MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  odadd2 Structured version   Visualization version   GIF version

Theorem odadd2 18898
Description: The order of a product in an abelian group is divisible by the LCM of the orders of the factors divided by the GCD. (Contributed by Mario Carneiro, 20-Oct-2015.)
Hypotheses
Ref Expression
odadd1.1 𝑂 = (od‘𝐺)
odadd1.2 𝑋 = (Base‘𝐺)
odadd1.3 + = (+g𝐺)
Assertion
Ref Expression
odadd2 ((𝐺 ∈ Abel ∧ 𝐴𝑋𝐵𝑋) → ((𝑂𝐴) · (𝑂𝐵)) ∥ ((𝑂‘(𝐴 + 𝐵)) · (((𝑂𝐴) gcd (𝑂𝐵))↑2)))

Proof of Theorem odadd2
StepHypRef Expression
1 odadd1.2 . . . . . . . . 9 𝑋 = (Base‘𝐺)
2 odadd1.1 . . . . . . . . 9 𝑂 = (od‘𝐺)
31, 2odcl 18593 . . . . . . . 8 (𝐴𝑋 → (𝑂𝐴) ∈ ℕ0)
433ad2ant2 1126 . . . . . . 7 ((𝐺 ∈ Abel ∧ 𝐴𝑋𝐵𝑋) → (𝑂𝐴) ∈ ℕ0)
54nn0zd 12073 . . . . . 6 ((𝐺 ∈ Abel ∧ 𝐴𝑋𝐵𝑋) → (𝑂𝐴) ∈ ℤ)
61, 2odcl 18593 . . . . . . . 8 (𝐵𝑋 → (𝑂𝐵) ∈ ℕ0)
763ad2ant3 1127 . . . . . . 7 ((𝐺 ∈ Abel ∧ 𝐴𝑋𝐵𝑋) → (𝑂𝐵) ∈ ℕ0)
87nn0zd 12073 . . . . . 6 ((𝐺 ∈ Abel ∧ 𝐴𝑋𝐵𝑋) → (𝑂𝐵) ∈ ℤ)
95, 8zmulcld 12081 . . . . 5 ((𝐺 ∈ Abel ∧ 𝐴𝑋𝐵𝑋) → ((𝑂𝐴) · (𝑂𝐵)) ∈ ℤ)
109adantr 481 . . . 4 (((𝐺 ∈ Abel ∧ 𝐴𝑋𝐵𝑋) ∧ ((𝑂𝐴) gcd (𝑂𝐵)) = 0) → ((𝑂𝐴) · (𝑂𝐵)) ∈ ℤ)
11 dvds0 15613 . . . 4 (((𝑂𝐴) · (𝑂𝐵)) ∈ ℤ → ((𝑂𝐴) · (𝑂𝐵)) ∥ 0)
1210, 11syl 17 . . 3 (((𝐺 ∈ Abel ∧ 𝐴𝑋𝐵𝑋) ∧ ((𝑂𝐴) gcd (𝑂𝐵)) = 0) → ((𝑂𝐴) · (𝑂𝐵)) ∥ 0)
13 simpr 485 . . . . . 6 (((𝐺 ∈ Abel ∧ 𝐴𝑋𝐵𝑋) ∧ ((𝑂𝐴) gcd (𝑂𝐵)) = 0) → ((𝑂𝐴) gcd (𝑂𝐵)) = 0)
1413sq0id 13545 . . . . 5 (((𝐺 ∈ Abel ∧ 𝐴𝑋𝐵𝑋) ∧ ((𝑂𝐴) gcd (𝑂𝐵)) = 0) → (((𝑂𝐴) gcd (𝑂𝐵))↑2) = 0)
1514oveq2d 7161 . . . 4 (((𝐺 ∈ Abel ∧ 𝐴𝑋𝐵𝑋) ∧ ((𝑂𝐴) gcd (𝑂𝐵)) = 0) → ((𝑂‘(𝐴 + 𝐵)) · (((𝑂𝐴) gcd (𝑂𝐵))↑2)) = ((𝑂‘(𝐴 + 𝐵)) · 0))
16 ablgrp 18840 . . . . . . . . . 10 (𝐺 ∈ Abel → 𝐺 ∈ Grp)
17 odadd1.3 . . . . . . . . . . 11 + = (+g𝐺)
181, 17grpcl 18049 . . . . . . . . . 10 ((𝐺 ∈ Grp ∧ 𝐴𝑋𝐵𝑋) → (𝐴 + 𝐵) ∈ 𝑋)
1916, 18syl3an1 1155 . . . . . . . . 9 ((𝐺 ∈ Abel ∧ 𝐴𝑋𝐵𝑋) → (𝐴 + 𝐵) ∈ 𝑋)
201, 2odcl 18593 . . . . . . . . 9 ((𝐴 + 𝐵) ∈ 𝑋 → (𝑂‘(𝐴 + 𝐵)) ∈ ℕ0)
2119, 20syl 17 . . . . . . . 8 ((𝐺 ∈ Abel ∧ 𝐴𝑋𝐵𝑋) → (𝑂‘(𝐴 + 𝐵)) ∈ ℕ0)
2221nn0zd 12073 . . . . . . 7 ((𝐺 ∈ Abel ∧ 𝐴𝑋𝐵𝑋) → (𝑂‘(𝐴 + 𝐵)) ∈ ℤ)
2322adantr 481 . . . . . 6 (((𝐺 ∈ Abel ∧ 𝐴𝑋𝐵𝑋) ∧ ((𝑂𝐴) gcd (𝑂𝐵)) = 0) → (𝑂‘(𝐴 + 𝐵)) ∈ ℤ)
2423zcnd 12076 . . . . 5 (((𝐺 ∈ Abel ∧ 𝐴𝑋𝐵𝑋) ∧ ((𝑂𝐴) gcd (𝑂𝐵)) = 0) → (𝑂‘(𝐴 + 𝐵)) ∈ ℂ)
2524mul01d 10827 . . . 4 (((𝐺 ∈ Abel ∧ 𝐴𝑋𝐵𝑋) ∧ ((𝑂𝐴) gcd (𝑂𝐵)) = 0) → ((𝑂‘(𝐴 + 𝐵)) · 0) = 0)
2615, 25eqtrd 2853 . . 3 (((𝐺 ∈ Abel ∧ 𝐴𝑋𝐵𝑋) ∧ ((𝑂𝐴) gcd (𝑂𝐵)) = 0) → ((𝑂‘(𝐴 + 𝐵)) · (((𝑂𝐴) gcd (𝑂𝐵))↑2)) = 0)
2712, 26breqtrrd 5085 . 2 (((𝐺 ∈ Abel ∧ 𝐴𝑋𝐵𝑋) ∧ ((𝑂𝐴) gcd (𝑂𝐵)) = 0) → ((𝑂𝐴) · (𝑂𝐵)) ∥ ((𝑂‘(𝐴 + 𝐵)) · (((𝑂𝐴) gcd (𝑂𝐵))↑2)))
285adantr 481 . . . . . . . 8 (((𝐺 ∈ Abel ∧ 𝐴𝑋𝐵𝑋) ∧ ((𝑂𝐴) gcd (𝑂𝐵)) ≠ 0) → (𝑂𝐴) ∈ ℤ)
298adantr 481 . . . . . . . 8 (((𝐺 ∈ Abel ∧ 𝐴𝑋𝐵𝑋) ∧ ((𝑂𝐴) gcd (𝑂𝐵)) ≠ 0) → (𝑂𝐵) ∈ ℤ)
3028, 29gcdcld 15845 . . . . . . 7 (((𝐺 ∈ Abel ∧ 𝐴𝑋𝐵𝑋) ∧ ((𝑂𝐴) gcd (𝑂𝐵)) ≠ 0) → ((𝑂𝐴) gcd (𝑂𝐵)) ∈ ℕ0)
