Theorem lsmcomx 19785

Description: Subgroup sum commutes (extended domain version). (Contributed by NM, 25-Feb-2014.) (Revised by Mario Carneiro, 19-Apr-2016.)

Hypotheses

Ref	Expression
lsmcomx.v	⊢ 𝐵 = (Base‘𝐺)
lsmcomx.s	⊢ ⊕ = (LSSum‘𝐺)

Assertion

Ref	Expression
lsmcomx	⊢ ((𝐺 ∈ Abel ∧ 𝑇 ⊆ 𝐵 ∧ 𝑈 ⊆ 𝐵) → (𝑇 ⊕ 𝑈) = (𝑈 ⊕ 𝑇))

Proof of Theorem lsmcomx

Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpl1 1192	. . . . . . 7 ⊢ (((𝐺 ∈ Abel ∧ 𝑇 ⊆ 𝐵 ∧ 𝑈 ⊆ 𝐵) ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ 𝑈)) → 𝐺 ∈ Abel)
2		simpl2 1193	. . . . . . . 8 ⊢ (((𝐺 ∈ Abel ∧ 𝑇 ⊆ 𝐵 ∧ 𝑈 ⊆ 𝐵) ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ 𝑈)) → 𝑇 ⊆ 𝐵)
3		simprl 770	. . . . . . . 8 ⊢ (((𝐺 ∈ Abel ∧ 𝑇 ⊆ 𝐵 ∧ 𝑈 ⊆ 𝐵) ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ 𝑈)) → 𝑦 ∈ 𝑇)
4	2, 3	sseldd 3934	. . . . . . 7 ⊢ (((𝐺 ∈ Abel ∧ 𝑇 ⊆ 𝐵 ∧ 𝑈 ⊆ 𝐵) ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ 𝑈)) → 𝑦 ∈ 𝐵)
5		simpl3 1194	. . . . . . . 8 ⊢ (((𝐺 ∈ Abel ∧ 𝑇 ⊆ 𝐵 ∧ 𝑈 ⊆ 𝐵) ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ 𝑈)) → 𝑈 ⊆ 𝐵)
6		simprr 772	. . . . . . . 8 ⊢ (((𝐺 ∈ Abel ∧ 𝑇 ⊆ 𝐵 ∧ 𝑈 ⊆ 𝐵) ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ 𝑈)) → 𝑧 ∈ 𝑈)
7	5, 6	sseldd 3934	. . . . . . 7 ⊢ (((𝐺 ∈ Abel ∧ 𝑇 ⊆ 𝐵 ∧ 𝑈 ⊆ 𝐵) ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ 𝑈)) → 𝑧 ∈ 𝐵)
8		lsmcomx.v	. . . . . . . 8 ⊢ 𝐵 = (Base‘𝐺)
9		eqid 2736	. . . . . . . 8 ⊢ (+g‘𝐺) = (+g‘𝐺)
10	8, 9	ablcom 19728	. . . . . . 7 ⊢ ((𝐺 ∈ Abel ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) → (𝑦(+g‘𝐺)𝑧) = (𝑧(+g‘𝐺)𝑦))
11	1, 4, 7, 10	syl3anc 1373	. . . . . 6 ⊢ (((𝐺 ∈ Abel ∧ 𝑇 ⊆ 𝐵 ∧ 𝑈 ⊆ 𝐵) ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ 𝑈)) → (𝑦(+g‘𝐺)𝑧) = (𝑧(+g‘𝐺)𝑦))
