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Theorem dprdcntz 19126
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 2fveq3 6663 . . 3 (𝑦 = 𝑌 → (𝑍‘(𝑆𝑦)) = (𝑍‘(𝑆𝑌)))
21sseq2d 3984 . 2 (𝑦 = 𝑌 → ((𝑆𝑋) ⊆ (𝑍‘(𝑆𝑦)) ↔ (𝑆𝑋) ⊆ (𝑍‘(𝑆𝑌))))
3 sneq 4559 . . . . 5 (𝑥 = 𝑋 → {𝑥} = {𝑋})
43difeq2d 4084 . . . 4 (𝑥 = 𝑋 → (𝐼 ∖ {𝑥}) = (𝐼 ∖ {𝑋}))
5 fveq2 6658 . . . . 5 (𝑥 = 𝑋 → (𝑆𝑥) = (𝑆𝑋))
65sseq1d 3983 . . . 4 (𝑥 = 𝑋 → ((𝑆𝑥) ⊆ (𝑍‘(𝑆𝑦)) ↔ (𝑆𝑋) ⊆ (𝑍‘(𝑆𝑦))))
74, 6raleqbidv 3393 . . 3 (𝑥 = 𝑋 → (∀𝑦 ∈ (𝐼 ∖ {𝑥})(𝑆𝑥) ⊆ (𝑍‘(𝑆𝑦)) ↔ ∀𝑦 ∈ (𝐼 ∖ {𝑋})(𝑆𝑋) ⊆ (𝑍‘(𝑆𝑦))))
8 dprdcntz.1 . . . . . 6 (𝜑𝐺dom DProd 𝑆)
9 dprdcntz.2 . . . . . . . 8 (𝜑 → dom 𝑆 = 𝐼)
108, 9dprddomcld 19119 . . . . . . 7 (𝜑𝐼 ∈ V)
11 dprdcntz.z . . . . . . . 8 𝑍 = (Cntz‘𝐺)
12 eqid 2824 . . . . . . . 8 (0g𝐺) = (0g𝐺)
13 eqid 2824 . . . . . . . 8 (mrCls‘(SubGrp‘𝐺)) = (mrCls‘(SubGrp‘𝐺))
1411, 12, 13dmdprd 19116 . . . . . . 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)}))))
1510, 9, 14syl2anc 587 . . . . . 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)}))))
168, 15mpbid 235 . . . . 5 (𝜑 → (𝐺 ∈ Grp ∧ 𝑆:𝐼⟶(SubGrp‘𝐺) ∧ ∀𝑥𝐼 (∀𝑦 ∈ (𝐼 ∖ {𝑥})(𝑆𝑥) ⊆ (𝑍‘(𝑆𝑦)) ∧ ((𝑆𝑥) ∩ ((mrCls‘(SubGrp‘𝐺))‘ (𝑆 “ (𝐼 ∖ {𝑥})))) = {(0g𝐺)})))
1716simp3d 1141 . . . 4 (𝜑 → ∀𝑥𝐼 (∀𝑦 ∈ (𝐼 ∖ {𝑥})(𝑆𝑥) ⊆ (𝑍‘(𝑆𝑦)) ∧ ((𝑆𝑥) ∩ ((mrCls‘(SubGrp‘𝐺))‘ (𝑆 “ (𝐼 ∖ {𝑥})))) = {(0g𝐺)}))
18 simpl 486 . . . . 5 ((∀𝑦 ∈ (𝐼 ∖ {𝑥})(𝑆𝑥) ⊆ (𝑍‘(𝑆𝑦)) ∧ ((𝑆𝑥) ∩ ((mrCls‘(SubGrp‘𝐺))‘ (𝑆 “ (𝐼 ∖ {𝑥})))) = {(0g𝐺)}) → ∀𝑦 ∈ (𝐼 ∖ {𝑥})(𝑆𝑥) ⊆ (𝑍‘(𝑆𝑦)))
1918ralimi 3155 . . . 4 (∀𝑥𝐼 (∀𝑦 ∈ (𝐼 ∖ {𝑥})(𝑆𝑥) ⊆ (𝑍‘(𝑆𝑦)) ∧ ((𝑆𝑥) ∩ ((mrCls‘(SubGrp‘𝐺))‘ (𝑆 “ (𝐼 ∖ {𝑥})))) = {(0g𝐺)}) → ∀𝑥𝐼𝑦 ∈ (𝐼 ∖ {𝑥})(𝑆𝑥) ⊆ (𝑍‘(𝑆𝑦)))
2017, 19syl 17 . . 3 (𝜑 → ∀𝑥𝐼𝑦 ∈ (𝐼 ∖ {𝑥})(𝑆𝑥) ⊆ (𝑍‘(𝑆𝑦)))
21 dprdcntz.3 . . 3 (𝜑𝑋𝐼)
227, 20, 21rspcdva 3611 . 2 (𝜑 → ∀𝑦 ∈ (𝐼 ∖ {𝑋})(𝑆𝑋) ⊆ (𝑍‘(𝑆𝑦)))
23 dprdcntz.4 . . 3 (𝜑𝑌𝐼)
24 dprdcntz.5 . . . 4 (𝜑𝑋𝑌)
2524necomd 3069 . . 3 (𝜑𝑌𝑋)
26 eldifsn 4703 . . 3 (𝑌 ∈ (𝐼 ∖ {𝑋}) ↔ (𝑌𝐼𝑌𝑋))
2723, 25, 26sylanbrc 586 . 2 (𝜑𝑌 ∈ (𝐼 ∖ {𝑋}))
282, 22, 27rspcdva 3611 1 (𝜑 → (𝑆𝑋) ⊆ (𝑍‘(𝑆𝑌)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  w3a 1084   = wceq 1538  wcel 2115  wne 3014  wral 3133  Vcvv 3480  cdif 3916  cin 3918  wss 3919  {csn 4549   cuni 4824   class class class wbr 5052  dom cdm 5542  cima 5545  wf 6339  cfv 6343  0gc0g 16709  mrClscmrc 16850  Grpcgrp 18099  SubGrpcsubg 18269  Cntzccntz 18441   DProd cdprd 19111
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5176  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317  ax-un 7451
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-nel 3119  df-ral 3138  df-rex 3139  df-reu 3140  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4276  df-if 4450  df-pw 4523  df-sn 4550  df-pr 4552  df-op 4556  df-uni 4825  df-iun 4907  df-br 5053  df-opab 5115  df-mpt 5133  df-id 5447  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-res 5554  df-ima 5555  df-iota 6302  df-fun 6345  df-fn 6346  df-f 6347  df-f1 6348  df-fo 6349  df-f1o 6350  df-fv 6351  df-oprab 7149  df-mpo 7150  df-1st 7679  df-2nd 7680  df-ixp 8452  df-dprd 19113
This theorem is referenced by:  dprdfcntz  19133  dprdfadd  19138  dprdres  19146  dprdss  19147  dprdf1o  19150  dprdcntz2  19156  dprd2da  19160  dmdprdsplit2lem  19163
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