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Theorem dpjfval 18448
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 18389 . . . 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 796 . . . . . 6 ((𝜑 ∧ (𝑔 = 𝐺𝑠 = 𝑆)) → 𝑠 = 𝑆)
54dmeqd 5324 . . . . 5 ((𝜑 ∧ (𝑔 = 𝐺𝑠 = 𝑆)) → dom 𝑠 = dom 𝑆)
6 dpjfval.2 . . . . . 6 (𝜑 → dom 𝑆 = 𝐼)
76adantr 481 . . . . 5 ((𝜑 ∧ (𝑔 = 𝐺𝑠 = 𝑆)) → dom 𝑆 = 𝐼)
85, 7eqtrd 2655 . . . 4 ((𝜑 ∧ (𝑔 = 𝐺𝑠 = 𝑆)) → dom 𝑠 = 𝐼)
9 simprl 794 . . . . . . 7 ((𝜑 ∧ (𝑔 = 𝐺𝑠 = 𝑆)) → 𝑔 = 𝐺)
109fveq2d 6193 . . . . . 6 ((𝜑 ∧ (𝑔 = 𝐺𝑠 = 𝑆)) → (proj1𝑔) = (proj1𝐺))
11 dpjfval.q . . . . . 6 𝑄 = (proj1𝐺)
1210, 11syl6eqr 2673 . . . . 5 ((𝜑 ∧ (𝑔 = 𝐺𝑠 = 𝑆)) → (proj1𝑔) = 𝑄)
134fveq1d 6191 . . . . 5 ((𝜑 ∧ (𝑔 = 𝐺𝑠 = 𝑆)) → (𝑠𝑖) = (𝑆𝑖))
148difeq1d 3725 . . . . . . 7 ((𝜑 ∧ (𝑔 = 𝐺𝑠 = 𝑆)) → (dom 𝑠 ∖ {𝑖}) = (𝐼 ∖ {𝑖}))
154, 14reseq12d 5395 . . . . . 6 ((𝜑 ∧ (𝑔 = 𝐺𝑠 = 𝑆)) → (𝑠 ↾ (dom 𝑠 ∖ {𝑖})) = (𝑆 ↾ (𝐼 ∖ {𝑖})))
169, 15oveq12d 6665 . . . . 5 ((𝜑 ∧ (𝑔 = 𝐺𝑠 = 𝑆)) → (𝑔 DProd (𝑠 ↾ (dom 𝑠 ∖ {𝑖}))) = (𝐺 DProd (𝑆 ↾ (𝐼 ∖ {𝑖}))))
1712, 13, 16oveq123d 6668 . . . 4 ((𝜑 ∧ (𝑔 = 𝐺𝑠 = 𝑆)) → ((𝑠𝑖)(proj1𝑔)(𝑔 DProd (𝑠 ↾ (dom 𝑠 ∖ {𝑖})))) = ((𝑆𝑖)𝑄(𝐺 DProd (𝑆 ↾ (𝐼 ∖ {𝑖})))))
188, 17mpteq12dv 4731 . . 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 477 . . . . 5 ((𝜑𝑔 = 𝐺) → 𝑔 = 𝐺)
2019sneqd 4187 . . . 4 ((𝜑𝑔 = 𝐺) → {𝑔} = {𝐺})
2120imaeq2d 5464 . . 3 ((𝜑𝑔 = 𝐺) → (dom DProd “ {𝑔}) = (dom DProd “ {𝐺}))
22 dpjfval.1 . . . 4 (𝜑𝐺dom DProd 𝑆)
23 dprdgrp 18398 . . . 4 (𝐺dom DProd 𝑆𝐺 ∈ Grp)
2422, 23syl 17 . . 3 (𝜑𝐺 ∈ Grp)
25 reldmdprd 18390 . . . . 5 Rel dom DProd
26 elrelimasn 5487 . . . . 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 18394 . . . 4 (𝜑𝐼 ∈ V)
30 mptexg 6481 . . . 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 6784 . 2 (𝜑 → (𝐺dProj𝑆) = (𝑖𝐼 ↦ ((𝑆𝑖)𝑄(𝐺 DProd (𝑆 ↾ (𝐼 ∖ {𝑖}))))))
331, 32syl5eq 2667 1 (𝜑𝑃 = (𝑖𝐼 ↦ ((𝑆𝑖)𝑄(𝐺 DProd (𝑆 ↾ (𝐼 ∖ {𝑖}))))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1482  wcel 1989  Vcvv 3198  cdif 3569  {csn 4175   class class class wbr 4651  cmpt 4727  dom cdm 5112  cres 5114  cima 5115  Rel wrel 5117  cfv 5886  (class class class)co 6647  cmpt2 6649  Grpcgrp 17416  proj1cpj1 18044   DProd cdprd 18386  dProjcdpj 18387
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-8 1991  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601  ax-rep 4769  ax-sep 4779  ax-nul 4787  ax-pow 4841  ax-pr 4904  ax-un 6946
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-ne 2794  df-nel 2897  df-ral 2916  df-rex 2917  df-reu 2918  df-rab 2920  df-v 3200  df-sbc 3434  df-csb 3532  df-dif 3575  df-un 3577  df-in 3579  df-ss 3586  df-nul 3914  df-if 4085  df-pw 4158  df-sn 4176  df-pr 4178  df-op 4182  df-uni 4435  df-iun 4520  df-br 4652  df-opab 4711  df-mpt 4728  df-id 5022  df-xp 5118  df-rel 5119  df-cnv 5120  df-co 5121  df-dm 5122  df-rn 5123  df-res 5124  df-ima 5125  df-iota 5849  df-fun 5888  df-fn 5889  df-f 5890  df-f1 5891  df-fo 5892  df-f1o 5893  df-fv 5894  df-ov 6650  df-oprab 6651  df-mpt2 6652  df-1st 7165  df-2nd 7166  df-ixp 7906  df-dprd 18388  df-dpj 18389
This theorem is referenced by:  dpjval  18449
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