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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 22019 | . . 3 ⊢ (𝜑 → 𝑉 = (𝑥 ∈ 𝐼 ↦ (𝑓 ∈ 𝐷 ↦ if(𝑓 = (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑥, 1, 0)), 1 , 0 )))) |
8 | 7 | fveq1d 6909 | . 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 2747 | . . . . . . . . 9 ⊢ (𝑥 = 𝑋 → (𝑦 = 𝑥 ↔ 𝑦 = 𝑋)) | |
11 | 10 | ifbid 4554 | . . . . . . . 8 ⊢ (𝑥 = 𝑋 → if(𝑦 = 𝑥, 1, 0) = if(𝑦 = 𝑋, 1, 0)) |
12 | 11 | mpteq2dv 5250 | . . . . . . 7 ⊢ (𝑥 = 𝑋 → (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑥, 1, 0)) = (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) |
13 | 12 | eqeq2d 2746 | . . . . . 6 ⊢ (𝑥 = 𝑋 → (𝑓 = (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑥, 1, 0)) ↔ 𝑓 = (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))) |
14 | 13 | ifbid 4554 | . . . . 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 5250 | . . . 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 2735 | . . . 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 7464 | . . . . . 6 ⊢ (ℕ0 ↑m 𝐼) ∈ V | |
18 | 2, 17 | rabex2 5347 | . . . . 5 ⊢ 𝐷 ∈ V |
19 | 18 | mptex 7243 | . . . 4 ⊢ (𝑓 ∈ 𝐷 ↦ if(𝑓 = (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)), 1 , 0 )) ∈ V |
20 | 15, 16, 19 | fvmpt 7016 | . . 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 2775 | 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 1537 ∈ wcel 2106 {crab 3433 ifcif 4531 ↦ cmpt 5231 ◡ccnv 5688 “ cima 5692 ‘cfv 6563 (class class class)co 7431 ↑m cmap 8865 Fincfn 8984 0cc0 11153 1c1 11154 ℕcn 12264 ℕ0cn0 12524 0gc0g 17486 1rcur 20199 mVar cmvr 21943 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-ov 7434 df-oprab 7435 df-mpo 7436 df-mvr 21948 |
This theorem is referenced by: mvrval2 22021 mplcoe3 22074 evlslem1 22124 rhmply1vr1 22407 |
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