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Theorem addcmpblnq0 6747
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)⟩))

Proof of Theorem addcmpblnq0
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nndi 6150 . . . . . . . 8 ((𝑥 ∈ ω ∧ 𝑦 ∈ ω ∧ 𝑧 ∈ ω) → (𝑥 ·𝑜 (𝑦 +𝑜 𝑧)) = ((𝑥 ·𝑜 𝑦) +𝑜 (𝑥 ·𝑜 𝑧)))
21adantl 271 . . . . . . 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 500 . . . . . . . 8 ((((𝐴 ∈ ω ∧ 𝐵N) ∧ (𝐶 ∈ ω ∧ 𝐷N)) ∧ ((𝐹 ∈ ω ∧ 𝐺N) ∧ (𝑅 ∈ ω ∧ 𝑆N))) → 𝐴 ∈ ω)
4 simprlr 505 . . . . . . . . 9 ((((𝐴 ∈ ω ∧ 𝐵N) ∧ (𝐶 ∈ ω ∧ 𝐷N)) ∧ ((𝐹 ∈ ω ∧ 𝐺N) ∧ (𝑅 ∈ ω ∧ 𝑆N))) → 𝐺N)
5 pinn 6613 . . . . . . . . 9 (𝐺N𝐺 ∈ ω)
64, 5syl 14 . . . . . . . 8 ((((𝐴 ∈ ω ∧ 𝐵N) ∧ (𝐶 ∈ ω ∧ 𝐷N)) ∧ ((𝐹 ∈ ω ∧ 𝐺N) ∧ (𝑅 ∈ ω ∧ 𝑆N))) → 𝐺 ∈ ω)
7 nnmcl 6145 . . . . . . . 8 ((𝐴 ∈ ω ∧ 𝐺 ∈ ω) → (𝐴 ·𝑜 𝐺) ∈ ω)
83, 6, 7syl2anc 403 . . . . . . 7 ((((𝐴 ∈ ω ∧ 𝐵N) ∧ (𝐶 ∈ ω ∧ 𝐷N)) ∧ ((𝐹 ∈ ω ∧ 𝐺N) ∧ (𝑅 ∈ ω ∧ 𝑆N))) → (𝐴 ·𝑜 𝐺) ∈ ω)
9 simpllr 501 . . . . . . . . 9 ((((𝐴 ∈ ω ∧ 𝐵N) ∧ (𝐶 ∈ ω ∧ 𝐷N)) ∧ ((𝐹 ∈ ω ∧ 𝐺N) ∧ (𝑅 ∈ ω ∧ 𝑆N))) → 𝐵N)
10 pinn 6613 . . . . . . . . 9 (𝐵N𝐵 ∈ ω)
119, 10syl 14 . . . . . . . 8 ((((𝐴 ∈ ω ∧ 𝐵N) ∧ (𝐶 ∈ ω ∧ 𝐷N)) ∧ ((𝐹 ∈ ω ∧ 𝐺N) ∧ (𝑅 ∈ ω ∧ 𝑆N))) → 𝐵 ∈ ω)
12 simprll 504 . . . . . . . 8 ((((𝐴 ∈ ω ∧ 𝐵N) ∧ (𝐶 ∈ ω ∧ 𝐷N)) ∧ ((𝐹 ∈ ω ∧ 𝐺N) ∧ (𝑅 ∈ ω ∧ 𝑆N))) → 𝐹 ∈ ω)
13 nnmcl 6145 . . . . . . . 8 ((𝐵 ∈ ω ∧ 𝐹 ∈ ω) → (𝐵 ·𝑜 𝐹) ∈ ω)
1411, 12, 13syl2anc 403 . . . . . . 7 ((((𝐴 ∈ ω ∧ 𝐵N) ∧ (𝐶 ∈ ω ∧ 𝐷N)) ∧ ((𝐹 ∈ ω ∧ 𝐺N) ∧ (𝑅 ∈ ω ∧ 𝑆N))) → (𝐵 ·𝑜 𝐹) ∈ ω)
15 simplrr 503 . . . . . . . . 9 ((((𝐴 ∈ ω ∧ 𝐵N) ∧ (𝐶 ∈ ω ∧ 𝐷N)) ∧ ((𝐹 ∈ ω ∧ 𝐺N) ∧ (𝑅 ∈ ω ∧ 𝑆N))) → 𝐷N)
16 pinn 6613 . . . . . . . . 9 (𝐷N𝐷 ∈ ω)
1715, 16syl 14 . . . . . . . 8 ((((𝐴 ∈ ω ∧ 𝐵N) ∧ (𝐶 ∈ ω ∧ 𝐷N)) ∧ ((𝐹 ∈ ω ∧ 𝐺N) ∧ (𝑅 ∈ ω ∧ 𝑆N))) → 𝐷 ∈ ω)
18 simprrr 507 . . . . . . . . 9 ((((𝐴 ∈ ω ∧ 𝐵N) ∧ (𝐶 ∈ ω ∧ 𝐷N)) ∧ ((𝐹 ∈ ω ∧ 𝐺N) ∧ (𝑅 ∈ ω ∧ 𝑆N))) → 𝑆N)
19 pinn 6613 . . . . . . . . 9 (𝑆N𝑆 ∈ ω)
