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Theorem odadd2 18176
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 17879 . . . . . . . 8 (𝐴𝑋 → (𝑂𝐴) ∈ ℕ0)
433ad2ant2 1081 . . . . . . 7 ((𝐺 ∈ Abel ∧ 𝐴𝑋𝐵𝑋) → (𝑂𝐴) ∈ ℕ0)
54nn0zd 11427 . . . . . 6 ((𝐺 ∈ Abel ∧ 𝐴𝑋𝐵𝑋) → (𝑂𝐴) ∈ ℤ)
61, 2odcl 17879 . . . . . . . 8 (𝐵𝑋 → (𝑂𝐵) ∈ ℕ0)
763ad2ant3 1082 . . . . . . 7 ((𝐺 ∈ Abel ∧ 𝐴𝑋𝐵𝑋) → (𝑂𝐵) ∈ ℕ0)
87nn0zd 11427 . . . . . 6 ((𝐺 ∈ Abel ∧ 𝐴𝑋𝐵𝑋) → (𝑂𝐵) ∈ ℤ)
95, 8zmulcld 11435 . . . . 5 ((𝐺 ∈ Abel ∧ 𝐴𝑋𝐵𝑋) → ((𝑂𝐴) · (𝑂𝐵)) ∈ ℤ)
109adantr 481 . . . 4 (((𝐺 ∈ Abel ∧ 𝐴𝑋𝐵𝑋) ∧ ((𝑂𝐴) gcd (𝑂𝐵)) = 0) → ((𝑂𝐴) · (𝑂𝐵)) ∈ ℤ)
11 dvds0 14924 . . . 4 (((𝑂𝐴) · (𝑂𝐵)) ∈ ℤ → ((𝑂𝐴) · (𝑂𝐵)) ∥ 0)
1210, 11syl 17 . . 3 (((𝐺 ∈ Abel ∧ 𝐴𝑋𝐵𝑋) ∧ ((𝑂𝐴) gcd (𝑂𝐵)) = 0) → ((𝑂𝐴) · (𝑂𝐵)) ∥ 0)
13 simpr 477 . . . . . 6 (((𝐺 ∈ Abel ∧ 𝐴𝑋𝐵𝑋) ∧ ((𝑂𝐴) gcd (𝑂𝐵)) = 0) → ((𝑂𝐴) gcd (𝑂𝐵)) = 0)
1413sq0id 12900 . . . . 5 (((𝐺 ∈ Abel ∧ 𝐴𝑋𝐵𝑋) ∧ ((𝑂𝐴) gcd (𝑂𝐵)) = 0) → (((𝑂𝐴) gcd (𝑂𝐵))↑2) = 0)
1514oveq2d 6623 . . . 4 (((𝐺 ∈ Abel ∧ 𝐴𝑋𝐵𝑋) ∧ ((𝑂𝐴) gcd (𝑂𝐵)) = 0) → ((𝑂‘(𝐴 + 𝐵)) · (((𝑂𝐴) gcd (𝑂𝐵))↑2)) = ((𝑂‘(𝐴 + 𝐵)) · 0))
16 ablgrp 18122 . . . . . . . . . 10 (𝐺 ∈ Abel → 𝐺 ∈ Grp)
17 odadd1.3 . . . . . . . . . . 11 + = (+g𝐺)
181, 17grpcl 17354 . . . . . . . . . 10 ((𝐺 ∈ Grp ∧ 𝐴𝑋𝐵𝑋) → (𝐴 + 𝐵) ∈ 𝑋)
1916, 18syl3an1 1356 . . . . . . . . 9 ((𝐺 ∈ Abel ∧ 𝐴𝑋𝐵𝑋) → (𝐴 + 𝐵) ∈ 𝑋)
201, 2odcl 17879 . . . . . . . . 9 ((𝐴 + 𝐵) ∈ 𝑋 → (𝑂‘(𝐴 + 𝐵)) ∈ ℕ0)
2119, 20syl 17 . . . . . . . 8 ((𝐺 ∈ Abel ∧ 𝐴𝑋𝐵𝑋) → (𝑂‘(𝐴 + 𝐵)) ∈ ℕ0)
2221nn0zd 11427 . . . . . . 7 ((𝐺 ∈ Abel ∧ 𝐴𝑋𝐵𝑋) → (𝑂‘(𝐴 + 𝐵)) ∈ ℤ)
2322adantr 481 . . . . . 6 (((𝐺 ∈ Abel ∧ 𝐴𝑋𝐵𝑋) ∧ ((𝑂𝐴) gcd (𝑂𝐵)) = 0) → (𝑂‘(𝐴 + 𝐵)) ∈ ℤ)
2423zcnd 11430 . . . . 5 (((𝐺 ∈ Abel ∧ 𝐴𝑋𝐵𝑋) ∧ ((𝑂𝐴) gcd (𝑂𝐵)) = 0) → (𝑂‘(𝐴 + 𝐵)) ∈ ℂ)
2524mul01d 10182 . . . 4 (((𝐺 ∈ Abel ∧ 𝐴𝑋𝐵𝑋) ∧ ((𝑂𝐴) gcd (𝑂𝐵)) = 0) → ((𝑂‘(𝐴 + 𝐵)) · 0) = 0)
2615, 25eqtrd 2655 . . 3 (((𝐺 ∈ Abel ∧ 𝐴𝑋𝐵𝑋) ∧ ((𝑂𝐴) gcd (𝑂𝐵)) = 0) → ((𝑂‘(𝐴 + 𝐵)) · (((𝑂𝐴) gcd (𝑂𝐵))↑2)) = 0)
2712, 26breqtrrd 4643 . 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 15157 . . . . . . 7 (((𝐺 ∈ Abel ∧ 𝐴𝑋𝐵𝑋) ∧ ((𝑂𝐴) gcd (𝑂𝐵)) ≠ 0) → ((𝑂𝐴) gcd (𝑂𝐵)) ∈ ℕ0)
