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Theorem dprddisj 18454
 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 18446 . . . . . 6 (𝜑𝐼 ∈ V)
5 eqid 2651 . . . . . . 7 (Cntz‘𝐺) = (Cntz‘𝐺)
6 dprddisj.0 . . . . . . 7 0 = (0g𝐺)
7 dprddisj.k . . . . . . 7 𝐾 = (mrCls‘(SubGrp‘𝐺))
85, 6, 7dmdprd 18443 . . . . . 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 694 . . . . 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 1095 . . 3 (𝜑 → ∀𝑥𝐼 (∀𝑦 ∈ (𝐼 ∖ {𝑥})(𝑆𝑥) ⊆ ((Cntz‘𝐺)‘(𝑆𝑦)) ∧ ((𝑆𝑥) ∩ (𝐾 (𝑆 “ (𝐼 ∖ {𝑥})))) = { 0 }))
12 simpr 476 . . . 4 ((∀𝑦 ∈ (𝐼 ∖ {𝑥})(𝑆𝑥) ⊆ ((Cntz‘𝐺)‘(𝑆𝑦)) ∧ ((𝑆𝑥) ∩ (𝐾 (𝑆 “ (𝐼 ∖ {𝑥})))) = { 0 }) → ((𝑆𝑥) ∩ (𝐾 (𝑆 “ (𝐼 ∖ {𝑥})))) = { 0 })
1312ralimi 2981 . . 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 6229 . . . . 5 (𝑥 = 𝑋 → (𝑆𝑥) = (𝑆𝑋))
16 sneq 4220 . . . . . . . . 9 (𝑥 = 𝑋 → {𝑥} = {𝑋})
1716difeq2d 3761 . . . . . . . 8 (𝑥 = 𝑋 → (𝐼 ∖ {𝑥}) = (𝐼 ∖ {𝑋}))
1817imaeq2d 5501 . . . . . . 7 (𝑥 = 𝑋 → (𝑆 “ (𝐼 ∖ {𝑥})) = (𝑆 “ (𝐼 ∖ {𝑋})))
1918unieqd 4478 . . . . . 6 (𝑥 = 𝑋 (𝑆 “ (𝐼 ∖ {𝑥})) = (𝑆 “ (𝐼 ∖ {𝑋})))
2019fveq2d 6233 . . . . 5 (𝑥 = 𝑋 → (𝐾 (𝑆 “ (𝐼 ∖ {𝑥}))) = (𝐾 (𝑆 “ (𝐼 ∖ {𝑋}))))
2115, 20ineq12d 3848 . . . 4 (𝑥 = 𝑋 → ((𝑆𝑥) ∩ (𝐾 (𝑆 “ (𝐼 ∖ {𝑥})))) = ((𝑆𝑋) ∩ (𝐾 (𝑆 “ (𝐼 ∖ {𝑋})))))
2221eqeq1d 2653 . . 3 (𝑥 = 𝑋 → (((𝑆𝑥) ∩ (𝐾 (𝑆 “ (𝐼 ∖ {𝑥})))) = { 0 } ↔ ((𝑆𝑋) ∩ (𝐾 (𝑆 “ (𝐼 ∖ {𝑋})))) = { 0 }))
2322rspcv 3336 . 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 383   ∧ w3a 1054   = wceq 1523   ∈ wcel 2030  ∀wral 2941  Vcvv 3231   ∖ cdif 3604   ∩ cin 3606   ⊆ wss 3607  {csn 4210  ∪ cuni 4468   class class class wbr 4685  dom cdm 5143   “ cima 5146  ⟶wf 5922  ‘cfv 5926  0gc0g 16147  mrClscmrc 16290  Grpcgrp 17469  SubGrpcsubg 17635  Cntzccntz 17794   DProd cdprd 18438 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-oprab 6694  df-mpt2 6695  df-1st 7210  df-2nd 7211  df-ixp 7951  df-dprd 18440 This theorem is referenced by:  dprdfeq0  18467  dprdres  18473  dprdss  18474  dprdf1o  18477  dprd2da  18487  dmdprdsplit2lem  18490
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