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Theorem dpjfval 20075
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 (𝜑𝐺dom DProd 𝑆)
dpjfval.2 (𝜑 → dom 𝑆 = 𝐼)
dpjfval.p 𝑃 = (𝐺dProj𝑆)
dpjfval.q 𝑄 = (proj1𝐺)
Assertion
Ref Expression
dpjfval (𝜑𝑃 = (𝑖𝐼 ↦ ((𝑆𝑖)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.
StepHypRef Expression
1 dpjfval.p . 2 𝑃 = (𝐺dProj𝑆)
2 df-dpj 20016 . . . 4 dProj = (𝑔 ∈ Grp, 𝑠 ∈ (dom DProd “ {𝑔}) ↦ (𝑖 ∈ dom 𝑠 ↦ ((𝑠𝑖)(proj1𝑔)(𝑔 DProd (𝑠 ↾ (dom 𝑠 ∖ {𝑖}))))))
32a1i 11 . . 3 (𝜑 → dProj = (𝑔 ∈ Grp, 𝑠 ∈ (dom DProd “ {𝑔}) ↦ (𝑖 ∈ dom 𝑠 ↦ ((𝑠𝑖)(proj1𝑔)(𝑔 DProd (𝑠 ↾ (dom 𝑠 ∖ {𝑖})))))))
4 simprr 773 . . . . . 6 ((𝜑 ∧ (𝑔 = 𝐺𝑠 = 𝑆)) → 𝑠 = 𝑆)
54dmeqd 5916 . . . . 5 ((𝜑 ∧ (𝑔 = 𝐺𝑠 = 𝑆)) → dom 𝑠 = dom 𝑆)
6 dpjfval.2 . . . . . 6 (𝜑 → dom 𝑆 = 𝐼)
76adantr 480 . . . . 5 ((𝜑 ∧ (𝑔 = 𝐺𝑠 = 𝑆)) → dom 𝑆 = 𝐼)
85, 7eqtrd 2777 . . . 4 ((𝜑 ∧ (𝑔 = 𝐺𝑠 = 𝑆)) → dom 𝑠 = 𝐼)
9 simprl 771 . . . . . . 7 ((𝜑 ∧ (𝑔 = 𝐺𝑠 = 𝑆)) → 𝑔 = 𝐺)
109fveq2d 6910 . . . . . 6 ((𝜑 ∧ (𝑔 = 𝐺𝑠 = 𝑆)) → (proj1𝑔) = (proj1𝐺))
11 dpjfval.q . . . . . 6 𝑄 = (proj1𝐺)
1210, 11eqtr4di 2795 . . . . 5 ((𝜑 ∧ (𝑔 = 𝐺𝑠 = 𝑆)) → (proj1𝑔) = 𝑄)
134fveq1d 6908 . . . . 5 ((𝜑 ∧ (𝑔 = 𝐺𝑠 = 𝑆)) → (𝑠𝑖) = (𝑆𝑖))
148difeq1d 4125 . . . . . . 7 ((𝜑 ∧ (𝑔 = 𝐺𝑠 = 𝑆)) → (dom 𝑠 ∖ {𝑖}) = (𝐼 ∖ {𝑖}))
154, 14reseq12d 5998 . . . . . 6 ((𝜑 ∧ (𝑔 = 𝐺𝑠 = 𝑆)) → (𝑠 ↾ (dom 𝑠 ∖ {𝑖})) = (𝑆 ↾ (𝐼 ∖ {𝑖})))
169, 15oveq12d 7449 . . . . 5 ((𝜑 ∧ (𝑔 = 𝐺𝑠 = 𝑆)) → (𝑔 DProd (𝑠 ↾ (dom 𝑠 ∖ {𝑖}))) = (𝐺 DProd (𝑆 ↾ (𝐼 ∖ {𝑖}))))
1712, 13, 16oveq123d 7452 . . . 4 ((𝜑 ∧ (𝑔 = 𝐺𝑠 = 𝑆)) → ((𝑠𝑖)(proj1𝑔)(𝑔 DProd (𝑠 ↾ (dom 𝑠 ∖ {𝑖})))) = ((𝑆𝑖)𝑄(𝐺 DProd (𝑆 ↾ (𝐼 ∖ {𝑖})))))
188, 17mpteq12dv 5233 . . 3 ((𝜑 ∧ (𝑔 = 𝐺𝑠 = 𝑆)) → (𝑖 ∈ dom 𝑠 ↦ ((𝑠𝑖)(proj1𝑔)(𝑔 DProd (𝑠 ↾ (dom 𝑠 ∖ {𝑖}))))) = (𝑖𝐼 ↦ ((𝑆𝑖)𝑄(𝐺 DProd (𝑆 ↾ (𝐼 ∖ {𝑖}))))))
19 simpr 484 . . . . 5 ((𝜑𝑔 = 𝐺) → 𝑔 = 𝐺)
2019sneqd 4638 . . . 4 ((𝜑𝑔 = 𝐺) → {𝑔} = {𝐺})
2120imaeq2d 6078 . . 3 ((𝜑𝑔 = 𝐺) → (dom DProd “ {𝑔}) = (dom DProd “ {𝐺}))
22 dpjfval.1 . . . 4 (𝜑𝐺dom DProd 𝑆)
23 dprdgrp 20025 . . . 4 (𝐺dom DProd 𝑆𝐺 ∈ Grp)
2422, 23syl 17 . . 3 (𝜑𝐺 ∈ Grp)
25 reldmdprd 20017 . . . . 5 Rel dom DProd
26 elrelimasn 6104 . . . . 5 (Rel dom DProd → (𝑆 ∈ (dom DProd “ {𝐺}) ↔ 𝐺dom DProd 𝑆))
2725, 26ax-mp 5 . . . 4 (𝑆 ∈ (dom DProd “ {𝐺}) ↔ 𝐺dom DProd 𝑆)
2822, 27sylibr 234 . . 3 (𝜑𝑆 ∈ (dom DProd “ {𝐺}))
2922, 6dprddomcld 20021 . . . 4 (𝜑𝐼 ∈ V)
3029mptexd 7244 . . 3 (𝜑 → (𝑖𝐼 ↦ ((𝑆𝑖)𝑄(𝐺 DProd (𝑆 ↾ (𝐼 ∖ {𝑖}))))) ∈ V)
313, 18, 21, 24, 28, 30ovmpodx 7584 . 2 (𝜑 → (𝐺dProj𝑆) = (𝑖𝐼 ↦ ((𝑆𝑖)𝑄(𝐺 DProd (𝑆 ↾ (𝐼 ∖ {𝑖}))))))
321, 31eqtrid 2789 1 (𝜑𝑃 = (𝑖𝐼 ↦ ((𝑆𝑖)𝑄(𝐺 DProd (𝑆 ↾ (𝐼 ∖ {𝑖}))))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1540  wcel 2108  Vcvv 3480  cdif 3948  {csn 4626   class class class wbr 5143  cmpt 5225  dom cdm 5685  cres 5687  cima 5688  Rel wrel 5690  cfv 6561  (class class class)co 7431  cmpo 7433  Grpcgrp 18951  proj1cpj1 19653   DProd cdprd 20013  dProjcdpj 20014
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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8014  df-2nd 8015  df-ixp 8938  df-dprd 20015  df-dpj 20016
This theorem is referenced by:  dpjval  20076
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