Intuitionistic Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >  addcmpblnq0 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 ∧ 𝐵N) ∧ (𝐶 ∈ ω ∧ 𝐷N)) ∧ ((𝐹 ∈ ω ∧ 𝐺N) ∧ (𝑅 ∈ ω ∧ 𝑆N))) → (((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 6065 . . . . . . . 8 ((𝑥 ∈ ω ∧ 𝑦 ∈ ω ∧ 𝑧 ∈ ω) → (𝑥 ·𝑜 (𝑦 +𝑜 𝑧)) = ((𝑥 ·𝑜 𝑦) +𝑜 (𝑥 ·𝑜 𝑧)))
21adantl 262 . . . . . . 7 (((((𝐴 ∈ ω ∧ 𝐵N) ∧ (𝐶 ∈ ω ∧ 𝐷N)) ∧ ((𝐹 ∈ ω ∧ 𝐺N) ∧ (𝑅 ∈ ω ∧ 𝑆N))) ∧ (𝑥 ∈ ω ∧ 𝑦 ∈ ω ∧ 𝑧 ∈ ω)) → (𝑥 ·𝑜 (𝑦 +𝑜 𝑧)) = ((𝑥 ·𝑜 𝑦) +𝑜 (𝑥 ·𝑜 𝑧)))
3 simplll 485 . . . . . . . 8 ((((𝐴 ∈ ω ∧ 𝐵N) ∧ (𝐶 ∈ ω ∧ 𝐷N)) ∧ ((𝐹 ∈ ω ∧ 𝐺N) ∧ (𝑅 ∈ ω ∧ 𝑆N))) → 𝐴 ∈ ω)
4 simprlr 490 . . . . . . . . 9 ((((𝐴 ∈ ω ∧ 𝐵N) ∧ (𝐶 ∈ ω ∧ 𝐷N)) ∧ ((𝐹 ∈ ω ∧ 𝐺N) ∧ (𝑅 ∈ ω ∧ 𝑆N))) → 𝐺N)
5 pinn 6407 . . . . . . . . 9 (𝐺N𝐺 ∈ ω)
64, 5syl 14 . . . . . . . 8 ((((𝐴 ∈ ω ∧ 𝐵N) ∧ (𝐶 ∈ ω ∧ 𝐷N)) ∧ ((𝐹 ∈ ω ∧ 𝐺N) ∧ (𝑅 ∈ ω ∧ 𝑆N))) → 𝐺 ∈ ω)
7 nnmcl 6060 . . . . . . . 8 ((𝐴 ∈ ω ∧ 𝐺 ∈ ω) → (𝐴 ·𝑜 𝐺) ∈ ω)
83, 6, 7syl2anc 391 . . . . . . 7 ((((𝐴 ∈ ω ∧ 𝐵N) ∧ (𝐶 ∈ ω ∧ 𝐷N)) ∧ ((𝐹 ∈ ω ∧ 𝐺N) ∧ (𝑅 ∈ ω ∧ 𝑆N))) → (𝐴 ·𝑜 𝐺) ∈ ω)
9 simpllr 486 . . . . . . . . 9 ((((𝐴 ∈ ω ∧ 𝐵N) ∧ (𝐶 ∈ ω ∧ 𝐷N)) ∧ ((𝐹 ∈ ω ∧ 𝐺N) ∧ (𝑅 ∈ ω ∧ 𝑆N))) → 𝐵N)
10 pinn 6407 . . . . . . . . 9 (𝐵N𝐵 ∈ ω)
119, 10syl 14 . . . . . . . 8 ((((𝐴 ∈ ω ∧ 𝐵N) ∧ (𝐶 ∈ ω ∧ 𝐷N)) ∧ ((𝐹 ∈ ω ∧ 𝐺N) ∧ (𝑅 ∈ ω ∧ 𝑆N))) → 𝐵 ∈ ω)
12 simprll 489 . . . . . . . 8 ((((𝐴 ∈ ω ∧ 𝐵N) ∧ (𝐶 ∈ ω ∧ 𝐷N)) ∧ ((𝐹 ∈ ω ∧ 𝐺N) ∧ (𝑅 ∈ ω ∧ 𝑆N))) → 𝐹 ∈ ω)
13 nnmcl 6060 . . . . . . . 8 ((𝐵 ∈ ω ∧ 𝐹 ∈ ω) → (𝐵 ·𝑜 𝐹) ∈ ω)