3130nn0cnd 11300 . . . . . 6 (((𝐺 ∈ Abel ∧ 𝐴𝑋𝐵𝑋) ∧ ((𝑂𝐴) gcd (𝑂𝐵)) ≠ 0) → ((𝑂𝐴) gcd (𝑂𝐵)) ∈ ℂ)
3231sqvald 12948 . . . . 5 (((𝐺 ∈ Abel ∧ 𝐴𝑋𝐵𝑋) ∧ ((𝑂𝐴) gcd (𝑂𝐵)) ≠ 0) → (((𝑂𝐴) gcd (𝑂𝐵))↑2) = (((𝑂𝐴) gcd (𝑂𝐵)) · ((𝑂𝐴) gcd (𝑂𝐵))))
3332oveq2d 6623 . . . 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 15152 . . . . . . . . 9 (((𝑂𝐴) ∈ ℤ ∧ (𝑂𝐵) ∈ ℤ) → (((𝑂𝐴) gcd (𝑂𝐵)) ∥ (𝑂𝐴) ∧ ((𝑂𝐴) gcd (𝑂𝐵)) ∥ (𝑂𝐵)))
3528, 29, 34syl2anc 692 . . . . . . . 8 (((𝐺 ∈ Abel ∧ 𝐴𝑋𝐵𝑋) ∧ ((𝑂𝐴) gcd (𝑂𝐵)) ≠ 0) → (((𝑂𝐴) gcd (𝑂𝐵)) ∥ (𝑂𝐴) ∧ ((𝑂𝐴) gcd (𝑂𝐵)) ∥ (𝑂𝐵)))
3635simpld 475 . . . . . . 7 (((𝐺 ∈ Abel ∧ 𝐴𝑋𝐵𝑋) ∧ ((𝑂𝐴) gcd (𝑂𝐵)) ≠ 0) → ((𝑂𝐴) gcd (𝑂𝐵)) ∥ (𝑂𝐴))
3730nn0zd 11427 . . . . . . . 8 (((𝐺 ∈ Abel ∧ 𝐴𝑋𝐵𝑋) ∧ ((𝑂𝐴) gcd (𝑂𝐵)) ≠ 0) → ((𝑂𝐴) gcd (𝑂𝐵)) ∈ ℤ)
38 simpr 477 . . . . . . . 8 (((𝐺 ∈ Abel ∧ 𝐴𝑋𝐵𝑋) ∧ ((𝑂𝐴) gcd (𝑂𝐵)) ≠ 0) → ((𝑂𝐴) gcd (𝑂𝐵)) ≠ 0)
39 dvdsval2 14913 . . . . . . . 8 ((((𝑂𝐴) gcd (𝑂𝐵)) ∈ ℤ ∧ ((𝑂𝐴) gcd (𝑂𝐵)) ≠ 0 ∧ (𝑂𝐴) ∈ ℤ) → (((𝑂𝐴) gcd (𝑂𝐵)) ∥ (𝑂𝐴) ↔ ((𝑂𝐴) / ((𝑂𝐴) gcd (𝑂𝐵))) ∈ ℤ))
4037, 38, 28, 39syl3anc 1323 . . . . . . 7 (((𝐺 ∈ Abel ∧ 𝐴𝑋𝐵𝑋) ∧ ((𝑂𝐴) gcd (𝑂𝐵)) ≠ 0) → (((𝑂𝐴) gcd (𝑂𝐵)) ∥ (𝑂𝐴) ↔ ((𝑂𝐴) / ((𝑂𝐴) gcd (𝑂𝐵))) ∈ ℤ))
4136, 40mpbid 222 . . . . . 6 (((𝐺 ∈ Abel ∧ 𝐴𝑋𝐵𝑋) ∧ ((𝑂𝐴) gcd (𝑂𝐵)) ≠ 0) → ((𝑂𝐴) / ((𝑂𝐴) gcd (𝑂𝐵))) ∈ ℤ)
4241zcnd 11430 . . . . 5 (((𝐺 ∈ Abel ∧ 𝐴𝑋𝐵𝑋) ∧ ((𝑂𝐴) gcd (𝑂𝐵)) ≠ 0) → ((𝑂𝐴) / ((𝑂𝐴) gcd (𝑂𝐵))) ∈ ℂ)
4335simprd 479 . . . . . . 7 (((𝐺 ∈ Abel ∧ 𝐴𝑋𝐵𝑋) ∧ ((𝑂𝐴) gcd (𝑂𝐵)) ≠ 0) → ((𝑂𝐴) gcd (𝑂𝐵)) ∥ (𝑂𝐵))
44 dvdsval2 14913 . . . . . . . 8 ((((𝑂𝐴) gcd (𝑂𝐵)) ∈ ℤ ∧ ((𝑂𝐴) gcd (𝑂𝐵)) ≠ 0 ∧ (𝑂𝐵) ∈ ℤ) → (((𝑂𝐴) gcd (𝑂𝐵)) ∥ (𝑂𝐵) ↔ ((𝑂𝐵) / ((𝑂𝐴) gcd (𝑂𝐵))) ∈ ℤ))
4537, 38, 29, 44syl3anc 1323 . . . . . . 7 (((𝐺 ∈ Abel ∧ 𝐴𝑋𝐵𝑋) ∧ ((𝑂𝐴) gcd (𝑂𝐵)) ≠ 0) → (((𝑂𝐴) gcd (𝑂𝐵)) ∥ (𝑂𝐵) ↔ ((𝑂𝐵) / ((𝑂𝐴) gcd (𝑂𝐵))) ∈ ℤ))
4643, 45mpbid 222 . . . . . 6 (((𝐺 ∈ Abel ∧ 𝐴𝑋𝐵𝑋) ∧ ((𝑂𝐴) gcd (𝑂𝐵)) ≠ 0) → ((𝑂𝐵) / ((𝑂𝐴) gcd (𝑂𝐵))) ∈ ℤ)
4746zcnd 11430 . . . . 5 (((𝐺 ∈ Abel ∧ 𝐴𝑋𝐵𝑋) ∧ ((𝑂𝐴) gcd (𝑂𝐵)) ≠ 0) → ((𝑂𝐵) / ((𝑂𝐴) gcd (𝑂𝐵))) ∈ ℂ)
