Theorem lsmcom2 18553

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 1119	. . . . . . . . 9 ⊢ ((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑇 ⊆ (𝑍‘𝑈)) → 𝑇 ⊆ (𝑍‘𝑈))
2	1	sselda 3851	. . . . . . . 8 ⊢ (((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑇 ⊆ (𝑍‘𝑈)) ∧ 𝑎 ∈ 𝑇) → 𝑎 ∈ (𝑍‘𝑈))
3	2	adantrr 705	. . . . . . 7 ⊢ (((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑇 ⊆ (𝑍‘𝑈)) ∧ (𝑎 ∈ 𝑇 ∧ 𝑏 ∈ 𝑈)) → 𝑎 ∈ (𝑍‘𝑈))
4		simprr 761	. . . . . . 7 ⊢ (((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑇 ⊆ (𝑍‘𝑈)) ∧ (𝑎 ∈ 𝑇 ∧ 𝑏 ∈ 𝑈)) → 𝑏 ∈ 𝑈)
5		eqid 2771	. . . . . . . 8 ⊢ (+g‘𝐺) = (+g‘𝐺)
6		lsmsubg.z	. . . . . . . 8 ⊢ 𝑍 = (Cntz‘𝐺)
7	5, 6	cntzi 18242	. . . . . . 7 ⊢ ((𝑎 ∈ (𝑍‘𝑈) ∧ 𝑏 ∈ 𝑈) → (𝑎(+g‘𝐺)𝑏) = (𝑏(+g‘𝐺)𝑎))
8	3, 4, 7	syl2anc 576	. . . . . 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 2781	. . . . 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 3237	. . . 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 3289	. . . 4 ⊢ (∃𝑎 ∈ 𝑇 ∃𝑏 ∈ 𝑈 𝑥 = (𝑏(+g‘𝐺)𝑎) ↔ ∃𝑏 ∈ 𝑈 ∃𝑎 ∈ 𝑇 𝑥 = (𝑏(+g‘𝐺)𝑎))
12	10, 11	syl6bb 279	. . 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 18547	. . . 4 ⊢ ((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺)) → (𝑥 ∈ (𝑇 ⊕ 𝑈) ↔ ∃𝑎 ∈ 𝑇 ∃𝑏 ∈ 𝑈 𝑥 = (𝑎(+g‘𝐺)𝑏)))
15	14	3adant3 1113	. . 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 18547	. . . . 5 ⊢ ((𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑇 ∈ (SubGrp‘𝐺)) → (𝑥 ∈ (𝑈 ⊕ 𝑇) ↔ ∃𝑏 ∈ 𝑈 ∃𝑎 ∈ 𝑇 𝑥 = (𝑏(+g‘𝐺)𝑎)))
17	16	ancoms 451	. . . 4 ⊢ ((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺)) → (𝑥 ∈ (𝑈 ⊕ 𝑇) ↔ ∃𝑏 ∈ 𝑈 ∃𝑎 ∈ 𝑇 𝑥 = (𝑏(+g‘𝐺)𝑎)))
18	17	3adant3 1113	. . 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 303	. 2 ⊢ ((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑇 ⊆ (𝑍‘𝑈)) → (𝑥 ∈ (𝑇 ⊕ 𝑈) ↔ 𝑥 ∈ (𝑈 ⊕ 𝑇)))
20	19	eqrdv 2769	1 ⊢ ((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑇 ⊆ (𝑍‘𝑈)) → (𝑇 ⊕ 𝑈) = (𝑈 ⊕ 𝑇))

Syntax hints:   → wi 4   ↔ wb 198   ∧ wa 387   ∧ w3a 1069   = wceq 1508   ∈ wcel 2051  ∃wrex 3082   ⊆ wss 3822  ‘cfv 6185  (class class class)co 6974  +_gcplusg 16419  SubGrpcsubg 18069  Cntzccntz 18228  LSSumclsm 18532

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1759  ax-4 1773  ax-5 1870  ax-6 1929  ax-7 1966  ax-8 2053  ax-9 2060  ax-10 2080  ax-11 2094  ax-12 2107  ax-13 2302  ax-ext 2743  ax-rep 5045  ax-sep 5056  ax-nul 5063  ax-pow 5115  ax-pr 5182  ax-un 7277

This theorem depends on definitions:  df-bi 199  df-an 388  df-or 835  df-3an 1071  df-tru 1511  df-ex 1744  df-nf 1748  df-sb 2017  df-mo 2548  df-eu 2585  df-clab 2752  df-cleq 2764  df-clel 2839  df-nfc 2911  df-ne 2961  df-ral 3086  df-rex 3087  df-reu 3088  df-rab 3090  df-v 3410  df-sbc 3675  df-csb 3780  df-dif 3825  df-un 3827  df-in 3829  df-ss 3836  df-nul 4173  df-if 4345  df-pw 4418  df-sn 4436  df-pr 4438  df-op 4442  df-uni 4709  df-iun 4790  df-br 4926  df-opab 4988  df-mpt 5005  df-id 5308  df-xp 5409  df-rel 5410  df-cnv 5411  df-co 5412  df-dm 5413  df-rn 5414  df-res 5415  df-ima 5416  df-iota 6149  df-fun 6187  df-fn 6188  df-f 6189  df-f1 6190  df-fo 6191  df-f1o 6192  df-fv 6193  df-ov 6977  df-oprab 6978  df-mpo 6979  df-1st 7499  df-2nd 7500  df-subg 18072  df-cntz 18230  df-lsm 18534

This theorem is referenced by:  lsmdisj3  18579  lsmdisj3r  18582  lsmdisj3a  18585  lsmdisj3b  18586  pj2f  18594  pj1id  18595