3130nn0cnd 11945 . . . . . 6 (((𝐺 ∈ Abel ∧ 𝐴𝑋𝐵𝑋) ∧ ((𝑂𝐴) gcd (𝑂𝐵)) ≠ 0) → ((𝑂𝐴) gcd (𝑂𝐵)) ∈ ℂ)
3231sqvald 13495 . . . . 5 (((𝐺 ∈ Abel ∧ 𝐴𝑋𝐵𝑋) ∧ ((𝑂𝐴) gcd (𝑂𝐵)) ≠ 0) → (((𝑂𝐴) gcd (𝑂𝐵))↑2) = (((𝑂𝐴) gcd (𝑂𝐵)) · ((𝑂𝐴) gcd (𝑂𝐵))))
3332oveq2d 7161 . . . 4 (((𝐺 ∈ Abel ∧ 𝐴𝑋𝐵𝑋) ∧ ((𝑂𝐴) gcd (𝑂𝐵)) ≠ 0) → ((((𝑂𝐴) / ((𝑂𝐴) gcd (𝑂𝐵))) · ((𝑂𝐵) / ((𝑂𝐴) gcd (𝑂𝐵)))) · (((𝑂𝐴) gcd (𝑂𝐵))↑2)) = ((((𝑂𝐴) / ((𝑂𝐴) gcd (𝑂𝐵))) · ((𝑂𝐵) / ((𝑂𝐴) gcd (𝑂𝐵)))) · (((𝑂𝐴) gcd (𝑂𝐵)) · ((𝑂𝐴) gcd (𝑂𝐵)))))
34 gcddvds 15840 . . . . . . . . 9 (((𝑂𝐴) ∈ ℤ ∧ (𝑂𝐵) ∈ ℤ) → (((𝑂𝐴) gcd (𝑂𝐵)) ∥ (𝑂𝐴) ∧ ((𝑂𝐴) gcd (𝑂𝐵)) ∥ (𝑂𝐵)))
3528, 29, 34syl2anc 584 . . . . . . . 8 (((𝐺 ∈ Abel ∧ 𝐴𝑋𝐵𝑋) ∧ ((𝑂𝐴) gcd (𝑂𝐵)) ≠ 0) → (((𝑂𝐴) gcd (𝑂𝐵)) ∥ (𝑂𝐴) ∧ ((𝑂𝐴) gcd (𝑂𝐵)) ∥ (𝑂𝐵)))
3635simpld 495 . . . . . . 7 (((𝐺 ∈ Abel ∧ 𝐴𝑋𝐵𝑋) ∧ ((𝑂𝐴) gcd (𝑂𝐵)) ≠ 0) → ((𝑂𝐴) gcd (𝑂𝐵)) ∥ (𝑂𝐴))
3730nn0zd 12073 . . . . . . . 8 (((𝐺 ∈ Abel ∧ 𝐴𝑋𝐵𝑋) ∧ ((𝑂𝐴) gcd (𝑂𝐵)) ≠ 0) → ((𝑂𝐴) gcd (𝑂𝐵)) ∈ ℤ)
38 simpr 485 . . . . . . . 8 (((𝐺 ∈ Abel ∧ 𝐴𝑋𝐵𝑋) ∧ ((𝑂𝐴) gcd (𝑂𝐵)) ≠ 0) → ((𝑂𝐴) gcd (𝑂𝐵)) ≠ 0)
39 dvdsval2 15598 . . . . . . . 8 ((((𝑂𝐴) gcd (𝑂𝐵)) ∈ ℤ ∧ ((𝑂𝐴) gcd (𝑂𝐵)) ≠ 0 ∧ (𝑂𝐴) ∈ ℤ) → (((𝑂𝐴) gcd (𝑂𝐵)) ∥ (𝑂𝐴) ↔ ((𝑂𝐴) / ((𝑂𝐴) gcd (𝑂𝐵))) ∈ ℤ))
4037, 38, 28, 39syl3anc 1363 . . . . . . 7 (((𝐺 ∈ Abel ∧ 𝐴𝑋𝐵𝑋) ∧ ((𝑂𝐴) gcd (𝑂𝐵)) ≠ 0) → (((𝑂𝐴) gcd (𝑂𝐵)) ∥ (𝑂𝐴) ↔ ((𝑂𝐴) / ((𝑂𝐴) gcd (𝑂𝐵))) ∈ ℤ))
4136, 40mpbid 233 . . . . . 6 (((𝐺 ∈ Abel ∧ 𝐴𝑋𝐵𝑋) ∧ ((𝑂𝐴) gcd (𝑂𝐵)) ≠ 0) → ((𝑂𝐴) / ((𝑂𝐴) gcd (𝑂𝐵))) ∈ ℤ)
4241zcnd 12076 . . . . 5 (((𝐺 ∈ Abel ∧ 𝐴𝑋𝐵𝑋) ∧ ((𝑂𝐴) gcd (𝑂𝐵)) ≠ 0) → ((𝑂𝐴) / ((𝑂𝐴) gcd (𝑂𝐵))) ∈ ℂ)
4335simprd 496 . . . . . . 7 (((𝐺 ∈ Abel ∧ 𝐴𝑋𝐵𝑋) ∧ ((𝑂𝐴) gcd (𝑂𝐵)) ≠ 0) → ((𝑂𝐴) gcd (𝑂𝐵)) ∥ (𝑂𝐵))
44 dvdsval2 15598 . . . . . . . 8 ((((𝑂𝐴) gcd (𝑂𝐵)) ∈ ℤ ∧ ((𝑂𝐴) gcd (𝑂𝐵)) ≠ 0 ∧ (𝑂𝐵) ∈ ℤ) → (((𝑂𝐴) gcd (𝑂𝐵)) ∥ (𝑂𝐵) ↔ ((𝑂𝐵) / ((𝑂𝐴) gcd (𝑂𝐵))) ∈ ℤ))
4537, 38, 29, 44syl3anc 1363 . . . . . . 7 (((𝐺 ∈ Abel ∧ 𝐴𝑋𝐵𝑋) ∧ ((𝑂𝐴) gcd (𝑂𝐵)) ≠ 0) → (((𝑂𝐴) gcd (𝑂𝐵)) ∥ (𝑂𝐵) ↔ ((𝑂𝐵) / ((𝑂𝐴) gcd (𝑂𝐵))) ∈ ℤ))
4643, 45mpbid 233 . . . . . 6 (((𝐺 ∈ Abel ∧ 𝐴𝑋𝐵𝑋) ∧ ((𝑂𝐴) gcd (𝑂𝐵)) ≠ 0) → ((𝑂𝐵) / ((𝑂𝐴) gcd (𝑂𝐵))) ∈ ℤ)
4746zcnd 12076 . . . . 5 (((𝐺 ∈ Abel ∧ 𝐴𝑋𝐵𝑋) ∧ ((𝑂𝐴) gcd (𝑂𝐵)) ≠ 0) → ((𝑂𝐵) / ((𝑂𝐴) gcd (𝑂𝐵))) ∈ ℂ)
4842, 31, 47, 31mul4d 10840 . . . 4 (((𝐺 ∈ Abel ∧ 𝐴𝑋𝐵𝑋) ∧ ((𝑂𝐴) gcd (𝑂𝐵)) ≠ 0) → ((((𝑂𝐴) / ((𝑂𝐴) gcd (𝑂𝐵))) · ((𝑂𝐴) gcd (𝑂𝐵))) · (((𝑂𝐵) / ((𝑂𝐴) gcd (𝑂𝐵))) · ((𝑂𝐴) gcd (𝑂𝐵)))) = ((((𝑂𝐴) / ((𝑂𝐴) gcd (𝑂𝐵))) · ((𝑂𝐵) / ((𝑂𝐴) gcd (𝑂𝐵)))) · (((𝑂𝐴) gcd (𝑂𝐵)) · ((𝑂𝐴) gcd (𝑂𝐵)))))
4928zcnd 12076 . . . . . 6 (((𝐺 ∈ Abel ∧ 𝐴𝑋𝐵𝑋) ∧ ((𝑂𝐴) gcd (𝑂𝐵)) ≠ 0) → (𝑂𝐴) ∈ ℂ)
5049, 31, 38divcan1d 11405 . . . . 5 (((𝐺 ∈ Abel ∧ 𝐴𝑋𝐵𝑋) ∧ ((𝑂𝐴) gcd (𝑂𝐵)) ≠ 0) → (((𝑂𝐴) / ((𝑂𝐴) gcd (𝑂𝐵))) · ((𝑂𝐴) gcd (𝑂𝐵))) = (𝑂𝐴))
5129zcnd 12076 . . . . . 6 (((𝐺 ∈ Abel ∧ 𝐴𝑋𝐵𝑋) ∧ ((𝑂𝐴) gcd (𝑂𝐵)) ≠ 0) → (𝑂𝐵) ∈ ℂ)
5251, 31, 38divcan1d 11405 . . . . 5 (((𝐺 ∈ Abel ∧ 𝐴𝑋𝐵𝑋) ∧ ((𝑂𝐴) gcd (𝑂𝐵)) ≠ 0) → (((𝑂𝐵) / ((𝑂𝐴) gcd (𝑂𝐵))) · ((𝑂𝐴) gcd (𝑂𝐵))) = (𝑂𝐵))
