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Theorem dprddisj 19053
Description: The function 𝑆 is a family having trivial intersections. (Contributed by Mario Carneiro, 25-Apr-2016.)
Hypotheses
Ref Expression
dprdcntz.1 (𝜑𝐺dom DProd 𝑆)
dprdcntz.2 (𝜑 → dom 𝑆 = 𝐼)
dprdcntz.3 (𝜑𝑋𝐼)
dprddisj.0 0 = (0g𝐺)
dprddisj.k 𝐾 = (mrCls‘(SubGrp‘𝐺))
Assertion
Ref Expression
dprddisj (𝜑 → ((𝑆𝑋) ∩ (𝐾 (𝑆 “ (𝐼 ∖ {𝑋})))) = { 0 })

Proof of Theorem dprddisj
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fveq2 6666 . . . 4 (𝑥 = 𝑋 → (𝑆𝑥) = (𝑆𝑋))
2 sneq 4573 . . . . . . . 8 (𝑥 = 𝑋 → {𝑥} = {𝑋})
32difeq2d 4102 . . . . . . 7 (𝑥 = 𝑋 → (𝐼 ∖ {𝑥}) = (𝐼 ∖ {𝑋}))
43imaeq2d 5926 . . . . . 6 (𝑥 = 𝑋 → (𝑆 “ (𝐼 ∖ {𝑥})) = (𝑆 “ (𝐼 ∖ {𝑋})))
54unieqd 4846 . . . . 5 (𝑥 = 𝑋 (𝑆 “ (𝐼 ∖ {𝑥})) = (𝑆 “ (𝐼 ∖ {𝑋})))
65fveq2d 6670 . . . 4 (𝑥 = 𝑋 → (𝐾 (𝑆 “ (𝐼 ∖ {𝑥}))) = (𝐾 (𝑆 “ (𝐼 ∖ {𝑋}))))
71, 6ineq12d 4193 . . 3 (𝑥 = 𝑋 → ((𝑆𝑥) ∩ (𝐾 (𝑆 “ (𝐼 ∖ {𝑥})))) = ((𝑆𝑋) ∩ (𝐾 (𝑆 “ (𝐼 ∖ {𝑋})))))
87eqeq1d 2827 . 2 (𝑥 = 𝑋 → (((𝑆𝑥) ∩ (𝐾 (𝑆 “ (𝐼 ∖ {𝑥})))) = { 0 } ↔ ((𝑆𝑋) ∩ (𝐾 (𝑆 “ (𝐼 ∖ {𝑋})))) = { 0 }))
9 dprdcntz.1 . . . . 5 (𝜑𝐺dom DProd 𝑆)
10 dprdcntz.2 . . . . . . 7 (𝜑 → dom 𝑆 = 𝐼)
119, 10dprddomcld 19045 . . . . . 6 (𝜑𝐼 ∈ V)
12 eqid 2825 . . . . . . 7 (Cntz‘𝐺) = (Cntz‘𝐺)
13 dprddisj.0 . . . . . . 7 0 = (0g𝐺)
14 dprddisj.k . . . . . . 7 𝐾 = (mrCls‘(SubGrp‘𝐺))
1512, 13, 14dmdprd 19042 . . . . . 6 ((𝐼 ∈ V ∧ dom 𝑆 = 𝐼) → (𝐺dom DProd 𝑆 ↔ (𝐺 ∈ Grp ∧ 𝑆:𝐼⟶(SubGrp‘𝐺) ∧ ∀𝑥𝐼 (∀𝑦 ∈ (𝐼 ∖ {𝑥})(𝑆𝑥) ⊆ ((Cntz‘𝐺)‘(𝑆𝑦)) ∧ ((𝑆𝑥) ∩ (𝐾 (𝑆 “ (𝐼 ∖ {𝑥})))) = { 0 }))))
1611, 10, 15syl2anc 584 . . . . 5 (𝜑 → (𝐺dom DProd 𝑆 ↔ (𝐺 ∈ Grp ∧ 𝑆:𝐼⟶(SubGrp‘𝐺) ∧ ∀𝑥𝐼 (∀𝑦 ∈ (𝐼 ∖ {𝑥})(𝑆𝑥) ⊆ ((Cntz‘𝐺)‘(𝑆𝑦)) ∧ ((𝑆𝑥) ∩ (𝐾 (𝑆 “ (𝐼 ∖ {𝑥})))) = { 0 }))))
179, 16mpbid 233 . . . 4 (𝜑 → (𝐺 ∈ Grp ∧ 𝑆:𝐼⟶(SubGrp‘𝐺) ∧ ∀𝑥𝐼 (∀𝑦 ∈ (𝐼 ∖ {𝑥})(𝑆𝑥) ⊆ ((Cntz‘𝐺)‘(𝑆𝑦)) ∧ ((𝑆𝑥) ∩ (𝐾 (𝑆 “ (𝐼 ∖ {𝑥})))) = { 0 })))
