Theorem lsmcomx 20076

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 1210	. . . . . . 7 ⊢ (((𝐺 ∈ Abel ∧ 𝑇 ⊆ 𝐵 ∧ 𝑈 ⊆ 𝐵) ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ 𝑈)) → 𝐺 ∈ Abel)
2		simpl2 1211	. . . . . . . 8 ⊢ (((𝐺 ∈ Abel ∧ 𝑇 ⊆ 𝐵 ∧ 𝑈 ⊆ 𝐵) ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ 𝑈)) → 𝑇 ⊆ 𝐵)
3		simprl 783	. . . . . . . 8 ⊢ (((𝐺 ∈ Abel ∧ 𝑇 ⊆ 𝐵 ∧ 𝑈 ⊆ 𝐵) ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ 𝑈)) → 𝑦 ∈ 𝑇)
4	2, 3	sseldd 3909	. . . . . . 7 ⊢ (((𝐺 ∈ Abel ∧ 𝑇 ⊆ 𝐵 ∧ 𝑈 ⊆ 𝐵) ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ 𝑈)) → 𝑦 ∈ 𝐵)
5		simpl3 1212	. . . . . . . 8 ⊢ (((𝐺 ∈ Abel ∧ 𝑇 ⊆ 𝐵 ∧ 𝑈 ⊆ 𝐵) ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ 𝑈)) → 𝑈 ⊆ 𝐵)
6		simprr 785	. . . . . . . 8 ⊢ (((𝐺 ∈ Abel ∧ 𝑇 ⊆ 𝐵 ∧ 𝑈 ⊆ 𝐵) ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ 𝑈)) → 𝑧 ∈ 𝑈)
7	5, 6	sseldd 3909	. . . . . . 7 ⊢ (((𝐺 ∈ Abel ∧ 𝑇 ⊆ 𝐵 ∧ 𝑈 ⊆ 𝐵) ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ 𝑈)) → 𝑧 ∈ 𝐵)
8		lsmcomx.v	. . . . . . . 8 ⊢ 𝐵 = (Base‘𝐺)
9		eqid 2760	. . . . . . . 8 ⊢ (+g‘𝐺) = (+g‘𝐺)
10	8, 9	ablcom 20019	. . . . . . 7 ⊢ ((𝐺 ∈ Abel ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) → (𝑦(+g‘𝐺)𝑧) = (𝑧(+g‘𝐺)𝑦))
11	1, 4, 7, 10	syl3anc 1398	. . . . . 6 ⊢ (((𝐺 ∈ Abel ∧ 𝑇 ⊆ 𝐵 ∧ 𝑈 ⊆ 𝐵) ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ 𝑈)) → (𝑦(+g‘𝐺)𝑧) = (𝑧(+g‘𝐺)𝑦))
12	11	eqeq2d 2771	. . . . 5 ⊢ (((𝐺 ∈ Abel ∧ 𝑇 ⊆ 𝐵 ∧ 𝑈 ⊆ 𝐵) ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ 𝑈)) → (𝑥 = (𝑦(+g‘𝐺)𝑧) ↔ 𝑥 = (𝑧(+g‘𝐺)𝑦)))
13	12	2rexbidva 3225	. . . 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 3291	. . . 4 ⊢ (∃𝑦 ∈ 𝑇 ∃𝑧 ∈ 𝑈 𝑥 = (𝑧(+g‘𝐺)𝑦) ↔ ∃𝑧 ∈ 𝑈 ∃𝑦 ∈ 𝑇 𝑥 = (𝑧(+g‘𝐺)𝑦))
15	13, 14	bitrdi 290	. . 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 19860	. . 3 ⊢ ((𝐺 ∈ Abel ∧ 𝑇 ⊆ 𝐵 ∧ 𝑈 ⊆ 𝐵) → (𝑥 ∈ (𝑇 ⊕ 𝑈) ↔ ∃𝑦 ∈ 𝑇 ∃𝑧 ∈ 𝑈 𝑥 = (𝑦(+g‘𝐺)𝑧)))
18	8, 9, 16	lsmelvalx 19860	. . . 4 ⊢ ((𝐺 ∈ Abel ∧ 𝑈 ⊆ 𝐵 ∧ 𝑇 ⊆ 𝐵) → (𝑥 ∈ (𝑈 ⊕ 𝑇) ↔ ∃𝑧 ∈ 𝑈 ∃𝑦 ∈ 𝑇 𝑥 = (𝑧(+g‘𝐺)𝑦)))
19	18	3com23 1144	. . 3 ⊢ ((𝐺 ∈ Abel ∧ 𝑇 ⊆ 𝐵 ∧ 𝑈 ⊆ 𝐵) → (𝑥 ∈ (𝑈 ⊕ 𝑇) ↔ ∃𝑧 ∈ 𝑈 ∃𝑦 ∈ 𝑇 𝑥 = (𝑧(+g‘𝐺)𝑦)))
20	15, 17, 19	3bitr4d 314	. 2 ⊢ ((𝐺 ∈ Abel ∧ 𝑇 ⊆ 𝐵 ∧ 𝑈 ⊆ 𝐵) → (𝑥 ∈ (𝑇 ⊕ 𝑈) ↔ 𝑥 ∈ (𝑈 ⊕ 𝑇)))
21	20	eqrdv 2758	1 ⊢ ((𝐺 ∈ Abel ∧ 𝑇 ⊆ 𝐵 ∧ 𝑈 ⊆ 𝐵) → (𝑇 ⊕ 𝑈) = (𝑈 ⊕ 𝑇))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1569   ∈ wcel 2144  ∃wrex 3086   ⊆ wss 3895  ‘cfv 6539  (class class class)co 7421  Basecbs 17393  +_gcplusg 17434  LSSumclsm 19854  Abelcabl 20001

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7752

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-iota 6494  df-fun 6541  df-fn 6542  df-f 6543  df-f1 6544  df-fo 6545  df-f1o 6546  df-fv 6547  df-ov 7424  df-oprab 7425  df-mpo 7426  df-1st 8002  df-2nd 8003  df-lsm 19856  df-cmn 20002  df-abl 20003

This theorem is used by:  lsmcom  20078