1411, 12, 13syl2anc 391 . . . . . . 7 ((((𝐴 ∈ ω ∧ 𝐵N) ∧ (𝐶 ∈ ω ∧ 𝐷N)) ∧ ((𝐹 ∈ ω ∧ 𝐺N) ∧ (𝑅 ∈ ω ∧ 𝑆N))) → (𝐵 ·𝑜 𝐹) ∈ ω)
15 simplrr 488 . . . . . . . . 9 ((((𝐴 ∈ ω ∧ 𝐵N) ∧ (𝐶 ∈ ω ∧ 𝐷N)) ∧ ((𝐹 ∈ ω ∧ 𝐺N) ∧ (𝑅 ∈ ω ∧ 𝑆N))) → 𝐷N)
16 pinn 6407 . . . . . . . . 9 (𝐷N𝐷 ∈ ω)
1715, 16syl 14 . . . . . . . 8 ((((𝐴 ∈ ω ∧ 𝐵N) ∧ (𝐶 ∈ ω ∧ 𝐷N)) ∧ ((𝐹 ∈ ω ∧ 𝐺N) ∧ (𝑅 ∈ ω ∧ 𝑆N))) → 𝐷 ∈ ω)
18 simprrr 492 . . . . . . . . 9 ((((𝐴 ∈ ω ∧ 𝐵N) ∧ (𝐶 ∈ ω ∧ 𝐷N)) ∧ ((𝐹 ∈ ω ∧ 𝐺N) ∧ (𝑅 ∈ ω ∧ 𝑆N))) → 𝑆N)
19 pinn 6407 . . . . . . . . 9 (𝑆N𝑆 ∈ ω)
2018, 19syl 14 . . . . . . . 8 ((((𝐴 ∈ ω ∧ 𝐵N) ∧ (𝐶 ∈ ω ∧ 𝐷N)) ∧ ((𝐹 ∈ ω ∧ 𝐺N) ∧ (𝑅 ∈ ω ∧ 𝑆N))) → 𝑆 ∈ ω)
21 nnmcl 6060 . . . . . . . 8 ((𝐷 ∈ ω ∧ 𝑆 ∈ ω) → (𝐷 ·𝑜 𝑆) ∈ ω)
2217, 20, 21syl2anc 391 . . . . . . 7 ((((𝐴 ∈ ω ∧ 𝐵N) ∧ (𝐶 ∈ ω ∧ 𝐷N)) ∧ ((𝐹 ∈ ω ∧ 𝐺N) ∧ (𝑅 ∈ ω ∧ 𝑆N))) → (𝐷 ·𝑜 𝑆) ∈ ω)
23 nnacl 6059 . . . . . . . 8 ((𝑥 ∈ ω ∧ 𝑦 ∈ ω) → (𝑥 +𝑜 𝑦) ∈ ω)
2423adantl 262 . . . . . . 7 (((((𝐴 ∈ ω ∧ 𝐵N) ∧ (𝐶 ∈ ω ∧ 𝐷N)) ∧ ((𝐹 ∈ ω ∧ 𝐺N) ∧ (𝑅 ∈ ω ∧ 𝑆N))) ∧ (𝑥 ∈ ω ∧ 𝑦 ∈ ω)) → (𝑥 +𝑜 𝑦) ∈ ω)
25 nnmcom 6068 . . . . . . . 8 ((𝑥 ∈ ω ∧ 𝑦 ∈ ω) → (𝑥 ·𝑜 𝑦) = (𝑦 ·𝑜 𝑥))
2625adantl 262 . . . . . . 7 (((((𝐴 ∈ ω ∧ 𝐵N) ∧ (𝐶 ∈ ω ∧ 𝐷N)) ∧ ((𝐹 ∈ ω ∧ 𝐺N) ∧ (𝑅 ∈ ω ∧ 𝑆N))) ∧ (𝑥 ∈ ω ∧ 𝑦 ∈ ω)) → (𝑥 ·𝑜 𝑦) = (𝑦 ·𝑜 𝑥))
272, 8, 14, 22, 24, 26caovdir2d 5677 . . . . . 6 ((((𝐴 ∈ ω ∧ 𝐵N) ∧ (𝐶 ∈ ω ∧ 𝐷N)) ∧ ((𝐹 ∈ ω ∧ 𝐺N) ∧ (𝑅 ∈ ω ∧ 𝑆N))) → (((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 6066 . . . . . . . . 9 ((𝑥 ∈ ω ∧ 𝑦 ∈ ω ∧ 𝑧 ∈ ω) → ((𝑥 ·𝑜 𝑦) ·𝑜 𝑧) = (𝑥 ·𝑜 (𝑦 ·𝑜 𝑧)))
2928adantl 262 . . . . . . . 8 (((((𝐴 ∈ ω ∧ 𝐵N) ∧ (𝐶 ∈ ω ∧ 𝐷N)) ∧ ((𝐹 ∈ ω ∧ 𝐺N) ∧ (𝑅 ∈ ω ∧ 𝑆N))) ∧ (𝑥 ∈ ω ∧ 𝑦 ∈ ω ∧ 𝑧 ∈ ω)) → ((𝑥 ·𝑜 𝑦) ·𝑜 𝑧) = (𝑥 ·𝑜 (𝑦 ·𝑜 𝑧)))
