Theorem ablo4 27404

Description: Commutative/associative law for Abelian groups. (Contributed by NM, 26-Apr-2007.) (New usage is discouraged.)

Hypothesis

Ref	Expression
ablcom.1	⊢ 𝑋 = ran 𝐺

Assertion

Ref	Expression
ablo4	⊢ ((𝐺 ∈ AbelOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋)) → ((𝐴𝐺𝐵)𝐺(𝐶𝐺𝐷)) = ((𝐴𝐺𝐶)𝐺(𝐵𝐺𝐷)))

Proof of Theorem ablo4

Step	Hyp	Ref	Expression
1		simprll 802	. . . . . 6 ⊢ ((𝐺 ∈ AbelOp ∧ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋))) → 𝐴 ∈ 𝑋)
2		simprlr 803	. . . . . 6 ⊢ ((𝐺 ∈ AbelOp ∧ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋))) → 𝐵 ∈ 𝑋)
3		simprrl 804	. . . . . 6 ⊢ ((𝐺 ∈ AbelOp ∧ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋))) → 𝐶 ∈ 𝑋)
4	1, 2, 3	3jca 1242	. . . . 5 ⊢ ((𝐺 ∈ AbelOp ∧ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋))) → (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋))
5		ablcom.1	. . . . . 6 ⊢ 𝑋 = ran 𝐺
6	5	ablo32 27403	. . . . 5 ⊢ ((𝐺 ∈ AbelOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → ((𝐴𝐺𝐵)𝐺𝐶) = ((𝐴𝐺𝐶)𝐺𝐵))
7	4, 6	syldan 487	. . . 4 ⊢ ((𝐺 ∈ AbelOp ∧ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋))) → ((𝐴𝐺𝐵)𝐺𝐶) = ((𝐴𝐺𝐶)𝐺𝐵))
8	7	oveq1d 6665	. . 3 ⊢ ((𝐺 ∈ AbelOp ∧ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋))) → (((𝐴𝐺𝐵)𝐺𝐶)𝐺𝐷) = (((𝐴𝐺𝐶)𝐺𝐵)𝐺𝐷))
9		ablogrpo 27401	. . . 4 ⊢ (𝐺 ∈ AbelOp → 𝐺 ∈ GrpOp)
10	5	grpocl 27354	. . . . . . . 8 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐺𝐵) ∈ 𝑋)
11	10	3expb 1266	. . . . . . 7 ⊢ ((𝐺 ∈ GrpOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → (𝐴𝐺𝐵) ∈ 𝑋)
12	11	adantrr 753	. . . . . 6 ⊢ ((𝐺 ∈ GrpOp ∧ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋))) → (𝐴𝐺𝐵) ∈ 𝑋)
13		simprrl 804	. . . . . 6 ⊢ ((𝐺 ∈ GrpOp ∧ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋))) → 𝐶 ∈ 𝑋)
14		simprrr 805	. . . . . 6 ⊢ ((𝐺 ∈ GrpOp ∧ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋))) → 𝐷 ∈ 𝑋)
15	12, 13, 14	3jca 1242	. . . . 5 ⊢ ((𝐺 ∈ GrpOp ∧ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋))) → ((𝐴𝐺𝐵) ∈ 𝑋 ∧ 𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋))
16	5	grpoass 27357	. . . . 5 ⊢ ((𝐺 ∈ GrpOp ∧ ((𝐴𝐺𝐵) ∈ 𝑋 ∧ 𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋)) → (((𝐴𝐺𝐵)𝐺𝐶)𝐺𝐷) = ((𝐴𝐺𝐵)𝐺(𝐶𝐺𝐷)))
17	15, 16	syldan 487	. . . 4 ⊢ ((𝐺 ∈ GrpOp ∧ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋))) → (((𝐴𝐺𝐵)𝐺𝐶)𝐺𝐷) = ((𝐴𝐺𝐵)𝐺(𝐶𝐺𝐷)))
18	9, 17	sylan 488	. . 3 ⊢ ((𝐺 ∈ AbelOp ∧ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋))) → (((𝐴𝐺𝐵)𝐺𝐶)𝐺𝐷) = ((𝐴𝐺𝐵)𝐺(𝐶𝐺𝐷)))
19	5	grpocl 27354	. . . . . . . . 9 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋) → (𝐴𝐺𝐶) ∈ 𝑋)
20	19	3expb 1266	. . . . . . . 8 ⊢ ((𝐺 ∈ GrpOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → (𝐴𝐺𝐶) ∈ 𝑋)
21	20	adantrlr 759	. . . . . . 7 ⊢ ((𝐺 ∈ GrpOp ∧ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ 𝐶 ∈ 𝑋)) → (𝐴𝐺𝐶) ∈ 𝑋)
22	21	adantrrr 761	. . . . . 6 ⊢ ((𝐺 ∈ GrpOp ∧ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋))) → (𝐴𝐺𝐶) ∈ 𝑋)
23		simprlr 803	. . . . . 6 ⊢ ((𝐺 ∈ GrpOp ∧ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋))) → 𝐵 ∈ 𝑋)
24	22, 23, 14	3jca 1242	. . . . 5 ⊢ ((𝐺 ∈ GrpOp ∧ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋))) → ((𝐴𝐺𝐶) ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋))
25	5	grpoass 27357	. . . . 5 ⊢ ((𝐺 ∈ GrpOp ∧ ((𝐴𝐺𝐶) ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋)) → (((𝐴𝐺𝐶)𝐺𝐵)𝐺𝐷) = ((𝐴𝐺𝐶)𝐺(𝐵𝐺𝐷)))
26	24, 25	syldan 487	. . . 4 ⊢ ((𝐺 ∈ GrpOp ∧ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋))) → (((𝐴𝐺𝐶)𝐺𝐵)𝐺𝐷) = ((𝐴𝐺𝐶)𝐺(𝐵𝐺𝐷)))
27	9, 26	sylan 488	. . 3 ⊢ ((𝐺 ∈ AbelOp ∧ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋))) → (((𝐴𝐺𝐶)𝐺𝐵)𝐺𝐷) = ((𝐴𝐺𝐶)𝐺(𝐵𝐺𝐷)))
28	8, 18, 27	3eqtr3d 2664	. 2 ⊢ ((𝐺 ∈ AbelOp ∧ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋))) → ((𝐴𝐺𝐵)𝐺(𝐶𝐺𝐷)) = ((𝐴𝐺𝐶)𝐺(𝐵𝐺𝐷)))
29	28	3impb 1260	1 ⊢ ((𝐺 ∈ AbelOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋)) → ((𝐴𝐺𝐵)𝐺(𝐶𝐺𝐷)) = ((𝐴𝐺𝐶)𝐺(𝐵𝐺𝐷)))

Syntax hints:   → wi 4   ∧ wa 384   ∧ w3a 1037   = wceq 1483   ∈ wcel 1990  ran crn 5115  (class class class)co 6650  GrpOpcgr 27343  AbelOpcablo 27398

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-fo 5894  df-fv 5896  df-ov 6653  df-grpo 27347  df-ablo 27399

This theorem is referenced by:  nvadd4  27480  ipdirilem  27684  rngoa4  33715