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Theorem dprdcntz 18628
Description: The function 𝑆 is a family having pairwise commuting values. (Contributed by Mario Carneiro, 25-Apr-2016.)
Hypotheses
Ref Expression
dprdcntz.1 (𝜑𝐺dom DProd 𝑆)
dprdcntz.2 (𝜑 → dom 𝑆 = 𝐼)
dprdcntz.3 (𝜑𝑋𝐼)
dprdcntz.4 (𝜑𝑌𝐼)
dprdcntz.5 (𝜑𝑋𝑌)
dprdcntz.z 𝑍 = (Cntz‘𝐺)
Assertion
Ref Expression
dprdcntz (𝜑 → (𝑆𝑋) ⊆ (𝑍‘(𝑆𝑌)))

Proof of Theorem dprdcntz
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fveq2 6354 . . . 4 (𝑦 = 𝑌 → (𝑆𝑦) = (𝑆𝑌))
21fveq2d 6358 . . 3 (𝑦 = 𝑌 → (𝑍‘(𝑆𝑦)) = (𝑍‘(𝑆𝑌)))
32sseq2d 3775 . 2 (𝑦 = 𝑌 → ((𝑆𝑋) ⊆ (𝑍‘(𝑆𝑦)) ↔ (𝑆𝑋) ⊆ (𝑍‘(𝑆𝑌))))
4 sneq 4332 . . . . 5 (𝑥 = 𝑋 → {𝑥} = {𝑋})
54difeq2d 3872 . . . 4 (𝑥 = 𝑋 → (𝐼 ∖ {𝑥}) = (𝐼 ∖ {𝑋}))
6 fveq2 6354 . . . . 5 (𝑥 = 𝑋 → (𝑆𝑥) = (𝑆𝑋))
76sseq1d 3774 . . . 4 (𝑥 = 𝑋 → ((𝑆𝑥) ⊆ (𝑍‘(𝑆𝑦)) ↔ (𝑆𝑋) ⊆ (𝑍‘(𝑆𝑦))))
85, 7raleqbidv 3292 . . 3 (𝑥 = 𝑋 → (∀𝑦 ∈ (𝐼 ∖ {𝑥})(𝑆𝑥) ⊆ (𝑍‘(𝑆𝑦)) ↔ ∀𝑦 ∈ (𝐼 ∖ {𝑋})(𝑆𝑋) ⊆ (𝑍‘(𝑆𝑦))))
9 dprdcntz.1 . . . . . 6 (𝜑𝐺dom DProd 𝑆)
10 dprdcntz.2 . . . . . . . 8 (𝜑 → dom 𝑆 = 𝐼)
119, 10dprddomcld 18621 . . . . . . 7 (𝜑𝐼 ∈ V)
12 dprdcntz.z . . . . . . . 8 𝑍 = (Cntz‘𝐺)
13 eqid 2761 . . . . . . . 8 (0g𝐺) = (0g𝐺)
14 eqid 2761 . . . . . . . 8 (mrCls‘(SubGrp‘𝐺)) = (mrCls‘(SubGrp‘𝐺))
1512, 13, 14dmdprd 18618 . . . . . . 7 ((𝐼 ∈ V ∧ dom 𝑆 = 𝐼) → (𝐺dom DProd 𝑆 ↔ (𝐺 ∈ Grp ∧ 𝑆:𝐼⟶(SubGrp‘𝐺) ∧ ∀𝑥𝐼 (∀𝑦 ∈ (𝐼 ∖ {𝑥})(𝑆𝑥) ⊆ (𝑍‘(𝑆𝑦)) ∧ ((𝑆𝑥) ∩ ((mrCls‘(SubGrp‘𝐺))‘ (𝑆 “ (𝐼 ∖ {𝑥})))) = {(0g𝐺)}))))
1611, 10, 15syl2anc 696 . . . . . 6 (𝜑 → (𝐺dom DProd 𝑆 ↔ (𝐺 ∈ Grp ∧ 𝑆:𝐼⟶(SubGrp‘𝐺) ∧ ∀𝑥𝐼 (∀𝑦 ∈ (𝐼 ∖ {𝑥})(𝑆𝑥) ⊆ (𝑍‘(𝑆𝑦)) ∧ ((𝑆𝑥) ∩ ((mrCls‘(SubGrp‘𝐺))‘ (𝑆 “ (𝐼 ∖ {𝑥})))) = {(0g𝐺)}))))
179, 16mpbid 222 . . . . 5 (𝜑 → (𝐺 ∈ Grp ∧ 𝑆:𝐼⟶(SubGrp‘𝐺) ∧ ∀𝑥𝐼 (∀𝑦 ∈ (𝐼 ∖ {𝑥})(𝑆𝑥) ⊆ (𝑍‘(𝑆𝑦)) ∧ ((𝑆𝑥) ∩ ((mrCls‘(SubGrp‘𝐺))‘ (𝑆 “ (𝐼 ∖ {𝑥})))) = {(0g𝐺)})))
1817simp3d 1139 . . . 4 (𝜑 → ∀𝑥𝐼 (∀𝑦 ∈ (𝐼 ∖ {𝑥})(𝑆𝑥) ⊆ (𝑍‘(𝑆𝑦)) ∧ ((𝑆𝑥) ∩ ((mrCls‘(SubGrp‘𝐺))‘ (𝑆 “ (𝐼 ∖ {𝑥})))) = {(0g𝐺)}))
19 simpl 474 . . . . 5 ((∀𝑦 ∈ (𝐼 ∖ {𝑥})(𝑆𝑥) ⊆ (𝑍‘(𝑆𝑦)) ∧ ((𝑆𝑥) ∩ ((mrCls‘(SubGrp‘𝐺))‘ (𝑆 “ (𝐼 ∖ {𝑥})))) = {(0g𝐺)}) → ∀𝑦 ∈ (𝐼 ∖ {𝑥})(𝑆𝑥) ⊆ (𝑍‘(𝑆𝑦)))
