Theorem dprdcntz 19907

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 6831	. . 3 ⊢ (𝑦 = 𝑌 → (𝑍‘(𝑆‘𝑦)) = (𝑍‘(𝑆‘𝑌)))
2	1	sseq2d 3970	. 2 ⊢ (𝑦 = 𝑌 → ((𝑆‘𝑋) ⊆ (𝑍‘(𝑆‘𝑦)) ↔ (𝑆‘𝑋) ⊆ (𝑍‘(𝑆‘𝑌))))
3		sneq 4589	. . . . 5 ⊢ (𝑥 = 𝑋 → {𝑥} = {𝑋})
4	3	difeq2d 4079	. . . 4 ⊢ (𝑥 = 𝑋 → (𝐼 ∖ {𝑥}) = (𝐼 ∖ {𝑋}))
5		fveq2 6826	. . . . 5 ⊢ (𝑥 = 𝑋 → (𝑆‘𝑥) = (𝑆‘𝑋))
6	5	sseq1d 3969	. . . 4 ⊢ (𝑥 = 𝑋 → ((𝑆‘𝑥) ⊆ (𝑍‘(𝑆‘𝑦)) ↔ (𝑆‘𝑋) ⊆ (𝑍‘(𝑆‘𝑦))))
7	4, 6	raleqbidv 3310	. . 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 19900	. . . . . . 7 ⊢ (𝜑 → 𝐼 ∈ V)
11		dprdcntz.z	. . . . . . . 8 ⊢ 𝑍 = (Cntz‘𝐺)
12		eqid 2729	. . . . . . . 8 ⊢ (0g‘𝐺) = (0g‘𝐺)
13		eqid 2729	. . . . . . . 8 ⊢ (mrCls‘(SubGrp‘𝐺)) = (mrCls‘(SubGrp‘𝐺))
14	11, 12, 13	dmdprd 19897	. . . . . . 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 232	. . . . 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 1144	. . . 4 ⊢ (𝜑 → ∀𝑥 ∈ 𝐼 (∀𝑦 ∈ (𝐼 ∖ {𝑥})(𝑆‘𝑥) ⊆ (𝑍‘(𝑆‘𝑦)) ∧ ((𝑆‘𝑥) ∩ ((mrCls‘(SubGrp‘𝐺))‘∪ (𝑆 “ (𝐼 ∖ {𝑥})))) = {(0g‘𝐺)}))
18		simpl 482	. . . . 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 3066	. . . 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 3580	. 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 2980	. . 3 ⊢ (𝜑 → 𝑌 ≠ 𝑋)
26		eldifsn 4740	. . 3 ⊢ (𝑌 ∈ (𝐼 ∖ {𝑋}) ↔ (𝑌 ∈ 𝐼 ∧ 𝑌 ≠ 𝑋))
27	23, 25, 26	sylanbrc 583	. 2 ⊢ (𝜑 → 𝑌 ∈ (𝐼 ∖ {𝑋}))
28	2, 22, 27	rspcdva 3580	1 ⊢ (𝜑 → (𝑆‘𝑋) ⊆ (𝑍‘(𝑆‘𝑌)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109   ≠ wne 2925  ∀wral 3044  Vcvv 3438   ∖ cdif 3902   ∩ cin 3904   ⊆ wss 3905  {csn 4579  ∪ cuni 4861   class class class wbr 5095  dom cdm 5623   “ cima 5626  ⟶wf 6482  ‘cfv 6486  0_gc0g 17361  mrClscmrc 17503  Grpcgrp 18830  SubGrpcsubg 19017  Cntzccntz 19212   DProd cdprd 19892

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5221  ax-sep 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374  ax-un 7675

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-reu 3346  df-rab 3397  df-v 3440  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4862  df-iun 4946  df-br 5096  df-opab 5158  df-mpt 5177  df-id 5518  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-oprab 7357  df-mpo 7358  df-1st 7931  df-2nd 7932  df-ixp 8832  df-dprd 19894

This theorem is referenced by:  dprdfcntz  19914  dprdfadd  19919  dprdres  19927  dprdss  19928  dprdf1o  19931  dprdcntz2  19937  dprd2da  19941  dmdprdsplit2lem  19944