2018, 19syl 14 . . . . . . . 8 ((((𝐴 ∈ ω ∧ 𝐵N) ∧ (𝐶 ∈ ω ∧ 𝐷N)) ∧ ((𝐹 ∈ ω ∧ 𝐺N) ∧ (𝑅 ∈ ω ∧ 𝑆N))) → 𝑆 ∈ ω)
21 nnmcl 6145 . . . . . . . 8 ((𝐷 ∈ ω ∧ 𝑆 ∈ ω) → (𝐷 ·𝑜 𝑆) ∈ ω)
2217, 20, 21syl2anc 403 . . . . . . 7 ((((𝐴 ∈ ω ∧ 𝐵N) ∧ (𝐶 ∈ ω ∧ 𝐷N)) ∧ ((𝐹 ∈ ω ∧ 𝐺N) ∧ (𝑅 ∈ ω ∧ 𝑆N))) → (𝐷 ·𝑜 𝑆) ∈ ω)
23 nnacl 6144 . . . . . . . 8 ((𝑥 ∈ ω ∧ 𝑦 ∈ ω) → (𝑥 +𝑜 𝑦) ∈ ω)
2423adantl 271 . . . . . . 7 (((((𝐴 ∈ ω ∧ 𝐵N) ∧ (𝐶 ∈ ω ∧ 𝐷N)) ∧ ((𝐹 ∈ ω ∧ 𝐺N) ∧ (𝑅 ∈ ω ∧ 𝑆N))) ∧ (𝑥 ∈ ω ∧ 𝑦 ∈ ω)) → (𝑥 +𝑜 𝑦) ∈ ω)
25 nnmcom 6153 . . . . . . . 8 ((𝑥 ∈ ω ∧ 𝑦 ∈ ω) → (𝑥 ·𝑜 𝑦) = (𝑦 ·𝑜 𝑥))
2625adantl 271 . . . . . . 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 5728 . . . . . 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 6151 . . . . . . . . 9 ((𝑥 ∈ ω ∧ 𝑦 ∈ ω ∧ 𝑧 ∈ ω) → ((𝑥 ·𝑜 𝑦) ·𝑜 𝑧) = (𝑥 ·𝑜 (𝑦 ·𝑜 𝑧)))
2928adantl 271 . . . . . . . 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 6145 . . . . . . . . 9 ((𝑥 ∈ ω ∧ 𝑦 ∈ ω) → (𝑥 ·𝑜 𝑦) ∈ ω)
3130adantl 271 . . . . . . . 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 5736 . . . . . . 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 5736 . . . . . . 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 5581 . . . . . 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 2115 . . . . 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 5570 . . . . . 6 ((𝐴 ·𝑜 𝐷) = (𝐵 ·𝑜 𝐶) → ((𝐴 ·𝑜 𝐷) ·𝑜 (𝐺 ·𝑜 𝑆)) = ((𝐵 ·𝑜 𝐶) ·𝑜 (𝐺 ·𝑜 𝑆)))
37 oveq2 5571 . . . . . 6 ((𝐹 ·𝑜 𝑆) = (𝐺 ·𝑜 𝑅) → ((𝐵 ·𝑜 𝐷) ·𝑜 (𝐹 ·𝑜 𝑆)) = ((𝐵 ·𝑜 𝐷) ·𝑜 (𝐺 ·𝑜 𝑅)))
3836, 37oveqan12d 5582 . . . . 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 2135 . . . 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 6145 . . . . . . . 8 ((𝐵 ∈ ω ∧ 𝐺 ∈ ω) → (𝐵 ·𝑜 𝐺) ∈ ω)
4111, 6, 40syl2anc 403 . . . . . . 7 ((((𝐴 ∈ ω ∧ 𝐵N) ∧ (𝐶 ∈ ω ∧ 𝐷N)) ∧ ((𝐹 ∈ ω ∧ 𝐺N) ∧ (𝑅 ∈ ω ∧ 𝑆N))) → (𝐵 ·𝑜 𝐺) ∈ ω)
42 simplrl 502 . . . . . . . 8 ((((𝐴 ∈ ω ∧ 𝐵N) ∧ (𝐶 ∈ ω ∧ 𝐷N)) ∧ ((𝐹 ∈ ω ∧ 𝐺N) ∧ (𝑅 ∈ ω ∧ 𝑆N))) → 𝐶 ∈ ω)
