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Mirrors > Home > MPE Home > Th. List > evlval | Structured version Visualization version GIF version |
Description: Value of the simple/same ring evaluation map. (Contributed by Stefan O'Rear, 19-Mar-2015.) (Revised by Mario Carneiro, 12-Jun-2015.) |
Ref | Expression |
---|---|
evlval.q | ⊢ 𝑄 = (𝐼 eval 𝑅) |
evlval.b | ⊢ 𝐵 = (Base‘𝑅) |
Ref | Expression |
---|---|
evlval | ⊢ 𝑄 = ((𝐼 evalSub 𝑅)‘𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | evlval.q | . 2 ⊢ 𝑄 = (𝐼 eval 𝑅) | |
2 | oveq12 6919 | . . . . 5 ⊢ ((𝑖 = 𝐼 ∧ 𝑟 = 𝑅) → (𝑖 evalSub 𝑟) = (𝐼 evalSub 𝑅)) | |
3 | fveq2 6437 | . . . . . . 7 ⊢ (𝑟 = 𝑅 → (Base‘𝑟) = (Base‘𝑅)) | |
4 | evlval.b | . . . . . . 7 ⊢ 𝐵 = (Base‘𝑅) | |
5 | 3, 4 | syl6eqr 2879 | . . . . . 6 ⊢ (𝑟 = 𝑅 → (Base‘𝑟) = 𝐵) |
6 | 5 | adantl 475 | . . . . 5 ⊢ ((𝑖 = 𝐼 ∧ 𝑟 = 𝑅) → (Base‘𝑟) = 𝐵) |
7 | 2, 6 | fveq12d 6444 | . . . 4 ⊢ ((𝑖 = 𝐼 ∧ 𝑟 = 𝑅) → ((𝑖 evalSub 𝑟)‘(Base‘𝑟)) = ((𝐼 evalSub 𝑅)‘𝐵)) |
8 | df-evl 19874 | . . . 4 ⊢ eval = (𝑖 ∈ V, 𝑟 ∈ V ↦ ((𝑖 evalSub 𝑟)‘(Base‘𝑟))) | |
9 | fvex 6450 | . . . 4 ⊢ ((𝐼 evalSub 𝑅)‘𝐵) ∈ V | |
10 | 7, 8, 9 | ovmpt2a 7056 | . . 3 ⊢ ((𝐼 ∈ V ∧ 𝑅 ∈ V) → (𝐼 eval 𝑅) = ((𝐼 evalSub 𝑅)‘𝐵)) |
11 | 8 | mpt2ndm0 7140 | . . . . 5 ⊢ (¬ (𝐼 ∈ V ∧ 𝑅 ∈ V) → (𝐼 eval 𝑅) = ∅) |
12 | 0fv 6477 | . . . . 5 ⊢ (∅‘𝐵) = ∅ | |
13 | 11, 12 | syl6eqr 2879 | . . . 4 ⊢ (¬ (𝐼 ∈ V ∧ 𝑅 ∈ V) → (𝐼 eval 𝑅) = (∅‘𝐵)) |
14 | reldmevls 19884 | . . . . . 6 ⊢ Rel dom evalSub | |
15 | 14 | ovprc 6947 | . . . . 5 ⊢ (¬ (𝐼 ∈ V ∧ 𝑅 ∈ V) → (𝐼 evalSub 𝑅) = ∅) |
16 | 15 | fveq1d 6439 | . . . 4 ⊢ (¬ (𝐼 ∈ V ∧ 𝑅 ∈ V) → ((𝐼 evalSub 𝑅)‘𝐵) = (∅‘𝐵)) |
17 | 13, 16 | eqtr4d 2864 | . . 3 ⊢ (¬ (𝐼 ∈ V ∧ 𝑅 ∈ V) → (𝐼 eval 𝑅) = ((𝐼 evalSub 𝑅)‘𝐵)) |
18 | 10, 17 | pm2.61i 177 | . 2 ⊢ (𝐼 eval 𝑅) = ((𝐼 evalSub 𝑅)‘𝐵) |
19 | 1, 18 | eqtri 2849 | 1 ⊢ 𝑄 = ((𝐼 evalSub 𝑅)‘𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∧ wa 386 = wceq 1656 ∈ wcel 2164 Vcvv 3414 ∅c0 4146 ‘cfv 6127 (class class class)co 6910 Basecbs 16229 evalSub ces 19871 eval cevl 19872 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-8 2166 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-sep 5007 ax-nul 5015 ax-pow 5067 ax-pr 5129 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3an 1113 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ral 3122 df-rex 3123 df-rab 3126 df-v 3416 df-sbc 3663 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-nul 4147 df-if 4309 df-sn 4400 df-pr 4402 df-op 4406 df-uni 4661 df-br 4876 df-opab 4938 df-id 5252 df-xp 5352 df-rel 5353 df-cnv 5354 df-co 5355 df-dm 5356 df-iota 6090 df-fun 6129 df-fv 6135 df-ov 6913 df-oprab 6914 df-mpt2 6915 df-evls 19873 df-evl 19874 |
This theorem is referenced by: evlrhm 19892 evlsscasrng 19893 evlsvarsrng 19895 evl1fval1lem 20061 evl1sca 20065 evl1var 20067 pf1rcl 20080 mpfpf1 20082 pf1ind 20086 mzpmfp 38149 |
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