Theorem dpjfval 20043

Description: Value of the direct product projection (defined in terms of binary projection). (Contributed by Mario Carneiro, 26-Apr-2016.)

Hypotheses

Ref	Expression
dpjfval.1	⊢ (𝜑 → 𝐺dom DProd 𝑆)
dpjfval.2	⊢ (𝜑 → dom 𝑆 = 𝐼)
dpjfval.p	⊢ 𝑃 = (𝐺dProj𝑆)
dpjfval.q	⊢ 𝑄 = (proj1‘𝐺)

Assertion

Ref	Expression
dpjfval	⊢ (𝜑 → 𝑃 = (𝑖 ∈ 𝐼 ↦ ((𝑆‘𝑖)𝑄(𝐺 DProd (𝑆 ↾ (𝐼 ∖ {𝑖}))))))

Distinct variable groups:   𝑖,𝐺   𝜑,𝑖   𝑖,𝐼   𝑆,𝑖

Allowed substitution hints:   𝑃(𝑖)   𝑄(𝑖)

Proof of Theorem dpjfval

Dummy variables 𝑔 𝑠 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		dpjfval.p	. 2 ⊢ 𝑃 = (𝐺dProj𝑆)
2		df-dpj 19984	. . . 4 ⊢ dProj = (𝑔 ∈ Grp, 𝑠 ∈ (dom DProd “ {𝑔}) ↦ (𝑖 ∈ dom 𝑠 ↦ ((𝑠‘𝑖)(proj1‘𝑔)(𝑔 DProd (𝑠 ↾ (dom 𝑠 ∖ {𝑖}))))))
3	2	a1i 11	. . 3 ⊢ (𝜑 → dProj = (𝑔 ∈ Grp, 𝑠 ∈ (dom DProd “ {𝑔}) ↦ (𝑖 ∈ dom 𝑠 ↦ ((𝑠‘𝑖)(proj1‘𝑔)(𝑔 DProd (𝑠 ↾ (dom 𝑠 ∖ {𝑖})))))))
4		simprr 772	. . . . . 6 ⊢ ((𝜑 ∧ (𝑔 = 𝐺 ∧ 𝑠 = 𝑆)) → 𝑠 = 𝑆)
5	4	dmeqd 5890	. . . . 5 ⊢ ((𝜑 ∧ (𝑔 = 𝐺 ∧ 𝑠 = 𝑆)) → dom 𝑠 = dom 𝑆)
6		dpjfval.2	. . . . . 6 ⊢ (𝜑 → dom 𝑆 = 𝐼)
7	6	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ (𝑔 = 𝐺 ∧ 𝑠 = 𝑆)) → dom 𝑆 = 𝐼)
8	5, 7	eqtrd 2771	. . . 4 ⊢ ((𝜑 ∧ (𝑔 = 𝐺 ∧ 𝑠 = 𝑆)) → dom 𝑠 = 𝐼)
9		simprl 770	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑔 = 𝐺 ∧ 𝑠 = 𝑆)) → 𝑔 = 𝐺)
10	9	fveq2d 6885	. . . . . 6 ⊢ ((𝜑 ∧ (𝑔 = 𝐺 ∧ 𝑠 = 𝑆)) → (proj1‘𝑔) = (proj1‘𝐺))
11		dpjfval.q	. . . . . 6 ⊢ 𝑄 = (proj1‘𝐺)
12	10, 11	eqtr4di 2789	. . . . 5 ⊢ ((𝜑 ∧ (𝑔 = 𝐺 ∧ 𝑠 = 𝑆)) → (proj1‘𝑔) = 𝑄)
13	4	fveq1d 6883	. . . . 5 ⊢ ((𝜑 ∧ (𝑔 = 𝐺 ∧ 𝑠 = 𝑆)) → (𝑠‘𝑖) = (𝑆‘𝑖))
14	8	difeq1d 4105	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑔 = 𝐺 ∧ 𝑠 = 𝑆)) → (dom 𝑠 ∖ {𝑖}) = (𝐼 ∖ {𝑖}))
15	4, 14	reseq12d 5972	. . . . . 6 ⊢ ((𝜑 ∧ (𝑔 = 𝐺 ∧ 𝑠 = 𝑆)) → (𝑠 ↾ (dom 𝑠 ∖ {𝑖})) = (𝑆 ↾ (𝐼 ∖ {𝑖})))
16	9, 15	oveq12d 7428	. . . . 5 ⊢ ((𝜑 ∧ (𝑔 = 𝐺 ∧ 𝑠 = 𝑆)) → (𝑔 DProd (𝑠 ↾ (dom 𝑠 ∖ {𝑖}))) = (𝐺 DProd (𝑆 ↾ (𝐼 ∖ {𝑖}))))
17	12, 13, 16	oveq123d 7431	. . . 4 ⊢ ((𝜑 ∧ (𝑔 = 𝐺 ∧ 𝑠 = 𝑆)) → ((𝑠‘𝑖)(proj1‘𝑔)(𝑔 DProd (𝑠 ↾ (dom 𝑠 ∖ {𝑖})))) = ((𝑆‘𝑖)𝑄(𝐺 DProd (𝑆 ↾ (𝐼 ∖ {𝑖})))))