4842, 31, 47, 31mul4d 10195 . . . 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 11430 . . . . . 6 (((𝐺 ∈ Abel ∧ 𝐴𝑋𝐵𝑋) ∧ ((𝑂𝐴) gcd (𝑂𝐵)) ≠ 0) → (𝑂𝐴) ∈ ℂ)
5049, 31, 38divcan1d 10749 . . . . 5 (((𝐺 ∈ Abel ∧ 𝐴𝑋𝐵𝑋) ∧ ((𝑂𝐴) gcd (𝑂𝐵)) ≠ 0) → (((𝑂𝐴) / ((𝑂𝐴) gcd (𝑂𝐵))) · ((𝑂𝐴) gcd (𝑂𝐵))) = (𝑂𝐴))
5129zcnd 11430 . . . . . 6 (((𝐺 ∈ Abel ∧ 𝐴𝑋𝐵𝑋) ∧ ((𝑂𝐴) gcd (𝑂𝐵)) ≠ 0) → (𝑂𝐵) ∈ ℂ)
5251, 31, 38divcan1d 10749 . . . . 5 (((𝐺 ∈ Abel ∧ 𝐴𝑋𝐵𝑋) ∧ ((𝑂𝐴) gcd (𝑂𝐵)) ≠ 0) → (((𝑂𝐵) / ((𝑂𝐴) gcd (𝑂𝐵))) · ((𝑂𝐴) gcd (𝑂𝐵))) = (𝑂𝐵))
5350, 52oveq12d 6625 . . . 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 2661 . . 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 14931 . . . . . . . . . 10 (((𝑂‘(𝐴 + 𝐵)) ∈ ℤ ∧ (𝑂𝐴) ∈ ℤ) → (𝑂𝐴) ∥ ((𝑂‘(𝐴 + 𝐵)) · (𝑂𝐴)))
5755, 28, 56syl2anc 692 . . . . . . . . 9 (((𝐺 ∈ Abel ∧ 𝐴𝑋𝐵𝑋) ∧ ((𝑂𝐴) gcd (𝑂𝐵)) ≠ 0) → (𝑂𝐴) ∥ ((𝑂‘(𝐴 + 𝐵)) · (𝑂𝐴)))
58 simpl1 1062 . . . . . . . . . . . . 13 (((𝐺 ∈ Abel ∧ 𝐴𝑋𝐵𝑋) ∧ ((𝑂𝐴) gcd (𝑂𝐵)) ≠ 0) → 𝐺 ∈ Abel)
5955, 29zmulcld 11435 . . . . . . . . . . . . 13 (((𝐺 ∈ Abel ∧ 𝐴𝑋𝐵𝑋) ∧ ((𝑂𝐴) gcd (𝑂𝐵)) ≠ 0) → ((𝑂‘(𝐴 + 𝐵)) · (𝑂𝐵)) ∈ ℤ)
60 simpl2 1063 . . . . . . . . . . . . 13 (((𝐺 ∈ Abel ∧ 𝐴𝑋𝐵𝑋) ∧ ((𝑂𝐴) gcd (𝑂𝐵)) ≠ 0) → 𝐴𝑋)
61 simpl3 1064 . . . . . . . . . . . . 13 (((𝐺 ∈ Abel ∧ 𝐴𝑋𝐵𝑋) ∧ ((𝑂𝐴) gcd (𝑂𝐵)) ≠ 0) → 𝐵𝑋)
62 eqid 2621 . . . . . . . . . . . . . 14 (.g𝐺) = (.g𝐺)
631, 62, 17mulgdi 18156 . . . . . . . . . . . . 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 1325 . . . . . . . . . . . 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 14931 . . . . . . . . . . . . . . 15 (((𝑂‘(𝐴 + 𝐵)) ∈ ℤ ∧ (𝑂𝐵) ∈ ℤ) → (𝑂𝐵) ∥ ((𝑂‘(𝐴 + 𝐵)) · (𝑂𝐵)))
6655, 29, 65syl2anc 692 . . . . . . . . . . . . . 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 2621 . . . . . . . . . . . . . . . 16 (0g𝐺) = (0g𝐺)
691, 2, 62, 68oddvds 17890 . . . . . . . . . . . . . . 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 1323 . . . . . . . . . . . . . 14 (((𝐺 ∈ Abel ∧ 𝐴𝑋𝐵𝑋) ∧ ((𝑂𝐴) gcd (𝑂𝐵)) ≠ 0) → ((𝑂𝐵) ∥ ((𝑂‘(𝐴 + 𝐵)) · (𝑂𝐵)) ↔ (((𝑂‘(𝐴 + 𝐵)) · (𝑂𝐵))(.g𝐺)𝐵) = (0g𝐺)))
