) (𝐶 𝜔 𝐷 N)) ((𝐹 𝜔 𝐺 N) (𝑅 𝜔 𝑆 N))) ((A ·𝑜 𝐷) = (B ·𝑜 𝐶) (𝐹 ·𝑜 𝑆) = (𝐺 ·𝑜 𝑅))) → (((A ·𝑜 𝐺) +𝑜 (B ·𝑜 𝐹)) ·𝑜 (𝐷 ·𝑜 𝑆)) = (((B ·𝑜 𝐶) ·𝑜 (𝐺 ·𝑜 𝑆)) +𝑜 ((B ·𝑜 𝐷) ·𝑜 (𝐺 ·𝑜 𝑅))))
40 nnmcl 5999 . . . . . . . 8 ((B 𝜔 𝐺 𝜔) → (B ·𝑜 𝐺) 𝜔)
4111, 6, 40syl2anc 391 . . . . . . 7 ((((A 𝜔 B N) (𝐶 𝜔 𝐷 N)) ((𝐹 𝜔 𝐺 N) (𝑅 𝜔 𝑆 N))) → (B ·𝑜 𝐺) 𝜔)
42 simplrl 487 . . . . . . . 8 ((((A 𝜔 B N) (𝐶 𝜔 𝐷 N)) ((𝐹 𝜔 𝐺 N) (𝑅 𝜔 𝑆 N))) → 𝐶 𝜔)
43 nnmcl 5999 . . . . . . . 8 ((𝐶 𝜔 𝑆 𝜔) → (𝐶 ·𝑜 𝑆) 𝜔)
4442, 20, 43syl2anc 391 . . . . . . 7 ((((A 𝜔 B N) (𝐶 𝜔 𝐷 N)) ((𝐹 𝜔 𝐺 N) (𝑅 𝜔 𝑆 N))) → (𝐶 ·𝑜 𝑆) 𝜔)
45 simprrl 491 . . . . . . . 8 ((((A 𝜔 B N) (𝐶 𝜔 𝐷 N)) ((𝐹 𝜔 𝐺 N) (𝑅 𝜔 𝑆 N))) → 𝑅 𝜔)
46 nnmcl 5999 . . . . . . . 8 ((𝐷 𝜔 𝑅 𝜔) → (𝐷 ·𝑜 𝑅) 𝜔)
4717, 45, 46syl2anc 391 . . . . . . 7 ((((A 𝜔 B N) (𝐶 𝜔 𝐷 N)) ((𝐹 𝜔 𝐺 N) (𝑅 𝜔 𝑆 N))) → (𝐷 ·𝑜 𝑅) 𝜔)
48 nndi 6004 . . . . . . 7 (((B ·𝑜 𝐺) 𝜔 (𝐶 ·𝑜 𝑆) 𝜔 (𝐷 ·𝑜 𝑅) 𝜔) → ((B ·𝑜 𝐺) ·𝑜 ((𝐶 ·𝑜 𝑆) +𝑜 (𝐷 ·𝑜 𝑅))) = (((B ·𝑜 𝐺) ·𝑜 (𝐶 ·𝑜 𝑆)) +𝑜 ((B ·𝑜 𝐺) ·𝑜 (𝐷 ·𝑜 𝑅))))
4941, 44, 47, 48syl3anc 1134 . . . . . 6 ((((A 𝜔 B N) (𝐶 𝜔 𝐷 N)) ((𝐹 𝜔 𝐺 N) (𝑅 𝜔 𝑆 N))) → ((B ·𝑜 𝐺) ·𝑜 ((𝐶 ·𝑜 𝑆) +𝑜 (𝐷 ·𝑜 𝑅))) = (((B ·𝑜 𝐺) ·𝑜 (𝐶 ·𝑜 𝑆)) +𝑜 ((B ·𝑜 𝐺) ·𝑜 (𝐷 ·𝑜 𝑅))))
5011, 6, 42, 26, 29, 20, 31caov4d 5627 . . . . . . 7 ((((A 𝜔 B N) (𝐶 𝜔 𝐷 N)) ((𝐹 𝜔 𝐺 N) (𝑅 𝜔 𝑆 N))) → ((B ·𝑜 𝐺) ·𝑜 (𝐶 ·𝑜 𝑆)) = ((B ·𝑜 𝐶) ·𝑜 (𝐺 ·𝑜 𝑆)))
5111, 6, 17, 26, 29, 45, 31caov4d 5627 . . . . . . 7 ((((A 𝜔 B N) (𝐶 𝜔 𝐷 N)) ((𝐹 𝜔 𝐺 N) (𝑅 𝜔 𝑆 N))) → ((B ·𝑜 𝐺) ·𝑜 (𝐷 ·𝑜 𝑅)) = ((B ·𝑜 𝐷) ·𝑜 (𝐺 ·𝑜 𝑅)))