30 nnmcl 6060 . . . . . . . . 9 ((𝑥 ∈ ω ∧ 𝑦 ∈ ω) → (𝑥 ·𝑜 𝑦) ∈ ω)
3130adantl 262 . . . . . . . 8 (((((𝐴 ∈ ω ∧ 𝐵N) ∧ (𝐶 ∈ ω ∧ 𝐷N)) ∧ ((𝐹 ∈ ω ∧ 𝐺N) ∧ (𝑅 ∈ ω ∧ 𝑆N))) ∧ (𝑥 ∈ ω ∧ 𝑦 ∈ ω)) → (𝑥 ·𝑜 𝑦) ∈ ω)
323, 6, 17, 26, 29, 20, 31caov4d 5685 . . . . . . 7 ((((𝐴 ∈ ω ∧ 𝐵N) ∧ (𝐶 ∈ ω ∧ 𝐷N)) ∧ ((𝐹 ∈ ω ∧ 𝐺N) ∧ (𝑅 ∈ ω ∧ 𝑆N))) → ((𝐴 ·𝑜 𝐺) ·𝑜 (𝐷 ·𝑜 𝑆)) = ((𝐴 ·𝑜 𝐷) ·𝑜 (𝐺 ·𝑜 𝑆)))
3311, 12, 17, 26, 29, 20, 31caov4d 5685 . . . . . . 7 ((((𝐴 ∈ ω ∧ 𝐵N) ∧ (𝐶 ∈ ω ∧ 𝐷N)) ∧ ((𝐹 ∈ ω ∧ 𝐺N) ∧ (𝑅 ∈ ω ∧ 𝑆N))) → ((𝐵 ·𝑜 𝐹) ·𝑜 (𝐷 ·𝑜 𝑆)) = ((𝐵 ·𝑜 𝐷) ·𝑜 (𝐹 ·𝑜 𝑆)))
3432, 33oveq12d 5530 . . . . . 6 ((((𝐴 ∈ ω ∧ 𝐵N) ∧ (𝐶 ∈ ω ∧ 𝐷N)) ∧ ((𝐹 ∈ ω ∧ 𝐺N) ∧ (𝑅 ∈ ω ∧ 𝑆N))) → (((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 2072 . . . . 5 ((((𝐴 ∈ ω ∧ 𝐵N) ∧ (𝐶 ∈ ω ∧ 𝐷N)) ∧ ((𝐹 ∈ ω ∧ 𝐺N) ∧ (𝑅 ∈ ω ∧ 𝑆N))) → (((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 5519 . . . . . 6 ((𝐴 ·𝑜 𝐷) = (𝐵 ·𝑜 𝐶) → ((𝐴 ·𝑜 𝐷) ·𝑜 (𝐺 ·𝑜 𝑆)) = ((𝐵 ·𝑜 𝐶) ·𝑜 (𝐺 ·𝑜 𝑆)))
37 oveq2 5520 . . . . . 6 ((𝐹 ·𝑜 𝑆) = (𝐺 ·𝑜 𝑅) → ((𝐵 ·𝑜 𝐷) ·𝑜 (𝐹 ·𝑜 𝑆)) = ((𝐵 ·𝑜 𝐷) ·𝑜 (𝐺 ·𝑜 𝑅)))
3836, 37oveqan12d 5531 . . . . 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 2092 . . . 4 (((((𝐴 ∈ ω ∧ 𝐵N) ∧ (𝐶 ∈ ω ∧ 𝐷N)) ∧ ((𝐹 ∈ ω ∧ 𝐺N) ∧ (𝑅 ∈ ω ∧ 𝑆N))) ∧ ((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 6060 . . . . . . . 8 ((𝐵 ∈ ω ∧ 𝐺 ∈ ω) → (𝐵 ·𝑜 𝐺) ∈ ω)
4111, 6, 40syl2anc 391 . . . . . . 7 ((((𝐴 ∈ ω ∧ 𝐵N) ∧ (𝐶 ∈ ω ∧ 𝐷N)) ∧ ((𝐹 ∈ ω ∧ 𝐺N) ∧ (𝑅 ∈ ω ∧ 𝑆N))) → (𝐵 ·𝑜 𝐺) ∈ ω)