1817simp3d 1138 . . 3 (𝜑 → ∀𝑥𝐼 (∀𝑦 ∈ (𝐼 ∖ {𝑥})(𝑆𝑥) ⊆ ((Cntz‘𝐺)‘(𝑆𝑦)) ∧ ((𝑆𝑥) ∩ (𝐾 (𝑆 “ (𝐼 ∖ {𝑥})))) = { 0 }))
19 simpr 485 . . . 4 ((∀𝑦 ∈ (𝐼 ∖ {𝑥})(𝑆𝑥) ⊆ ((Cntz‘𝐺)‘(𝑆𝑦)) ∧ ((𝑆𝑥) ∩ (𝐾 (𝑆 “ (𝐼 ∖ {𝑥})))) = { 0 }) → ((𝑆𝑥) ∩ (𝐾 (𝑆 “ (𝐼 ∖ {𝑥})))) = { 0 })
2019ralimi 3164 . . 3 (∀𝑥𝐼 (∀𝑦 ∈ (𝐼 ∖ {𝑥})(𝑆𝑥) ⊆ ((Cntz‘𝐺)‘(𝑆𝑦)) ∧ ((𝑆𝑥) ∩ (𝐾 (𝑆 “ (𝐼 ∖ {𝑥})))) = { 0 }) → ∀𝑥𝐼 ((𝑆𝑥) ∩ (𝐾 (𝑆 “ (𝐼 ∖ {𝑥})))) = { 0 })
2118, 20syl 17 . 2 (𝜑 → ∀𝑥𝐼 ((𝑆𝑥) ∩ (𝐾 (𝑆 “ (𝐼 ∖ {𝑥})))) = { 0 })
22 dprdcntz.3 . 2 (𝜑𝑋𝐼)
238, 21, 22rspcdva 3628 1 (𝜑 → ((𝑆𝑋) ∩ (𝐾 (𝑆 “ (𝐼 ∖ {𝑋})))) = { 0 })
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396  w3a 1081   = wceq 1530  wcel 2107  wral 3142  Vcvv 3499  cdif 3936  cin 3938  wss 3939  {csn 4563   cuni 4836   class class class wbr 5062  dom cdm 5553  cima 5556  wf 6347  cfv 6351  0gc0g 16705  mrClscmrc 16846  Grpcgrp 18035  SubGrpcsubg 18205  Cntzccntz 18377   DProd cdprd 19037
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2797  ax-rep 5186  ax-sep 5199  ax-nul 5206  ax-pow 5262  ax-pr 5325  ax-un 7454
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2619  df-eu 2651  df-clab 2804  df-cleq 2818  df-clel 2897  df-nfc 2967  df-ne 3021  df-nel 3128  df-ral 3147  df-rex 3148  df-reu 3149  df-rab 3151  df-v 3501  df-sbc 3776  df-csb 3887  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4470  df-pw 4543  df-sn 4564  df-pr 4566  df-op 4570  df-uni 4837  df-iun 4918  df-br 5063  df-opab 5125  df-mpt 5143  df-id 5458  df-xp 5559  df-rel 5560  df-cnv 5561  df-co 5562  df-dm 5563  df-rn 5564  df-res 5565  df-ima 5566  df-iota 6311  df-fun 6353  df-fn 6354  df-f 6355  df-f1 6356  df-fo 6357  df-f1o 6358  df-fv 6359  df-oprab 7155  df-mpo 7156  df-1st 7683  df-2nd 7684  df-ixp 8454  df-dprd 19039
This theorem is referenced by:  dprdfeq0  19066  dprdres  19072  dprdss  19073  dprdf1o  19076  dprd2da  19086  dmdprdsplit2lem  19089
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