5250, 51oveq12d 5473 . . . . . 6 ((((A 𝜔 B N) (𝐶 𝜔 𝐷 N)) ((𝐹 𝜔 𝐺 N) (𝑅 𝜔 𝑆 N))) → (((B ·𝑜 𝐺) ·𝑜 (𝐶 ·𝑜 𝑆)) +𝑜 ((B ·𝑜 𝐺) ·𝑜 (𝐷 ·𝑜 𝑅))) = (((B ·𝑜 𝐶) ·𝑜 (𝐺 ·𝑜 𝑆)) +𝑜 ((B ·𝑜 𝐷) ·𝑜 (𝐺 ·𝑜 𝑅))))
5349, 52eqtrd 2069 . . . . 5 ((((A 𝜔 B N) (𝐶 𝜔 𝐷 N)) ((𝐹 𝜔 𝐺 N) (𝑅 𝜔 𝑆 N))) → ((B ·𝑜 𝐺) ·𝑜 ((𝐶 ·𝑜 𝑆) +𝑜 (𝐷 ·𝑜 𝑅))) = (((B ·𝑜 𝐶) ·𝑜 (𝐺 ·𝑜 𝑆)) +𝑜 ((B ·𝑜 𝐷) ·𝑜 (𝐺 ·𝑜 𝑅))))
5453adantr 261 . . . 4 (((((A 𝜔 B N) (𝐶 𝜔 𝐷 N)) ((𝐹 𝜔 𝐺 N) (𝑅 𝜔 𝑆 N))) ((A ·𝑜 𝐷) = (B ·𝑜 𝐶) (𝐹 ·𝑜 𝑆) = (𝐺 ·𝑜 𝑅))) → ((B ·𝑜 𝐺) ·𝑜 ((𝐶 ·𝑜 𝑆) +𝑜 (𝐷 ·𝑜 𝑅))) = (((B ·𝑜 𝐶) ·𝑜 (𝐺 ·𝑜 𝑆)) +𝑜 ((B ·𝑜 𝐷) ·𝑜 (𝐺 ·𝑜 𝑅))))
5539, 54eqtr4d 2072 . . 3 (((((A 𝜔 B N) (𝐶 𝜔 𝐷 N)) ((𝐹 𝜔 𝐺 N) (𝑅 𝜔 𝑆 N))) ((A ·𝑜 𝐷) = (B ·𝑜 𝐶) (𝐹 ·𝑜 𝑆) = (𝐺 ·𝑜 𝑅))) → (((A ·𝑜 𝐺) +𝑜 (B ·𝑜 𝐹)) ·𝑜 (𝐷 ·𝑜 𝑆)) = ((B ·𝑜 𝐺) ·𝑜 ((𝐶 ·𝑜 𝑆) +𝑜 (𝐷 ·𝑜 𝑅))))
56 nnacl 5998 . . . . . 6 (((A ·𝑜 𝐺) 𝜔 (B ·𝑜 𝐹) 𝜔) → ((A ·𝑜 𝐺) +𝑜 (B ·𝑜 𝐹)) 𝜔)
578, 14, 56syl2anc 391 . . . . 5 ((((A 𝜔 B N) (𝐶 𝜔 𝐷 N)) ((𝐹 𝜔 𝐺 N) (𝑅 𝜔 𝑆 N))) → ((A ·𝑜 𝐺) +𝑜 (B ·𝑜 𝐹)) 𝜔)
58 mulpiord 6301 . . . . . . . 8 ((B N 𝐺 N) → (B ·N 𝐺) = (B ·𝑜 𝐺))
59 mulclpi 6312 . . . . . . . 8 ((B N 𝐺 N) → (B ·N 𝐺) N)
6058, 59eqeltrrd 2112 . . . . . . 7 ((B N 𝐺 N) → (B ·𝑜 𝐺) N)
6160ad2ant2l 477 . . . . . 6 (((A 𝜔 B N) (𝐹 𝜔 𝐺 N)) → (B ·𝑜 𝐺) N)
6261ad2ant2r 478 . . . . 5 ((((A 𝜔 B N) (𝐶 𝜔 𝐷 N)) ((𝐹 𝜔 𝐺 N) (𝑅 𝜔 𝑆 N))) → (B ·𝑜 𝐺) N)
63 nnacl 5998 . . . . . 6 (((𝐶 ·𝑜 𝑆) 𝜔 (𝐷 ·𝑜 𝑅) 𝜔) → ((𝐶 ·𝑜 𝑆) +𝑜 (𝐷 ·𝑜 𝑅)) 𝜔)
6444, 47, 63syl2anc 391 . . . . 5 ((((A 𝜔 B N) (𝐶 𝜔 𝐷 N)) ((𝐹 𝜔 𝐺 N) (𝑅 𝜔 𝑆 N))) → ((𝐶 ·𝑜 𝑆) +𝑜 (𝐷 ·𝑜 𝑅)) 𝜔)
