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Theorem dpjfval 20090
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 20031 . . . 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 5919 . . . . 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 2775 . . . 4 ((𝜑 ∧ (𝑔 = 𝐺𝑠 = 𝑆)) → dom 𝑠 = 𝐼)
9 simprl 771 . . . . . . 7 ((𝜑 ∧ (𝑔 = 𝐺𝑠 = 𝑆)) → 𝑔 = 𝐺)
109fveq2d 6911 . . . . . 6 ((𝜑 ∧ (𝑔 = 𝐺𝑠 = 𝑆)) → (proj1𝑔) = (proj1𝐺))
11 dpjfval.q . . . . . 6 𝑄 = (proj1𝐺)
1210, 11eqtr4di 2793 . . . . 5 ((𝜑 ∧ (𝑔 = 𝐺𝑠 = 𝑆)) → (proj1𝑔) = 𝑄)
134fveq1d 6909 . . . . 5 ((𝜑 ∧ (𝑔 = 𝐺𝑠 = 𝑆)) → (𝑠𝑖) = (𝑆𝑖))
148difeq1d 4135 . . . . . . 7 ((𝜑 ∧ (𝑔 = 𝐺𝑠 = 𝑆)) → (dom 𝑠 ∖ {𝑖}) = (𝐼 ∖ {𝑖}))
154, 14reseq12d 6001 . . . . . 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 5239 . . 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 4643 . . . 4 ((𝜑𝑔 = 𝐺) → {𝑔} = {𝐺})
2120imaeq2d 6080 . . 3 ((𝜑𝑔 = 𝐺) → (dom DProd “ {𝑔}) = (dom DProd “ {𝐺}))
22 dpjfval.1 . . . 4 (𝜑𝐺dom DProd 𝑆)
23 dprdgrp 20040 . . . 4 (𝐺dom DProd 𝑆𝐺 ∈ Grp)
2422, 23syl 17 . . 3 (𝜑𝐺 ∈ Grp)
25 reldmdprd 20032 . . . . 5 Rel dom DProd
26 elrelimasn 6106 . . . . 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 20036 . . . 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 2787 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 1537  wcel 2106  Vcvv 3478  cdif 3960  {csn 4631   class class class wbr 5148  cmpt 5231  dom cdm 5689  cres 5691  cima 5692  Rel wrel 5694  cfv 6563  (class class class)co 7431  cmpo 7433  Grpcgrp 18964  proj1cpj1 19668   DProd cdprd 20028  dProjcdpj 20029
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8013  df-2nd 8014  df-ixp 8937  df-dprd 20030  df-dpj 20031
This theorem is referenced by:  dpjval  20091
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