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Theorem dpjfval 18654
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 18595 . . . 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 813 . . . . . 6 ((𝜑 ∧ (𝑔 = 𝐺𝑠 = 𝑆)) → 𝑠 = 𝑆)
54dmeqd 5481 . . . . 5 ((𝜑 ∧ (𝑔 = 𝐺𝑠 = 𝑆)) → dom 𝑠 = dom 𝑆)
6 dpjfval.2 . . . . . 6 (𝜑 → dom 𝑆 = 𝐼)
76adantr 472 . . . . 5 ((𝜑 ∧ (𝑔 = 𝐺𝑠 = 𝑆)) → dom 𝑆 = 𝐼)
85, 7eqtrd 2794 . . . 4 ((𝜑 ∧ (𝑔 = 𝐺𝑠 = 𝑆)) → dom 𝑠 = 𝐼)
9 simprl 811 . . . . . . 7 ((𝜑 ∧ (𝑔 = 𝐺𝑠 = 𝑆)) → 𝑔 = 𝐺)
109fveq2d 6356 . . . . . 6 ((𝜑 ∧ (𝑔 = 𝐺𝑠 = 𝑆)) → (proj1𝑔) = (proj1𝐺))
11 dpjfval.q . . . . . 6 𝑄 = (proj1𝐺)
1210, 11syl6eqr 2812 . . . . 5 ((𝜑 ∧ (𝑔 = 𝐺𝑠 = 𝑆)) → (proj1𝑔) = 𝑄)
134fveq1d 6354 . . . . 5 ((𝜑 ∧ (𝑔 = 𝐺𝑠 = 𝑆)) → (𝑠𝑖) = (𝑆𝑖))
148difeq1d 3870 . . . . . . 7 ((𝜑 ∧ (𝑔 = 𝐺𝑠 = 𝑆)) → (dom 𝑠 ∖ {𝑖}) = (𝐼 ∖ {𝑖}))
154, 14reseq12d 5552 . . . . . 6 ((𝜑 ∧ (𝑔 = 𝐺𝑠 = 𝑆)) → (𝑠 ↾ (dom 𝑠 ∖ {𝑖})) = (𝑆 ↾ (𝐼 ∖ {𝑖})))
169, 15oveq12d 6831 . . . . 5 ((𝜑 ∧ (𝑔 = 𝐺𝑠 = 𝑆)) → (𝑔 DProd (𝑠 ↾ (dom 𝑠 ∖ {𝑖}))) = (𝐺 DProd (𝑆 ↾ (𝐼 ∖ {𝑖}))))
1712, 13, 16oveq123d 6834 . . . 4 ((𝜑 ∧ (𝑔 = 𝐺𝑠 = 𝑆)) → ((𝑠𝑖)(proj1𝑔)(𝑔 DProd (𝑠 ↾ (dom 𝑠 ∖ {𝑖})))) = ((𝑆𝑖)𝑄(𝐺 DProd (𝑆 ↾ (𝐼 ∖ {𝑖})))))
188, 17mpteq12dv 4885 . . 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 479 . . . . 5 ((𝜑𝑔 = 𝐺) → 𝑔 = 𝐺)
2019sneqd 4333 . . . 4 ((𝜑𝑔 = 𝐺) → {𝑔} = {𝐺})
2120imaeq2d 5624 . . 3 ((𝜑𝑔 = 𝐺) → (dom DProd “ {𝑔}) = (dom DProd “ {𝐺}))
22 dpjfval.1 . . . 4 (𝜑𝐺dom DProd 𝑆)
23 dprdgrp 18604 . . . 4 (𝐺dom DProd 𝑆𝐺 ∈ Grp)
2422, 23syl 17 . . 3 (𝜑𝐺 ∈ Grp)
25 reldmdprd 18596 . . . . 5 Rel dom DProd
26 elrelimasn 5647 . . . . 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 224 . . 3 (𝜑𝑆 ∈ (dom DProd “ {𝐺}))
2922, 6dprddomcld 18600 . . . 4 (𝜑𝐼 ∈ V)
30 mptexg 6648 . . . 4 (𝐼 ∈ V → (𝑖𝐼 ↦ ((𝑆𝑖)𝑄(𝐺 DProd (𝑆 ↾ (𝐼 ∖ {𝑖}))))) ∈ V)
3129, 30syl 17 . . 3 (𝜑 → (𝑖𝐼 ↦ ((𝑆𝑖)𝑄(𝐺 DProd (𝑆 ↾ (𝐼 ∖ {𝑖}))))) ∈ V)
323, 18, 21, 24, 28, 31ovmpt2dx 6952 . 2 (𝜑 → (𝐺dProj𝑆) = (𝑖𝐼 ↦ ((𝑆𝑖)𝑄(𝐺 DProd (𝑆 ↾ (𝐼 ∖ {𝑖}))))))
331, 32syl5eq 2806 1 (𝜑𝑃 = (𝑖𝐼 ↦ ((𝑆𝑖)𝑄(𝐺 DProd (𝑆 ↾ (𝐼 ∖ {𝑖}))))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1632  wcel 2139  Vcvv 3340  cdif 3712  {csn 4321   class class class wbr 4804  cmpt 4881  dom cdm 5266  cres 5268  cima 5269  Rel wrel 5271  cfv 6049  (class class class)co 6813  cmpt2 6815  Grpcgrp 17623  proj1cpj1 18250   DProd cdprd 18592  dProjcdpj 18593
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-nel 3036  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-ov 6816  df-oprab 6817  df-mpt2 6818  df-1st 7333  df-2nd 7334  df-ixp 8075  df-dprd 18594  df-dpj 18595
This theorem is referenced by:  dpjval  18655
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