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Mirrors > Home > MPE Home > Th. List > mvrval | 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) |
Ref | Expression |
---|---|
mvrval | ⊢ (𝜑 → (𝑉‘𝑋) = (𝑓 ∈ 𝐷 ↦ 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 | 1, 2, 3, 4, 5, 6 | mvrfval 20945 | . . 3 ⊢ (𝜑 → 𝑉 = (𝑥 ∈ 𝐼 ↦ (𝑓 ∈ 𝐷 ↦ if(𝑓 = (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑥, 1, 0)), 1 , 0 )))) |
8 | 7 | fveq1d 6719 | . 2 ⊢ (𝜑 → (𝑉‘𝑋) = ((𝑥 ∈ 𝐼 ↦ (𝑓 ∈ 𝐷 ↦ if(𝑓 = (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑥, 1, 0)), 1 , 0 )))‘𝑋)) |
9 | mvrval.x | . . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐼) | |
10 | eqeq2 2749 | . . . . . . . . 9 ⊢ (𝑥 = 𝑋 → (𝑦 = 𝑥 ↔ 𝑦 = 𝑋)) | |
11 | 10 | ifbid 4462 | . . . . . . . 8 ⊢ (𝑥 = 𝑋 → if(𝑦 = 𝑥, 1, 0) = if(𝑦 = 𝑋, 1, 0)) |
12 | 11 | mpteq2dv 5151 | . . . . . . 7 ⊢ (𝑥 = 𝑋 → (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑥, 1, 0)) = (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) |
13 | 12 | eqeq2d 2748 | . . . . . 6 ⊢ (𝑥 = 𝑋 → (𝑓 = (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑥, 1, 0)) ↔ 𝑓 = (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))) |
14 | 13 | ifbid 4462 | . . . . 5 ⊢ (𝑥 = 𝑋 → if(𝑓 = (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑥, 1, 0)), 1 , 0 ) = if(𝑓 = (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)), 1 , 0 )) |
15 | 14 | mpteq2dv 5151 | . . . 4 ⊢ (𝑥 = 𝑋 → (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 ))) |
16 | eqid 2737 | . . . 4 ⊢ (𝑥 ∈ 𝐼 ↦ (𝑓 ∈ 𝐷 ↦ if(𝑓 = (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑥, 1, 0)), 1 , 0 ))) = (𝑥 ∈ 𝐼 ↦ (𝑓 ∈ 𝐷 ↦ if(𝑓 = (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑥, 1, 0)), 1 , 0 ))) | |
17 | ovex 7246 | . . . . . 6 ⊢ (ℕ0 ↑m 𝐼) ∈ V | |
18 | 2, 17 | rabex2 5227 | . . . . 5 ⊢ 𝐷 ∈ V |
19 | 18 | mptex 7039 | . . . 4 ⊢ (𝑓 ∈ 𝐷 ↦ if(𝑓 = (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)), 1 , 0 )) ∈ V |
20 | 15, 16, 19 | fvmpt 6818 | . . 3 ⊢ (𝑋 ∈ 𝐼 → ((𝑥 ∈ 𝐼 ↦ (𝑓 ∈ 𝐷 ↦ if(𝑓 = (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑥, 1, 0)), 1 , 0 )))‘𝑋) = (𝑓 ∈ 𝐷 ↦ if(𝑓 = (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)), 1 , 0 ))) |
21 | 9, 20 | syl 17 | . 2 ⊢ (𝜑 → ((𝑥 ∈ 𝐼 ↦ (𝑓 ∈ 𝐷 ↦ if(𝑓 = (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑥, 1, 0)), 1 , 0 )))‘𝑋) = (𝑓 ∈ 𝐷 ↦ if(𝑓 = (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)), 1 , 0 ))) |
22 | 8, 21 | eqtrd 2777 | 1 ⊢ (𝜑 → (𝑉‘𝑋) = (𝑓 ∈ 𝐷 ↦ if(𝑓 = (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)), 1 , 0 ))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1543 ∈ wcel 2110 {crab 3065 ifcif 4439 ↦ cmpt 5135 ◡ccnv 5550 “ cima 5554 ‘cfv 6380 (class class class)co 7213 ↑m cmap 8508 Fincfn 8626 0cc0 10729 1c1 10730 ℕcn 11830 ℕ0cn0 12090 0gc0g 16944 1rcur 19516 mVar cmvr 20864 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-rep 5179 ax-sep 5192 ax-nul 5199 ax-pr 5322 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2941 df-ral 3066 df-rex 3067 df-reu 3068 df-rab 3070 df-v 3410 df-sbc 3695 df-csb 3812 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-nul 4238 df-if 4440 df-sn 4542 df-pr 4544 df-op 4548 df-uni 4820 df-iun 4906 df-br 5054 df-opab 5116 df-mpt 5136 df-id 5455 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-rn 5562 df-res 5563 df-ima 5564 df-iota 6338 df-fun 6382 df-fn 6383 df-f 6384 df-f1 6385 df-fo 6386 df-f1o 6387 df-fv 6388 df-ov 7216 df-oprab 7217 df-mpo 7218 df-mvr 20869 |
This theorem is referenced by: mvrval2 20947 mplcoe3 20995 evlslem1 21042 |
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