Theorem dprdcntz 19611

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 6779	. . 3 ⊢ (𝑦 = 𝑌 → (𝑍‘(𝑆‘𝑦)) = (𝑍‘(𝑆‘𝑌)))
2	1	sseq2d 3953	. 2 ⊢ (𝑦 = 𝑌 → ((𝑆‘𝑋) ⊆ (𝑍‘(𝑆‘𝑦)) ↔ (𝑆‘𝑋) ⊆ (𝑍‘(𝑆‘𝑌))))
3		sneq 4571	. . . . 5 ⊢ (𝑥 = 𝑋 → {𝑥} = {𝑋})
4	3	difeq2d 4057	. . . 4 ⊢ (𝑥 = 𝑋 → (𝐼 ∖ {𝑥}) = (𝐼 ∖ {𝑋}))
5		fveq2 6774	. . . . 5 ⊢ (𝑥 = 𝑋 → (𝑆‘𝑥) = (𝑆‘𝑋))
6	5	sseq1d 3952	. . . 4 ⊢ (𝑥 = 𝑋 → ((𝑆‘𝑥) ⊆ (𝑍‘(𝑆‘𝑦)) ↔ (𝑆‘𝑋) ⊆ (𝑍‘(𝑆‘𝑦))))
7	4, 6	raleqbidv 3336	. . 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 19604	. . . . . . 7 ⊢ (𝜑 → 𝐼 ∈ V)
11		dprdcntz.z	. . . . . . . 8 ⊢ 𝑍 = (Cntz‘𝐺)
12		eqid 2738	. . . . . . . 8 ⊢ (0g‘𝐺) = (0g‘𝐺)
13		eqid 2738	. . . . . . . 8 ⊢ (mrCls‘(SubGrp‘𝐺)) = (mrCls‘(SubGrp‘𝐺))
14	11, 12, 13	dmdprd 19601	. . . . . . 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 584	. . . . . 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 231	. . . . 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 1143	. . . 4 ⊢ (𝜑 → ∀𝑥 ∈ 𝐼 (∀𝑦 ∈ (𝐼 ∖ {𝑥})(𝑆‘𝑥) ⊆ (𝑍‘(𝑆‘𝑦)) ∧ ((𝑆‘𝑥) ∩ ((mrCls‘(SubGrp‘𝐺))‘∪ (𝑆 “ (𝐼 ∖ {𝑥})))) = {(0g‘𝐺)}))
18		simpl 483	. . . . 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 3087	. . . 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 3562	. 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 2999	. . 3 ⊢ (𝜑 → 𝑌 ≠ 𝑋)
26		eldifsn 4720	. . 3 ⊢ (𝑌 ∈ (𝐼 ∖ {𝑋}) ↔ (𝑌 ∈ 𝐼 ∧ 𝑌 ≠ 𝑋))
27	23, 25, 26	sylanbrc 583	. 2 ⊢ (𝜑 → 𝑌 ∈ (𝐼 ∖ {𝑋}))
28	2, 22, 27	rspcdva 3562	1 ⊢ (𝜑 → (𝑆‘𝑋) ⊆ (𝑍‘(𝑆‘𝑌)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1086   = wceq 1539   ∈ wcel 2106   ≠ wne 2943  ∀wral 3064  Vcvv 3432   ∖ cdif 3884   ∩ cin 3886   ⊆ wss 3887  {csn 4561  ∪ cuni 4839   class class class wbr 5074  dom cdm 5589   “ cima 5592  ⟶wf 6429  ‘cfv 6433  0_gc0g 17150  mrClscmrc 17292  Grpcgrp 18577  SubGrpcsubg 18749  Cntzccntz 18921   DProd cdprd 19596

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-oprab 7279  df-mpo 7280  df-1st 7831  df-2nd 7832  df-ixp 8686  df-dprd 19598

This theorem is referenced by:  dprdfcntz  19618  dprdfadd  19623  dprdres  19631  dprdss  19632  dprdf1o  19635  dprdcntz2  19641  dprd2da  19645  dmdprdsplit2lem  19648