42 simplrl 487 . . . . . . . 8 ((((𝐴 ∈ ω ∧ 𝐵N) ∧ (𝐶 ∈ ω ∧ 𝐷N)) ∧ ((𝐹 ∈ ω ∧ 𝐺N) ∧ (𝑅 ∈ ω ∧ 𝑆N))) → 𝐶 ∈ ω)
43 nnmcl 6060 . . . . . . . 8 ((𝐶 ∈ ω ∧ 𝑆 ∈ ω) → (𝐶 ·𝑜 𝑆) ∈ ω)
4442, 20, 43syl2anc 391 . . . . . . 7 ((((𝐴 ∈ ω ∧ 𝐵N) ∧ (𝐶 ∈ ω ∧ 𝐷N)) ∧ ((𝐹 ∈ ω ∧ 𝐺N) ∧ (𝑅 ∈ ω ∧ 𝑆N))) → (𝐶 ·𝑜 𝑆) ∈ ω)
45 simprrl 491 . . . . . . . 8 ((((𝐴 ∈ ω ∧ 𝐵N) ∧ (𝐶 ∈ ω ∧ 𝐷N)) ∧ ((𝐹 ∈ ω ∧ 𝐺N) ∧ (𝑅 ∈ ω ∧ 𝑆N))) → 𝑅 ∈ ω)
46 nnmcl 6060 . . . . . . . 8 ((𝐷 ∈ ω ∧ 𝑅 ∈ ω) → (𝐷 ·𝑜 𝑅) ∈ ω)
4717, 45, 46syl2anc 391 . . . . . . 7 ((((𝐴 ∈ ω ∧ 𝐵N) ∧ (𝐶 ∈ ω ∧ 𝐷N)) ∧ ((𝐹 ∈ ω ∧ 𝐺N) ∧ (𝑅 ∈ ω ∧ 𝑆N))) → (𝐷 ·𝑜 𝑅) ∈ ω)
48 nndi 6065 . . . . . . 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 1135 . . . . . 6 ((((𝐴 ∈ ω ∧ 𝐵N) ∧ (𝐶 ∈ ω ∧ 𝐷N)) ∧ ((𝐹 ∈ ω ∧ 𝐺N) ∧ (𝑅 ∈ ω ∧ 𝑆N))) → ((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 5685 . . . . . . 7 ((((𝐴 ∈ ω ∧ 𝐵N) ∧ (𝐶 ∈ ω ∧ 𝐷N)) ∧ ((𝐹 ∈ ω ∧ 𝐺N) ∧ (𝑅 ∈ ω ∧ 𝑆N))) → ((𝐵 ·𝑜 𝐺) ·𝑜 (𝐶 ·𝑜 𝑆)) = ((𝐵 ·𝑜 𝐶) ·𝑜 (𝐺 ·𝑜 𝑆)))
5111, 6, 17, 26, 29, 45, 31caov4d 5685 . . . . . . 7 ((((𝐴 ∈ ω ∧ 𝐵N) ∧ (𝐶 ∈ ω ∧ 𝐷N)) ∧ ((𝐹 ∈ ω ∧ 𝐺N) ∧ (𝑅 ∈ ω ∧ 𝑆N))) → ((𝐵 ·𝑜 𝐺) ·𝑜 (𝐷 ·𝑜 𝑅)) = ((𝐵 ·𝑜 𝐷) ·𝑜 (𝐺 ·𝑜 𝑅)))
5250, 51oveq12d 5530 . . . . . 6 ((((𝐴 ∈ ω ∧ 𝐵N) ∧ (𝐶 ∈ ω ∧ 𝐷N)) ∧ ((𝐹 ∈ ω ∧ 𝐺N) ∧ (𝑅 ∈ ω ∧ 𝑆N))) → (((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 2072 . . . . 5 ((((𝐴 ∈ ω ∧ 𝐵N) ∧ (𝐶 ∈ ω ∧ 𝐷N)) ∧ ((𝐹 ∈ ω ∧ 𝐺N) ∧ (𝑅 ∈ ω ∧ 𝑆N))) → ((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 ∧ 𝐵N) ∧ (𝐶 ∈ ω ∧ 𝐷N)) ∧ ((𝐹 ∈ ω ∧ 𝐺N) ∧ (𝑅 ∈ ω ∧ 𝑆N))) ∧ ((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 2075 . . 3 (((((𝐴 ∈ ω ∧ 𝐵N) ∧ (𝐶 ∈ ω ∧ 𝐷N)) ∧ ((𝐹 ∈ ω ∧ 𝐺N) ∧ (𝑅 ∈ ω ∧ 𝑆N))) ∧ ((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 6059 . . . . . 6 (((𝐴 ·𝑜 𝐺) ∈ ω ∧ (𝐵 ·𝑜 𝐹) ∈ ω) → ((𝐴 ·𝑜 𝐺) +𝑜 (𝐵 ·𝑜 𝐹)) ∈ ω)
