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Mirrors > Home > MPE Home > Th. List > dpjval | Structured version Visualization version GIF version |
Description: Value of the direct product projection (defined in terms of binary projection). (Contributed by Mario Carneiro, 26-Apr-2016.) |
Ref | Expression |
---|---|
dpjfval.1 | ⊢ (𝜑 → 𝐺dom DProd 𝑆) |
dpjfval.2 | ⊢ (𝜑 → dom 𝑆 = 𝐼) |
dpjfval.p | ⊢ 𝑃 = (𝐺dProj𝑆) |
dpjfval.q | ⊢ 𝑄 = (proj1‘𝐺) |
dpjval.3 | ⊢ (𝜑 → 𝑋 ∈ 𝐼) |
Ref | Expression |
---|---|
dpjval | ⊢ (𝜑 → (𝑃‘𝑋) = ((𝑆‘𝑋)𝑄(𝐺 DProd (𝑆 ↾ (𝐼 ∖ {𝑋}))))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dpjfval.1 | . . 3 ⊢ (𝜑 → 𝐺dom DProd 𝑆) | |
2 | dpjfval.2 | . . 3 ⊢ (𝜑 → dom 𝑆 = 𝐼) | |
3 | dpjfval.p | . . 3 ⊢ 𝑃 = (𝐺dProj𝑆) | |
4 | dpjfval.q | . . 3 ⊢ 𝑄 = (proj1‘𝐺) | |
5 | 1, 2, 3, 4 | dpjfval 19925 | . 2 ⊢ (𝜑 → 𝑃 = (𝑥 ∈ 𝐼 ↦ ((𝑆‘𝑥)𝑄(𝐺 DProd (𝑆 ↾ (𝐼 ∖ {𝑥})))))) |
6 | simpr 486 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → 𝑥 = 𝑋) | |
7 | 6 | fveq2d 6896 | . . 3 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → (𝑆‘𝑥) = (𝑆‘𝑋)) |
8 | 6 | sneqd 4641 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → {𝑥} = {𝑋}) |
9 | 8 | difeq2d 4123 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → (𝐼 ∖ {𝑥}) = (𝐼 ∖ {𝑋})) |
10 | 9 | reseq2d 5982 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → (𝑆 ↾ (𝐼 ∖ {𝑥})) = (𝑆 ↾ (𝐼 ∖ {𝑋}))) |
11 | 10 | oveq2d 7425 | . . 3 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → (𝐺 DProd (𝑆 ↾ (𝐼 ∖ {𝑥}))) = (𝐺 DProd (𝑆 ↾ (𝐼 ∖ {𝑋})))) |
12 | 7, 11 | oveq12d 7427 | . 2 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → ((𝑆‘𝑥)𝑄(𝐺 DProd (𝑆 ↾ (𝐼 ∖ {𝑥})))) = ((𝑆‘𝑋)𝑄(𝐺 DProd (𝑆 ↾ (𝐼 ∖ {𝑋}))))) |
13 | dpjval.3 | . 2 ⊢ (𝜑 → 𝑋 ∈ 𝐼) | |
14 | ovexd 7444 | . 2 ⊢ (𝜑 → ((𝑆‘𝑋)𝑄(𝐺 DProd (𝑆 ↾ (𝐼 ∖ {𝑋})))) ∈ V) | |
15 | 5, 12, 13, 14 | fvmptd 7006 | 1 ⊢ (𝜑 → (𝑃‘𝑋) = ((𝑆‘𝑋)𝑄(𝐺 DProd (𝑆 ↾ (𝐼 ∖ {𝑋}))))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 = wceq 1542 ∈ wcel 2107 Vcvv 3475 ∖ cdif 3946 {csn 4629 class class class wbr 5149 dom cdm 5677 ↾ cres 5679 ‘cfv 6544 (class class class)co 7409 proj1cpj1 19503 DProd cdprd 19863 dProjcdpj 19864 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5575 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-ov 7412 df-oprab 7413 df-mpo 7414 df-1st 7975 df-2nd 7976 df-ixp 8892 df-dprd 19865 df-dpj 19866 |
This theorem is referenced by: dpjf 19927 dpjidcl 19928 dpjlid 19931 dpjghm 19933 |
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