Theorem dpjfval 19177

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 19118	. . . 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 771	. . . . . 6 ⊢ ((𝜑 ∧ (𝑔 = 𝐺 ∧ 𝑠 = 𝑆)) → 𝑠 = 𝑆)
5	4	dmeqd 5774	. . . . 5 ⊢ ((𝜑 ∧ (𝑔 = 𝐺 ∧ 𝑠 = 𝑆)) → dom 𝑠 = dom 𝑆)
6		dpjfval.2	. . . . . 6 ⊢ (𝜑 → dom 𝑆 = 𝐼)
7	6	adantr 483	. . . . 5 ⊢ ((𝜑 ∧ (𝑔 = 𝐺 ∧ 𝑠 = 𝑆)) → dom 𝑆 = 𝐼)
8	5, 7	eqtrd 2856	. . . 4 ⊢ ((𝜑 ∧ (𝑔 = 𝐺 ∧ 𝑠 = 𝑆)) → dom 𝑠 = 𝐼)
9		simprl 769	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑔 = 𝐺 ∧ 𝑠 = 𝑆)) → 𝑔 = 𝐺)
10	9	fveq2d 6674	. . . . . 6 ⊢ ((𝜑 ∧ (𝑔 = 𝐺 ∧ 𝑠 = 𝑆)) → (proj1‘𝑔) = (proj1‘𝐺))
11		dpjfval.q	. . . . . 6 ⊢ 𝑄 = (proj1‘𝐺)
12	10, 11	syl6eqr 2874	. . . . 5 ⊢ ((𝜑 ∧ (𝑔 = 𝐺 ∧ 𝑠 = 𝑆)) → (proj1‘𝑔) = 𝑄)
13	4	fveq1d 6672	. . . . 5 ⊢ ((𝜑 ∧ (𝑔 = 𝐺 ∧ 𝑠 = 𝑆)) → (𝑠‘𝑖) = (𝑆‘𝑖))
14	8	difeq1d 4098	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑔 = 𝐺 ∧ 𝑠 = 𝑆)) → (dom 𝑠 ∖ {𝑖}) = (𝐼 ∖ {𝑖}))
15	4, 14	reseq12d 5854	. . . . . 6 ⊢ ((𝜑 ∧ (𝑔 = 𝐺 ∧ 𝑠 = 𝑆)) → (𝑠 ↾ (dom 𝑠 ∖ {𝑖})) = (𝑆 ↾ (𝐼 ∖ {𝑖})))
16	9, 15	oveq12d 7174	. . . . 5 ⊢ ((𝜑 ∧ (𝑔 = 𝐺 ∧ 𝑠 = 𝑆)) → (𝑔 DProd (𝑠 ↾ (dom 𝑠 ∖ {𝑖}))) = (𝐺 DProd (𝑆 ↾ (𝐼 ∖ {𝑖}))))
17	12, 13, 16	oveq123d 7177	. . . 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 5151	. . 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 487	. . . . 5 ⊢ ((𝜑 ∧ 𝑔 = 𝐺) → 𝑔 = 𝐺)
20	19	sneqd 4579	. . . 4 ⊢ ((𝜑 ∧ 𝑔 = 𝐺) → {𝑔} = {𝐺})
21	20	imaeq2d 5929	. . 3 ⊢ ((𝜑 ∧ 𝑔 = 𝐺) → (dom DProd “ {𝑔}) = (dom DProd “ {𝐺}))
22		dpjfval.1	. . . 4 ⊢ (𝜑 → 𝐺dom DProd 𝑆)
23		dprdgrp 19127	. . . 4 ⊢ (𝐺dom DProd 𝑆 → 𝐺 ∈ Grp)
24	22, 23	syl 17	. . 3 ⊢ (𝜑 → 𝐺 ∈ Grp)
25		reldmdprd 19119	. . . . 5 ⊢ Rel dom DProd
26		elrelimasn 5953	. . . . 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 236	. . 3 ⊢ (𝜑 → 𝑆 ∈ (dom DProd “ {𝐺}))
29	22, 6	dprddomcld 19123	. . . 4 ⊢ (𝜑 → 𝐼 ∈ V)
30	29	mptexd 6987	. . 3 ⊢ (𝜑 → (𝑖 ∈ 𝐼 ↦ ((𝑆‘𝑖)𝑄(𝐺 DProd (𝑆 ↾ (𝐼 ∖ {𝑖}))))) ∈ V)
31	3, 18, 21, 24, 28, 30	ovmpodx 7301	. 2 ⊢ (𝜑 → (𝐺dProj𝑆) = (𝑖 ∈ 𝐼 ↦ ((𝑆‘𝑖)𝑄(𝐺 DProd (𝑆 ↾ (𝐼 ∖ {𝑖}))))))
32	1, 31	syl5eq 2868	1 ⊢ (𝜑 → 𝑃 = (𝑖 ∈ 𝐼 ↦ ((𝑆‘𝑖)𝑄(𝐺 DProd (𝑆 ↾ (𝐼 ∖ {𝑖}))))))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   = wceq 1537   ∈ wcel 2114  Vcvv 3494   ∖ cdif 3933  {csn 4567   class class class wbr 5066   ↦ cmpt 5146  dom cdm 5555   ↾ cres 5557   “ cima 5558  Rel wrel 5560  ‘cfv 6355  (class class class)co 7156   ∈ cmpo 7158  Grpcgrp 18103  proj₁cpj1 18760   DProd cdprd 19115  dProjcdpj 19116

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 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-rep 5190  ax-sep 5203  ax-nul 5210  ax-pow 5266  ax-pr 5330  ax-un 7461

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-nel 3124  df-ral 3143  df-rex 3144  df-reu 3145  df-rab 3147  df-v 3496  df-sbc 3773  df-csb 3884  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4839  df-iun 4921  df-br 5067  df-opab 5129  df-mpt 5147  df-id 5460  df-xp 5561  df-rel 5562  df-cnv 5563  df-co 5564  df-dm 5565  df-rn 5566  df-res 5567  df-ima 5568  df-iota 6314  df-fun 6357  df-fn 6358  df-f 6359  df-f1 6360  df-fo 6361  df-f1o 6362  df-fv 6363  df-ov 7159  df-oprab 7160  df-mpo 7161  df-1st 7689  df-2nd 7690  df-ixp 8462  df-dprd 19117  df-dpj 19118

This theorem is referenced by:  dpjval  19178