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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 7165 | . . . . 5 ⊢ ((𝑖 = 𝐼 ∧ 𝑟 = 𝑅) → (𝑖 evalSub 𝑟) = (𝐼 evalSub 𝑅)) | |
3 | fveq2 6670 | . . . . . . 7 ⊢ (𝑟 = 𝑅 → (Base‘𝑟) = (Base‘𝑅)) | |
4 | evlval.b | . . . . . . 7 ⊢ 𝐵 = (Base‘𝑅) | |
5 | 3, 4 | syl6eqr 2874 | . . . . . 6 ⊢ (𝑟 = 𝑅 → (Base‘𝑟) = 𝐵) |
6 | 5 | adantl 484 | . . . . 5 ⊢ ((𝑖 = 𝐼 ∧ 𝑟 = 𝑅) → (Base‘𝑟) = 𝐵) |
7 | 2, 6 | fveq12d 6677 | . . . 4 ⊢ ((𝑖 = 𝐼 ∧ 𝑟 = 𝑅) → ((𝑖 evalSub 𝑟)‘(Base‘𝑟)) = ((𝐼 evalSub 𝑅)‘𝐵)) |
8 | df-evl 20287 | . . . 4 ⊢ eval = (𝑖 ∈ V, 𝑟 ∈ V ↦ ((𝑖 evalSub 𝑟)‘(Base‘𝑟))) | |
9 | fvex 6683 | . . . 4 ⊢ ((𝐼 evalSub 𝑅)‘𝐵) ∈ V | |
10 | 7, 8, 9 | ovmpoa 7305 | . . 3 ⊢ ((𝐼 ∈ V ∧ 𝑅 ∈ V) → (𝐼 eval 𝑅) = ((𝐼 evalSub 𝑅)‘𝐵)) |
11 | 8 | mpondm0 7386 | . . . . 5 ⊢ (¬ (𝐼 ∈ V ∧ 𝑅 ∈ V) → (𝐼 eval 𝑅) = ∅) |
12 | 0fv 6709 | . . . . 5 ⊢ (∅‘𝐵) = ∅ | |
13 | 11, 12 | syl6eqr 2874 | . . . 4 ⊢ (¬ (𝐼 ∈ V ∧ 𝑅 ∈ V) → (𝐼 eval 𝑅) = (∅‘𝐵)) |
14 | reldmevls 20297 | . . . . . 6 ⊢ Rel dom evalSub | |
15 | 14 | ovprc 7194 | . . . . 5 ⊢ (¬ (𝐼 ∈ V ∧ 𝑅 ∈ V) → (𝐼 evalSub 𝑅) = ∅) |
16 | 15 | fveq1d 6672 | . . . 4 ⊢ (¬ (𝐼 ∈ V ∧ 𝑅 ∈ V) → ((𝐼 evalSub 𝑅)‘𝐵) = (∅‘𝐵)) |
17 | 13, 16 | eqtr4d 2859 | . . 3 ⊢ (¬ (𝐼 ∈ V ∧ 𝑅 ∈ V) → (𝐼 eval 𝑅) = ((𝐼 evalSub 𝑅)‘𝐵)) |
18 | 10, 17 | pm2.61i 184 | . 2 ⊢ (𝐼 eval 𝑅) = ((𝐼 evalSub 𝑅)‘𝐵) |
19 | 1, 18 | eqtri 2844 | 1 ⊢ 𝑄 = ((𝐼 evalSub 𝑅)‘𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∧ wa 398 = wceq 1537 ∈ wcel 2114 Vcvv 3494 ∅c0 4291 ‘cfv 6355 (class class class)co 7156 Basecbs 16483 evalSub ces 20284 eval cevl 20285 |
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-sep 5203 ax-nul 5210 ax-pow 5266 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-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3773 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-br 5067 df-opab 5129 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-iota 6314 df-fun 6357 df-fv 6363 df-ov 7159 df-oprab 7160 df-mpo 7161 df-evls 20286 df-evl 20287 |
This theorem is referenced by: evlrhm 20309 evlsscasrng 20310 evlsvarsrng 20312 evl1fval1lem 20493 evl1sca 20497 evl1var 20499 pf1rcl 20512 mpfpf1 20514 pf1ind 20518 mzpmfp 39393 |
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