Theorem dprdcntz 20033

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 6868	. . 3 ⊢ (𝑦 = 𝑌 → (𝑍‘(𝑆‘𝑦)) = (𝑍‘(𝑆‘𝑌)))
2	1	sseq2d 3968	. 2 ⊢ (𝑦 = 𝑌 → ((𝑆‘𝑋) ⊆ (𝑍‘(𝑆‘𝑦)) ↔ (𝑆‘𝑋) ⊆ (𝑍‘(𝑆‘𝑌))))
3		sneq 4591	. . . . 5 ⊢ (𝑥 = 𝑋 → {𝑥} = {𝑋})
4	3	difeq2d 4080	. . . 4 ⊢ (𝑥 = 𝑋 → (𝐼 ∖ {𝑥}) = (𝐼 ∖ {𝑋}))
5		fveq2 6863	. . . . 5 ⊢ (𝑥 = 𝑋 → (𝑆‘𝑥) = (𝑆‘𝑋))
6	5	sseq1d 3967	. . . 4 ⊢ (𝑥 = 𝑋 → ((𝑆‘𝑥) ⊆ (𝑍‘(𝑆‘𝑦)) ↔ (𝑆‘𝑋) ⊆ (𝑍‘(𝑆‘𝑦))))
7	4, 6	raleqbidv 3335	. . 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 20026	. . . . . . 7 ⊢ (𝜑 → 𝐼 ∈ V)
11		dprdcntz.z	. . . . . . . 8 ⊢ 𝑍 = (Cntz‘𝐺)
12		eqid 2761	. . . . . . . 8 ⊢ (0g‘𝐺) = (0g‘𝐺)
13		eqid 2761	. . . . . . . 8 ⊢ (mrCls‘(SubGrp‘𝐺)) = (mrCls‘(SubGrp‘𝐺))
14	11, 12, 13	dmdprd 20023	. . . . . . 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 593	. . . . . 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 1156	. . . 4 ⊢ (𝜑 → ∀𝑥 ∈ 𝐼 (∀𝑦 ∈ (𝐼 ∖ {𝑥})(𝑆‘𝑥) ⊆ (𝑍‘(𝑆‘𝑦)) ∧ ((𝑆‘𝑥) ∩ ((mrCls‘(SubGrp‘𝐺))‘∪ (𝑆 “ (𝐼 ∖ {𝑥})))) = {(0g‘𝐺)}))
18		simpl 486	. . . . 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 3098	. . . 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 3582	. 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 3011	. . 3 ⊢ (𝜑 → 𝑌 ≠ 𝑋)
26		eldifsn 4745	. . 3 ⊢ (𝑌 ∈ (𝐼 ∖ {𝑋}) ↔ (𝑌 ∈ 𝐼 ∧ 𝑌 ≠ 𝑋))
27	23, 25, 26	sylanbrc 592	. 2 ⊢ (𝜑 → 𝑌 ∈ (𝐼 ∖ {𝑋}))
28	2, 22, 27	rspcdva 3582	1 ⊢ (𝜑 → (𝑆‘𝑋) ⊆ (𝑍‘(𝑆‘𝑌)))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399   ∧ w3a 1097   = wceq 1559   ∈ wcel 2141   ≠ wne 2956  ∀wral 3075  Vcvv 3453   ∖ cdif 3901   ∩ cin 3903   ⊆ wss 3904  {csn 4581  ∪ cuni 4864   class class class wbr 5099  dom cdm 5645   “ cima 5648  ⟶wf 6513  ‘cfv 6517  0_gc0g 17451  mrClscmrc 17594  Grpcgrp 18958  SubGrpcsubg 19145  Cntzccntz 19338   DProd cdprd 20018

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-rep 5226  ax-sep 5245  ax-nul 5255  ax-pow 5321  ax-pr 5389  ax-un 7714

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-nel 3061  df-ral 3076  df-rex 3086  df-reu 3367  df-rab 3414  df-v 3455  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-iun 4950  df-br 5100  df-opab 5162  df-mpt 5181  df-id 5540  df-xp 5651  df-rel 5652  df-cnv 5653  df-co 5654  df-dm 5655  df-rn 5656  df-res 5657  df-ima 5658  df-iota 6473  df-fun 6519  df-fn 6520  df-f 6521  df-f1 6522  df-fo 6523  df-f1o 6524  df-fv 6525  df-oprab 7396  df-mpo 7397  df-1st 7966  df-2nd 7967  df-ixp 8876  df-dprd 20020

This theorem is referenced by:  dprdfcntz  20040  dprdfadd  20045  dprdres  20053  dprdss  20054  dprdf1o  20057  dprdcntz2  20063  dprd2da  20067  dmdprdsplit2lem  20070