5350, 52oveq12d 7163 . . . 4 (((𝐺 ∈ Abel ∧ 𝐴𝑋𝐵𝑋) ∧ ((𝑂𝐴) gcd (𝑂𝐵)) ≠ 0) → ((((𝑂𝐴) / ((𝑂𝐴) gcd (𝑂𝐵))) · ((𝑂𝐴) gcd (𝑂𝐵))) · (((𝑂𝐵) / ((𝑂𝐴) gcd (𝑂𝐵))) · ((𝑂𝐴) gcd (𝑂𝐵)))) = ((𝑂𝐴) · (𝑂𝐵)))
5433, 48, 533eqtr2d 2859 . . 3 (((𝐺 ∈ Abel ∧ 𝐴𝑋𝐵𝑋) ∧ ((𝑂𝐴) gcd (𝑂𝐵)) ≠ 0) → ((((𝑂𝐴) / ((𝑂𝐴) gcd (𝑂𝐵))) · ((𝑂𝐵) / ((𝑂𝐴) gcd (𝑂𝐵)))) · (((𝑂𝐴) gcd (𝑂𝐵))↑2)) = ((𝑂𝐴) · (𝑂𝐵)))
5522adantr 481 . . . . . . . . . 10 (((𝐺 ∈ Abel ∧ 𝐴𝑋𝐵𝑋) ∧ ((𝑂𝐴) gcd (𝑂𝐵)) ≠ 0) → (𝑂‘(𝐴 + 𝐵)) ∈ ℤ)
56 dvdsmul2 15620 . . . . . . . . . 10 (((𝑂‘(𝐴 + 𝐵)) ∈ ℤ ∧ (𝑂𝐴) ∈ ℤ) → (𝑂𝐴) ∥ ((𝑂‘(𝐴 + 𝐵)) · (𝑂𝐴)))
5755, 28, 56syl2anc 584 . . . . . . . . 9 (((𝐺 ∈ Abel ∧ 𝐴𝑋𝐵𝑋) ∧ ((𝑂𝐴) gcd (𝑂𝐵)) ≠ 0) → (𝑂𝐴) ∥ ((𝑂‘(𝐴 + 𝐵)) · (𝑂𝐴)))
58 simpl1 1183 . . . . . . . . . . . . 13 (((𝐺 ∈ Abel ∧ 𝐴𝑋𝐵𝑋) ∧ ((𝑂𝐴) gcd (𝑂𝐵)) ≠ 0) → 𝐺 ∈ Abel)
5955, 29zmulcld 12081 . . . . . . . . . . . . 13 (((𝐺 ∈ Abel ∧ 𝐴𝑋𝐵𝑋) ∧ ((𝑂𝐴) gcd (𝑂𝐵)) ≠ 0) → ((𝑂‘(𝐴 + 𝐵)) · (𝑂𝐵)) ∈ ℤ)
60 simpl2 1184 . . . . . . . . . . . . 13 (((𝐺 ∈ Abel ∧ 𝐴𝑋𝐵𝑋) ∧ ((𝑂𝐴) gcd (𝑂𝐵)) ≠ 0) → 𝐴𝑋)
61 simpl3 1185 . . . . . . . . . . . . 13 (((𝐺 ∈ Abel ∧ 𝐴𝑋𝐵𝑋) ∧ ((𝑂𝐴) gcd (𝑂𝐵)) ≠ 0) → 𝐵𝑋)
62 eqid 2818 . . . . . . . . . . . . . 14 (.g𝐺) = (.g𝐺)
631, 62, 17mulgdi 18876 . . . . . . . . . . . . 13 ((𝐺 ∈ Abel ∧ (((𝑂‘(𝐴 + 𝐵)) · (𝑂𝐵)) ∈ ℤ ∧ 𝐴𝑋𝐵𝑋)) → (((𝑂‘(𝐴 + 𝐵)) · (𝑂𝐵))(.g𝐺)(𝐴 + 𝐵)) = ((((𝑂‘(𝐴 + 𝐵)) · (𝑂𝐵))(.g𝐺)𝐴) + (((𝑂‘(𝐴 + 𝐵)) · (𝑂𝐵))(.g𝐺)𝐵)))
6458, 59, 60, 61, 63syl13anc 1364 . . . . . . . . . . . 12 (((𝐺 ∈ Abel ∧ 𝐴𝑋𝐵𝑋) ∧ ((𝑂𝐴) gcd (𝑂𝐵)) ≠ 0) → (((𝑂‘(𝐴 + 𝐵)) · (𝑂𝐵))(.g𝐺)(𝐴 + 𝐵)) = ((((𝑂‘(𝐴 + 𝐵)) · (𝑂𝐵))(.g𝐺)𝐴) + (((𝑂‘(𝐴 + 𝐵)) · (𝑂𝐵))(.g𝐺)𝐵)))
65 dvdsmul2 15620 . . . . . . . . . . . . . . 15 (((𝑂‘(𝐴 + 𝐵)) ∈ ℤ ∧ (𝑂𝐵) ∈ ℤ) → (𝑂𝐵) ∥ ((𝑂‘(𝐴 + 𝐵)) · (𝑂𝐵)))
6655, 29, 65syl2anc 584 . . . . . . . . . . . . . 14 (((𝐺 ∈ Abel ∧ 𝐴𝑋𝐵𝑋) ∧ ((𝑂𝐴) gcd (𝑂𝐵)) ≠ 0) → (𝑂𝐵) ∥ ((𝑂‘(𝐴 + 𝐵)) · (𝑂𝐵)))
6758, 16syl 17 . . . . . . . . . . . . . . 15 (((𝐺 ∈ Abel ∧ 𝐴𝑋𝐵𝑋) ∧ ((𝑂𝐴) gcd (𝑂𝐵)) ≠ 0) → 𝐺 ∈ Grp)
68 eqid 2818 . . . . . . . . . . . . . . . 16 (0g𝐺) = (0g𝐺)
691, 2, 62, 68oddvds 18604 . . . . . . . . . . . . . . 15 ((𝐺 ∈ Grp ∧ 𝐵𝑋 ∧ ((𝑂‘(𝐴 + 𝐵)) · (𝑂𝐵)) ∈ ℤ) → ((𝑂𝐵) ∥ ((𝑂‘(𝐴 + 𝐵)) · (𝑂𝐵)) ↔ (((𝑂‘(𝐴 + 𝐵)) · (𝑂𝐵))(.g𝐺)𝐵) = (0g𝐺)))
7067, 61, 59, 69syl3anc 1363 . . . . . . . . . . . . . 14 (((𝐺 ∈ Abel ∧ 𝐴𝑋𝐵𝑋) ∧ ((𝑂𝐴) gcd (𝑂𝐵)) ≠ 0) → ((𝑂𝐵) ∥ ((𝑂‘(𝐴 + 𝐵)) · (𝑂𝐵)) ↔ (((𝑂‘(𝐴 + 𝐵)) · (𝑂𝐵))(.g𝐺)𝐵) = (0g𝐺)))
7166, 70mpbid 233 . . . . . . . . . . . . 13 (((𝐺 ∈ Abel ∧ 𝐴𝑋𝐵𝑋) ∧ ((𝑂𝐴) gcd (𝑂𝐵)) ≠ 0) → (((𝑂‘(𝐴 + 𝐵)) · (𝑂𝐵))(.g𝐺)𝐵) = (0g𝐺))
7271oveq2d 7161 . . . . . . . . . . . 12 (((𝐺 ∈ Abel ∧ 𝐴𝑋𝐵𝑋) ∧ ((𝑂𝐴) gcd (𝑂𝐵)) ≠ 0) → ((((𝑂‘(𝐴 + 𝐵)) · (𝑂𝐵))(.g𝐺)𝐴) + (((𝑂‘(𝐴 + 𝐵)) · (𝑂𝐵))(.g𝐺)𝐵)) = ((((𝑂‘(𝐴 + 𝐵)) · (𝑂𝐵))(.g𝐺)𝐴) + (0g𝐺)))
7364, 72eqtrd 2853 . . . . . . . . . . 11 (((𝐺 ∈ Abel ∧ 𝐴𝑋𝐵𝑋) ∧ ((𝑂𝐴) gcd (𝑂𝐵)) ≠ 0) → (((𝑂‘(𝐴 + 𝐵)) · (𝑂𝐵))(.g𝐺)(𝐴 + 𝐵)) = ((((𝑂‘(𝐴 + 𝐵)) · (𝑂𝐵))(.g𝐺)𝐴) + (0g𝐺)))