43 nnmcl 6145 . . . . . . . 8 ((𝐶 ∈ ω ∧ 𝑆 ∈ ω) → (𝐶 ·𝑜 𝑆) ∈ ω)
4442, 20, 43syl2anc 403 . . . . . . 7 ((((𝐴 ∈ ω ∧ 𝐵N) ∧ (𝐶 ∈ ω ∧ 𝐷N)) ∧ ((𝐹 ∈ ω ∧ 𝐺N) ∧ (𝑅 ∈ ω ∧ 𝑆N))) → (𝐶 ·𝑜 𝑆) ∈ ω)
45 simprrl 506 . . . . . . . 8 ((((𝐴 ∈ ω ∧ 𝐵N) ∧ (𝐶 ∈ ω ∧ 𝐷N)) ∧ ((𝐹 ∈ ω ∧ 𝐺N) ∧ (𝑅 ∈ ω ∧ 𝑆N))) → 𝑅 ∈ ω)
46 nnmcl 6145 . . . . . . . 8 ((𝐷 ∈ ω ∧ 𝑅 ∈ ω) → (𝐷 ·𝑜 𝑅) ∈ ω)
4717, 45, 46syl2anc 403 . . . . . . 7 ((((𝐴 ∈ ω ∧ 𝐵N) ∧ (𝐶 ∈ ω ∧ 𝐷N)) ∧ ((𝐹 ∈ ω ∧ 𝐺N) ∧ (𝑅 ∈ ω ∧ 𝑆N))) → (𝐷 ·𝑜 𝑅) ∈ ω)
48 nndi 6150 . . . . . . 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 1170 . . . . . 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 5736 . . . . . . 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 5736 . . . . . . 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 5581 . . . . . 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 2115 . . . . 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 270 . . . 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 2118 . . 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 6144 . . . . . 6 (((𝐴 ·𝑜 𝐺) ∈ ω ∧ (𝐵 ·𝑜 𝐹) ∈ ω) → ((𝐴 ·𝑜 𝐺) +𝑜 (𝐵 ·𝑜 𝐹)) ∈ ω)
578, 14, 56syl2anc 403 . . . . 5 ((((𝐴 ∈ ω ∧ 𝐵N) ∧ (𝐶 ∈ ω ∧ 𝐷N)) ∧ ((𝐹 ∈ ω ∧ 𝐺N) ∧ (𝑅 ∈ ω ∧ 𝑆N))) → ((𝐴 ·𝑜 𝐺) +𝑜 (𝐵 ·𝑜 𝐹)) ∈ ω)
58 mulpiord 6621 . . . . . . . 8 ((𝐵N𝐺N) → (𝐵 ·N 𝐺) = (𝐵 ·𝑜 𝐺))
59 mulclpi 6632 . . . . . . . 8 ((𝐵N𝐺N) → (𝐵 ·N 𝐺) ∈ N)
6058, 59eqeltrrd 2160 . . . . . . 7 ((𝐵N𝐺N) → (𝐵 ·𝑜 𝐺) ∈ N)
6160ad2ant2l 492 . . . . . 6 (((𝐴 ∈ ω ∧ 𝐵N) ∧ (𝐹 ∈ ω ∧ 𝐺N)) → (𝐵 ·𝑜 𝐺) ∈ N)
6261ad2ant2r 493 . . . . 5 ((((𝐴 ∈ ω ∧ 𝐵N) ∧ (𝐶 ∈ ω ∧ 𝐷N)) ∧ ((𝐹 ∈ ω ∧ 𝐺N) ∧ (𝑅 ∈ ω ∧ 𝑆N))) → (𝐵 ·𝑜 𝐺) ∈ N)
63 nnacl 6144 . . . . . 6 (((𝐶 ·𝑜 𝑆) ∈ ω ∧ (𝐷 ·𝑜 𝑅) ∈ ω) → ((𝐶 ·𝑜 𝑆) +𝑜 (𝐷 ·𝑜 𝑅)) ∈ ω)
6444, 47, 63syl2anc 403 . . . . 5 ((((𝐴 ∈ ω ∧ 𝐵N) ∧ (𝐶 ∈ ω ∧ 𝐷N)) ∧ ((𝐹 ∈ ω ∧ 𝐺N) ∧ (𝑅 ∈ ω ∧ 𝑆N))) → ((𝐶 ·𝑜 𝑆) +𝑜 (𝐷 ·𝑜 𝑅)) ∈ ω)
65 mulpiord 6621 . . . . . . . 8 ((𝐷N𝑆N) → (𝐷 ·N 𝑆) = (𝐷 ·𝑜 𝑆))
66 mulclpi 6632 . . . . . . . 8 ((𝐷N𝑆N) → (𝐷 ·N 𝑆) ∈ N)