2019ralimi 3091 . . . 4 (∀𝑥𝐼 (∀𝑦 ∈ (𝐼 ∖ {𝑥})(𝑆𝑥) ⊆ (𝑍‘(𝑆𝑦)) ∧ ((𝑆𝑥) ∩ ((mrCls‘(SubGrp‘𝐺))‘ (𝑆 “ (𝐼 ∖ {𝑥})))) = {(0g𝐺)}) → ∀𝑥𝐼𝑦 ∈ (𝐼 ∖ {𝑥})(𝑆𝑥) ⊆ (𝑍‘(𝑆𝑦)))
2118, 20syl 17 . . 3 (𝜑 → ∀𝑥𝐼𝑦 ∈ (𝐼 ∖ {𝑥})(𝑆𝑥) ⊆ (𝑍‘(𝑆𝑦)))
22 dprdcntz.3 . . 3 (𝜑𝑋𝐼)
238, 21, 22rspcdva 3456 . 2 (𝜑 → ∀𝑦 ∈ (𝐼 ∖ {𝑋})(𝑆𝑋) ⊆ (𝑍‘(𝑆𝑦)))
24 dprdcntz.4 . . 3 (𝜑𝑌𝐼)
25 dprdcntz.5 . . . 4 (𝜑𝑋𝑌)
2625necomd 2988 . . 3 (𝜑𝑌𝑋)
27 eldifsn 4463 . . 3 (𝑌 ∈ (𝐼 ∖ {𝑋}) ↔ (𝑌𝐼𝑌𝑋))
2824, 26, 27sylanbrc 701 . 2 (𝜑𝑌 ∈ (𝐼 ∖ {𝑋}))
293, 23, 28rspcdva 3456 1 (𝜑 → (𝑆𝑋) ⊆ (𝑍‘(𝑆𝑌)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  w3a 1072   = wceq 1632  wcel 2140  wne 2933  wral 3051  Vcvv 3341  cdif 3713  cin 3715  wss 3716  {csn 4322   cuni 4589   class class class wbr 4805  dom cdm 5267  cima 5270  wf 6046  cfv 6050  0gc0g 16323  mrClscmrc 16466  Grpcgrp 17644  SubGrpcsubg 17810  Cntzccntz 17969   DProd cdprd 18613
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1989  ax-6 2055  ax-7 2091  ax-8 2142  ax-9 2149  ax-10 2169  ax-11 2184  ax-12 2197  ax-13 2392  ax-ext 2741  ax-rep 4924  ax-sep 4934  ax-nul 4942  ax-pow 4993  ax-pr 5056  ax-un 7116
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2048  df-eu 2612  df-mo 2613  df-clab 2748  df-cleq 2754  df-clel 2757  df-nfc 2892  df-ne 2934  df-nel 3037  df-ral 3056  df-rex 3057  df-reu 3058  df-rab 3060  df-v 3343  df-sbc 3578  df-csb 3676  df-dif 3719  df-un 3721  df-in 3723  df-ss 3730  df-nul 4060  df-if 4232  df-pw 4305  df-sn 4323  df-pr 4325  df-op 4329  df-uni 4590  df-iun 4675  df-br 4806  df-opab 4866  df-mpt 4883  df-id 5175  df-xp 5273  df-rel 5274  df-cnv 5275  df-co 5276  df-dm 5277  df-rn 5278  df-res 5279  df-ima 5280  df-iota 6013  df-fun 6052  df-fn 6053  df-f 6054  df-f1 6055  df-fo 6056  df-f1o 6057  df-fv 6058  df-oprab 6819  df-mpt2 6820  df-1st 7335  df-2nd 7336  df-ixp 8078  df-dprd 18615
This theorem is referenced by:  dprdfcntz  18635  dprdfadd  18640  dprdres  18648  dprdss  18649  dprdf1o  18652  dprdcntz2  18658  dprd2da  18662  dmdprdsplit2lem  18665
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