Theorem lsmcomx 18975

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 1187	. . . . . . 7 ⊢ (((𝐺 ∈ Abel ∧ 𝑇 ⊆ 𝐵 ∧ 𝑈 ⊆ 𝐵) ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ 𝑈)) → 𝐺 ∈ Abel)
2		simpl2 1188	. . . . . . . 8 ⊢ (((𝐺 ∈ Abel ∧ 𝑇 ⊆ 𝐵 ∧ 𝑈 ⊆ 𝐵) ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ 𝑈)) → 𝑇 ⊆ 𝐵)
3		simprl 769	. . . . . . . 8 ⊢ (((𝐺 ∈ Abel ∧ 𝑇 ⊆ 𝐵 ∧ 𝑈 ⊆ 𝐵) ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ 𝑈)) → 𝑦 ∈ 𝑇)
4	2, 3	sseldd 3967	. . . . . . 7 ⊢ (((𝐺 ∈ Abel ∧ 𝑇 ⊆ 𝐵 ∧ 𝑈 ⊆ 𝐵) ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ 𝑈)) → 𝑦 ∈ 𝐵)
5		simpl3 1189	. . . . . . . 8 ⊢ (((𝐺 ∈ Abel ∧ 𝑇 ⊆ 𝐵 ∧ 𝑈 ⊆ 𝐵) ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ 𝑈)) → 𝑈 ⊆ 𝐵)
6		simprr 771	. . . . . . . 8 ⊢ (((𝐺 ∈ Abel ∧ 𝑇 ⊆ 𝐵 ∧ 𝑈 ⊆ 𝐵) ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ 𝑈)) → 𝑧 ∈ 𝑈)
7	5, 6	sseldd 3967	. . . . . . 7 ⊢ (((𝐺 ∈ Abel ∧ 𝑇 ⊆ 𝐵 ∧ 𝑈 ⊆ 𝐵) ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ 𝑈)) → 𝑧 ∈ 𝐵)
8		lsmcomx.v	. . . . . . . 8 ⊢ 𝐵 = (Base‘𝐺)
9		eqid 2821	. . . . . . . 8 ⊢ (+g‘𝐺) = (+g‘𝐺)
10	8, 9	ablcom 18923	. . . . . . 7 ⊢ ((𝐺 ∈ Abel ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) → (𝑦(+g‘𝐺)𝑧) = (𝑧(+g‘𝐺)𝑦))
11	1, 4, 7, 10	syl3anc 1367	. . . . . 6 ⊢ (((𝐺 ∈ Abel ∧ 𝑇 ⊆ 𝐵 ∧ 𝑈 ⊆ 𝐵) ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ 𝑈)) → (𝑦(+g‘𝐺)𝑧) = (𝑧(+g‘𝐺)𝑦))
12	11	eqeq2d 2832	. . . . 5 ⊢ (((𝐺 ∈ Abel ∧ 𝑇 ⊆ 𝐵 ∧ 𝑈 ⊆ 𝐵) ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ 𝑈)) → (𝑥 = (𝑦(+g‘𝐺)𝑧) ↔ 𝑥 = (𝑧(+g‘𝐺)𝑦)))
13	12	2rexbidva 3299	. . . 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 3355	. . . 4 ⊢ (∃𝑦 ∈ 𝑇 ∃𝑧 ∈ 𝑈 𝑥 = (𝑧(+g‘𝐺)𝑦) ↔ ∃𝑧 ∈ 𝑈 ∃𝑦 ∈ 𝑇 𝑥 = (𝑧(+g‘𝐺)𝑦))
15	13, 14	syl6bb 289	. . 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 18764	. . 3 ⊢ ((𝐺 ∈ Abel ∧ 𝑇 ⊆ 𝐵 ∧ 𝑈 ⊆ 𝐵) → (𝑥 ∈ (𝑇 ⊕ 𝑈) ↔ ∃𝑦 ∈ 𝑇 ∃𝑧 ∈ 𝑈 𝑥 = (𝑦(+g‘𝐺)𝑧)))
18	8, 9, 16	lsmelvalx 18764	. . . 4 ⊢ ((𝐺 ∈ Abel ∧ 𝑈 ⊆ 𝐵 ∧ 𝑇 ⊆ 𝐵) → (𝑥 ∈ (𝑈 ⊕ 𝑇) ↔ ∃𝑧 ∈ 𝑈 ∃𝑦 ∈ 𝑇 𝑥 = (𝑧(+g‘𝐺)𝑦)))
19	18	3com23 1122	. . 3 ⊢ ((𝐺 ∈ Abel ∧ 𝑇 ⊆ 𝐵 ∧ 𝑈 ⊆ 𝐵) → (𝑥 ∈ (𝑈 ⊕ 𝑇) ↔ ∃𝑧 ∈ 𝑈 ∃𝑦 ∈ 𝑇 𝑥 = (𝑧(+g‘𝐺)𝑦)))
20	15, 17, 19	3bitr4d 313	. 2 ⊢ ((𝐺 ∈ Abel ∧ 𝑇 ⊆ 𝐵 ∧ 𝑈 ⊆ 𝐵) → (𝑥 ∈ (𝑇 ⊕ 𝑈) ↔ 𝑥 ∈ (𝑈 ⊕ 𝑇)))
21	20	eqrdv 2819	1 ⊢ ((𝐺 ∈ Abel ∧ 𝑇 ⊆ 𝐵 ∧ 𝑈 ⊆ 𝐵) → (𝑇 ⊕ 𝑈) = (𝑈 ⊕ 𝑇))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   ∧ w3a 1083   = wceq 1533   ∈ wcel 2110  ∃wrex 3139   ⊆ wss 3935  ‘cfv 6354  (class class class)co 7155  Basecbs 16482  +_gcplusg 16564  LSSumclsm 18758  Abelcabl 18906

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-rep 5189  ax-sep 5202  ax-nul 5209  ax-pow 5265  ax-pr 5329  ax-un 7460

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rab 3147  df-v 3496  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-pw 4540  df-sn 4567  df-pr 4569  df-op 4573  df-uni 4838  df-iun 4920  df-br 5066  df-opab 5128  df-mpt 5146  df-id 5459  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-iota 6313  df-fun 6356  df-fn 6357  df-f 6358  df-f1 6359  df-fo 6360  df-f1o 6361  df-fv 6362  df-ov 7158  df-oprab 7159  df-mpo 7160  df-1st 7688  df-2nd 7689  df-lsm 18760  df-cmn 18907  df-abl 18908

This theorem is referenced by:  lsmcom  18977