6765, 66eqeltrrd 2160 . . . . . . 7 ((𝐷N𝑆N) → (𝐷 ·𝑜 𝑆) ∈ N)
6867ad2ant2l 492 . . . . . 6 (((𝐶 ∈ ω ∧ 𝐷N) ∧ (𝑅 ∈ ω ∧ 𝑆N)) → (𝐷 ·𝑜 𝑆) ∈ N)
6968ad2ant2l 492 . . . . 5 ((((𝐴 ∈ ω ∧ 𝐵N) ∧ (𝐶 ∈ ω ∧ 𝐷N)) ∧ ((𝐹 ∈ ω ∧ 𝐺N) ∧ (𝑅 ∈ ω ∧ 𝑆N))) → (𝐷 ·𝑜 𝑆) ∈ N)
70 enq0breq 6740 . . . . 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 1171 . . . 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 270 . . 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 165 . 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 113 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 102  wb 103  w3a 920   = wceq 1285  wcel 1434  cop 3419   class class class wbr 3805  ωcom 4359  (class class class)co 5563   +𝑜 coa 6082   ·𝑜 comu 6083  Ncnpi 6576   ·N cmi 6578   ~Q0 ceq0 6590
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-coll 3913  ax-sep 3916  ax-nul 3924  ax-pow 3968  ax-pr 3992  ax-un 4216  ax-setind 4308  ax-iinf 4357
This theorem depends on definitions:  df-bi 115  df-dc 777  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ne 2250  df-ral 2358  df-rex 2359  df-reu 2360  df-rab 2362  df-v 2612  df-sbc 2825  df-csb 2918  df-dif 2984  df-un 2986  df-in 2988  df-ss 2995  df-nul 3268  df-pw 3402  df-sn 3422  df-pr 3423  df-op 3425  df-uni 3622  df-int 3657  df-iun 3700  df-br 3806  df-opab 3860  df-mpt 3861  df-tr 3896  df-id 4076  df-iord 4149  df-on 4151  df-suc 4154  df-iom 4360  df-xp 4397  df-rel 4398  df-cnv 4399  df-co 4400  df-dm 4401  df-rn 4402  df-res 4403  df-ima 4404  df-iota 4917  df-fun 4954  df-fn 4955  df-f 4956  df-f1 4957  df-fo 4958  df-f1o 4959  df-fv 4960  df-ov 5566  df-oprab 5567  df-mpt2 5568  df-1st 5818  df-2nd 5819  df-recs 5974  df-irdg 6039  df-oadd 6089  df-omul 6090  df-ni 6608  df-mi 6610  df-enq0 6728
This theorem is referenced by:  addnq0mo  6751
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