74 dvdsmul1 15619 . . . . . . . . . . . . 13 (((𝑂‘(𝐴 + 𝐵)) ∈ ℤ ∧ (𝑂𝐵) ∈ ℤ) → (𝑂‘(𝐴 + 𝐵)) ∥ ((𝑂‘(𝐴 + 𝐵)) · (𝑂𝐵)))
7555, 29, 74syl2anc 584 . . . . . . . . . . . 12 (((𝐺 ∈ Abel ∧ 𝐴𝑋𝐵𝑋) ∧ ((𝑂𝐴) gcd (𝑂𝐵)) ≠ 0) → (𝑂‘(𝐴 + 𝐵)) ∥ ((𝑂‘(𝐴 + 𝐵)) · (𝑂𝐵)))
7619adantr 481 . . . . . . . . . . . . 13 (((𝐺 ∈ Abel ∧ 𝐴𝑋𝐵𝑋) ∧ ((𝑂𝐴) gcd (𝑂𝐵)) ≠ 0) → (𝐴 + 𝐵) ∈ 𝑋)
771, 2, 62, 68oddvds 18604 . . . . . . . . . . . . 13 ((𝐺 ∈ Grp ∧ (𝐴 + 𝐵) ∈ 𝑋 ∧ ((𝑂‘(𝐴 + 𝐵)) · (𝑂𝐵)) ∈ ℤ) → ((𝑂‘(𝐴 + 𝐵)) ∥ ((𝑂‘(𝐴 + 𝐵)) · (𝑂𝐵)) ↔ (((𝑂‘(𝐴 + 𝐵)) · (𝑂𝐵))(.g𝐺)(𝐴 + 𝐵)) = (0g𝐺)))
7867, 76, 59, 77syl3anc 1363 . . . . . . . . . . . 12 (((𝐺 ∈ Abel ∧ 𝐴𝑋𝐵𝑋) ∧ ((𝑂𝐴) gcd (𝑂𝐵)) ≠ 0) → ((𝑂‘(𝐴 + 𝐵)) ∥ ((𝑂‘(𝐴 + 𝐵)) · (𝑂𝐵)) ↔ (((𝑂‘(𝐴 + 𝐵)) · (𝑂𝐵))(.g𝐺)(𝐴 + 𝐵)) = (0g𝐺)))
7975, 78mpbid 233 . . . . . . . . . . 11 (((𝐺 ∈ Abel ∧ 𝐴𝑋𝐵𝑋) ∧ ((𝑂𝐴) gcd (𝑂𝐵)) ≠ 0) → (((𝑂‘(𝐴 + 𝐵)) · (𝑂𝐵))(.g𝐺)(𝐴 + 𝐵)) = (0g𝐺))
801, 62mulgcl 18183 . . . . . . . . . . . . 13 ((𝐺 ∈ Grp ∧ ((𝑂‘(𝐴 + 𝐵)) · (𝑂𝐵)) ∈ ℤ ∧ 𝐴𝑋) → (((𝑂‘(𝐴 + 𝐵)) · (𝑂𝐵))(.g𝐺)𝐴) ∈ 𝑋)
8167, 59, 60, 80syl3anc 1363 . . . . . . . . . . . 12 (((𝐺 ∈ Abel ∧ 𝐴𝑋𝐵𝑋) ∧ ((𝑂𝐴) gcd (𝑂𝐵)) ≠ 0) → (((𝑂‘(𝐴 + 𝐵)) · (𝑂𝐵))(.g𝐺)𝐴) ∈ 𝑋)
821, 17, 68grprid 18072 . . . . . . . . . . . 12 ((𝐺 ∈ Grp ∧ (((𝑂‘(𝐴 + 𝐵)) · (𝑂𝐵))(.g𝐺)𝐴) ∈ 𝑋) → ((((𝑂‘(𝐴 + 𝐵)) · (𝑂𝐵))(.g𝐺)𝐴) + (0g𝐺)) = (((𝑂‘(𝐴 + 𝐵)) · (𝑂𝐵))(.g𝐺)𝐴))
8367, 81, 82syl2anc 584 . . . . . . . . . . 11 (((𝐺 ∈ Abel ∧ 𝐴𝑋𝐵𝑋) ∧ ((𝑂𝐴) gcd (𝑂𝐵)) ≠ 0) → ((((𝑂‘(𝐴 + 𝐵)) · (𝑂𝐵))(.g𝐺)𝐴) + (0g𝐺)) = (((𝑂‘(𝐴 + 𝐵)) · (𝑂𝐵))(.g𝐺)𝐴))
8473, 79, 833eqtr3rd 2862 . . . . . . . . . 10 (((𝐺 ∈ Abel ∧ 𝐴𝑋𝐵𝑋) ∧ ((𝑂𝐴) gcd (𝑂𝐵)) ≠ 0) → (((𝑂‘(𝐴 + 𝐵)) · (𝑂𝐵))(.g𝐺)𝐴) = (0g𝐺))
851, 2, 62, 68oddvds 18604 . . . . . . . . . . 11 ((𝐺 ∈ Grp ∧ 𝐴𝑋 ∧ ((𝑂‘(𝐴 + 𝐵)) · (𝑂𝐵)) ∈ ℤ) → ((𝑂𝐴) ∥ ((𝑂‘(𝐴 + 𝐵)) · (𝑂𝐵)) ↔ (((𝑂‘(𝐴 + 𝐵)) · (𝑂𝐵))(.g𝐺)𝐴) = (0g𝐺)))
8667, 60, 59, 85syl3anc 1363 . . . . . . . . . 10 (((𝐺 ∈ Abel ∧ 𝐴𝑋𝐵𝑋) ∧ ((𝑂𝐴) gcd (𝑂𝐵)) ≠ 0) → ((𝑂𝐴) ∥ ((𝑂‘(𝐴 + 𝐵)) · (𝑂𝐵)) ↔ (((𝑂‘(𝐴 + 𝐵)) · (𝑂𝐵))(.g𝐺)𝐴) = (0g𝐺)))
8784, 86mpbird 258 . . . . . . . . 9 (((𝐺 ∈ Abel ∧ 𝐴𝑋𝐵𝑋) ∧ ((𝑂𝐴) gcd (𝑂𝐵)) ≠ 0) → (𝑂𝐴) ∥ ((𝑂‘(𝐴 + 𝐵)) · (𝑂𝐵)))
8855, 28zmulcld 12081 . . . . . . . . . 10 (((𝐺 ∈ Abel ∧ 𝐴𝑋𝐵𝑋) ∧ ((𝑂𝐴) gcd (𝑂𝐵)) ≠ 0) → ((𝑂‘(𝐴 + 𝐵)) · (𝑂𝐴)) ∈ ℤ)
89 dvdsgcd 15880 . . . . . . . . . 10 (((𝑂𝐴) ∈ ℤ ∧ ((𝑂‘(𝐴 + 𝐵)) · (𝑂𝐴)) ∈ ℤ ∧ ((𝑂‘(𝐴 + 𝐵)) · (𝑂𝐵)) ∈ ℤ) → (((𝑂𝐴) ∥ ((𝑂‘(𝐴 + 𝐵)) · (𝑂𝐴)) ∧ (𝑂𝐴) ∥ ((𝑂‘(𝐴 + 𝐵)) · (𝑂𝐵))) → (𝑂𝐴) ∥ (((𝑂‘(𝐴 + 𝐵)) · (𝑂𝐴)) gcd ((𝑂‘(𝐴 + 𝐵)) · (𝑂𝐵)))))
9028, 88, 59, 89syl3anc 1363 . . . . . . . . 9 (((𝐺 ∈ Abel ∧ 𝐴𝑋𝐵𝑋) ∧ ((𝑂𝐴) gcd (𝑂𝐵)) ≠ 0) → (((𝑂𝐴) ∥ ((𝑂‘(𝐴 + 𝐵)) · (𝑂𝐴)) ∧ (𝑂𝐴) ∥ ((𝑂‘(𝐴 + 𝐵)) · (𝑂𝐵))) → (𝑂𝐴) ∥ (((𝑂‘(𝐴 + 𝐵)) · (𝑂𝐴)) gcd ((𝑂‘(𝐴 + 𝐵)) · (𝑂𝐵)))))
9157, 87, 90mp2and 695 . . . . . . . 8 (((𝐺 ∈ Abel ∧ 𝐴𝑋𝐵𝑋) ∧ ((𝑂𝐴) gcd (𝑂𝐵)) ≠ 0) → (𝑂𝐴) ∥ (((𝑂‘(𝐴 + 𝐵)) · (𝑂𝐴)) gcd ((𝑂‘(𝐴 + 𝐵)) · (𝑂𝐵))))
9221adantr 481 . . . . . . . . 9 (((𝐺 ∈ Abel ∧ 𝐴𝑋𝐵𝑋) ∧ ((𝑂𝐴) gcd (𝑂𝐵)) ≠ 0) → (𝑂‘(𝐴 + 𝐵)) ∈ ℕ0)