578, 14, 56syl2anc 391 . . . . 5 ((((𝐴 ∈ ω ∧ 𝐵N) ∧ (𝐶 ∈ ω ∧ 𝐷N)) ∧ ((𝐹 ∈ ω ∧ 𝐺N) ∧ (𝑅 ∈ ω ∧ 𝑆N))) → ((𝐴 ·𝑜 𝐺) +𝑜 (𝐵 ·𝑜 𝐹)) ∈ ω)
58 mulpiord 6415 . . . . . . . 8 ((𝐵N𝐺N) → (𝐵 ·N 𝐺) = (𝐵 ·𝑜 𝐺))
59 mulclpi 6426 . . . . . . . 8 ((𝐵N𝐺N) → (𝐵 ·N 𝐺) ∈ N)
6058, 59eqeltrrd 2115 . . . . . . 7 ((𝐵N𝐺N) → (𝐵 ·𝑜 𝐺) ∈ N)
6160ad2ant2l 477 . . . . . 6 (((𝐴 ∈ ω ∧ 𝐵N) ∧ (𝐹 ∈ ω ∧ 𝐺N)) → (𝐵 ·𝑜 𝐺) ∈ N)
6261ad2ant2r 478 . . . . 5 ((((𝐴 ∈ ω ∧ 𝐵N) ∧ (𝐶 ∈ ω ∧ 𝐷N)) ∧ ((𝐹 ∈ ω ∧ 𝐺N) ∧ (𝑅 ∈ ω ∧ 𝑆N))) → (𝐵 ·𝑜 𝐺) ∈ N)
63 nnacl 6059 . . . . . 6 (((𝐶 ·𝑜 𝑆) ∈ ω ∧ (𝐷 ·𝑜 𝑅) ∈ ω) → ((𝐶 ·𝑜 𝑆) +𝑜 (𝐷 ·𝑜 𝑅)) ∈ ω)
6444, 47, 63syl2anc 391 . . . . 5 ((((𝐴 ∈ ω ∧ 𝐵N) ∧ (𝐶 ∈ ω ∧ 𝐷N)) ∧ ((𝐹 ∈ ω ∧ 𝐺N) ∧ (𝑅 ∈ ω ∧ 𝑆N))) → ((𝐶 ·𝑜 𝑆) +𝑜 (𝐷 ·𝑜 𝑅)) ∈ ω)
65 mulpiord 6415 . . . . . . . 8 ((𝐷N𝑆N) → (𝐷 ·N 𝑆) = (𝐷 ·𝑜 𝑆))
66 mulclpi 6426 . . . . . . . 8 ((𝐷N𝑆N) → (𝐷 ·N 𝑆) ∈ N)
6765, 66eqeltrrd 2115 . . . . . . 7 ((𝐷N𝑆N) → (𝐷 ·𝑜 𝑆) ∈ N)
6867ad2ant2l 477 . . . . . 6 (((𝐶 ∈ ω ∧ 𝐷N) ∧ (𝑅 ∈ ω ∧ 𝑆N)) → (𝐷 ·𝑜 𝑆) ∈ N)
6968ad2ant2l 477 . . . . 5 ((((𝐴 ∈ ω ∧ 𝐵N) ∧ (𝐶 ∈ ω ∧ 𝐷N)) ∧ ((𝐹 ∈ ω ∧ 𝐺N) ∧ (𝑅 ∈ ω ∧ 𝑆N))) → (𝐷 ·𝑜 𝑆) ∈ N)