65 mulpiord 6301 . . . . . . . 8 ((𝐷 N 𝑆 N) → (𝐷 ·N 𝑆) = (𝐷 ·𝑜 𝑆))
66 mulclpi 6312 . . . . . . . 8 ((𝐷 N 𝑆 N) → (𝐷 ·N 𝑆) N)
6765, 66eqeltrrd 2112 . . . . . . 7 ((𝐷 N 𝑆 N) → (𝐷 ·𝑜 𝑆) N)
6867ad2ant2l 477 . . . . . 6 (((𝐶 𝜔 𝐷 N) (𝑅 𝜔 𝑆 N)) → (𝐷 ·𝑜 𝑆) N)
6968ad2ant2l 477 . . . . 5 ((((A 𝜔 B N) (𝐶 𝜔 𝐷 N)) ((𝐹 𝜔 𝐺 N) (𝑅 𝜔 𝑆 N))) → (𝐷 ·𝑜 𝑆) N)
70 enq0breq 6419 . . . . 5 (((((A ·𝑜 𝐺) +𝑜 (B ·𝑜 𝐹)) 𝜔 (B ·𝑜 𝐺) N) (((𝐶 ·𝑜 𝑆) +𝑜 (𝐷 ·𝑜 𝑅)) 𝜔 (𝐷 ·𝑜 𝑆) N)) → (⟨((A ·𝑜 𝐺) +𝑜 (B ·𝑜 𝐹)), (B ·𝑜 𝐺)⟩ ~Q0 ⟨((𝐶 ·𝑜 𝑆) +𝑜 (𝐷 ·𝑜 𝑅)), (𝐷 ·𝑜 𝑆)⟩ ↔ (((A ·𝑜 𝐺) +𝑜 (B ·𝑜 𝐹)) ·𝑜 (𝐷 ·𝑜 𝑆)) = ((B ·𝑜 𝐺) ·𝑜 ((𝐶 ·𝑜 𝑆) +𝑜 (𝐷 ·𝑜 𝑅)))))
7157, 62, 64, 69, 70syl22anc 1135 . . . 4 ((((A 𝜔 B N) (𝐶 𝜔 𝐷 N)) ((𝐹 𝜔 𝐺 N) (𝑅 𝜔 𝑆 N))) → (⟨((A ·𝑜 𝐺) +𝑜 (B ·𝑜 𝐹)), (B ·𝑜 𝐺)⟩ ~Q0 ⟨((𝐶 ·𝑜 𝑆) +𝑜 (𝐷 ·𝑜 𝑅)), (𝐷 ·𝑜 𝑆)⟩ ↔ (((A ·𝑜 𝐺) +𝑜 (B ·𝑜 𝐹)) ·𝑜 (𝐷 ·𝑜 𝑆)) = ((B ·𝑜 𝐺) ·𝑜 ((𝐶 ·𝑜 𝑆) +𝑜 (𝐷 ·𝑜 𝑅)))))
7271adantr 261 . . 3 (((((A 𝜔 B N) (𝐶 𝜔 𝐷 N)) ((𝐹 𝜔 𝐺 N) (𝑅 𝜔 𝑆 N))) ((A ·𝑜 𝐷) = (B ·𝑜 𝐶) (𝐹 ·𝑜 𝑆) = (𝐺 ·𝑜 𝑅))) → (⟨((A ·𝑜 𝐺) +𝑜 (B ·𝑜 𝐹)), (B ·𝑜 𝐺)⟩ ~Q0 ⟨((𝐶 ·𝑜 𝑆) +𝑜 (𝐷 ·𝑜 𝑅)), (𝐷 ·𝑜 𝑆)⟩ ↔ (((A ·𝑜 𝐺) +𝑜 (B ·𝑜 𝐹)) ·𝑜 (𝐷 ·𝑜 𝑆)) = ((B ·𝑜 𝐺) ·𝑜 ((𝐶 ·𝑜 𝑆) +𝑜 (𝐷 ·𝑜 𝑅)))))
7355, 72mpbird 156 . 2 (((((A 𝜔 B N) (𝐶 𝜔 𝐷 N)) ((𝐹 𝜔 𝐺 N) (𝑅 𝜔 𝑆 N))) ((A ·𝑜 𝐷) = (B ·𝑜 𝐶) (𝐹 ·𝑜 𝑆) = (𝐺 ·𝑜 𝑅))) → ⟨((A ·𝑜 𝐺) +𝑜 (B ·𝑜 𝐹)), (B ·𝑜 𝐺)⟩ ~Q0 ⟨((𝐶 ·𝑜 𝑆) +𝑜 (𝐷 ·𝑜 𝑅)), (𝐷 ·𝑜 𝑆)⟩)