93 mulgcd 15884 . . . . . . . . 9 (((𝑂‘(𝐴 + 𝐵)) ∈ ℕ0 ∧ (𝑂𝐴) ∈ ℤ ∧ (𝑂𝐵) ∈ ℤ) → (((𝑂‘(𝐴 + 𝐵)) · (𝑂𝐴)) gcd ((𝑂‘(𝐴 + 𝐵)) · (𝑂𝐵))) = ((𝑂‘(𝐴 + 𝐵)) · ((𝑂𝐴) gcd (𝑂𝐵))))
9492, 28, 29, 93syl3anc 1363 . . . . . . . 8 (((𝐺 ∈ Abel ∧ 𝐴𝑋𝐵𝑋) ∧ ((𝑂𝐴) gcd (𝑂𝐵)) ≠ 0) → (((𝑂‘(𝐴 + 𝐵)) · (𝑂𝐴)) gcd ((𝑂‘(𝐴 + 𝐵)) · (𝑂𝐵))) = ((𝑂‘(𝐴 + 𝐵)) · ((𝑂𝐴) gcd (𝑂𝐵))))
9591, 94breqtrd 5083 . . . . . . 7 (((𝐺 ∈ Abel ∧ 𝐴𝑋𝐵𝑋) ∧ ((𝑂𝐴) gcd (𝑂𝐵)) ≠ 0) → (𝑂𝐴) ∥ ((𝑂‘(𝐴 + 𝐵)) · ((𝑂𝐴) gcd (𝑂𝐵))))
9650, 95eqbrtrd 5079 . . . . . 6 (((𝐺 ∈ Abel ∧ 𝐴𝑋𝐵𝑋) ∧ ((𝑂𝐴) gcd (𝑂𝐵)) ≠ 0) → (((𝑂𝐴) / ((𝑂𝐴) gcd (𝑂𝐵))) · ((𝑂𝐴) gcd (𝑂𝐵))) ∥ ((𝑂‘(𝐴 + 𝐵)) · ((𝑂𝐴) gcd (𝑂𝐵))))
97 dvdsmulcr 15627 . . . . . . 7 ((((𝑂𝐴) / ((𝑂𝐴) gcd (𝑂𝐵))) ∈ ℤ ∧ (𝑂‘(𝐴 + 𝐵)) ∈ ℤ ∧ (((𝑂𝐴) gcd (𝑂𝐵)) ∈ ℤ ∧ ((𝑂𝐴) gcd (𝑂𝐵)) ≠ 0)) → ((((𝑂𝐴) / ((𝑂𝐴) gcd (𝑂𝐵))) · ((𝑂𝐴) gcd (𝑂𝐵))) ∥ ((𝑂‘(𝐴 + 𝐵)) · ((𝑂𝐴) gcd (𝑂𝐵))) ↔ ((𝑂𝐴) / ((𝑂𝐴) gcd (𝑂𝐵))) ∥ (𝑂‘(𝐴 + 𝐵))))
9841, 55, 37, 38, 97syl112anc 1366 . . . . . 6 (((𝐺 ∈ Abel ∧ 𝐴𝑋𝐵𝑋) ∧ ((𝑂𝐴) gcd (𝑂𝐵)) ≠ 0) → ((((𝑂𝐴) / ((𝑂𝐴) gcd (𝑂𝐵))) · ((𝑂𝐴) gcd (𝑂𝐵))) ∥ ((𝑂‘(𝐴 + 𝐵)) · ((𝑂𝐴) gcd (𝑂𝐵))) ↔ ((𝑂𝐴) / ((𝑂𝐴) gcd (𝑂𝐵))) ∥ (𝑂‘(𝐴 + 𝐵))))
9996, 98mpbid 233 . . . . 5 (((𝐺 ∈ Abel ∧ 𝐴𝑋𝐵𝑋) ∧ ((𝑂𝐴) gcd (𝑂𝐵)) ≠ 0) → ((𝑂𝐴) / ((𝑂𝐴) gcd (𝑂𝐵))) ∥ (𝑂‘(𝐴 + 𝐵)))
1001, 62, 17mulgdi 18876 . . . . . . . . . . . . 13 ((𝐺 ∈ Abel ∧ (((𝑂‘(𝐴 + 𝐵)) · (𝑂𝐴)) ∈ ℤ ∧ 𝐴𝑋𝐵𝑋)) → (((𝑂‘(𝐴 + 𝐵)) · (𝑂𝐴))(.g𝐺)(𝐴 + 𝐵)) = ((((𝑂‘(𝐴 + 𝐵)) · (𝑂𝐴))(.g𝐺)𝐴) + (((𝑂‘(𝐴 + 𝐵)) · (𝑂𝐴))(.g𝐺)𝐵)))
10158, 88, 60, 61, 100syl13anc 1364 . . . . . . . . . . . 12 (((𝐺 ∈ Abel ∧ 𝐴𝑋𝐵𝑋) ∧ ((𝑂𝐴) gcd (𝑂𝐵)) ≠ 0) → (((𝑂‘(𝐴 + 𝐵)) · (𝑂𝐴))(.g𝐺)(𝐴 + 𝐵)) = ((((𝑂‘(𝐴 + 𝐵)) · (𝑂𝐴))(.g𝐺)𝐴) + (((𝑂‘(𝐴 + 𝐵)) · (𝑂𝐴))(.g𝐺)𝐵)))
1021, 2, 62, 68oddvds 18604 . . . . . . . . . . . . . . 15 ((𝐺 ∈ Grp ∧ 𝐴𝑋 ∧ ((𝑂‘(𝐴 + 𝐵)) · (𝑂𝐴)) ∈ ℤ) → ((𝑂𝐴) ∥ ((𝑂‘(𝐴 + 𝐵)) · (𝑂𝐴)) ↔ (((𝑂‘(𝐴 + 𝐵)) · (𝑂𝐴))(.g𝐺)𝐴) = (0g𝐺)))
10367, 60, 88, 102syl3anc 1363 . . . . . . . . . . . . . 14 (((𝐺 ∈ Abel ∧ 𝐴𝑋𝐵𝑋) ∧ ((𝑂𝐴) gcd (𝑂𝐵)) ≠ 0) → ((𝑂𝐴) ∥ ((𝑂‘(𝐴 + 𝐵)) · (𝑂𝐴)) ↔ (((𝑂‘(𝐴 + 𝐵)) · (𝑂𝐴))(.g𝐺)𝐴) = (0g𝐺)))
10457, 103mpbid 233 . . . . . . . . . . . . 13 (((𝐺 ∈ Abel ∧ 𝐴𝑋𝐵𝑋) ∧ ((𝑂𝐴) gcd (𝑂𝐵)) ≠ 0) → (((𝑂‘(𝐴 + 𝐵)) · (𝑂𝐴))(.g𝐺)𝐴) = (0g𝐺))
105104oveq1d 7160 . . . . . . . . . . . 12 (((𝐺 ∈ Abel ∧ 𝐴𝑋𝐵𝑋) ∧ ((𝑂𝐴) gcd (𝑂𝐵)) ≠ 0) → ((((𝑂‘(𝐴 + 𝐵)) · (𝑂𝐴))(.g𝐺)𝐴) + (((𝑂‘(𝐴 + 𝐵)) · (𝑂𝐴))(.g𝐺)𝐵)) = ((0g𝐺) + (((𝑂‘(𝐴 + 𝐵)) · (𝑂𝐴))(.g𝐺)𝐵)))
106101, 105eqtrd 2853 . . . . . . . . . . 11 (((𝐺 ∈ Abel ∧ 𝐴𝑋𝐵𝑋) ∧ ((𝑂𝐴) gcd (𝑂𝐵)) ≠ 0) → (((𝑂‘(𝐴 + 𝐵)) · (𝑂𝐴))(.g𝐺)(𝐴 + 𝐵)) = ((0g𝐺) + (((𝑂‘(𝐴 + 𝐵)) · (𝑂𝐴))(.g𝐺)𝐵)))
107 dvdsmul1 15619 . . . . . . . . . . . . 13 (((𝑂‘(𝐴 + 𝐵)) ∈ ℤ ∧ (𝑂𝐴) ∈ ℤ) → (𝑂‘(𝐴 + 𝐵)) ∥ ((𝑂‘(𝐴 + 𝐵)) · (𝑂𝐴)))
10855, 28, 107syl2anc 584 . . . . . . . . . . . 12 (((𝐺 ∈ Abel ∧ 𝐴𝑋𝐵𝑋) ∧ ((𝑂𝐴) gcd (𝑂𝐵)) ≠ 0) → (𝑂‘(𝐴 + 𝐵)) ∥ ((𝑂‘(𝐴 + 𝐵)) · (𝑂𝐴)))
1091, 2, 62, 68oddvds 18604 . . . . . . . . . . . . 13 ((𝐺 ∈ Grp ∧ (𝐴 + 𝐵) ∈ 𝑋 ∧ ((𝑂‘(𝐴 + 𝐵)) · (𝑂𝐴)) ∈ ℤ) → ((𝑂‘(𝐴 + 𝐵)) ∥ ((𝑂‘(𝐴 + 𝐵)) · (𝑂𝐴)) ↔ (((𝑂‘(𝐴 + 𝐵)) · (𝑂𝐴))(.g𝐺)(𝐴 + 𝐵)) = (0g𝐺)))
