![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > mvrval2 | Structured version Visualization version GIF version |
Description: Value of the generating elements of the power series structure. (Contributed by Mario Carneiro, 7-Jan-2015.) |
Ref | Expression |
---|---|
mvrfval.v | ⊢ 𝑉 = (𝐼 mVar 𝑅) |
mvrfval.d | ⊢ 𝐷 = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} |
mvrfval.z | ⊢ 0 = (0g‘𝑅) |
mvrfval.o | ⊢ 1 = (1r‘𝑅) |
mvrfval.i | ⊢ (𝜑 → 𝐼 ∈ 𝑊) |
mvrfval.r | ⊢ (𝜑 → 𝑅 ∈ 𝑌) |
mvrval.x | ⊢ (𝜑 → 𝑋 ∈ 𝐼) |
mvrval2.f | ⊢ (𝜑 → 𝐹 ∈ 𝐷) |
Ref | Expression |
---|---|
mvrval2 | ⊢ (𝜑 → ((𝑉‘𝑋)‘𝐹) = if(𝐹 = (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)), 1 , 0 )) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mvrfval.v | . . . 4 ⊢ 𝑉 = (𝐼 mVar 𝑅) | |
2 | mvrfval.d | . . . 4 ⊢ 𝐷 = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} | |
3 | mvrfval.z | . . . 4 ⊢ 0 = (0g‘𝑅) | |
4 | mvrfval.o | . . . 4 ⊢ 1 = (1r‘𝑅) | |
5 | mvrfval.i | . . . 4 ⊢ (𝜑 → 𝐼 ∈ 𝑊) | |
6 | mvrfval.r | . . . 4 ⊢ (𝜑 → 𝑅 ∈ 𝑌) | |
7 | mvrval.x | . . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝐼) | |
8 | 1, 2, 3, 4, 5, 6, 7 | mvrval 21413 | . . 3 ⊢ (𝜑 → (𝑉‘𝑋) = (𝑓 ∈ 𝐷 ↦ if(𝑓 = (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)), 1 , 0 ))) |
9 | 8 | fveq1d 6848 | . 2 ⊢ (𝜑 → ((𝑉‘𝑋)‘𝐹) = ((𝑓 ∈ 𝐷 ↦ if(𝑓 = (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)), 1 , 0 ))‘𝐹)) |
10 | mvrval2.f | . . 3 ⊢ (𝜑 → 𝐹 ∈ 𝐷) | |
11 | eqeq1 2737 | . . . . 5 ⊢ (𝑓 = 𝐹 → (𝑓 = (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)) ↔ 𝐹 = (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))) | |
12 | 11 | ifbid 4513 | . . . 4 ⊢ (𝑓 = 𝐹 → if(𝑓 = (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)), 1 , 0 ) = if(𝐹 = (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)), 1 , 0 )) |
13 | eqid 2733 | . . . 4 ⊢ (𝑓 ∈ 𝐷 ↦ if(𝑓 = (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)), 1 , 0 )) = (𝑓 ∈ 𝐷 ↦ if(𝑓 = (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)), 1 , 0 )) | |
14 | 4 | fvexi 6860 | . . . . 5 ⊢ 1 ∈ V |
15 | 3 | fvexi 6860 | . . . . 5 ⊢ 0 ∈ V |
16 | 14, 15 | ifex 4540 | . . . 4 ⊢ if(𝐹 = (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)), 1 , 0 ) ∈ V |
17 | 12, 13, 16 | fvmpt 6952 | . . 3 ⊢ (𝐹 ∈ 𝐷 → ((𝑓 ∈ 𝐷 ↦ if(𝑓 = (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)), 1 , 0 ))‘𝐹) = if(𝐹 = (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)), 1 , 0 )) |
18 | 10, 17 | syl 17 | . 2 ⊢ (𝜑 → ((𝑓 ∈ 𝐷 ↦ if(𝑓 = (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)), 1 , 0 ))‘𝐹) = if(𝐹 = (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)), 1 , 0 )) |
19 | 9, 18 | eqtrd 2773 | 1 ⊢ (𝜑 → ((𝑉‘𝑋)‘𝐹) = if(𝐹 = (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)), 1 , 0 )) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2107 {crab 3406 ifcif 4490 ↦ cmpt 5192 ◡ccnv 5636 “ cima 5640 ‘cfv 6500 (class class class)co 7361 ↑m cmap 8771 Fincfn 8889 0cc0 11059 1c1 11060 ℕcn 12161 ℕ0cn0 12421 0gc0g 17329 1rcur 19921 mVar cmvr 21330 |
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 5246 ax-sep 5260 ax-nul 5267 ax-pr 5388 |
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 2941 df-ral 3062 df-rex 3071 df-reu 3353 df-rab 3407 df-v 3449 df-sbc 3744 df-csb 3860 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-nul 4287 df-if 4491 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4870 df-iun 4960 df-br 5110 df-opab 5172 df-mpt 5193 df-id 5535 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-rn 5648 df-res 5649 df-ima 5650 df-iota 6452 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-ov 7364 df-oprab 7365 df-mpo 7366 df-mvr 21335 |
This theorem is referenced by: mvrid 21415 mvrf1 21417 mvrcl 21444 mhpvarcl 21561 |
Copyright terms: Public domain | W3C validator |