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Theorem dprddisj 18389
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 dprdcntz.3 . 2 (𝜑𝑋𝐼)
2 dprdcntz.1 . . . . 5 (𝜑𝐺dom DProd 𝑆)
3 dprdcntz.2 . . . . . . 7 (𝜑 → dom 𝑆 = 𝐼)
42, 3dprddomcld 18381 . . . . . 6 (𝜑𝐼 ∈ V)
5 eqid 2620 . . . . . . 7 (Cntz‘𝐺) = (Cntz‘𝐺)
6 dprddisj.0 . . . . . . 7 0 = (0g𝐺)
7 dprddisj.k . . . . . . 7 𝐾 = (mrCls‘(SubGrp‘𝐺))
85, 6, 7dmdprd 18378 . . . . . 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 }))))
94, 3, 8syl2anc 692 . . . . 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 }))))
102, 9mpbid 222 . . . 4 (𝜑 → (𝐺 ∈ Grp ∧ 𝑆:𝐼⟶(SubGrp‘𝐺) ∧ ∀𝑥𝐼 (∀𝑦 ∈ (𝐼 ∖ {𝑥})(𝑆𝑥) ⊆ ((Cntz‘𝐺)‘(𝑆𝑦)) ∧ ((𝑆𝑥) ∩ (𝐾 (𝑆 “ (𝐼 ∖ {𝑥})))) = { 0 })))
1110simp3d 1073 . . 3 (𝜑 → ∀𝑥𝐼 (∀𝑦 ∈ (𝐼 ∖ {𝑥})(𝑆𝑥) ⊆ ((Cntz‘𝐺)‘(𝑆𝑦)) ∧ ((𝑆𝑥) ∩ (𝐾 (𝑆 “ (𝐼 ∖ {𝑥})))) = { 0 }))
12 simpr 477 . . . 4 ((∀𝑦 ∈ (𝐼 ∖ {𝑥})(𝑆𝑥) ⊆ ((Cntz‘𝐺)‘(𝑆𝑦)) ∧ ((𝑆𝑥) ∩ (𝐾 (𝑆 “ (𝐼 ∖ {𝑥})))) = { 0 }) → ((𝑆𝑥) ∩ (𝐾 (𝑆 “ (𝐼 ∖ {𝑥})))) = { 0 })
1312ralimi 2949 . . 3 (∀𝑥𝐼 (∀𝑦 ∈ (𝐼 ∖ {𝑥})(𝑆𝑥) ⊆ ((Cntz‘𝐺)‘(𝑆𝑦)) ∧ ((𝑆𝑥) ∩ (𝐾 (𝑆 “ (𝐼 ∖ {𝑥})))) = { 0 }) → ∀𝑥𝐼 ((𝑆𝑥) ∩ (𝐾 (𝑆 “ (𝐼 ∖ {𝑥})))) = { 0 })
1411, 13syl 17 . 2 (𝜑 → ∀𝑥𝐼 ((𝑆𝑥) ∩ (𝐾 (𝑆 “ (𝐼 ∖ {𝑥})))) = { 0 })
15 fveq2 6178 . . . . 5 (𝑥 = 𝑋 → (𝑆𝑥) = (𝑆𝑋))
16 sneq 4178 . . . . . . . . 9 (𝑥 = 𝑋 → {𝑥} = {𝑋})
1716difeq2d 3720 . . . . . . . 8 (𝑥 = 𝑋 → (𝐼 ∖ {𝑥}) = (𝐼 ∖ {𝑋}))
1817imaeq2d 5454 . . . . . . 7 (𝑥 = 𝑋 → (𝑆 “ (𝐼 ∖ {𝑥})) = (𝑆 “ (𝐼 ∖ {𝑋})))
1918unieqd 4437 . . . . . 6 (𝑥 = 𝑋 (𝑆 “ (𝐼 ∖ {𝑥})) = (𝑆 “ (𝐼 ∖ {𝑋})))
2019fveq2d 6182 . . . . 5 (𝑥 = 𝑋 → (𝐾 (𝑆 “ (𝐼 ∖ {𝑥}))) = (𝐾 (𝑆 “ (𝐼 ∖ {𝑋}))))
2115, 20ineq12d 3807 . . . 4 (𝑥 = 𝑋 → ((𝑆𝑥) ∩ (𝐾 (𝑆 “ (𝐼 ∖ {𝑥})))) = ((𝑆𝑋) ∩ (𝐾 (𝑆 “ (𝐼 ∖ {𝑋})))))
2221eqeq1d 2622 . . 3 (𝑥 = 𝑋 → (((𝑆𝑥) ∩ (𝐾 (𝑆 “ (𝐼 ∖ {𝑥})))) = { 0 } ↔ ((𝑆𝑋) ∩ (𝐾 (𝑆 “ (𝐼 ∖ {𝑋})))) = { 0 }))
2322rspcv 3300 . 2 (𝑋𝐼 → (∀𝑥𝐼 ((𝑆𝑥) ∩ (𝐾 (𝑆 “ (𝐼 ∖ {𝑥})))) = { 0 } → ((𝑆𝑋) ∩ (𝐾 (𝑆 “ (𝐼 ∖ {𝑋})))) = { 0 }))
241, 14, 23sylc 65 1 (𝜑 → ((𝑆𝑋) ∩ (𝐾 (𝑆 “ (𝐼 ∖ {𝑋})))) = { 0 })
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  w3a 1036   = wceq 1481  wcel 1988  wral 2909  Vcvv 3195  cdif 3564  cin 3566  wss 3567  {csn 4168   cuni 4427   class class class wbr 4644  dom cdm 5104  cima 5107  wf 5872  cfv 5876  0gc0g 16081  mrClscmrc 16224  Grpcgrp 17403  SubGrpcsubg 17569  Cntzccntz 17729   DProd cdprd 18373
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-rep 4762  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-nel 2895  df-ral 2914  df-rex 2915  df-reu 2916  df-rab 2918  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-op 4175  df-uni 4428  df-iun 4513  df-br 4645  df-opab 4704  df-mpt 4721  df-id 5014  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-f1 5881  df-fo 5882  df-f1o 5883  df-fv 5884  df-oprab 6639  df-mpt2 6640  df-1st 7153  df-2nd 7154  df-ixp 7894  df-dprd 18375
This theorem is referenced by:  dprdfeq0  18402  dprdres  18408  dprdss  18409  dprdf1o  18412  dprd2da  18422  dmdprdsplit2lem  18425
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