7166, 70mpbid 222 . . . . . . . . . . . . 13 (((𝐺 ∈ Abel ∧ 𝐴𝑋𝐵𝑋) ∧ ((𝑂𝐴) gcd (𝑂𝐵)) ≠ 0) → (((𝑂‘(𝐴 + 𝐵)) · (𝑂𝐵))(.g𝐺)𝐵) = (0g𝐺))
7271oveq2d 6623 . . . . . . . . . . . 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 2655 . . . . . . . . . . 11 (((𝐺 ∈ Abel ∧ 𝐴𝑋𝐵𝑋) ∧ ((𝑂𝐴) gcd (𝑂𝐵)) ≠ 0) → (((𝑂‘(𝐴 + 𝐵)) · (𝑂𝐵))(.g𝐺)(𝐴 + 𝐵)) = ((((𝑂‘(𝐴 + 𝐵)) · (𝑂𝐵))(.g𝐺)𝐴) + (0g𝐺)))
74 dvdsmul1 14930 . . . . . . . . . . . . 13 (((𝑂‘(𝐴 + 𝐵)) ∈ ℤ ∧ (𝑂𝐵) ∈ ℤ) → (𝑂‘(𝐴 + 𝐵)) ∥ ((𝑂‘(𝐴 + 𝐵)) · (𝑂𝐵)))
7555, 29, 74syl2anc 692 . . . . . . . . . . . 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 17890 . . . . . . . . . . . . 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 1323 . . . . . . . . . . . 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 222 . . . . . . . . . . 11 (((𝐺 ∈ Abel ∧ 𝐴𝑋𝐵𝑋) ∧ ((𝑂𝐴) gcd (𝑂𝐵)) ≠ 0) → (((𝑂‘(𝐴 + 𝐵)) · (𝑂𝐵))(.g𝐺)(𝐴 + 𝐵)) = (0g𝐺))
801, 62mulgcl 17483 . . . . . . . . . . . . 13 ((𝐺 ∈ Grp ∧ ((𝑂‘(𝐴 + 𝐵)) · (𝑂𝐵)) ∈ ℤ ∧ 𝐴𝑋) → (((𝑂‘(𝐴 + 𝐵)) · (𝑂𝐵))(.g𝐺)𝐴) ∈ 𝑋)
8167, 59, 60, 80syl3anc 1323 . . . . . . . . . . . 12 (((𝐺 ∈ Abel ∧ 𝐴𝑋𝐵𝑋) ∧ ((𝑂𝐴) gcd (𝑂𝐵)) ≠ 0) → (((𝑂‘(𝐴 + 𝐵)) · (𝑂𝐵))(.g𝐺)𝐴) ∈ 𝑋)
821, 17, 68grprid 17377 . . . . . . . . . . . 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 692 . . . . . . . . . . 11 (((𝐺 ∈ Abel ∧ 𝐴𝑋𝐵𝑋) ∧ ((𝑂𝐴) gcd (𝑂𝐵)) ≠ 0) → ((((𝑂‘(𝐴 + 𝐵)) · (𝑂𝐵))(.g𝐺)𝐴) + (0g𝐺)) = (((𝑂‘(𝐴 + 𝐵)) · (𝑂𝐵))(.g𝐺)𝐴))
8473, 79, 833eqtr3rd 2664 . . . . . . . . . 10 (((𝐺 ∈ Abel ∧ 𝐴𝑋𝐵𝑋) ∧ ((𝑂𝐴) gcd (𝑂𝐵)) ≠ 0) → (((𝑂‘(𝐴 + 𝐵)) · (𝑂𝐵))(.g𝐺)𝐴) = (0g𝐺))
851, 2, 62, 68oddvds 17890 . . . . . . . . . . 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 1323 . . . . . . . . . 10 (((𝐺 ∈ Abel ∧ 𝐴𝑋𝐵𝑋) ∧ ((𝑂𝐴) gcd (𝑂𝐵)) ≠ 0) → ((𝑂𝐴) ∥ ((𝑂‘(𝐴 + 𝐵)) · (𝑂𝐵)) ↔ (((𝑂‘(𝐴 + 𝐵)) · (𝑂𝐵))(.g𝐺)𝐴) = (0g𝐺)))
8784, 86mpbird 247 . . . . . . . . 9 (((𝐺 ∈ Abel ∧ 𝐴𝑋𝐵𝑋) ∧ ((𝑂𝐴) gcd (𝑂𝐵)) ≠ 0) → (𝑂𝐴) ∥ ((𝑂‘(𝐴 + 𝐵)) · (𝑂𝐵)))
8855, 28zmulcld 11435 . . . . . . . . . 10 (((𝐺 ∈ Abel ∧ 𝐴𝑋𝐵𝑋) ∧ ((𝑂𝐴) gcd (𝑂𝐵)) ≠ 0) → ((𝑂‘(𝐴 + 𝐵)) · (𝑂𝐴)) ∈ ℤ)
