Theorem dpjfval 19656

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 19597	. . . 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 770	. . . . . 6 ⊢ ((𝜑 ∧ (𝑔 = 𝐺 ∧ 𝑠 = 𝑆)) → 𝑠 = 𝑆)
5	4	dmeqd 5813	. . . . 5 ⊢ ((𝜑 ∧ (𝑔 = 𝐺 ∧ 𝑠 = 𝑆)) → dom 𝑠 = dom 𝑆)
6		dpjfval.2	. . . . . 6 ⊢ (𝜑 → dom 𝑆 = 𝐼)
7	6	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ (𝑔 = 𝐺 ∧ 𝑠 = 𝑆)) → dom 𝑆 = 𝐼)
8	5, 7	eqtrd 2780	. . . 4 ⊢ ((𝜑 ∧ (𝑔 = 𝐺 ∧ 𝑠 = 𝑆)) → dom 𝑠 = 𝐼)
9		simprl 768	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑔 = 𝐺 ∧ 𝑠 = 𝑆)) → 𝑔 = 𝐺)
10	9	fveq2d 6775	. . . . . 6 ⊢ ((𝜑 ∧ (𝑔 = 𝐺 ∧ 𝑠 = 𝑆)) → (proj1‘𝑔) = (proj1‘𝐺))
11		dpjfval.q	. . . . . 6 ⊢ 𝑄 = (proj1‘𝐺)
12	10, 11	eqtr4di 2798	. . . . 5 ⊢ ((𝜑 ∧ (𝑔 = 𝐺 ∧ 𝑠 = 𝑆)) → (proj1‘𝑔) = 𝑄)
13	4	fveq1d 6773	. . . . 5 ⊢ ((𝜑 ∧ (𝑔 = 𝐺 ∧ 𝑠 = 𝑆)) → (𝑠‘𝑖) = (𝑆‘𝑖))
14	8	difeq1d 4061	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑔 = 𝐺 ∧ 𝑠 = 𝑆)) → (dom 𝑠 ∖ {𝑖}) = (𝐼 ∖ {𝑖}))
15	4, 14	reseq12d 5891	. . . . . 6 ⊢ ((𝜑 ∧ (𝑔 = 𝐺 ∧ 𝑠 = 𝑆)) → (𝑠 ↾ (dom 𝑠 ∖ {𝑖})) = (𝑆 ↾ (𝐼 ∖ {𝑖})))
16	9, 15	oveq12d 7289	. . . . 5 ⊢ ((𝜑 ∧ (𝑔 = 𝐺 ∧ 𝑠 = 𝑆)) → (𝑔 DProd (𝑠 ↾ (dom 𝑠 ∖ {𝑖}))) = (𝐺 DProd (𝑆 ↾ (𝐼 ∖ {𝑖}))))
17	12, 13, 16	oveq123d 7292	. . . 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 5170	. . 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 485	. . . . 5 ⊢ ((𝜑 ∧ 𝑔 = 𝐺) → 𝑔 = 𝐺)
20	19	sneqd 4579	. . . 4 ⊢ ((𝜑 ∧ 𝑔 = 𝐺) → {𝑔} = {𝐺})
21	20	imaeq2d 5968	. . 3 ⊢ ((𝜑 ∧ 𝑔 = 𝐺) → (dom DProd “ {𝑔}) = (dom DProd “ {𝐺}))
22		dpjfval.1	. . . 4 ⊢ (𝜑 → 𝐺dom DProd 𝑆)
23		dprdgrp 19606	. . . 4 ⊢ (𝐺dom DProd 𝑆 → 𝐺 ∈ Grp)
24	22, 23	syl 17	. . 3 ⊢ (𝜑 → 𝐺 ∈ Grp)
25		reldmdprd 19598	. . . . 5 ⊢ Rel dom DProd
26		elrelimasn 5992	. . . . 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 233	. . 3 ⊢ (𝜑 → 𝑆 ∈ (dom DProd “ {𝐺}))
29	22, 6	dprddomcld 19602	. . . 4 ⊢ (𝜑 → 𝐼 ∈ V)
30	29	mptexd 7097	. . 3 ⊢ (𝜑 → (𝑖 ∈ 𝐼 ↦ ((𝑆‘𝑖)𝑄(𝐺 DProd (𝑆 ↾ (𝐼 ∖ {𝑖}))))) ∈ V)
31	3, 18, 21, 24, 28, 30	ovmpodx 7418	. 2 ⊢ (𝜑 → (𝐺dProj𝑆) = (𝑖 ∈ 𝐼 ↦ ((𝑆‘𝑖)𝑄(𝐺 DProd (𝑆 ↾ (𝐼 ∖ {𝑖}))))))
32	1, 31	eqtrid 2792	1 ⊢ (𝜑 → 𝑃 = (𝑖 ∈ 𝐼 ↦ ((𝑆‘𝑖)𝑄(𝐺 DProd (𝑆 ↾ (𝐼 ∖ {𝑖}))))))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1542   ∈ wcel 2110  Vcvv 3431   ∖ cdif 3889  {csn 4567   class class class wbr 5079   ↦ cmpt 5162  dom cdm 5590   ↾ cres 5592   “ cima 5593  Rel wrel 5595  ‘cfv 6432  (class class class)co 7271   ∈ cmpo 7273  Grpcgrp 18575  proj₁cpj1 19238   DProd cdprd 19594  dProjcdpj 19595

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 1975  ax-7 2015  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2711  ax-rep 5214  ax-sep 5227  ax-nul 5234  ax-pow 5292  ax-pr 5356  ax-un 7582

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2072  df-mo 2542  df-eu 2571  df-clab 2718  df-cleq 2732  df-clel 2818  df-nfc 2891  df-ne 2946  df-nel 3052  df-ral 3071  df-rex 3072  df-reu 3073  df-rab 3075  df-v 3433  df-sbc 3721  df-csb 3838  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-nul 4263  df-if 4466  df-pw 4541  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4846  df-iun 4932  df-br 5080  df-opab 5142  df-mpt 5163  df-id 5490  df-xp 5596  df-rel 5597  df-cnv 5598  df-co 5599  df-dm 5600  df-rn 5601  df-res 5602  df-ima 5603  df-iota 6390  df-fun 6434  df-fn 6435  df-f 6436  df-f1 6437  df-fo 6438  df-f1o 6439  df-fv 6440  df-ov 7274  df-oprab 7275  df-mpo 7276  df-1st 7824  df-2nd 7825  df-ixp 8669  df-dprd 19596  df-dpj 19597

This theorem is referenced by:  dpjval  19657