Theorem lsmcom2 19902

Description: Subgroup sum commutes. (Contributed by Mario Carneiro, 22-Apr-2016.)

Hypotheses

Ref	Expression
lsmsubg.p	⊢ ⊕ = (LSSum‘𝐺)
lsmsubg.z	⊢ 𝑍 = (Cntz‘𝐺)

Assertion

Ref	Expression
lsmcom2	⊢ ((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑇 ⊆ (𝑍‘𝑈)) → (𝑇 ⊕ 𝑈) = (𝑈 ⊕ 𝑇))

Proof of Theorem lsmcom2

Dummy variables 𝑎 𝑏 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simp3 1156	. . . . . . . . 9 ⊢ ((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑇 ⊆ (𝑍‘𝑈)) → 𝑇 ⊆ (𝑍‘𝑈))
2	1	sselda 3913	. . . . . . . 8 ⊢ (((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑇 ⊆ (𝑍‘𝑈)) ∧ 𝑎 ∈ 𝑇) → 𝑎 ∈ (𝑍‘𝑈))
3	2	adantrr 730	. . . . . . 7 ⊢ (((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑇 ⊆ (𝑍‘𝑈)) ∧ (𝑎 ∈ 𝑇 ∧ 𝑏 ∈ 𝑈)) → 𝑎 ∈ (𝑍‘𝑈))
4		simprr 785	. . . . . . 7 ⊢ (((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑇 ⊆ (𝑍‘𝑈)) ∧ (𝑎 ∈ 𝑇 ∧ 𝑏 ∈ 𝑈)) → 𝑏 ∈ 𝑈)
5		eqid 2763	. . . . . . . 8 ⊢ (+g‘𝐺) = (+g‘𝐺)
6		lsmsubg.z	. . . . . . . 8 ⊢ 𝑍 = (Cntz‘𝐺)
7	5, 6	cntzi 19576	. . . . . . 7 ⊢ ((𝑎 ∈ (𝑍‘𝑈) ∧ 𝑏 ∈ 𝑈) → (𝑎(+g‘𝐺)𝑏) = (𝑏(+g‘𝐺)𝑎))
8	3, 4, 7	syl2anc 596	. . . . . 6 ⊢ (((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑇 ⊆ (𝑍‘𝑈)) ∧ (𝑎 ∈ 𝑇 ∧ 𝑏 ∈ 𝑈)) → (𝑎(+g‘𝐺)𝑏) = (𝑏(+g‘𝐺)𝑎))
9	8	eqeq2d 2774	. . . . 5 ⊢ (((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑇 ⊆ (𝑍‘𝑈)) ∧ (𝑎 ∈ 𝑇 ∧ 𝑏 ∈ 𝑈)) → (𝑥 = (𝑎(+g‘𝐺)𝑏) ↔ 𝑥 = (𝑏(+g‘𝐺)𝑎)))
10	9	2rexbidva 3229	. . . 4 ⊢ ((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑇 ⊆ (𝑍‘𝑈)) → (∃𝑎 ∈ 𝑇 ∃𝑏 ∈ 𝑈 𝑥 = (𝑎(+g‘𝐺)𝑏) ↔ ∃𝑎 ∈ 𝑇 ∃𝑏 ∈ 𝑈 𝑥 = (𝑏(+g‘𝐺)𝑎)))
11		rexcom 3295	. . . 4 ⊢ (∃𝑎 ∈ 𝑇 ∃𝑏 ∈ 𝑈 𝑥 = (𝑏(+g‘𝐺)𝑎) ↔ ∃𝑏 ∈ 𝑈 ∃𝑎 ∈ 𝑇 𝑥 = (𝑏(+g‘𝐺)𝑎))
12	10, 11	bitrdi 290	. . 3 ⊢ ((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑇 ⊆ (𝑍‘𝑈)) → (∃𝑎 ∈ 𝑇 ∃𝑏 ∈ 𝑈 𝑥 = (𝑎(+g‘𝐺)𝑏) ↔ ∃𝑏 ∈ 𝑈 ∃𝑎 ∈ 𝑇 𝑥 = (𝑏(+g‘𝐺)𝑎)))
13		lsmsubg.p	. . . . 5 ⊢ ⊕ = (LSSum‘𝐺)
14	5, 13	lsmelval 19896	. . . 4 ⊢ ((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺)) → (𝑥 ∈ (𝑇 ⊕ 𝑈) ↔ ∃𝑎 ∈ 𝑇 ∃𝑏 ∈ 𝑈 𝑥 = (𝑎(+g‘𝐺)𝑏)))
15	14	3adant3 1150	. . 3 ⊢ ((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑇 ⊆ (𝑍‘𝑈)) → (𝑥 ∈ (𝑇 ⊕ 𝑈) ↔ ∃𝑎 ∈ 𝑇 ∃𝑏 ∈ 𝑈 𝑥 = (𝑎(+g‘𝐺)𝑏)))
16	5, 13	lsmelval 19896	. . . . 5 ⊢ ((𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑇 ∈ (SubGrp‘𝐺)) → (𝑥 ∈ (𝑈 ⊕ 𝑇) ↔ ∃𝑏 ∈ 𝑈 ∃𝑎 ∈ 𝑇 𝑥 = (𝑏(+g‘𝐺)𝑎)))
17	16	ancoms 464	. . . 4 ⊢ ((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺)) → (𝑥 ∈ (𝑈 ⊕ 𝑇) ↔ ∃𝑏 ∈ 𝑈 ∃𝑎 ∈ 𝑇 𝑥 = (𝑏(+g‘𝐺)𝑎)))
18	17	3adant3 1150	. . 3 ⊢ ((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑇 ⊆ (𝑍‘𝑈)) → (𝑥 ∈ (𝑈 ⊕ 𝑇) ↔ ∃𝑏 ∈ 𝑈 ∃𝑎 ∈ 𝑇 𝑥 = (𝑏(+g‘𝐺)𝑎)))
19	12, 15, 18	3bitr4d 314	. 2 ⊢ ((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑇 ⊆ (𝑍‘𝑈)) → (𝑥 ∈ (𝑇 ⊕ 𝑈) ↔ 𝑥 ∈ (𝑈 ⊕ 𝑇)))
20	19	eqrdv 2761	1 ⊢ ((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑇 ⊆ (𝑍‘𝑈)) → (𝑇 ⊕ 𝑈) = (𝑈 ⊕ 𝑇))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2147  ∃wrex 3090   ⊆ wss 3900  ‘cfv 6561  (class class class)co 7446  +_gcplusg 17460  SubGrpcsubg 19362  Cntzccntz 19562  LSSumclsm 19881

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5236  ax-sep 5253  ax-nul 5264  ax-pow 5332  ax-pr 5396  ax-un 7778

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-iun 4956  df-br 5107  df-opab 5172  df-mpt 5191  df-id 5551  df-xp 5662  df-rel 5663  df-cnv 5664  df-co 5665  df-dm 5666  df-rn 5667  df-res 5668  df-ima 5669  df-iota 6515  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7449  df-oprab 7450  df-mpo 7451  df-1st 8028  df-2nd 8029  df-subg 19365  df-cntz 19564  df-lsm 19883

This theorem is used by:  lsmdisj3  19930  lsmdisj3r  19933  lsmdisj3a  19936  lsmdisj3b  19937  pj2f  19945  pj1id  19946