89 dvdsgcd 15188 . . . . . . . . . 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 1323 . . . . . . . . 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 714 . . . . . . . 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 15192 . . . . . . . . 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 1323 . . . . . . . 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 4641 . . . . . . 7 (((𝐺 ∈ Abel ∧ 𝐴𝑋𝐵𝑋) ∧ ((𝑂𝐴) gcd (𝑂𝐵)) ≠ 0) → (𝑂𝐴) ∥ ((𝑂‘(𝐴 + 𝐵)) · ((𝑂𝐴) gcd (𝑂𝐵))))
9650, 95eqbrtrd 4637 . . . . . 6 (((𝐺 ∈ Abel ∧ 𝐴𝑋𝐵𝑋) ∧ ((𝑂𝐴) gcd (𝑂𝐵)) ≠ 0) → (((𝑂𝐴) / ((𝑂𝐴) gcd (𝑂𝐵))) · ((𝑂𝐴) gcd (𝑂𝐵))) ∥ ((𝑂‘(𝐴 + 𝐵)) · ((𝑂𝐴) gcd (𝑂𝐵))))
97 dvdsmulcr 14938 . . . . . . 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 1327 . . . . . 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 222 . . . . 5 (((𝐺 ∈ Abel ∧ 𝐴𝑋𝐵𝑋) ∧ ((𝑂𝐴) gcd (𝑂𝐵)) ≠ 0) → ((𝑂𝐴) / ((𝑂𝐴) gcd (𝑂𝐵))) ∥ (𝑂‘(𝐴 + 𝐵)))
1001, 62, 17mulgdi 18156 . . . . . . . . . . . . 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 1325 . . . . . . . . . . . 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 17890 . . . . . . . . . . . . . . 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 1323 . . . . . . . . . . . . . 14 (((𝐺 ∈ Abel ∧ 𝐴𝑋𝐵𝑋) ∧ ((𝑂𝐴) gcd (𝑂𝐵)) ≠ 0) → ((𝑂𝐴) ∥ ((𝑂‘(𝐴 + 𝐵)) · (𝑂𝐴)) ↔ (((𝑂‘(𝐴 + 𝐵)) · (𝑂𝐴))(.g𝐺)𝐴) = (0g𝐺)))
10457, 103mpbid 222 . . . . . . . . . . . . 13 (((𝐺 ∈ Abel ∧ 𝐴𝑋𝐵𝑋) ∧ ((𝑂𝐴) gcd (𝑂𝐵)) ≠ 0) → (((𝑂‘(𝐴 + 𝐵)) · (𝑂𝐴))(.g𝐺)𝐴) = (0g𝐺))
105104oveq1d 6622 . . . . . . . . . . . 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 2655 . . . . . . . . . . 11 (((𝐺 ∈ Abel ∧ 𝐴𝑋𝐵𝑋) ∧ ((𝑂𝐴) gcd (𝑂𝐵)) ≠ 0) → (((𝑂‘(𝐴 + 𝐵)) · (𝑂𝐴))(.g𝐺)(𝐴 + 𝐵)) = ((0g𝐺) + (((𝑂‘(𝐴 + 𝐵)) · (𝑂𝐴))(.g𝐺)𝐵)))
107 dvdsmul1 14930 . . . . . . . . . . . . 13 (((𝑂‘(𝐴 + 𝐵)) ∈ ℤ ∧ (𝑂𝐴) ∈ ℤ) → (𝑂‘(𝐴 + 𝐵)) ∥ ((𝑂‘(𝐴 + 𝐵)) · (𝑂𝐴)))
10855, 28, 107syl2anc 692 . . . . . . . . . . . 12 (((𝐺 ∈ Abel ∧ 𝐴𝑋𝐵𝑋) ∧ ((𝑂𝐴) gcd (𝑂𝐵)) ≠ 0) → (𝑂‘(𝐴 + 𝐵)) ∥ ((𝑂‘(𝐴 + 𝐵)) · (𝑂𝐴)))
1091, 2, 62, 68oddvds 17890 . . . . . . . . . . . . 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 1323 . . . . . . . . . . . 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 222 . . . . . . . . . . 11 (((𝐺 ∈ Abel ∧ 𝐴𝑋𝐵𝑋) ∧ ((𝑂𝐴) gcd (𝑂𝐵)) ≠ 0) → (((𝑂‘(𝐴 + 𝐵)) · (𝑂𝐴))(.g𝐺)(𝐴 + 𝐵)) = (0g𝐺))