18	8, 17	mpteq12dv 5212	. . 3 ⊢ ((𝜑 ∧ (𝑔 = 𝐺 ∧ 𝑠 = 𝑆)) → (𝑖 ∈ dom 𝑠 ↦ ((𝑠‘𝑖)(proj1‘𝑔)(𝑔 DProd (𝑠 ↾ (dom 𝑠 ∖ {𝑖}))))) = (𝑖 ∈ 𝐼 ↦ ((𝑆‘𝑖)𝑄(𝐺 DProd (𝑆 ↾ (𝐼 ∖ {𝑖}))))))
19		simpr 484	. . . . 5 ⊢ ((𝜑 ∧ 𝑔 = 𝐺) → 𝑔 = 𝐺)
20	19	sneqd 4618	. . . 4 ⊢ ((𝜑 ∧ 𝑔 = 𝐺) → {𝑔} = {𝐺})
21	20	imaeq2d 6052	. . 3 ⊢ ((𝜑 ∧ 𝑔 = 𝐺) → (dom DProd “ {𝑔}) = (dom DProd “ {𝐺}))
22		dpjfval.1	. . . 4 ⊢ (𝜑 → 𝐺dom DProd 𝑆)
23		dprdgrp 19993	. . . 4 ⊢ (𝐺dom DProd 𝑆 → 𝐺 ∈ Grp)
24	22, 23	syl 17	. . 3 ⊢ (𝜑 → 𝐺 ∈ Grp)
25		reldmdprd 19985	. . . . 5 ⊢ Rel dom DProd
26		elrelimasn 6078	. . . . 5 ⊢ (Rel dom DProd → (𝑆 ∈ (dom DProd “ {𝐺}) ↔ 𝐺dom DProd 𝑆))
27	25, 26	ax-mp 5	. . . 4 ⊢ (𝑆 ∈ (dom DProd “ {𝐺}) ↔ 𝐺dom DProd 𝑆)
28	22, 27	sylibr 234	. . 3 ⊢ (𝜑 → 𝑆 ∈ (dom DProd “ {𝐺}))
29	22, 6	dprddomcld 19989	. . . 4 ⊢ (𝜑 → 𝐼 ∈ V)
30	29	mptexd 7221	. . 3 ⊢ (𝜑 → (𝑖 ∈ 𝐼 ↦ ((𝑆‘𝑖)𝑄(𝐺 DProd (𝑆 ↾ (𝐼 ∖ {𝑖}))))) ∈ V)
31	3, 18, 21, 24, 28, 30	ovmpodx 7563	. 2 ⊢ (𝜑 → (𝐺dProj𝑆) = (𝑖 ∈ 𝐼 ↦ ((𝑆‘𝑖)𝑄(𝐺 DProd (𝑆 ↾ (𝐼 ∖ {𝑖}))))))
32	1, 31	eqtrid 2783	1 ⊢ (𝜑 → 𝑃 = (𝑖 ∈ 𝐼 ↦ ((𝑆‘𝑖)𝑄(𝐺 DProd (𝑆 ↾ (𝐼 ∖ {𝑖}))))))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109  Vcvv 3464   ∖ cdif 3928  {csn 4606   class class class wbr 5124   ↦ cmpt 5206  dom cdm 5659   ↾ cres 5661   “ cima 5662  Rel wrel 5664  ‘cfv 6536  (class class class)co 7410   ∈ cmpo 7412  Grpcgrp 18921  proj₁cpj1 19621   DProd cdprd 19981  dProjcdpj 19982

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2708  ax-rep 5254  ax-sep 5271  ax-nul 5281  ax-pow 5340  ax-pr 5407  ax-un 7734

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2810  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3062  df-reu 3365  df-rab 3421  df-v 3466  df-sbc 3771  df-csb 3880  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4889  df-iun 4974  df-br 5125  df-opab 5187  df-mpt 5207  df-id 5553  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-iota 6489  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-ov 7413  df-oprab 7414  df-mpo 7415  df-1st 7993  df-2nd 7994  df-ixp 8917  df-dprd 19983  df-dpj 19984

This theorem is referenced by:  dpjval  20044