11067, 76, 88, 109syl3anc 1363 . . . . . . . . . . . 12 (((𝐺 ∈ Abel ∧ 𝐴𝑋𝐵𝑋) ∧ ((𝑂𝐴) gcd (𝑂𝐵)) ≠ 0) → ((𝑂‘(𝐴 + 𝐵)) ∥ ((𝑂‘(𝐴 + 𝐵)) · (𝑂𝐴)) ↔ (((𝑂‘(𝐴 + 𝐵)) · (𝑂𝐴))(.g𝐺)(𝐴 + 𝐵)) = (0g𝐺)))
111108, 110mpbid 233 . . . . . . . . . . 11 (((𝐺 ∈ Abel ∧ 𝐴𝑋𝐵𝑋) ∧ ((𝑂𝐴) gcd (𝑂𝐵)) ≠ 0) → (((𝑂‘(𝐴 + 𝐵)) · (𝑂𝐴))(.g𝐺)(𝐴 + 𝐵)) = (0g𝐺))
1121, 62mulgcl 18183 . . . . . . . . . . . . 13 ((𝐺 ∈ Grp ∧ ((𝑂‘(𝐴 + 𝐵)) · (𝑂𝐴)) ∈ ℤ ∧ 𝐵𝑋) → (((𝑂‘(𝐴 + 𝐵)) · (𝑂𝐴))(.g𝐺)𝐵) ∈ 𝑋)
11367, 88, 61, 112syl3anc 1363 . . . . . . . . . . . 12 (((𝐺 ∈ Abel ∧ 𝐴𝑋𝐵𝑋) ∧ ((𝑂𝐴) gcd (𝑂𝐵)) ≠ 0) → (((𝑂‘(𝐴 + 𝐵)) · (𝑂𝐴))(.g𝐺)𝐵) ∈ 𝑋)
1141, 17, 68grplid 18071 . . . . . . . . . . . 12 ((𝐺 ∈ Grp ∧ (((𝑂‘(𝐴 + 𝐵)) · (𝑂𝐴))(.g𝐺)𝐵) ∈ 𝑋) → ((0g𝐺) + (((𝑂‘(𝐴 + 𝐵)) · (𝑂𝐴))(.g𝐺)𝐵)) = (((𝑂‘(𝐴 + 𝐵)) · (𝑂𝐴))(.g𝐺)𝐵))
11567, 113, 114syl2anc 584 . . . . . . . . . . 11 (((𝐺 ∈ Abel ∧ 𝐴𝑋𝐵𝑋) ∧ ((𝑂𝐴) gcd (𝑂𝐵)) ≠ 0) → ((0g𝐺) + (((𝑂‘(𝐴 + 𝐵)) · (𝑂𝐴))(.g𝐺)𝐵)) = (((𝑂‘(𝐴 + 𝐵)) · (𝑂𝐴))(.g𝐺)𝐵))
116106, 111, 1153eqtr3rd 2862 . . . . . . . . . 10 (((𝐺 ∈ Abel ∧ 𝐴𝑋𝐵𝑋) ∧ ((𝑂𝐴) gcd (𝑂𝐵)) ≠ 0) → (((𝑂‘(𝐴 + 𝐵)) · (𝑂𝐴))(.g𝐺)𝐵) = (0g𝐺))
1171, 2, 62, 68oddvds 18604 . . . . . . . . . . 11 ((𝐺 ∈ Grp ∧ 𝐵𝑋 ∧ ((𝑂‘(𝐴 + 𝐵)) · (𝑂𝐴)) ∈ ℤ) → ((𝑂𝐵) ∥ ((𝑂‘(𝐴 + 𝐵)) · (𝑂𝐴)) ↔ (((𝑂‘(𝐴 + 𝐵)) · (𝑂𝐴))(.g𝐺)𝐵) = (0g𝐺)))
11867, 61, 88, 117syl3anc 1363 . . . . . . . . . 10 (((𝐺 ∈ Abel ∧ 𝐴𝑋𝐵𝑋) ∧ ((𝑂𝐴) gcd (𝑂𝐵)) ≠ 0) → ((𝑂𝐵) ∥ ((𝑂‘(𝐴 + 𝐵)) · (𝑂𝐴)) ↔ (((𝑂‘(𝐴 + 𝐵)) · (𝑂𝐴))(.g𝐺)𝐵) = (0g𝐺)))
119116, 118mpbird 258 . . . . . . . . 9 (((𝐺 ∈ Abel ∧ 𝐴𝑋𝐵𝑋) ∧ ((𝑂𝐴) gcd (𝑂𝐵)) ≠ 0) → (𝑂𝐵) ∥ ((𝑂‘(𝐴 + 𝐵)) · (𝑂𝐴)))
120 dvdsgcd 15880 . . . . . . . . . 10 (((𝑂𝐵) ∈ ℤ ∧ ((𝑂‘(𝐴 + 𝐵)) · (𝑂𝐴)) ∈ ℤ ∧ ((𝑂‘(𝐴 + 𝐵)) · (𝑂𝐵)) ∈ ℤ) → (((𝑂𝐵) ∥ ((𝑂‘(𝐴 + 𝐵)) · (𝑂𝐴)) ∧ (𝑂𝐵) ∥ ((𝑂‘(𝐴 + 𝐵)) · (𝑂𝐵))) → (𝑂𝐵) ∥ (((𝑂‘(𝐴 + 𝐵)) · (𝑂𝐴)) gcd ((𝑂‘(𝐴 + 𝐵)) · (𝑂𝐵)))))
12129, 88, 59, 120syl3anc 1363 . . . . . . . . 9 (((𝐺 ∈ Abel ∧ 𝐴𝑋𝐵𝑋) ∧ ((𝑂𝐴) gcd (𝑂𝐵)) ≠ 0) → (((𝑂𝐵) ∥ ((𝑂‘(𝐴 + 𝐵)) · (𝑂𝐴)) ∧ (𝑂𝐵) ∥ ((𝑂‘(𝐴 + 𝐵)) · (𝑂𝐵))) → (𝑂𝐵) ∥ (((𝑂‘(𝐴 + 𝐵)) · (𝑂𝐴)) gcd ((𝑂‘(𝐴 + 𝐵)) · (𝑂𝐵)))))
122119, 66, 121mp2and 695 . . . . . . . 8 (((𝐺 ∈ Abel ∧ 𝐴𝑋𝐵𝑋) ∧ ((𝑂𝐴) gcd (𝑂𝐵)) ≠ 0) → (𝑂𝐵) ∥ (((𝑂‘(𝐴 + 𝐵)) · (𝑂𝐴)) gcd ((𝑂‘(𝐴 + 𝐵)) · (𝑂𝐵))))
123122, 94breqtrd 5083 . . . . . . 7 (((𝐺 ∈ Abel ∧ 𝐴𝑋𝐵𝑋) ∧ ((𝑂𝐴) gcd (𝑂𝐵)) ≠ 0) → (𝑂𝐵) ∥ ((𝑂‘(𝐴 + 𝐵)) · ((𝑂𝐴) gcd (𝑂𝐵))))
12452, 123eqbrtrd 5079 . . . . . 6 (((𝐺 ∈ Abel ∧ 𝐴𝑋𝐵𝑋) ∧ ((𝑂𝐴) gcd (𝑂𝐵)) ≠ 0) → (((𝑂𝐵) / ((𝑂𝐴) gcd (𝑂𝐵))) · ((𝑂𝐴) gcd (𝑂𝐵))) ∥ ((𝑂‘(𝐴 + 𝐵)) · ((𝑂𝐴) gcd (𝑂𝐵))))
125 dvdsmulcr 15627 . . . . . . 7 ((((𝑂𝐵) / ((𝑂𝐴) gcd (𝑂𝐵))) ∈ ℤ ∧ (𝑂‘(𝐴 + 𝐵)) ∈ ℤ ∧ (((𝑂𝐴) gcd (𝑂𝐵)) ∈ ℤ ∧ ((𝑂𝐴) gcd (𝑂𝐵)) ≠ 0)) → ((((𝑂𝐵) / ((𝑂𝐴) gcd (𝑂𝐵))) · ((𝑂𝐴) gcd (𝑂𝐵))) ∥ ((𝑂‘(𝐴 + 𝐵)) · ((𝑂𝐴) gcd (𝑂𝐵))) ↔ ((𝑂𝐵) / ((𝑂𝐴) gcd (𝑂𝐵))) ∥ (𝑂‘(𝐴 + 𝐵))))
12646, 55, 37, 38, 125syl112anc 1366 . . . . . 6 (((𝐺 ∈ Abel ∧ 𝐴𝑋𝐵𝑋) ∧ ((𝑂𝐴) gcd (𝑂𝐵)) ≠ 0) → ((((𝑂𝐵) / ((𝑂𝐴) gcd (𝑂𝐵))) · ((𝑂𝐴) gcd (𝑂𝐵))) ∥ ((𝑂‘(𝐴 + 𝐵)) · ((𝑂𝐴) gcd (𝑂𝐵))) ↔ ((𝑂𝐵) / ((𝑂𝐴) gcd (𝑂𝐵))) ∥ (𝑂‘(𝐴 + 𝐵))))