1121, 62mulgcl 17483 . . . . . . . . . . . . 13 ((𝐺 ∈ Grp ∧ ((𝑂‘(𝐴 + 𝐵)) · (𝑂𝐴)) ∈ ℤ ∧ 𝐵𝑋) → (((𝑂‘(𝐴 + 𝐵)) · (𝑂𝐴))(.g𝐺)𝐵) ∈ 𝑋)
11367, 88, 61, 112syl3anc 1323 . . . . . . . . . . . 12 (((𝐺 ∈ Abel ∧ 𝐴𝑋𝐵𝑋) ∧ ((𝑂𝐴) gcd (𝑂𝐵)) ≠ 0) → (((𝑂‘(𝐴 + 𝐵)) · (𝑂𝐴))(.g𝐺)𝐵) ∈ 𝑋)
1141, 17, 68grplid 17376 . . . . . . . . . . . 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 692 . . . . . . . . . . 11 (((𝐺 ∈ Abel ∧ 𝐴𝑋𝐵𝑋) ∧ ((𝑂𝐴) gcd (𝑂𝐵)) ≠ 0) → ((0g𝐺) + (((𝑂‘(𝐴 + 𝐵)) · (𝑂𝐴))(.g𝐺)𝐵)) = (((𝑂‘(𝐴 + 𝐵)) · (𝑂𝐴))(.g𝐺)𝐵))
116106, 111, 1153eqtr3rd 2664 . . . . . . . . . 10 (((𝐺 ∈ Abel ∧ 𝐴𝑋𝐵𝑋) ∧ ((𝑂𝐴) gcd (𝑂𝐵)) ≠ 0) → (((𝑂‘(𝐴 + 𝐵)) · (𝑂𝐴))(.g𝐺)𝐵) = (0g𝐺))
1171, 2, 62, 68oddvds 17890 . . . . . . . . . . 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 1323 . . . . . . . . . 10 (((𝐺 ∈ Abel ∧ 𝐴𝑋𝐵𝑋) ∧ ((𝑂𝐴) gcd (𝑂𝐵)) ≠ 0) → ((𝑂𝐵) ∥ ((𝑂‘(𝐴 + 𝐵)) · (𝑂𝐴)) ↔ (((𝑂‘(𝐴 + 𝐵)) · (𝑂𝐴))(.g𝐺)𝐵) = (0g𝐺)))
119116, 118mpbird 247 . . . . . . . . 9 (((𝐺 ∈ Abel ∧ 𝐴𝑋𝐵𝑋) ∧ ((𝑂𝐴) gcd (𝑂𝐵)) ≠ 0) → (𝑂𝐵) ∥ ((𝑂‘(𝐴 + 𝐵)) · (𝑂𝐴)))
120 dvdsgcd 15188 . . . . . . . . . 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 1323 . . . . . . . . 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 714 . . . . . . . 8 (((𝐺 ∈ Abel ∧ 𝐴𝑋𝐵𝑋) ∧ ((𝑂𝐴) gcd (𝑂𝐵)) ≠ 0) → (𝑂𝐵) ∥ (((𝑂‘(𝐴 + 𝐵)) · (𝑂𝐴)) gcd ((𝑂‘(𝐴 + 𝐵)) · (𝑂𝐵))))
123122, 94breqtrd 4641 . . . . . . 7 (((𝐺 ∈ Abel ∧ 𝐴𝑋𝐵𝑋) ∧ ((𝑂𝐴) gcd (𝑂𝐵)) ≠ 0) → (𝑂𝐵) ∥ ((𝑂‘(𝐴 + 𝐵)) · ((𝑂𝐴) gcd (𝑂𝐵))))
12452, 123eqbrtrd 4637 . . . . . 6 (((𝐺 ∈ Abel ∧ 𝐴𝑋𝐵𝑋) ∧ ((𝑂𝐴) gcd (𝑂𝐵)) ≠ 0) → (((𝑂𝐵) / ((𝑂𝐴) gcd (𝑂𝐵))) · ((𝑂𝐴) gcd (𝑂𝐵))) ∥ ((𝑂‘(𝐴 + 𝐵)) · ((𝑂𝐴) gcd (𝑂𝐵))))
125 dvdsmulcr 14938 . . . . . . 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 1327 . . . . . 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 222 . . . . 5 (((𝐺 ∈ Abel ∧ 𝐴𝑋𝐵𝑋) ∧ ((𝑂𝐴) gcd (𝑂𝐵)) ≠ 0) → ((𝑂𝐵) / ((𝑂𝐴) gcd (𝑂𝐵))) ∥ (𝑂‘(𝐴 + 𝐵)))
12841, 46gcdcld 15157 . . . . . . . 8 (((𝐺 ∈ Abel ∧ 𝐴𝑋𝐵𝑋) ∧ ((𝑂𝐴) gcd (𝑂𝐵)) ≠ 0) → (((𝑂𝐴) / ((𝑂𝐴) gcd (𝑂𝐵))) gcd ((𝑂𝐵) / ((𝑂𝐴) gcd (𝑂𝐵)))) ∈ ℕ0)
