Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 20200 | . . 3 ⊢ (𝜑 → 𝑉 = (𝑥 ∈ 𝐼 ↦ (𝑓 ∈ 𝐷 ↦ if(𝑓 = (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑥, 1, 0)), 1 , 0 )))) |
8 | 7 | fveq1d 6672 | . 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 2833 | . . . . . . . . 9 ⊢ (𝑥 = 𝑋 → (𝑦 = 𝑥 ↔ 𝑦 = 𝑋)) | |
11 | 10 | ifbid 4489 | . . . . . . . 8 ⊢ (𝑥 = 𝑋 → if(𝑦 = 𝑥, 1, 0) = if(𝑦 = 𝑋, 1, 0)) |
12 | 11 | mpteq2dv 5162 | . . . . . . 7 ⊢ (𝑥 = 𝑋 → (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑥, 1, 0)) = (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) |
13 | 12 | eqeq2d 2832 | . . . . . 6 ⊢ (𝑥 = 𝑋 → (𝑓 = (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑥, 1, 0)) ↔ 𝑓 = (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))) |
14 | 13 | ifbid 4489 | . . . . 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 5162 | . . . 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 2821 | . . . 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 7189 | . . . . . 6 ⊢ (ℕ0 ↑m 𝐼) ∈ V | |
18 | 2, 17 | rabex2 5237 | . . . . 5 ⊢ 𝐷 ∈ V |
19 | 18 | mptex 6986 | . . . 4 ⊢ (𝑓 ∈ 𝐷 ↦ if(𝑓 = (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)), 1 , 0 )) ∈ V |
20 | 15, 16, 19 | fvmpt 6768 | . . 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 2856 | 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 2114 {crab 3142 ifcif 4467 ↦ cmpt 5146 ◡ccnv 5554 “ cima 5558 ‘cfv 6355 (class class class)co 7156 ↑m cmap 8406 Fincfn 8509 0cc0 10537 1c1 10538 ℕcn 11638 ℕ0cn0 11898 0gc0g 16713 1rcur 19251 mVar cmvr 20132 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pr 5330 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-ov 7159 df-oprab 7160 df-mpo 7161 df-mvr 20137 |
This theorem is referenced by: mvrval2 20202 mplcoe3 20247 evlslem1 20295 |
Copyright terms: Public domain | W3C validator |