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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 20204 | . . 3 ⊢ (𝜑 → (𝑉‘𝑋) = (𝑓 ∈ 𝐷 ↦ if(𝑓 = (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)), 1 , 0 ))) |
9 | 8 | fveq1d 6675 | . 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 2828 | . . . . 5 ⊢ (𝑓 = 𝐹 → (𝑓 = (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)) ↔ 𝐹 = (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))) | |
12 | 11 | ifbid 4492 | . . . 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 2824 | . . . 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 6687 | . . . . 5 ⊢ 1 ∈ V |
15 | 3 | fvexi 6687 | . . . . 5 ⊢ 0 ∈ V |
16 | 14, 15 | ifex 4518 | . . . 4 ⊢ if(𝐹 = (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)), 1 , 0 ) ∈ V |
17 | 12, 13, 16 | fvmpt 6771 | . . 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 2859 | 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 1536 ∈ wcel 2113 {crab 3145 ifcif 4470 ↦ cmpt 5149 ◡ccnv 5557 “ cima 5561 ‘cfv 6358 (class class class)co 7159 ↑m cmap 8409 Fincfn 8512 0cc0 10540 1c1 10541 ℕcn 11641 ℕ0cn0 11900 0gc0g 16716 1rcur 19254 mVar cmvr 20135 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-rep 5193 ax-sep 5206 ax-nul 5213 ax-pr 5333 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-ral 3146 df-rex 3147 df-reu 3148 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-nul 4295 df-if 4471 df-sn 4571 df-pr 4573 df-op 4577 df-uni 4842 df-iun 4924 df-br 5070 df-opab 5132 df-mpt 5150 df-id 5463 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-ov 7162 df-oprab 7163 df-mpo 7164 df-mvr 20140 |
This theorem is referenced by: mvrid 20206 mvrf1 20208 mvrcl 20232 |
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