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Theorem dpjfval 19168
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 19109 . . . 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 772 . . . . . 6 ((𝜑 ∧ (𝑔 = 𝐺𝑠 = 𝑆)) → 𝑠 = 𝑆)
54dmeqd 5757 . . . . 5 ((𝜑 ∧ (𝑔 = 𝐺𝑠 = 𝑆)) → dom 𝑠 = dom 𝑆)
6 dpjfval.2 . . . . . 6 (𝜑 → dom 𝑆 = 𝐼)
76adantr 484 . . . . 5 ((𝜑 ∧ (𝑔 = 𝐺𝑠 = 𝑆)) → dom 𝑆 = 𝐼)
85, 7eqtrd 2859 . . . 4 ((𝜑 ∧ (𝑔 = 𝐺𝑠 = 𝑆)) → dom 𝑠 = 𝐼)
9 simprl 770 . . . . . . 7 ((𝜑 ∧ (𝑔 = 𝐺𝑠 = 𝑆)) → 𝑔 = 𝐺)
109fveq2d 6657 . . . . . 6 ((𝜑 ∧ (𝑔 = 𝐺𝑠 = 𝑆)) → (proj1𝑔) = (proj1𝐺))
11 dpjfval.q . . . . . 6 𝑄 = (proj1𝐺)
1210, 11syl6eqr 2877 . . . . 5 ((𝜑 ∧ (𝑔 = 𝐺𝑠 = 𝑆)) → (proj1𝑔) = 𝑄)
134fveq1d 6655 . . . . 5 ((𝜑 ∧ (𝑔 = 𝐺𝑠 = 𝑆)) → (𝑠𝑖) = (𝑆𝑖))
148difeq1d 4082 . . . . . . 7 ((𝜑 ∧ (𝑔 = 𝐺𝑠 = 𝑆)) → (dom 𝑠 ∖ {𝑖}) = (𝐼 ∖ {𝑖}))
154, 14reseq12d 5837 . . . . . 6 ((𝜑 ∧ (𝑔 = 𝐺𝑠 = 𝑆)) → (𝑠 ↾ (dom 𝑠 ∖ {𝑖})) = (𝑆 ↾ (𝐼 ∖ {𝑖})))
169, 15oveq12d 7158 . . . . 5 ((𝜑 ∧ (𝑔 = 𝐺𝑠 = 𝑆)) → (𝑔 DProd (𝑠 ↾ (dom 𝑠 ∖ {𝑖}))) = (𝐺 DProd (𝑆 ↾ (𝐼 ∖ {𝑖}))))
1712, 13, 16oveq123d 7161 . . . 4 ((𝜑 ∧ (𝑔 = 𝐺𝑠 = 𝑆)) → ((𝑠𝑖)(proj1𝑔)(𝑔 DProd (𝑠 ↾ (dom 𝑠 ∖ {𝑖})))) = ((𝑆𝑖)𝑄(𝐺 DProd (𝑆 ↾ (𝐼 ∖ {𝑖})))))
188, 17mpteq12dv 5134 . . 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 488 . . . . 5 ((𝜑𝑔 = 𝐺) → 𝑔 = 𝐺)
2019sneqd 4560 . . . 4 ((𝜑𝑔 = 𝐺) → {𝑔} = {𝐺})
2120imaeq2d 5912 . . 3 ((𝜑𝑔 = 𝐺) → (dom DProd “ {𝑔}) = (dom DProd “ {𝐺}))
22 dpjfval.1 . . . 4 (𝜑𝐺dom DProd 𝑆)
23 dprdgrp 19118 . . . 4 (𝐺dom DProd 𝑆𝐺 ∈ Grp)
2422, 23syl 17 . . 3 (𝜑𝐺 ∈ Grp)
25 reldmdprd 19110 . . . . 5 Rel dom DProd
26 elrelimasn 5936 . . . . 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 237 . . 3 (𝜑𝑆 ∈ (dom DProd “ {𝐺}))
2922, 6dprddomcld 19114 . . . 4 (𝜑𝐼 ∈ V)
3029mptexd 6970 . . 3 (𝜑 → (𝑖𝐼 ↦ ((𝑆𝑖)𝑄(𝐺 DProd (𝑆 ↾ (𝐼 ∖ {𝑖}))))) ∈ V)
313, 18, 21, 24, 28, 30ovmpodx 7285 . 2 (𝜑 → (𝐺dProj𝑆) = (𝑖𝐼 ↦ ((𝑆𝑖)𝑄(𝐺 DProd (𝑆 ↾ (𝐼 ∖ {𝑖}))))))
321, 31syl5eq 2871 1 (𝜑𝑃 = (𝑖𝐼 ↦ ((𝑆𝑖)𝑄(𝐺 DProd (𝑆 ↾ (𝐼 ∖ {𝑖}))))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1538  wcel 2115  Vcvv 3479  cdif 3915  {csn 4548   class class class wbr 5049  cmpt 5129  dom cdm 5538  cres 5540  cima 5541  Rel wrel 5543  cfv 6338  (class class class)co 7140  cmpo 7142  Grpcgrp 18094  proj1cpj1 18751   DProd cdprd 19106  dProjcdpj 19107
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5173  ax-sep 5186  ax-nul 5193  ax-pow 5249  ax-pr 5313  ax-un 7446
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3014  df-nel 3118  df-ral 3137  df-rex 3138  df-reu 3139  df-rab 3141  df-v 3481  df-sbc 3758  df-csb 3866  df-dif 3921  df-un 3923  df-in 3925  df-ss 3935  df-nul 4275  df-if 4449  df-pw 4522  df-sn 4549  df-pr 4551  df-op 4555  df-uni 4822  df-iun 4904  df-br 5050  df-opab 5112  df-mpt 5130  df-id 5443  df-xp 5544  df-rel 5545  df-cnv 5546  df-co 5547  df-dm 5548  df-rn 5549  df-res 5550  df-ima 5551  df-iota 6297  df-fun 6340  df-fn 6341  df-f 6342  df-f1 6343  df-fo 6344  df-f1o 6345  df-fv 6346  df-ov 7143  df-oprab 7144  df-mpo 7145  df-1st 7674  df-2nd 7675  df-ixp 8447  df-dprd 19108  df-dpj 19109
This theorem is referenced by:  dpjval  19169
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