Theorem dprdcntz 19124

Description: The function 𝑆 is a family having pairwise commuting values. (Contributed by Mario Carneiro, 25-Apr-2016.)

Hypotheses

Ref	Expression
dprdcntz.1	⊢ (𝜑 → 𝐺dom DProd 𝑆)
dprdcntz.2	⊢ (𝜑 → dom 𝑆 = 𝐼)
dprdcntz.3	⊢ (𝜑 → 𝑋 ∈ 𝐼)
dprdcntz.4	⊢ (𝜑 → 𝑌 ∈ 𝐼)
dprdcntz.5	⊢ (𝜑 → 𝑋 ≠ 𝑌)
dprdcntz.z	⊢ 𝑍 = (Cntz‘𝐺)

Assertion

Ref	Expression
dprdcntz	⊢ (𝜑 → (𝑆‘𝑋) ⊆ (𝑍‘(𝑆‘𝑌)))

Proof of Theorem dprdcntz

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

Step	Hyp	Ref	Expression
1		2fveq3 6669	. . 3 ⊢ (𝑦 = 𝑌 → (𝑍‘(𝑆‘𝑦)) = (𝑍‘(𝑆‘𝑌)))
2	1	sseq2d 3998	. 2 ⊢ (𝑦 = 𝑌 → ((𝑆‘𝑋) ⊆ (𝑍‘(𝑆‘𝑦)) ↔ (𝑆‘𝑋) ⊆ (𝑍‘(𝑆‘𝑌))))
3		sneq 4570	. . . . 5 ⊢ (𝑥 = 𝑋 → {𝑥} = {𝑋})
4	3	difeq2d 4098	. . . 4 ⊢ (𝑥 = 𝑋 → (𝐼 ∖ {𝑥}) = (𝐼 ∖ {𝑋}))
5		fveq2 6664	. . . . 5 ⊢ (𝑥 = 𝑋 → (𝑆‘𝑥) = (𝑆‘𝑋))
6	5	sseq1d 3997	. . . 4 ⊢ (𝑥 = 𝑋 → ((𝑆‘𝑥) ⊆ (𝑍‘(𝑆‘𝑦)) ↔ (𝑆‘𝑋) ⊆ (𝑍‘(𝑆‘𝑦))))
7	4, 6	raleqbidv 3401	. . 3 ⊢ (𝑥 = 𝑋 → (∀𝑦 ∈ (𝐼 ∖ {𝑥})(𝑆‘𝑥) ⊆ (𝑍‘(𝑆‘𝑦)) ↔ ∀𝑦 ∈ (𝐼 ∖ {𝑋})(𝑆‘𝑋) ⊆ (𝑍‘(𝑆‘𝑦))))
8		dprdcntz.1	. . . . . 6 ⊢ (𝜑 → 𝐺dom DProd 𝑆)
9		dprdcntz.2	. . . . . . . 8 ⊢ (𝜑 → dom 𝑆 = 𝐼)
10	8, 9	dprddomcld 19117	. . . . . . 7 ⊢ (𝜑 → 𝐼 ∈ V)
11		dprdcntz.z	. . . . . . . 8 ⊢ 𝑍 = (Cntz‘𝐺)
12		eqid 2821	. . . . . . . 8 ⊢ (0g‘𝐺) = (0g‘𝐺)
13		eqid 2821	. . . . . . . 8 ⊢ (mrCls‘(SubGrp‘𝐺)) = (mrCls‘(SubGrp‘𝐺))
14	11, 12, 13	dmdprd 19114	. . . . . . 7 ⊢ ((𝐼 ∈ V ∧ dom 𝑆 = 𝐼) → (𝐺dom DProd 𝑆 ↔ (𝐺 ∈ Grp ∧ 𝑆:𝐼⟶(SubGrp‘𝐺) ∧ ∀𝑥 ∈ 𝐼 (∀𝑦 ∈ (𝐼 ∖ {𝑥})(𝑆‘𝑥) ⊆ (𝑍‘(𝑆‘𝑦)) ∧ ((𝑆‘𝑥) ∩ ((mrCls‘(SubGrp‘𝐺))‘∪ (𝑆 “ (𝐼 ∖ {𝑥})))) = {(0g‘𝐺)}))))
15	10, 9, 14	syl2anc 586	. . . . . 6 ⊢ (𝜑 → (𝐺dom DProd 𝑆 ↔ (𝐺 ∈ Grp ∧ 𝑆:𝐼⟶(SubGrp‘𝐺) ∧ ∀𝑥 ∈ 𝐼 (∀𝑦 ∈ (𝐼 ∖ {𝑥})(𝑆‘𝑥) ⊆ (𝑍‘(𝑆‘𝑦)) ∧ ((𝑆‘𝑥) ∩ ((mrCls‘(SubGrp‘𝐺))‘∪ (𝑆 “ (𝐼 ∖ {𝑥})))) = {(0g‘𝐺)}))))
16	8, 15	mpbid 234	. . . . 5 ⊢ (𝜑 → (𝐺 ∈ Grp ∧ 𝑆:𝐼⟶(SubGrp‘𝐺) ∧ ∀𝑥 ∈ 𝐼 (∀𝑦 ∈ (𝐼 ∖ {𝑥})(𝑆‘𝑥) ⊆ (𝑍‘(𝑆‘𝑦)) ∧ ((𝑆‘𝑥) ∩ ((mrCls‘(SubGrp‘𝐺))‘∪ (𝑆 “ (𝐼 ∖ {𝑥})))) = {(0g‘𝐺)})))
17	16	simp3d 1140	. . . 4 ⊢ (𝜑 → ∀𝑥 ∈ 𝐼 (∀𝑦 ∈ (𝐼 ∖ {𝑥})(𝑆‘𝑥) ⊆ (𝑍‘(𝑆‘𝑦)) ∧ ((𝑆‘𝑥) ∩ ((mrCls‘(SubGrp‘𝐺))‘∪ (𝑆 “ (𝐼 ∖ {𝑥})))) = {(0g‘𝐺)}))
