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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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