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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 6931 | . . . . 5 ⊢ ((𝑖 = 𝐼 ∧ 𝑟 = 𝑅) → (𝑖 evalSub 𝑟) = (𝐼 evalSub 𝑅)) | |
3 | fveq2 6446 | . . . . . . 7 ⊢ (𝑟 = 𝑅 → (Base‘𝑟) = (Base‘𝑅)) | |
4 | evlval.b | . . . . . . 7 ⊢ 𝐵 = (Base‘𝑅) | |
5 | 3, 4 | syl6eqr 2832 | . . . . . 6 ⊢ (𝑟 = 𝑅 → (Base‘𝑟) = 𝐵) |
6 | 5 | adantl 475 | . . . . 5 ⊢ ((𝑖 = 𝐼 ∧ 𝑟 = 𝑅) → (Base‘𝑟) = 𝐵) |
7 | 2, 6 | fveq12d 6453 | . . . 4 ⊢ ((𝑖 = 𝐼 ∧ 𝑟 = 𝑅) → ((𝑖 evalSub 𝑟)‘(Base‘𝑟)) = ((𝐼 evalSub 𝑅)‘𝐵)) |
8 | df-evl 19903 | . . . 4 ⊢ eval = (𝑖 ∈ V, 𝑟 ∈ V ↦ ((𝑖 evalSub 𝑟)‘(Base‘𝑟))) | |
9 | fvex 6459 | . . . 4 ⊢ ((𝐼 evalSub 𝑅)‘𝐵) ∈ V | |
10 | 7, 8, 9 | ovmpt2a 7068 | . . 3 ⊢ ((𝐼 ∈ V ∧ 𝑅 ∈ V) → (𝐼 eval 𝑅) = ((𝐼 evalSub 𝑅)‘𝐵)) |
11 | 8 | mpt2ndm0 7152 | . . . . 5 ⊢ (¬ (𝐼 ∈ V ∧ 𝑅 ∈ V) → (𝐼 eval 𝑅) = ∅) |
12 | 0fv 6486 | . . . . 5 ⊢ (∅‘𝐵) = ∅ | |
13 | 11, 12 | syl6eqr 2832 | . . . 4 ⊢ (¬ (𝐼 ∈ V ∧ 𝑅 ∈ V) → (𝐼 eval 𝑅) = (∅‘𝐵)) |
14 | reldmevls 19913 | . . . . . 6 ⊢ Rel dom evalSub | |
15 | 14 | ovprc 6959 | . . . . 5 ⊢ (¬ (𝐼 ∈ V ∧ 𝑅 ∈ V) → (𝐼 evalSub 𝑅) = ∅) |
16 | 15 | fveq1d 6448 | . . . 4 ⊢ (¬ (𝐼 ∈ V ∧ 𝑅 ∈ V) → ((𝐼 evalSub 𝑅)‘𝐵) = (∅‘𝐵)) |
17 | 13, 16 | eqtr4d 2817 | . . 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 2802 | 1 ⊢ 𝑄 = ((𝐼 evalSub 𝑅)‘𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∧ wa 386 = wceq 1601 ∈ wcel 2107 Vcvv 3398 ∅c0 4141 ‘cfv 6135 (class class class)co 6922 Basecbs 16255 evalSub ces 19900 eval cevl 19901 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ral 3095 df-rex 3096 df-rab 3099 df-v 3400 df-sbc 3653 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-nul 4142 df-if 4308 df-sn 4399 df-pr 4401 df-op 4405 df-uni 4672 df-br 4887 df-opab 4949 df-id 5261 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-iota 6099 df-fun 6137 df-fv 6143 df-ov 6925 df-oprab 6926 df-mpt2 6927 df-evls 19902 df-evl 19903 |
This theorem is referenced by: evlrhm 19921 evlsscasrng 19922 evlsvarsrng 19924 evl1fval1lem 20090 evl1sca 20094 evl1var 20096 pf1rcl 20109 mpfpf1 20111 pf1ind 20115 mzpmfp 38270 |
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