Theorem dprddisj 19977

Description: The function 𝑆 is a family having trivial intersections. (Contributed by Mario Carneiro, 25-Apr-2016.)

Hypotheses

Ref	Expression
dprdcntz.1	⊢ (𝜑 → 𝐺dom DProd 𝑆)
dprdcntz.2	⊢ (𝜑 → dom 𝑆 = 𝐼)
dprdcntz.3	⊢ (𝜑 → 𝑋 ∈ 𝐼)
dprddisj.0	⊢ 0 = (0g‘𝐺)
dprddisj.k	⊢ 𝐾 = (mrCls‘(SubGrp‘𝐺))

Assertion

Ref	Expression
dprddisj	⊢ (𝜑 → ((𝑆‘𝑋) ∩ (𝐾‘∪ (𝑆 “ (𝐼 ∖ {𝑋})))) = { 0 })

Proof of Theorem dprddisj

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq2 6827	. . . 4 ⊢ (𝑥 = 𝑋 → (𝑆‘𝑥) = (𝑆‘𝑋))
2		sneq 4565	. . . . . . . 8 ⊢ (𝑥 = 𝑋 → {𝑥} = {𝑋})
3	2	difeq2d 4057	. . . . . . 7 ⊢ (𝑥 = 𝑋 → (𝐼 ∖ {𝑥}) = (𝐼 ∖ {𝑋}))
4	3	imaeq2d 6012	. . . . . 6 ⊢ (𝑥 = 𝑋 → (𝑆 “ (𝐼 ∖ {𝑥})) = (𝑆 “ (𝐼 ∖ {𝑋})))
5	4	unieqd 4851	. . . . 5 ⊢ (𝑥 = 𝑋 → ∪ (𝑆 “ (𝐼 ∖ {𝑥})) = ∪ (𝑆 “ (𝐼 ∖ {𝑋})))
6	5	fveq2d 6831	. . . 4 ⊢ (𝑥 = 𝑋 → (𝐾‘∪ (𝑆 “ (𝐼 ∖ {𝑥}))) = (𝐾‘∪ (𝑆 “ (𝐼 ∖ {𝑋}))))
7	1, 6	ineq12d 4150	. . 3 ⊢ (𝑥 = 𝑋 → ((𝑆‘𝑥) ∩ (𝐾‘∪ (𝑆 “ (𝐼 ∖ {𝑥})))) = ((𝑆‘𝑋) ∩ (𝐾‘∪ (𝑆 “ (𝐼 ∖ {𝑋})))))
8	7	eqeq1d 2741	. 2 ⊢ (𝑥 = 𝑋 → (((𝑆‘𝑥) ∩ (𝐾‘∪ (𝑆 “ (𝐼 ∖ {𝑥})))) = { 0 } ↔ ((𝑆‘𝑋) ∩ (𝐾‘∪ (𝑆 “ (𝐼 ∖ {𝑋})))) = { 0 }))
9		dprdcntz.1	. . . . 5 ⊢ (𝜑 → 𝐺dom DProd 𝑆)
10		dprdcntz.2	. . . . . . 7 ⊢ (𝜑 → dom 𝑆 = 𝐼)
11	9, 10	dprddomcld 19969	. . . . . 6 ⊢ (𝜑 → 𝐼 ∈ V)
12		eqid 2739	. . . . . . 7 ⊢ (Cntz‘𝐺) = (Cntz‘𝐺)
13		dprddisj.0	. . . . . . 7 ⊢ 0 = (0g‘𝐺)
14		dprddisj.k	. . . . . . 7 ⊢ 𝐾 = (mrCls‘(SubGrp‘𝐺))
15	12, 13, 14	dmdprd 19966	. . . . . 6 ⊢ ((𝐼 ∈ V ∧ dom 𝑆 = 𝐼) → (𝐺dom DProd 𝑆 ↔ (𝐺 ∈ Grp ∧ 𝑆:𝐼⟶(SubGrp‘𝐺) ∧ ∀𝑥 ∈ 𝐼 (∀𝑦 ∈ (𝐼 ∖ {𝑥})(𝑆‘𝑥) ⊆ ((Cntz‘𝐺)‘(𝑆‘𝑦)) ∧ ((𝑆‘𝑥) ∩ (𝐾‘∪ (𝑆 “ (𝐼 ∖ {𝑥})))) = { 0 }))))
16	11, 10, 15	syl2anc 590	. . . . 5 ⊢ (𝜑 → (𝐺dom DProd 𝑆 ↔ (𝐺 ∈ Grp ∧ 𝑆:𝐼⟶(SubGrp‘𝐺) ∧ ∀𝑥 ∈ 𝐼 (∀𝑦 ∈ (𝐼 ∖ {𝑥})(𝑆‘𝑥) ⊆ ((Cntz‘𝐺)‘(𝑆‘𝑦)) ∧ ((𝑆‘𝑥) ∩ (𝐾‘∪ (𝑆 “ (𝐼 ∖ {𝑥})))) = { 0 }))))
17	9, 16	mpbid 233	. . . 4 ⊢ (𝜑 → (𝐺 ∈ Grp ∧ 𝑆:𝐼⟶(SubGrp‘𝐺) ∧ ∀𝑥 ∈ 𝐼 (∀𝑦 ∈ (𝐼 ∖ {𝑥})(𝑆‘𝑥) ⊆ ((Cntz‘𝐺)‘(𝑆‘𝑦)) ∧ ((𝑆‘𝑥) ∩ (𝐾‘∪ (𝑆 “ (𝐼 ∖ {𝑥})))) = { 0 })))
18	17	simp3d 1150	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝐼 (∀𝑦 ∈ (𝐼 ∖ {𝑥})(𝑆‘𝑥) ⊆ ((Cntz‘𝐺)‘(𝑆‘𝑦)) ∧ ((𝑆‘𝑥) ∩ (𝐾‘∪ (𝑆 “ (𝐼 ∖ {𝑥})))) = { 0 }))
19		simpr 485	. . . 4 ⊢ ((∀𝑦 ∈ (𝐼 ∖ {𝑥})(𝑆‘𝑥) ⊆ ((Cntz‘𝐺)‘(𝑆‘𝑦)) ∧ ((𝑆‘𝑥) ∩ (𝐾‘∪ (𝑆 “ (𝐼 ∖ {𝑥})))) = { 0 }) → ((𝑆‘𝑥) ∩ (𝐾‘∪ (𝑆 “ (𝐼 ∖ {𝑥})))) = { 0 })
20	19	ralimi 3076	. . 3 ⊢ (∀𝑥 ∈ 𝐼 (∀𝑦 ∈ (𝐼 ∖ {𝑥})(𝑆‘𝑥) ⊆ ((Cntz‘𝐺)‘(𝑆‘𝑦)) ∧ ((𝑆‘𝑥) ∩ (𝐾‘∪ (𝑆 “ (𝐼 ∖ {𝑥})))) = { 0 }) → ∀𝑥 ∈ 𝐼 ((𝑆‘𝑥) ∩ (𝐾‘∪ (𝑆 “ (𝐼 ∖ {𝑥})))) = { 0 })
21	18, 20	syl 17	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐼 ((𝑆‘𝑥) ∩ (𝐾‘∪ (𝑆 “ (𝐼 ∖ {𝑥})))) = { 0 })
22		dprdcntz.3	. 2 ⊢ (𝜑 → 𝑋 ∈ 𝐼)
23	8, 21, 22	rspcdva 3561	1 ⊢ (𝜑 → ((𝑆‘𝑋) ∩ (𝐾‘∪ (𝑆 “ (𝐼 ∖ {𝑋})))) = { 0 })

Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396   ∧ w3a 1092   = wceq 1547   ∈ wcel 2119  ∀wral 3053  Vcvv 3431   ∖ cdif 3880   ∩ cin 3882   ⊆ wss 3883  {csn 4555  ∪ cuni 4838   class class class wbr 5072  dom cdm 5618   “ cima 5621  ⟶wf 6481  ‘cfv 6485  0_gc0g 17393  mrClscmrc 17536  Grpcgrp 18900  SubGrpcsubg 19087  Cntzccntz 19281   DProd cdprd 19961

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-rep 5199  ax-sep 5218  ax-nul 5228  ax-pow 5294  ax-pr 5362  ax-un 7678

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-nel 3039  df-ral 3054  df-rex 3064  df-reu 3345  df-rab 3392  df-v 3433  df-sbc 3724  df-csb 3832  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4262  df-if 4455  df-pw 4531  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-iun 4923  df-br 5073  df-opab 5135  df-mpt 5154  df-id 5513  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-rn 5629  df-res 5630  df-ima 5631  df-iota 6441  df-fun 6487  df-fn 6488  df-f 6489  df-f1 6490  df-fo 6491  df-f1o 6492  df-fv 6493  df-oprab 7360  df-mpo 7361  df-1st 7931  df-2nd 7932  df-ixp 8836  df-dprd 19963

This theorem is referenced by:  dprdfeq0  19990  dprdres  19996  dprdss  19997  dprdf1o  20000  dprd2da  20010  dmdprdsplit2lem  20013