18		simpl 485	. . . . 5 ⊢ ((∀𝑦 ∈ (𝐼 ∖ {𝑥})(𝑆‘𝑥) ⊆ (𝑍‘(𝑆‘𝑦)) ∧ ((𝑆‘𝑥) ∩ ((mrCls‘(SubGrp‘𝐺))‘∪ (𝑆 “ (𝐼 ∖ {𝑥})))) = {(0g‘𝐺)}) → ∀𝑦 ∈ (𝐼 ∖ {𝑥})(𝑆‘𝑥) ⊆ (𝑍‘(𝑆‘𝑦)))
19	18	ralimi 3160	. . . 4 ⊢ (∀𝑥 ∈ 𝐼 (∀𝑦 ∈ (𝐼 ∖ {𝑥})(𝑆‘𝑥) ⊆ (𝑍‘(𝑆‘𝑦)) ∧ ((𝑆‘𝑥) ∩ ((mrCls‘(SubGrp‘𝐺))‘∪ (𝑆 “ (𝐼 ∖ {𝑥})))) = {(0g‘𝐺)}) → ∀𝑥 ∈ 𝐼 ∀𝑦 ∈ (𝐼 ∖ {𝑥})(𝑆‘𝑥) ⊆ (𝑍‘(𝑆‘𝑦)))
20	17, 19	syl 17	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝐼 ∀𝑦 ∈ (𝐼 ∖ {𝑥})(𝑆‘𝑥) ⊆ (𝑍‘(𝑆‘𝑦)))
21		dprdcntz.3	. . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐼)
22	7, 20, 21	rspcdva 3624	. 2 ⊢ (𝜑 → ∀𝑦 ∈ (𝐼 ∖ {𝑋})(𝑆‘𝑋) ⊆ (𝑍‘(𝑆‘𝑦)))
23		dprdcntz.4	. . 3 ⊢ (𝜑 → 𝑌 ∈ 𝐼)
24		dprdcntz.5	. . . 4 ⊢ (𝜑 → 𝑋 ≠ 𝑌)
25	24	necomd 3071	. . 3 ⊢ (𝜑 → 𝑌 ≠ 𝑋)
26		eldifsn 4712	. . 3 ⊢ (𝑌 ∈ (𝐼 ∖ {𝑋}) ↔ (𝑌 ∈ 𝐼 ∧ 𝑌 ≠ 𝑋))
27	23, 25, 26	sylanbrc 585	. 2 ⊢ (𝜑 → 𝑌 ∈ (𝐼 ∖ {𝑋}))
28	2, 22, 27	rspcdva 3624	1 ⊢ (𝜑 → (𝑆‘𝑋) ⊆ (𝑍‘(𝑆‘𝑌)))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   ∧ w3a 1083   = wceq 1533   ∈ wcel 2110   ≠ wne 3016  ∀wral 3138  Vcvv 3494   ∖ cdif 3932   ∩ cin 3934   ⊆ wss 3935  {csn 4560  ∪ cuni 4831   class class class wbr 5058  dom cdm 5549   “ cima 5552  ⟶wf 6345  ‘cfv 6349  0_gc0g 16707  mrClscmrc 16848  Grpcgrp 18097  SubGrpcsubg 18267  Cntzccntz 18439   DProd cdprd 19109

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 5182  ax-sep 5195  ax-nul 5202  ax-pow 5258  ax-pr 5321  ax-un 7455

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-nel 3124  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 4561  df-pr 4563  df-op 4567  df-uni 4832  df-iun 4913  df-br 5059  df-opab 5121  df-mpt 5139  df-id 5454  df-xp 5555  df-rel 5556  df-cnv 5557  df-co 5558  df-dm 5559  df-rn 5560  df-res 5561  df-ima 5562  df-iota 6308  df-fun 6351  df-fn 6352  df-f 6353  df-f1 6354  df-fo 6355  df-f1o 6356  df-fv 6357  df-oprab 7154  df-mpo 7155  df-1st 7683  df-2nd 7684  df-ixp 8456  df-dprd 19111

This theorem is referenced by:  dprdfcntz  19131  dprdfadd  19136  dprdres  19144  dprdss  19145  dprdf1o  19148  dprdcntz2  19154  dprd2da  19158  dmdprdsplit2lem  19161