127124, 126mpbid 233 . . . . 5 (((𝐺 ∈ Abel ∧ 𝐴𝑋𝐵𝑋) ∧ ((𝑂𝐴) gcd (𝑂𝐵)) ≠ 0) → ((𝑂𝐵) / ((𝑂𝐴) gcd (𝑂𝐵))) ∥ (𝑂‘(𝐴 + 𝐵)))
12841, 46gcdcld 15845 . . . . . . . 8 (((𝐺 ∈ Abel ∧ 𝐴𝑋𝐵𝑋) ∧ ((𝑂𝐴) gcd (𝑂𝐵)) ≠ 0) → (((𝑂𝐴) / ((𝑂𝐴) gcd (𝑂𝐵))) gcd ((𝑂𝐵) / ((𝑂𝐴) gcd (𝑂𝐵)))) ∈ ℕ0)
129128nn0cnd 11945 . . . . . . 7 (((𝐺 ∈ Abel ∧ 𝐴𝑋𝐵𝑋) ∧ ((𝑂𝐴) gcd (𝑂𝐵)) ≠ 0) → (((𝑂𝐴) / ((𝑂𝐴) gcd (𝑂𝐵))) gcd ((𝑂𝐵) / ((𝑂𝐴) gcd (𝑂𝐵)))) ∈ ℂ)
130 1cnd 10624 . . . . . . 7 (((𝐺 ∈ Abel ∧ 𝐴𝑋𝐵𝑋) ∧ ((𝑂𝐴) gcd (𝑂𝐵)) ≠ 0) → 1 ∈ ℂ)
13131mulid2d 10647 . . . . . . . 8 (((𝐺 ∈ Abel ∧ 𝐴𝑋𝐵𝑋) ∧ ((𝑂𝐴) gcd (𝑂𝐵)) ≠ 0) → (1 · ((𝑂𝐴) gcd (𝑂𝐵))) = ((𝑂𝐴) gcd (𝑂𝐵)))
13250, 52oveq12d 7163 . . . . . . . 8 (((𝐺 ∈ Abel ∧ 𝐴𝑋𝐵𝑋) ∧ ((𝑂𝐴) gcd (𝑂𝐵)) ≠ 0) → ((((𝑂𝐴) / ((𝑂𝐴) gcd (𝑂𝐵))) · ((𝑂𝐴) gcd (𝑂𝐵))) gcd (((𝑂𝐵) / ((𝑂𝐴) gcd (𝑂𝐵))) · ((𝑂𝐴) gcd (𝑂𝐵)))) = ((𝑂𝐴) gcd (𝑂𝐵)))
133 mulgcdr 15886 . . . . . . . . 9 ((((𝑂𝐴) / ((𝑂𝐴) gcd (𝑂𝐵))) ∈ ℤ ∧ ((𝑂𝐵) / ((𝑂𝐴) gcd (𝑂𝐵))) ∈ ℤ ∧ ((𝑂𝐴) gcd (𝑂𝐵)) ∈ ℕ0) → ((((𝑂𝐴) / ((𝑂𝐴) gcd (𝑂𝐵))) · ((𝑂𝐴) gcd (𝑂𝐵))) gcd (((𝑂𝐵) / ((𝑂𝐴) gcd (𝑂𝐵))) · ((𝑂𝐴) gcd (𝑂𝐵)))) = ((((𝑂𝐴) / ((𝑂𝐴) gcd (𝑂𝐵))) gcd ((𝑂𝐵) / ((𝑂𝐴) gcd (𝑂𝐵)))) · ((𝑂𝐴) gcd (𝑂𝐵))))
13441, 46, 30, 133syl3anc 1363 . . . . . . . 8 (((𝐺 ∈ Abel ∧ 𝐴𝑋𝐵𝑋) ∧ ((𝑂𝐴) gcd (𝑂𝐵)) ≠ 0) → ((((𝑂𝐴) / ((𝑂𝐴) gcd (𝑂𝐵))) · ((𝑂𝐴) gcd (𝑂𝐵))) gcd (((𝑂𝐵) / ((𝑂𝐴) gcd (𝑂𝐵))) · ((𝑂𝐴) gcd (𝑂𝐵)))) = ((((𝑂𝐴) / ((𝑂𝐴) gcd (𝑂𝐵))) gcd ((𝑂𝐵) / ((𝑂𝐴) gcd (𝑂𝐵)))) · ((𝑂𝐴) gcd (𝑂𝐵))))
135131, 132, 1343eqtr2rd 2860 . . . . . . 7 (((𝐺 ∈ Abel ∧ 𝐴𝑋𝐵𝑋) ∧ ((𝑂𝐴) gcd (𝑂𝐵)) ≠ 0) → ((((𝑂𝐴) / ((𝑂𝐴) gcd (𝑂𝐵))) gcd ((𝑂𝐵) / ((𝑂𝐴) gcd (𝑂𝐵)))) · ((𝑂𝐴) gcd (𝑂𝐵))) = (1 · ((𝑂𝐴) gcd (𝑂𝐵))))
136129, 130, 31, 38, 135mulcan2ad 11264 . . . . . 6 (((𝐺 ∈ Abel ∧ 𝐴𝑋𝐵𝑋) ∧ ((𝑂𝐴) gcd (𝑂𝐵)) ≠ 0) → (((𝑂𝐴) / ((𝑂𝐴) gcd (𝑂𝐵))) gcd ((𝑂𝐵) / ((𝑂𝐴) gcd (𝑂𝐵)))) = 1)
137 coprmdvds2 15986 . . . . . 6 (((((𝑂𝐴) / ((𝑂𝐴) gcd (𝑂𝐵))) ∈ ℤ ∧ ((𝑂𝐵) / ((𝑂𝐴) gcd (𝑂𝐵))) ∈ ℤ ∧ (𝑂‘(𝐴 + 𝐵)) ∈ ℤ) ∧ (((𝑂𝐴) / ((𝑂𝐴) gcd (𝑂𝐵))) gcd ((𝑂𝐵) / ((𝑂𝐴) gcd (𝑂𝐵)))) = 1) → ((((𝑂𝐴) / ((𝑂𝐴) gcd (𝑂𝐵))) ∥ (𝑂‘(𝐴 + 𝐵)) ∧ ((𝑂𝐵) / ((𝑂𝐴) gcd (𝑂𝐵))) ∥ (𝑂‘(𝐴 + 𝐵))) → (((𝑂𝐴) / ((𝑂𝐴) gcd (𝑂𝐵))) · ((𝑂𝐵) / ((𝑂𝐴) gcd (𝑂𝐵)))) ∥ (𝑂‘(𝐴 + 𝐵))))
13841, 46, 55, 136, 137syl31anc 1365 . . . . 5 (((𝐺 ∈ Abel ∧ 𝐴𝑋𝐵𝑋) ∧ ((𝑂𝐴) gcd (𝑂𝐵)) ≠ 0) → ((((𝑂𝐴) / ((𝑂𝐴) gcd (𝑂𝐵))) ∥ (𝑂‘(𝐴 + 𝐵)) ∧ ((𝑂𝐵) / ((𝑂𝐴) gcd (𝑂𝐵))) ∥ (𝑂‘(𝐴 + 𝐵))) → (((𝑂𝐴) / ((𝑂𝐴) gcd (𝑂𝐵))) · ((𝑂𝐵) / ((𝑂𝐴) gcd (𝑂𝐵)))) ∥ (𝑂‘(𝐴 + 𝐵))))
13999, 127, 138mp2and 695 . . . 4 (((𝐺 ∈ Abel ∧ 𝐴𝑋𝐵𝑋) ∧ ((𝑂𝐴) gcd (𝑂𝐵)) ≠ 0) → (((𝑂𝐴) / ((𝑂𝐴) gcd (𝑂𝐵))) · ((𝑂𝐵) / ((𝑂𝐴) gcd (𝑂𝐵)))) ∥ (𝑂‘(𝐴 + 𝐵)))
14041, 46zmulcld 12081 . . . . 5 (((𝐺 ∈ Abel ∧ 𝐴𝑋𝐵𝑋) ∧ ((𝑂𝐴) gcd (𝑂𝐵)) ≠ 0) → (((𝑂𝐴) / ((𝑂𝐴) gcd (𝑂𝐵))) · ((𝑂𝐵) / ((𝑂𝐴) gcd (𝑂𝐵)))) ∈ ℤ)
141 zsqcl 13482 . . . . . 6 (((𝑂𝐴) gcd (𝑂𝐵)) ∈ ℤ → (((𝑂𝐴) gcd (𝑂𝐵))↑2) ∈ ℤ)
14237, 141syl 17 . . . . 5 (((𝐺 ∈ Abel ∧ 𝐴𝑋𝐵𝑋) ∧ ((𝑂𝐴) gcd (𝑂𝐵)) ≠ 0) → (((𝑂𝐴) gcd (𝑂𝐵))↑2) ∈ ℤ)
