Theorem dprddisj 19924

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 6822	. . . 4 ⊢ (𝑥 = 𝑋 → (𝑆‘𝑥) = (𝑆‘𝑋))
2		sneq 4586	. . . . . . . 8 ⊢ (𝑥 = 𝑋 → {𝑥} = {𝑋})
3	2	difeq2d 4076	. . . . . . 7 ⊢ (𝑥 = 𝑋 → (𝐼 ∖ {𝑥}) = (𝐼 ∖ {𝑋}))
4	3	imaeq2d 6009	. . . . . 6 ⊢ (𝑥 = 𝑋 → (𝑆 “ (𝐼 ∖ {𝑥})) = (𝑆 “ (𝐼 ∖ {𝑋})))
5	4	unieqd 4872	. . . . 5 ⊢ (𝑥 = 𝑋 → ∪ (𝑆 “ (𝐼 ∖ {𝑥})) = ∪ (𝑆 “ (𝐼 ∖ {𝑋})))
6	5	fveq2d 6826	. . . 4 ⊢ (𝑥 = 𝑋 → (𝐾‘∪ (𝑆 “ (𝐼 ∖ {𝑥}))) = (𝐾‘∪ (𝑆 “ (𝐼 ∖ {𝑋}))))
7	1, 6	ineq12d 4171	. . 3 ⊢ (𝑥 = 𝑋 → ((𝑆‘𝑥) ∩ (𝐾‘∪ (𝑆 “ (𝐼 ∖ {𝑥})))) = ((𝑆‘𝑋) ∩ (𝐾‘∪ (𝑆 “ (𝐼 ∖ {𝑋})))))
8	7	eqeq1d 2733	. 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 19916	. . . . . 6 ⊢ (𝜑 → 𝐼 ∈ V)
12		eqid 2731	. . . . . . 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 19913	. . . . . 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 584	. . . . 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 232	. . . 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 1144	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝐼 (∀𝑦 ∈ (𝐼 ∖ {𝑥})(𝑆‘𝑥) ⊆ ((Cntz‘𝐺)‘(𝑆‘𝑦)) ∧ ((𝑆‘𝑥) ∩ (𝐾‘∪ (𝑆 “ (𝐼 ∖ {𝑥})))) = { 0 }))
19		simpr 484	. . . 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 3069	. . 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 3578	1 ⊢ (𝜑 → ((𝑆‘𝑋) ∩ (𝐾‘∪ (𝑆 “ (𝐼 ∖ {𝑋})))) = { 0 })

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1541   ∈ wcel 2111  ∀wral 3047  Vcvv 3436   ∖ cdif 3899   ∩ cin 3901   ⊆ wss 3902  {csn 4576  ∪ cuni 4859   class class class wbr 5091  dom cdm 5616   “ cima 5619  ⟶wf 6477  ‘cfv 6481  0_gc0g 17343  mrClscmrc 17485  Grpcgrp 18846  SubGrpcsubg 19033  Cntzccntz 19228   DProd cdprd 19908

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5217  ax-sep 5234  ax-nul 5244  ax-pow 5303  ax-pr 5370  ax-un 7668

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-nel 3033  df-ral 3048  df-rex 3057  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4476  df-pw 4552  df-sn 4577  df-pr 4579  df-op 4583  df-uni 4860  df-iun 4943  df-br 5092  df-opab 5154  df-mpt 5173  df-id 5511  df-xp 5622  df-rel 5623  df-cnv 5624  df-co 5625  df-dm 5626  df-rn 5627  df-res 5628  df-ima 5629  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488  df-fv 6489  df-oprab 7350  df-mpo 7351  df-1st 7921  df-2nd 7922  df-ixp 8822  df-dprd 19910

This theorem is referenced by:  dprdfeq0  19937  dprdres  19943  dprdss  19944  dprdf1o  19947  dprd2da  19957  dmdprdsplit2lem  19960