129128nn0cnd 11300 . . . . . . 7 (((𝐺 ∈ Abel ∧ 𝐴𝑋𝐵𝑋) ∧ ((𝑂𝐴) gcd (𝑂𝐵)) ≠ 0) → (((𝑂𝐴) / ((𝑂𝐴) gcd (𝑂𝐵))) gcd ((𝑂𝐵) / ((𝑂𝐴) gcd (𝑂𝐵)))) ∈ ℂ)
130 1cnd 10003 . . . . . . 7 (((𝐺 ∈ Abel ∧ 𝐴𝑋𝐵𝑋) ∧ ((𝑂𝐴) gcd (𝑂𝐵)) ≠ 0) → 1 ∈ ℂ)
13131mulid2d 10005 . . . . . . . 8 (((𝐺 ∈ Abel ∧ 𝐴𝑋𝐵𝑋) ∧ ((𝑂𝐴) gcd (𝑂𝐵)) ≠ 0) → (1 · ((𝑂𝐴) gcd (𝑂𝐵))) = ((𝑂𝐴) gcd (𝑂𝐵)))
13250, 52oveq12d 6625 . . . . . . . 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 15194 . . . . . . . . 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 1323 . . . . . . . 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 2662 . . . . . . 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 10610 . . . . . 6 (((𝐺 ∈ Abel ∧ 𝐴𝑋𝐵𝑋) ∧ ((𝑂𝐴) gcd (𝑂𝐵)) ≠ 0) → (((𝑂𝐴) / ((𝑂𝐴) gcd (𝑂𝐵))) gcd ((𝑂𝐵) / ((𝑂𝐴) gcd (𝑂𝐵)))) = 1)
137 coprmdvds2 15295 . . . . . 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 1326 . . . . 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 714 . . . 4 (((𝐺 ∈ Abel ∧ 𝐴𝑋𝐵𝑋) ∧ ((𝑂𝐴) gcd (𝑂𝐵)) ≠ 0) → (((𝑂𝐴) / ((𝑂𝐴) gcd (𝑂𝐵))) · ((𝑂𝐵) / ((𝑂𝐴) gcd (𝑂𝐵)))) ∥ (𝑂‘(𝐴 + 𝐵)))
14041, 46zmulcld 11435 . . . . 5 (((𝐺 ∈ Abel ∧ 𝐴𝑋𝐵𝑋) ∧ ((𝑂𝐴) gcd (𝑂𝐵)) ≠ 0) → (((𝑂𝐴) / ((𝑂𝐴) gcd (𝑂𝐵))) · ((𝑂𝐵) / ((𝑂𝐴) gcd (𝑂𝐵)))) ∈ ℤ)
141 zsqcl 12877 . . . . . 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 14936 . . . . 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 1323 . . . 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 4639 . 2 (((𝐺 ∈ Abel ∧ 𝐴𝑋𝐵𝑋) ∧ ((𝑂𝐴) gcd (𝑂𝐵)) ≠ 0) → ((𝑂𝐴) · (𝑂𝐵)) ∥ ((𝑂‘(𝐴 + 𝐵)) · (((𝑂𝐴) gcd (𝑂𝐵))↑2)))
14727, 146pm2.61dane 2877 1 ((𝐺 ∈ Abel ∧ 𝐴𝑋𝐵𝑋) → ((𝑂𝐴) · (𝑂𝐵)) ∥ ((𝑂‘(𝐴 + 𝐵)) · (((𝑂𝐴) gcd (𝑂𝐵))↑2)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  w3a 1036   = wceq 1480  wcel 1987  wne 2790   class class class wbr 4615  cfv 5849  (class class class)co 6607  0cc0 9883  1c1 9884   · cmul 9888   / cdiv 10631  2c2 11017  0cn0 11239  cz 11324  cexp 12803  cdvds 14910   gcd cgcd 15143  Basecbs 15784  +gcplusg 15865  0gc0g 16024  Grpcgrp 17346  .gcmg 17464  odcod 17868  Abelcabl 18118