12	11	eqeq2d 2747	. . . . 5 ⊢ (((𝐺 ∈ Abel ∧ 𝑇 ⊆ 𝐵 ∧ 𝑈 ⊆ 𝐵) ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ 𝑈)) → (𝑥 = (𝑦(+g‘𝐺)𝑧) ↔ 𝑥 = (𝑧(+g‘𝐺)𝑦)))
13	12	2rexbidva 3199	. . . 4 ⊢ ((𝐺 ∈ Abel ∧ 𝑇 ⊆ 𝐵 ∧ 𝑈 ⊆ 𝐵) → (∃𝑦 ∈ 𝑇 ∃𝑧 ∈ 𝑈 𝑥 = (𝑦(+g‘𝐺)𝑧) ↔ ∃𝑦 ∈ 𝑇 ∃𝑧 ∈ 𝑈 𝑥 = (𝑧(+g‘𝐺)𝑦)))
14		rexcom 3265	. . . 4 ⊢ (∃𝑦 ∈ 𝑇 ∃𝑧 ∈ 𝑈 𝑥 = (𝑧(+g‘𝐺)𝑦) ↔ ∃𝑧 ∈ 𝑈 ∃𝑦 ∈ 𝑇 𝑥 = (𝑧(+g‘𝐺)𝑦))
15	13, 14	bitrdi 287	. . 3 ⊢ ((𝐺 ∈ Abel ∧ 𝑇 ⊆ 𝐵 ∧ 𝑈 ⊆ 𝐵) → (∃𝑦 ∈ 𝑇 ∃𝑧 ∈ 𝑈 𝑥 = (𝑦(+g‘𝐺)𝑧) ↔ ∃𝑧 ∈ 𝑈 ∃𝑦 ∈ 𝑇 𝑥 = (𝑧(+g‘𝐺)𝑦)))
16		lsmcomx.s	. . . 4 ⊢ ⊕ = (LSSum‘𝐺)
17	8, 9, 16	lsmelvalx 19569	. . 3 ⊢ ((𝐺 ∈ Abel ∧ 𝑇 ⊆ 𝐵 ∧ 𝑈 ⊆ 𝐵) → (𝑥 ∈ (𝑇 ⊕ 𝑈) ↔ ∃𝑦 ∈ 𝑇 ∃𝑧 ∈ 𝑈 𝑥 = (𝑦(+g‘𝐺)𝑧)))
18	8, 9, 16	lsmelvalx 19569	. . . 4 ⊢ ((𝐺 ∈ Abel ∧ 𝑈 ⊆ 𝐵 ∧ 𝑇 ⊆ 𝐵) → (𝑥 ∈ (𝑈 ⊕ 𝑇) ↔ ∃𝑧 ∈ 𝑈 ∃𝑦 ∈ 𝑇 𝑥 = (𝑧(+g‘𝐺)𝑦)))
19	18	3com23 1126	. . 3 ⊢ ((𝐺 ∈ Abel ∧ 𝑇 ⊆ 𝐵 ∧ 𝑈 ⊆ 𝐵) → (𝑥 ∈ (𝑈 ⊕ 𝑇) ↔ ∃𝑧 ∈ 𝑈 ∃𝑦 ∈ 𝑇 𝑥 = (𝑧(+g‘𝐺)𝑦)))
20	15, 17, 19	3bitr4d 311	. 2 ⊢ ((𝐺 ∈ Abel ∧ 𝑇 ⊆ 𝐵 ∧ 𝑈 ⊆ 𝐵) → (𝑥 ∈ (𝑇 ⊕ 𝑈) ↔ 𝑥 ∈ (𝑈 ⊕ 𝑇)))
21	20	eqrdv 2734	1 ⊢ ((𝐺 ∈ Abel ∧ 𝑇 ⊆ 𝐵 ∧ 𝑈 ⊆ 𝐵) → (𝑇 ⊕ 𝑈) = (𝑈 ⊕ 𝑇))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1541   ∈ wcel 2113  ∃wrex 3060   ⊆ wss 3901  ‘cfv 6492  (class class class)co 7358  Basecbs 17136  +_gcplusg 17177  LSSumclsm 19563  Abelcabl 19710

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-rep 5224  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377  ax-un 7680

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-iun 4948  df-br 5099  df-opab 5161  df-mpt 5180  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-ov 7361  df-oprab 7362  df-mpo 7363  df-1st 7933  df-2nd 7934  df-lsm 19565  df-cmn 19711  df-abl 19712

This theorem is referenced by:  lsmcom  19787