Theorem dprdcntz 19698

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 6824	. . 3 ⊢ (𝑦 = 𝑌 → (𝑍‘(𝑆‘𝑦)) = (𝑍‘(𝑆‘𝑌)))
2	1	sseq2d 3963	. 2 ⊢ (𝑦 = 𝑌 → ((𝑆‘𝑋) ⊆ (𝑍‘(𝑆‘𝑦)) ↔ (𝑆‘𝑋) ⊆ (𝑍‘(𝑆‘𝑌))))
3		sneq 4582	. . . . 5 ⊢ (𝑥 = 𝑋 → {𝑥} = {𝑋})
4	3	difeq2d 4068	. . . 4 ⊢ (𝑥 = 𝑋 → (𝐼 ∖ {𝑥}) = (𝐼 ∖ {𝑋}))
5		fveq2 6819	. . . . 5 ⊢ (𝑥 = 𝑋 → (𝑆‘𝑥) = (𝑆‘𝑋))
6	5	sseq1d 3962	. . . 4 ⊢ (𝑥 = 𝑋 → ((𝑆‘𝑥) ⊆ (𝑍‘(𝑆‘𝑦)) ↔ (𝑆‘𝑋) ⊆ (𝑍‘(𝑆‘𝑦))))
7	4, 6	raleqbidv 3315	. . 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 19691	. . . . . . 7 ⊢ (𝜑 → 𝐼 ∈ V)
11		dprdcntz.z	. . . . . . . 8 ⊢ 𝑍 = (Cntz‘𝐺)
12		eqid 2736	. . . . . . . 8 ⊢ (0g‘𝐺) = (0g‘𝐺)
13		eqid 2736	. . . . . . . 8 ⊢ (mrCls‘(SubGrp‘𝐺)) = (mrCls‘(SubGrp‘𝐺))
14	11, 12, 13	dmdprd 19688	. . . . . . 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 3082	. . . 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 3571	. 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 2996	. . 3 ⊢ (𝜑 → 𝑌 ≠ 𝑋)
26		eldifsn 4733	. . 3 ⊢ (𝑌 ∈ (𝐼 ∖ {𝑋}) ↔ (𝑌 ∈ 𝐼 ∧ 𝑌 ≠ 𝑋))
27	23, 25, 26	sylanbrc 583	. 2 ⊢ (𝜑 → 𝑌 ∈ (𝐼 ∖ {𝑋}))
28	2, 22, 27	rspcdva 3571	1 ⊢ (𝜑 → (𝑆‘𝑋) ⊆ (𝑍‘(𝑆‘𝑌)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1086   = wceq 1540   ∈ wcel 2105   ≠ wne 2940  ∀wral 3061  Vcvv 3441   ∖ cdif 3894   ∩ cin 3896   ⊆ wss 3897  {csn 4572  ∪ cuni 4851   class class class wbr 5089  dom cdm 5614   “ cima 5617  ⟶wf 6469  ‘cfv 6473  0_gc0g 17239  mrClscmrc 17381  Grpcgrp 18665  SubGrpcsubg 18837  Cntzccntz 19009   DProd cdprd 19683

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2707  ax-rep 5226  ax-sep 5240  ax-nul 5247  ax-pow 5305  ax-pr 5369  ax-un 7642

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3350  df-rab 3404  df-v 3443  df-sbc 3727  df-csb 3843  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4269  df-if 4473  df-pw 4548  df-sn 4573  df-pr 4575  df-op 4579  df-uni 4852  df-iun 4940  df-br 5090  df-opab 5152  df-mpt 5173  df-id 5512  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-iota 6425  df-fun 6475  df-fn 6476  df-f 6477  df-f1 6478  df-fo 6479  df-f1o 6480  df-fv 6481  df-oprab 7333  df-mpo 7334  df-1st 7891  df-2nd 7892  df-ixp 8749  df-dprd 19685

This theorem is referenced by:  dprdfcntz  19705  dprdfadd  19710  dprdres  19718  dprdss  19719  dprdf1o  19722  dprdcntz2  19728  dprd2da  19732  dmdprdsplit2lem  19735