Theorem ablomuldiv 30885

Description: Law for group multiplication and division. (Contributed by NM, 15-Feb-2008.) (New usage is discouraged.)

Hypotheses

Ref	Expression
abldiv.1	⊢ 𝑋 = ran 𝐺
abldiv.3	⊢ 𝐷 = ( /𝑔 ‘𝐺)

Assertion

Ref	Expression
ablomuldiv	⊢ ((𝐺 ∈ AbelOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → ((𝐴𝐺𝐵)𝐷𝐶) = ((𝐴𝐷𝐶)𝐺𝐵))

Proof of Theorem ablomuldiv

Step	Hyp	Ref	Expression
1		abldiv.1	. . . . 5 ⊢ 𝑋 = ran 𝐺
2	1	ablocom 30881	. . . 4 ⊢ ((𝐺 ∈ AbelOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐺𝐵) = (𝐵𝐺𝐴))
3	2	3adant3r3 1203	. . 3 ⊢ ((𝐺 ∈ AbelOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → (𝐴𝐺𝐵) = (𝐵𝐺𝐴))
4	3	oveq1d 7427	. 2 ⊢ ((𝐺 ∈ AbelOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → ((𝐴𝐺𝐵)𝐷𝐶) = ((𝐵𝐺𝐴)𝐷𝐶))
5		3ancoma 1115	. . 3 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋) ↔ (𝐵 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋))
6		ablogrpo 30880	. . . 4 ⊢ (𝐺 ∈ AbelOp → 𝐺 ∈ GrpOp)
7		abldiv.3	. . . . 5 ⊢ 𝐷 = ( /𝑔 ‘𝐺)
8	1, 7	grpomuldivass 30874	. . . 4 ⊢ ((𝐺 ∈ GrpOp ∧ (𝐵 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → ((𝐵𝐺𝐴)𝐷𝐶) = (𝐵𝐺(𝐴𝐷𝐶)))
9	6, 8	sylan 591	. . 3 ⊢ ((𝐺 ∈ AbelOp ∧ (𝐵 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → ((𝐵𝐺𝐴)𝐷𝐶) = (𝐵𝐺(𝐴𝐷𝐶)))
10	5, 9	sylan2b 605	. 2 ⊢ ((𝐺 ∈ AbelOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → ((𝐵𝐺𝐴)𝐷𝐶) = (𝐵𝐺(𝐴𝐷𝐶)))
11		simpr2 1214	. . . 4 ⊢ ((𝐺 ∈ AbelOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → 𝐵 ∈ 𝑋)
12	1, 7	grpodivcl 30872	. . . . . 6 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋) → (𝐴𝐷𝐶) ∈ 𝑋)
13	6, 12	syl3an1 1181	. . . . 5 ⊢ ((𝐺 ∈ AbelOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋) → (𝐴𝐷𝐶) ∈ 𝑋)
14	13	3adant3r2 1202	. . . 4 ⊢ ((𝐺 ∈ AbelOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → (𝐴𝐷𝐶) ∈ 𝑋)
15	11, 14	jca 520	. . 3 ⊢ ((𝐺 ∈ AbelOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → (𝐵 ∈ 𝑋 ∧ (𝐴𝐷𝐶) ∈ 𝑋))
16	1	ablocom 30881	. . . 4 ⊢ ((𝐺 ∈ AbelOp ∧ 𝐵 ∈ 𝑋 ∧ (𝐴𝐷𝐶) ∈ 𝑋) → (𝐵𝐺(𝐴𝐷𝐶)) = ((𝐴𝐷𝐶)𝐺𝐵))
17	16	3expb 1138	. . 3 ⊢ ((𝐺 ∈ AbelOp ∧ (𝐵 ∈ 𝑋 ∧ (𝐴𝐷𝐶) ∈ 𝑋)) → (𝐵𝐺(𝐴𝐷𝐶)) = ((𝐴𝐷𝐶)𝐺𝐵))
18	15, 17	syldan 602	. 2 ⊢ ((𝐺 ∈ AbelOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → (𝐵𝐺(𝐴𝐷𝐶)) = ((𝐴𝐷𝐶)𝐺𝐵))
19	4, 10, 18	3eqtrd 2802	1 ⊢ ((𝐺 ∈ AbelOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → ((𝐴𝐺𝐵)𝐷𝐶) = ((𝐴𝐷𝐶)𝐺𝐵))

Syntax hints:   → wi 4   ∧ wa 400   ∧ w3a 1103   = wceq 1570   ∈ wcel 2143  ran crn 5664  ‘cfv 6538  (class class class)co 7412  GrpOpcgr 30822   /_𝑔 cgs 30825  AbelOpcablo 30877

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5239  ax-sep 5258  ax-nul 5270  ax-pow 5338  ax-pr 5406  ax-un 7734

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3746  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4489  df-pw 4565  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-iun 4959  df-br 5111  df-opab 5175  df-mpt 5194  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-riota 7369  df-ov 7415  df-oprab 7416  df-mpo 7417  df-1st 7987  df-2nd 7988  df-grpo 30826  df-gid 30827  df-ginv 30828  df-gdiv 30829  df-ablo 30878

This theorem is referenced by:  ablodivdiv  30886  nvaddsub  30988  ablo4pnp  38512