143 dvdsmulc 15625 . . . . 5 (((((𝑂𝐴) / ((𝑂𝐴) gcd (𝑂𝐵))) · ((𝑂𝐵) / ((𝑂𝐴) gcd (𝑂𝐵)))) ∈ ℤ ∧ (𝑂‘(𝐴 + 𝐵)) ∈ ℤ ∧ (((𝑂𝐴) gcd (𝑂𝐵))↑2) ∈ ℤ) → ((((𝑂𝐴) / ((𝑂𝐴) gcd (𝑂𝐵))) · ((𝑂𝐵) / ((𝑂𝐴) gcd (𝑂𝐵)))) ∥ (𝑂‘(𝐴 + 𝐵)) → ((((𝑂𝐴) / ((𝑂𝐴) gcd (𝑂𝐵))) · ((𝑂𝐵) / ((𝑂𝐴) gcd (𝑂𝐵)))) · (((𝑂𝐴) gcd (𝑂𝐵))↑2)) ∥ ((𝑂‘(𝐴 + 𝐵)) · (((𝑂𝐴) gcd (𝑂𝐵))↑2))))
144140, 55, 142, 143syl3anc 1363 . . . 4 (((𝐺 ∈ Abel ∧ 𝐴𝑋𝐵𝑋) ∧ ((𝑂𝐴) gcd (𝑂𝐵)) ≠ 0) → ((((𝑂𝐴) / ((𝑂𝐴) gcd (𝑂𝐵))) · ((𝑂𝐵) / ((𝑂𝐴) gcd (𝑂𝐵)))) ∥ (𝑂‘(𝐴 + 𝐵)) → ((((𝑂𝐴) / ((𝑂𝐴) gcd (𝑂𝐵))) · ((𝑂𝐵) / ((𝑂𝐴) gcd (𝑂𝐵)))) · (((𝑂𝐴) gcd (𝑂𝐵))↑2)) ∥ ((𝑂‘(𝐴 + 𝐵)) · (((𝑂𝐴) gcd (𝑂𝐵))↑2))))
145139, 144mpd 15 . . 3 (((𝐺 ∈ Abel ∧ 𝐴𝑋𝐵𝑋) ∧ ((𝑂𝐴) gcd (𝑂𝐵)) ≠ 0) → ((((𝑂𝐴) / ((𝑂𝐴) gcd (𝑂𝐵))) · ((𝑂𝐵) / ((𝑂𝐴) gcd (𝑂𝐵)))) · (((𝑂𝐴) gcd (𝑂𝐵))↑2)) ∥ ((𝑂‘(𝐴 + 𝐵)) · (((𝑂𝐴) gcd (𝑂𝐵))↑2)))
14654, 145eqbrtrrd 5081 . 2 (((𝐺 ∈ Abel ∧ 𝐴𝑋𝐵𝑋) ∧ ((𝑂𝐴) gcd (𝑂𝐵)) ≠ 0) → ((𝑂𝐴) · (𝑂𝐵)) ∥ ((𝑂‘(𝐴 + 𝐵)) · (((𝑂𝐴) gcd (𝑂𝐵))↑2)))
14727, 146pm2.61dane 3101 1 ((𝐺 ∈ Abel ∧ 𝐴𝑋𝐵𝑋) → ((𝑂𝐴) · (𝑂𝐵)) ∥ ((𝑂‘(𝐴 + 𝐵)) · (((𝑂𝐴) gcd (𝑂𝐵))↑2)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396  w3a 1079   = wceq 1528  wcel 2105  wne 3013   class class class wbr 5057  cfv 6348  (class class class)co 7145  0cc0 10525  1c1 10526   · cmul 10530   / cdiv 11285  2c2 11680  0cn0 11885  cz 11969  cexp 13417  cdvds 15595   gcd cgcd 15831  Basecbs 16471  +gcplusg 16553  0gc0g 16701  Grpcgrp 18041  .gcmg 18162  odcod 18581  Abelcabl 18836
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450  ax-cnex 10581  ax-resscn 10582  ax-1cn 10583  ax-icn 10584  ax-addcl 10585  ax-addrcl 10586  ax-mulcl 10587  ax-mulrcl 10588  ax-mulcom 10589  ax-addass 10590  ax-mulass 10591  ax-distr 10592  ax-i2m1 10593  ax-1ne0 10594  ax-1rid 10595  ax-rnegex 10596  ax-rrecex 10597  ax-cnre 10598  ax-pre-lttri 10599  ax-pre-lttrn 10600  ax-pre-ltadd 10601  ax-pre-mulgt0 10602  ax-pre-sup 10603
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-nel 3121  df-ral 3140  df-rex 3141  df-reu 3142  df-rmo 3143  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-riota 7103  df-ov 7148  df-oprab 7149  df-mpo 7150  df-om 7570  df-1st 7678  df-2nd 7679  df-wrecs 7936  df-recs 7997  df-rdg 8035  df-er 8278  df-en 8498  df-dom 8499  df-sdom 8500  df-sup 8894  df-inf 8895  df-pnf 10665  df-mnf 10666  df-xr 10667  df-ltxr 10668  df-le 10669  df-sub 10860  df-neg 10861  df-div 11286  df-nn 11627  df-2 11688  df-3 11689  df-n0 11886  df-z 11970  df-uz 12232  df-rp 12378  df-fz 12881  df-fzo 13022  df-fl 13150  df-mod 13226  df-seq 13358  df-exp 13418  df-cj 14446  df-re 14447  df-im 14448  df-sqrt 14582  df-abs 14583  df-dvds 15596  df-gcd 15832  df-0g 16703  df-mgm 17840  df-sgrp 17889  df-mnd 17900  df-grp 18044  df-minusg 18045  df-sbg 18046  df-mulg 18163  df-od 18585  df-cmn 18837  df-abl 18838
This theorem is referenced by:  odadd  18899
  Copyright terms: Public domain W3C validator