Intuitionistic Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >  addcmpblnq0 Structured version   GIF version

 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)⟩))

Dummy variables x y z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nndi 5980 . . . . . . . 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 473 . . . . . . . 8 ((((A 𝜔 B N) (𝐶 𝜔 𝐷 N)) ((𝐹 𝜔 𝐺 N) (𝑅 𝜔 𝑆 N))) → A 𝜔)
4 simprlr 478 . . . . . . . . 9 ((((A 𝜔 B N) (𝐶 𝜔 𝐷 N)) ((𝐹 𝜔 𝐺 N) (𝑅 𝜔 𝑆 N))) → 𝐺 N)
5 pinn 6169 . . . . . . . . 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 5975 . . . . . . . 8 ((A 𝜔 𝐺 𝜔) → (A ·𝑜 𝐺) 𝜔)
83, 6, 7syl2anc 393 . . . . . . 7 ((((A 𝜔 B N) (𝐶 𝜔 𝐷 N)) ((𝐹 𝜔 𝐺 N) (𝑅 𝜔 𝑆 N))) → (A ·𝑜 𝐺) 𝜔)
9 simpllr 474 . . . . . . . . 9 ((((A 𝜔 B N) (𝐶 𝜔 𝐷 N)) ((𝐹 𝜔 𝐺 N) (𝑅 𝜔 𝑆 N))) → B N)
10 pinn 6169 . . . . . . . . 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 477 . . . . . . . 8 ((((A 𝜔 B N) (𝐶 𝜔 𝐷 N)) ((𝐹 𝜔 𝐺 N) (𝑅 𝜔 𝑆 N))) → 𝐹 𝜔)
13 nnmcl 5975 . . . . . . . 8 ((B 𝜔 𝐹 𝜔) → (B ·𝑜 𝐹) 𝜔)
1411, 12, 13syl2anc 393 . . . . . . 7 ((((A 𝜔 B N) (𝐶 𝜔 𝐷 N)) ((𝐹 𝜔 𝐺 N) (𝑅 𝜔 𝑆 N))) → (B ·𝑜 𝐹) 𝜔)
15 simplrr 476 . . . . . . . . 9 ((((A 𝜔 B N) (𝐶 𝜔 𝐷 N)) ((𝐹 𝜔 𝐺 N) (𝑅 𝜔 𝑆 N))) → 𝐷 N)
16 pinn 6169 . . . . . . . . 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 480 . . . . . . . . 9 ((((A 𝜔 B N) (𝐶 𝜔 𝐷 N)) ((𝐹 𝜔 𝐺 N) (𝑅 𝜔 𝑆 N))) → 𝑆 N)
19 pinn 6169 . . . . . . . . 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 5975 . . . . . . . 8 ((𝐷 𝜔 𝑆 𝜔) → (𝐷 ·𝑜 𝑆) 𝜔)
2217, 20, 21syl2anc 393 . . . . . . 7 ((((A 𝜔 B N) (𝐶 𝜔 𝐷 N)) ((𝐹 𝜔 𝐺 N) (𝑅 𝜔 𝑆 N))) → (𝐷 ·𝑜 𝑆) 𝜔)
23 nnacl 5974 . . . . . . . 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 5983 . . . . . . . 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 5600 . . . . . 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 5981 . . . . . . . . 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 5975 . . . . . . . . 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 5608 . . . . . . 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 5608 . . . . . . 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 5454 . . . . . 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 2054 . . . . 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 5443 . . . . . 6 ((A ·𝑜 𝐷) = (B ·𝑜 𝐶) → ((A ·𝑜 𝐷) ·𝑜 (𝐺 ·𝑜 𝑆)) = ((B ·𝑜 𝐶) ·𝑜 (𝐺 ·𝑜 𝑆)))
37 oveq2 5444 . . . . . 6 ((𝐹 ·𝑜 𝑆) = (𝐺 ·𝑜 𝑅) → ((B ·𝑜 𝐷) ·𝑜 (𝐹 ·𝑜 𝑆)) = ((B ·𝑜 𝐷) ·𝑜 (𝐺 ·𝑜 𝑅)))
3836, 37oveqan12d 5455 . . . . 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 2074 . . . 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 5975 . . . . . . . 8 ((B 𝜔 𝐺 𝜔) → (B ·𝑜 𝐺) 𝜔)
4111, 6, 40syl2anc 393 . . . . . . 7 ((((A 𝜔 B N) (𝐶 𝜔 𝐷 N)) ((𝐹 𝜔 𝐺 N) (𝑅 𝜔 𝑆 N))) → (B ·𝑜 𝐺) 𝜔)
42 simplrl 475 . . . . . . . 8 ((((A 𝜔 B N) (𝐶 𝜔 𝐷 N)) ((𝐹 𝜔 𝐺 N) (𝑅 𝜔 𝑆 N))) → 𝐶 𝜔)
43 nnmcl 5975 . . . . . . . 8 ((𝐶 𝜔 𝑆 𝜔) → (𝐶 ·𝑜 𝑆) 𝜔)
4442, 20, 43syl2anc 393 . . . . . . 7 ((((A 𝜔 B N) (𝐶 𝜔 𝐷 N)) ((𝐹 𝜔 𝐺 N) (𝑅 𝜔 𝑆 N))) → (𝐶 ·𝑜 𝑆) 𝜔)
45 simprrl 479 . . . . . . . 8 ((((A 𝜔 B N) (𝐶 𝜔 𝐷 N)) ((𝐹 𝜔 𝐺 N) (𝑅 𝜔 𝑆 N))) → 𝑅 𝜔)
46 nnmcl 5975 . . . . . . . 8 ((𝐷 𝜔 𝑅 𝜔) → (𝐷 ·𝑜 𝑅) 𝜔)
4717, 45, 46syl2anc 393 . . . . . . 7 ((((A 𝜔 B N) (𝐶 𝜔 𝐷 N)) ((𝐹 𝜔 𝐺 N) (𝑅 𝜔 𝑆 N))) → (𝐷 ·𝑜 𝑅) 𝜔)
