Theorem dpjfval 20233

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 20174	. . . 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 785	. . . . . 6 ⊢ ((𝜑 ∧ (𝑔 = 𝐺 ∧ 𝑠 = 𝑆)) → 𝑠 = 𝑆)
5	4	dmeqd 5883	. . . . 5 ⊢ ((𝜑 ∧ (𝑔 = 𝐺 ∧ 𝑠 = 𝑆)) → dom 𝑠 = dom 𝑆)
6		dpjfval.2	. . . . . 6 ⊢ (𝜑 → dom 𝑆 = 𝐼)
7	6	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ (𝑔 = 𝐺 ∧ 𝑠 = 𝑆)) → dom 𝑆 = 𝐼)
8	5, 7	eqtrd 2795	. . . 4 ⊢ ((𝜑 ∧ (𝑔 = 𝐺 ∧ 𝑠 = 𝑆)) → dom 𝑠 = 𝐼)
9		simprl 783	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑔 = 𝐺 ∧ 𝑠 = 𝑆)) → 𝑔 = 𝐺)
10	9	fveq2d 6877	. . . . . 6 ⊢ ((𝜑 ∧ (𝑔 = 𝐺 ∧ 𝑠 = 𝑆)) → (proj1‘𝑔) = (proj1‘𝐺))
11		dpjfval.q	. . . . . 6 ⊢ 𝑄 = (proj1‘𝐺)
12	10, 11	eqtr4di 2813	. . . . 5 ⊢ ((𝜑 ∧ (𝑔 = 𝐺 ∧ 𝑠 = 𝑆)) → (proj1‘𝑔) = 𝑄)
13	4	fveq1d 6875	. . . . 5 ⊢ ((𝜑 ∧ (𝑔 = 𝐺 ∧ 𝑠 = 𝑆)) → (𝑠‘𝑖) = (𝑆‘𝑖))
14	8	difeq1d 4072	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑔 = 𝐺 ∧ 𝑠 = 𝑆)) → (dom 𝑠 ∖ {𝑖}) = (𝐼 ∖ {𝑖}))
15	4, 14	reseq12d 5967	. . . . . 6 ⊢ ((𝜑 ∧ (𝑔 = 𝐺 ∧ 𝑠 = 𝑆)) → (𝑠 ↾ (dom 𝑠 ∖ {𝑖})) = (𝑆 ↾ (𝐼 ∖ {𝑖})))
16	9, 15	oveq12d 7426	. . . . 5 ⊢ ((𝜑 ∧ (𝑔 = 𝐺 ∧ 𝑠 = 𝑆)) → (𝑔 DProd (𝑠 ↾ (dom 𝑠 ∖ {𝑖}))) = (𝐺 DProd (𝑆 ↾ (𝐼 ∖ {𝑖}))))
17	12, 13, 16	oveq123d 7429	. . . 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 5191	. . 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 490	. . . . 5 ⊢ ((𝜑 ∧ 𝑔 = 𝐺) → 𝑔 = 𝐺)
20	19	sneqd 4595	. . . 4 ⊢ ((𝜑 ∧ 𝑔 = 𝐺) → {𝑔} = {𝐺})
21	20	imaeq2d 6050	. . 3 ⊢ ((𝜑 ∧ 𝑔 = 𝐺) → (dom DProd “ {𝑔}) = (dom DProd “ {𝐺}))
22		dpjfval.1	. . . 4 ⊢ (𝜑 → 𝐺dom DProd 𝑆)
23		dprdgrp 20183	. . . 4 ⊢ (𝐺dom DProd 𝑆 → 𝐺 ∈ Grp)
24	22, 23	syl 18	. . 3 ⊢ (𝜑 → 𝐺 ∈ Grp)
25		reldmdprd 20175	. . . . 5 ⊢ Rel dom DProd
26		elrelimasn 6076	. . . . 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 237	. . 3 ⊢ (𝜑 → 𝑆 ∈ (dom DProd “ {𝐺}))
29	22, 6	dprddomcld 20179	. . . 4 ⊢ (𝜑 → 𝐼 ∈ V)
30	29	mptexd 7218	. . 3 ⊢ (𝜑 → (𝑖 ∈ 𝐼 ↦ ((𝑆‘𝑖)𝑄(𝐺 DProd (𝑆 ↾ (𝐼 ∖ {𝑖}))))) ∈ V)
31	3, 18, 21, 24, 28, 30	ovmpodx 7559	. 2 ⊢ (𝜑 → (𝐺dProj𝑆) = (𝑖 ∈ 𝐼 ↦ ((𝑆‘𝑖)𝑄(𝐺 DProd (𝑆 ↾ (𝐼 ∖ {𝑖}))))))
32	1, 31	eqtrid 2807	1 ⊢ (𝜑 → 𝑃 = (𝑖 ∈ 𝐼 ↦ ((𝑆‘𝑖)𝑄(𝐺 DProd (𝑆 ↾ (𝐼 ∖ {𝑖}))))))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2145  Vcvv 3450   ∖ cdif 3895  {csn 4583   class class class wbr 5102   ↦ cmpt 5185  dom cdm 5647   ↾ cres 5649   “ cima 5650  Rel wrel 5652  ‘cfv 6527  (class class class)co 7408   ∈ cmpo 7410  Grpcgrp 19106  proj₁cpj1 19811   DProd cdprd 20171  dProjcdpj 20172

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7734

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-nel 3062  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5542  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660  df-iota 6483  df-fun 6529  df-fn 6530  df-f 6531  df-f1 6532  df-fo 6533  df-f1o 6534  df-fv 6535  df-ov 7411  df-oprab 7412  df-mpo 7413  df-1st 7984  df-2nd 7985  df-ixp 8904  df-dprd 20173  df-dpj 20174

This theorem is used by:  dpjval  20234