7473ex 108 1 ((((A 𝜔 B N) (𝐶 𝜔 𝐷 N)) ((𝐹 𝜔 𝐺 N) (𝑅 𝜔 𝑆 N))) → (((A ·𝑜 𝐷) = (B ·𝑜 𝐶) (𝐹 ·𝑜 𝑆) = (𝐺 ·𝑜 𝑅)) → ⟨((A ·𝑜 𝐺) +𝑜 (B ·𝑜 𝐹)), (B ·𝑜 𝐺)⟩ ~Q0 ⟨((𝐶 ·𝑜 𝑆) +𝑜 (𝐷 ·𝑜 𝑅)), (𝐷 ·𝑜 𝑆)⟩))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98   w3a 884   = wceq 1242   wcel 1390  cop 3370   class class class wbr 3755  𝜔com 4256  (class class class)co 5455   +𝑜 coa 5937   ·𝑜 comu 5938  Ncnpi 6256   ·N cmi 6258   ~Q0 ceq0 6270
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-coll 3863  ax-sep 3866  ax-nul 3874  ax-pow 3918  ax-pr 3935  ax-un 4136  ax-setind 4220  ax-iinf 4254
This theorem depends on definitions:  df-bi 110  df-dc 742  df-3an 886  df-tru 1245  df-fal 1248  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ne 2203  df-ral 2305  df-rex 2306  df-reu 2307  df-rab 2309  df-v 2553  df-sbc 2759  df-csb 2847  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-nul 3219  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-int 3607  df-iun 3650  df-br 3756  df-opab 3810  df-mpt 3811  df-tr 3846  df-id 4021  df-iord 4069  df-on 4071  df-suc 4074  df-iom 4257  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301  df-iota 4810  df-fun 4847  df-fn 4848  df-f 4849  df-f1 4850  df-fo 4851  df-f1o 4852  df-fv 4853  df-ov 5458  df-oprab 5459  df-mpt2 5460  df-1st 5709  df-2nd 5710  df-recs 5861  df-irdg 5897  df-oadd 5944  df-omul 5945  df-ni 6288  df-mi 6290  df-enq0 6407
This theorem is referenced by:  addnq0mo  6430
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