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Theorem dprddisj 19212
 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 6663 . . . 4 (𝑥 = 𝑋 → (𝑆𝑥) = (𝑆𝑋))
2 sneq 4535 . . . . . . . 8 (𝑥 = 𝑋 → {𝑥} = {𝑋})
32difeq2d 4030 . . . . . . 7 (𝑥 = 𝑋 → (𝐼 ∖ {𝑥}) = (𝐼 ∖ {𝑋}))
43imaeq2d 5906 . . . . . 6 (𝑥 = 𝑋 → (𝑆 “ (𝐼 ∖ {𝑥})) = (𝑆 “ (𝐼 ∖ {𝑋})))
54unieqd 4815 . . . . 5 (𝑥 = 𝑋 (𝑆 “ (𝐼 ∖ {𝑥})) = (𝑆 “ (𝐼 ∖ {𝑋})))
65fveq2d 6667 . . . 4 (𝑥 = 𝑋 → (𝐾 (𝑆 “ (𝐼 ∖ {𝑥}))) = (𝐾 (𝑆 “ (𝐼 ∖ {𝑋}))))
71, 6ineq12d 4120 . . 3 (𝑥 = 𝑋 → ((𝑆𝑥) ∩ (𝐾 (𝑆 “ (𝐼 ∖ {𝑥})))) = ((𝑆𝑋) ∩ (𝐾 (𝑆 “ (𝐼 ∖ {𝑋})))))
87eqeq1d 2760 . 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 19204 . . . . . 6 (𝜑𝐼 ∈ V)
12 eqid 2758 . . . . . . 7 (Cntz‘𝐺) = (Cntz‘𝐺)
13 dprddisj.0 . . . . . . 7 0 = (0g𝐺)
14 dprddisj.k . . . . . . 7 𝐾 = (mrCls‘(SubGrp‘𝐺))
1512, 13, 14dmdprd 19201 . . . . . 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 587 . . . . 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 235 . . . 4 (𝜑 → (𝐺 ∈ Grp ∧ 𝑆:𝐼⟶(SubGrp‘𝐺) ∧ ∀𝑥𝐼 (∀𝑦 ∈ (𝐼 ∖ {𝑥})(𝑆𝑥) ⊆ ((Cntz‘𝐺)‘(𝑆𝑦)) ∧ ((𝑆𝑥) ∩ (𝐾 (𝑆 “ (𝐼 ∖ {𝑥})))) = { 0 })))
1817simp3d 1141 . . 3 (𝜑 → ∀𝑥𝐼 (∀𝑦 ∈ (𝐼 ∖ {𝑥})(𝑆𝑥) ⊆ ((Cntz‘𝐺)‘(𝑆𝑦)) ∧ ((𝑆𝑥) ∩ (𝐾 (𝑆 “ (𝐼 ∖ {𝑥})))) = { 0 }))
19 simpr 488 . . . 4 ((∀𝑦 ∈ (𝐼 ∖ {𝑥})(𝑆𝑥) ⊆ ((Cntz‘𝐺)‘(𝑆𝑦)) ∧ ((𝑆𝑥) ∩ (𝐾 (𝑆 “ (𝐼 ∖ {𝑥})))) = { 0 }) → ((𝑆𝑥) ∩ (𝐾 (𝑆 “ (𝐼 ∖ {𝑥})))) = { 0 })
2019ralimi 3092 . . 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 3545 1 (𝜑 → ((𝑆𝑋) ∩ (𝐾 (𝑆 “ (𝐼 ∖ {𝑋})))) = { 0 })
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111  ∀wral 3070  Vcvv 3409   ∖ cdif 3857   ∩ cin 3859   ⊆ wss 3860  {csn 4525  ∪ cuni 4801   class class class wbr 5036  dom cdm 5528   “ cima 5531  ⟶wf 6336  ‘cfv 6340  0gc0g 16784  mrClscmrc 16925  Grpcgrp 18182  SubGrpcsubg 18353  Cntzccntz 18525   DProd cdprd 19196 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-rep 5160  ax-sep 5173  ax-nul 5180  ax-pow 5238  ax-pr 5302  ax-un 7465 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-nel 3056  df-ral 3075  df-rex 3076  df-reu 3077  df-rab 3079  df-v 3411  df-sbc 3699  df-csb 3808  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-nul 4228  df-if 4424  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-iun 4888  df-br 5037  df-opab 5099  df-mpt 5117  df-id 5434  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-rn 5539  df-res 5540  df-ima 5541  df-iota 6299  df-fun 6342  df-fn 6343  df-f 6344  df-f1 6345  df-fo 6346  df-f1o 6347  df-fv 6348  df-oprab 7160  df-mpo 7161  df-1st 7699  df-2nd 7700  df-ixp 8493  df-dprd 19198 This theorem is referenced by:  dprdfeq0  19225  dprdres  19231  dprdss  19232  dprdf1o  19235  dprd2da  19245  dmdprdsplit2lem  19248
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