48 nndi 5980 . . . . . . 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 1121 . . . . . 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 5608 . . . . . . 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 5608 . . . . . . 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 5454 . . . . . 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 2054 . . . . 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 2057 . . 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 5974 . . . . . 6 (((A ·𝑜 𝐺) 𝜔 (B ·𝑜 𝐹) 𝜔) → ((A ·𝑜 𝐺) +𝑜 (B ·𝑜 𝐹)) 𝜔)
578, 14, 56syl2anc 393 . . . . 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 6177 . . . . . . . 8 ((B N 𝐺 N) → (B ·N 𝐺) = (B ·𝑜 𝐺))
59 mulclpi 6188 . . . . . . . 8 ((B N 𝐺 N) → (B ·N 𝐺) N)
6058, 59eqeltrrd 2097 . . . . . . 7 ((B N 𝐺 N) → (B ·𝑜 𝐺) N)
6160ad2ant2l 465 . . . . . 6 (((A 𝜔 B N) (𝐹 𝜔 𝐺 N)) → (B ·𝑜 𝐺) N)
6261ad2ant2r 466 . . . . 5 ((((A 𝜔 B N) (𝐶 𝜔 𝐷 N)) ((𝐹 𝜔 𝐺 N) (𝑅 𝜔 𝑆 N))) → (B ·𝑜 𝐺) N)
63 nnacl 5974 . . . . . 6 (((𝐶 ·𝑜 𝑆) 𝜔 (𝐷 ·𝑜 𝑅) 𝜔) → ((𝐶 ·𝑜 𝑆) +𝑜 (𝐷 ·𝑜 𝑅)) 𝜔)
6444, 47, 63syl2anc 393 . . . . 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 6177 . . . . . . . 8 ((𝐷 N 𝑆 N) → (𝐷 ·N 𝑆) = (𝐷 ·𝑜 𝑆))
66 mulclpi 6188 . . . . . . . 8 ((𝐷 N 𝑆 N) → (𝐷 ·N 𝑆) N)
6765, 66eqeltrrd 2097 . . . . . . 7 ((𝐷 N 𝑆 N) → (𝐷 ·𝑜 𝑆) N)
6867ad2ant2l 465 . . . . . 6 (((𝐶 𝜔 𝐷 N) (𝑅 𝜔 𝑆 N)) → (𝐷 ·𝑜 𝑆) N)
6968ad2ant2l 465 . . . . 5 ((((A 𝜔 B N) (𝐶 𝜔 𝐷 N)) ((𝐹 𝜔 𝐺 N) (𝑅 𝜔 𝑆 N))) → (𝐷 ·𝑜 𝑆) N)
70 enq0breq 6291 . . . . 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 1122 . . . 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 873   = wceq 1228   ∈ wcel 1374  ⟨cop 3353   class class class wbr 3738  𝜔com 4240  (class class class)co 5436   +𝑜 coa 5913   ·𝑜 comu 5914  Ncnpi 6130   ·N cmi 6132   ~Q0 ceq0 6144 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 532  ax-in2 533  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-13 1385  ax-14 1386  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-coll 3846  ax-sep 3849  ax-nul 3857  ax-pow 3901  ax-pr 3918  ax-un 4120  ax-setind 4204  ax-iinf 4238 This theorem depends on definitions:  df-bi 110  df-dc 734  df-3an 875  df-tru 1231  df-fal 1234  df-nf 1330  df-sb 1628  df-eu 1885  df-mo 1886  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ne 2188  df-ral 2289  df-rex 2290  df-reu 2291  df-rab 2293  df-v 2537  df-sbc 2742  df-csb 2830  df-dif 2897  df-un 2899  df-in 2901  df-ss 2908  df-nul 3202  df-pw 3336  df-sn 3356  df-pr 3357  df-op 3359  df-uni 3555  df-int 3590  df-iun 3633  df-br 3739  df-opab 3793  df-mpt 3794  df-tr 3829  df-id 4004  df-iord 4052  df-on 4054  df-suc 4057  df-iom 4241  df-xp 4278  df-rel 4279  df-cnv 4280  df-co 4281  df-dm 4282  df-rn 4283  df-res 4284  df-ima 4285  df-iota 4794  df-fun 4831  df-fn 4832  df-f 4833  df-f1 4834  df-fo 4835  df-f1o 4836  df-fv 4837  df-ov 5439  df-oprab 5440  df-mpt2 5441  df-1st 5690  df-2nd 5691  df-recs 5842  df-irdg 5878  df-oadd 5920  df-omul 5921  df-ni 6164  df-mi 6166  df-enq0 6279 This theorem is referenced by:  addnq0mo  6302
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