70 enq0breq 6534 . . . . 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 1136 . . . 4 ((((𝐴 ∈ ω ∧ 𝐵N) ∧ (𝐶 ∈ ω ∧ 𝐷N)) ∧ ((𝐹 ∈ ω ∧ 𝐺N) ∧ (𝑅 ∈ ω ∧ 𝑆N))) → (⟨((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 ∧ 𝐵N) ∧ (𝐶 ∈ ω ∧ 𝐷N)) ∧ ((𝐹 ∈ ω ∧ 𝐺N) ∧ (𝑅 ∈ ω ∧ 𝑆N))) ∧ ((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 ∧ 𝐵N) ∧ (𝐶 ∈ ω ∧ 𝐷N)) ∧ ((𝐹 ∈ ω ∧ 𝐺N) ∧ (𝑅 ∈ ω ∧ 𝑆N))) ∧ ((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 ∧ 𝐵N) ∧ (𝐶 ∈ ω ∧ 𝐷N)) ∧ ((𝐹 ∈ ω ∧ 𝐺N) ∧ (𝑅 ∈ ω ∧ 𝑆N))) → (((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 885   = wceq 1243   ∈ wcel 1393  ⟨cop 3378   class class class wbr 3764  ωcom 4313  (class class class)co 5512   +𝑜 coa 5998   ·𝑜 comu 5999  Ncnpi 6370   ·N cmi 6372   ~Q0 ceq0 6384 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 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-coll 3872  ax-sep 3875  ax-nul 3883  ax-pow 3927  ax-pr 3944  ax-un 4170  ax-setind 4262  ax-iinf 4311 This theorem depends on definitions:  df-bi 110  df-dc 743  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-ral 2311  df-rex 2312  df-reu 2313  df-rab 2315  df-v 2559  df-sbc 2765  df-csb 2853  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-int 3616  df-iun 3659  df-br 3765  df-opab 3819  df-mpt 3820  df-tr 3855  df-id 4030  df-iord 4103  df-on 4105  df-suc 4108  df-iom 4314  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-f1 4907  df-fo 4908  df-f1o 4909  df-fv 4910  df-ov 5515  df-oprab 5516  df-mpt2 5517  df-1st 5767  df-2nd 5768  df-recs 5920  df-irdg 5957  df-oadd 6005  df-omul 6006  df-ni 6402  df-mi 6404  df-enq0 6522 This theorem is referenced by:  addnq0mo  6545
 Copyright terms: Public domain W3C validator