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4733  ax-sep 4743  ax-nul 4751  ax-pow 4805  ax-pr 4869  ax-un 6905  ax-inf2 8485  ax-cnex 9939  ax-resscn 9940  ax-1cn 9941  ax-icn 9942  ax-addcl 9943  ax-addrcl 9944  ax-mulcl 9945  ax-mulrcl 9946  ax-mulcom 9947  ax-addass 9948  ax-mulass 9949  ax-distr 9950  ax-i2m1 9951  ax-1ne0 9952  ax-1rid 9953  ax-rnegex 9954  ax-rrecex 9955  ax-cnre 9956  ax-pre-lttri 9957  ax-pre-lttrn 9958  ax-pre-ltadd 9959  ax-pre-mulgt0 9960  ax-pre-sup 9961
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3188  df-sbc 3419  df-csb 3516  df-dif 3559  df-un 3561  df-in 3563  df-ss 3570  df-pss 3572  df-nul 3894  df-if 4061  df-pw 4134  df-sn 4151  df-pr 4153  df-tp 4155  df-op 4157  df-uni 4405  df-iun 4489  df-br 4616  df-opab 4676  df-mpt 4677  df-tr 4715  df-eprel 4987  df-id 4991  df-po 4997  df-so 4998  df-fr 5035  df-we 5037  df-xp 5082  df-rel 5083  df-cnv 5084  df-co 5085  df-dm 5086  df-rn 5087  df-res 5088  df-ima 5089  df-pred 5641  df-ord 5687  df-on 5688  df-lim 5689  df-suc 5690  df-iota 5812  df-fun 5851  df-fn 5852  df-f 5853  df-f1 5854  df-fo 5855  df-f1o 5856  df-fv 5857  df-riota 6568  df-ov 6610  df-oprab 6611  df-mpt2 6612  df-om 7016  df-1st 7116  df-2nd 7117  df-wrecs 7355  df-recs 7416  df-rdg 7454  df-er 7690  df-en 7903  df-dom 7904  df-sdom 7905  df-sup 8295  df-inf 8296  df-pnf 10023  df-mnf 10024  df-xr 10025  df-ltxr 10026  df-le 10027  df-sub 10215  df-neg 10216  df-div 10632  df-nn 10968  df-2 11026  df-3 11027  df-n0 11240  df-z 11325  df-uz 11635  df-rp 11780  df-fz 12272  df-fzo 12410  df-fl 12536  df-mod 12612  df-seq 12745  df-exp 12804  df-cj 13776  df-re 13777  df-im 13778  df-sqrt 13912  df-abs 13913  df-dvds 14911  df-gcd 15144  df-0g 16026  df-mgm 17166  df-sgrp 17208  df-mnd 17219  df-grp 17349  df-minusg 17350  df-sbg 17351  df-mulg 17465  df-od 17872  df-cmn 18119  df-abl 18120
This theorem is referenced by:  odadd  18177
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