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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 6558 | . . . . 5 ⊢ ((𝑖 = 𝐼 ∧ 𝑟 = 𝑅) → (𝑖 evalSub 𝑟) = (𝐼 evalSub 𝑅)) | |
3 | fveq2 6103 | . . . . . . 7 ⊢ (𝑟 = 𝑅 → (Base‘𝑟) = (Base‘𝑅)) | |
4 | evlval.b | . . . . . . 7 ⊢ 𝐵 = (Base‘𝑅) | |
5 | 3, 4 | syl6eqr 2662 | . . . . . 6 ⊢ (𝑟 = 𝑅 → (Base‘𝑟) = 𝐵) |
6 | 5 | adantl 481 | . . . . 5 ⊢ ((𝑖 = 𝐼 ∧ 𝑟 = 𝑅) → (Base‘𝑟) = 𝐵) |
7 | 2, 6 | fveq12d 6109 | . . . 4 ⊢ ((𝑖 = 𝐼 ∧ 𝑟 = 𝑅) → ((𝑖 evalSub 𝑟)‘(Base‘𝑟)) = ((𝐼 evalSub 𝑅)‘𝐵)) |
8 | df-evl 19328 | . . . 4 ⊢ eval = (𝑖 ∈ V, 𝑟 ∈ V ↦ ((𝑖 evalSub 𝑟)‘(Base‘𝑟))) | |
9 | fvex 6113 | . . . 4 ⊢ ((𝐼 evalSub 𝑅)‘𝐵) ∈ V | |
10 | 7, 8, 9 | ovmpt2a 6689 | . . 3 ⊢ ((𝐼 ∈ V ∧ 𝑅 ∈ V) → (𝐼 eval 𝑅) = ((𝐼 evalSub 𝑅)‘𝐵)) |
11 | 8 | mpt2ndm0 6773 | . . . . 5 ⊢ (¬ (𝐼 ∈ V ∧ 𝑅 ∈ V) → (𝐼 eval 𝑅) = ∅) |
12 | 0fv 6137 | . . . . 5 ⊢ (∅‘𝐵) = ∅ | |
13 | 11, 12 | syl6eqr 2662 | . . . 4 ⊢ (¬ (𝐼 ∈ V ∧ 𝑅 ∈ V) → (𝐼 eval 𝑅) = (∅‘𝐵)) |
14 | reldmevls 19338 | . . . . . 6 ⊢ Rel dom evalSub | |
15 | 14 | ovprc 6581 | . . . . 5 ⊢ (¬ (𝐼 ∈ V ∧ 𝑅 ∈ V) → (𝐼 evalSub 𝑅) = ∅) |
16 | 15 | fveq1d 6105 | . . . 4 ⊢ (¬ (𝐼 ∈ V ∧ 𝑅 ∈ V) → ((𝐼 evalSub 𝑅)‘𝐵) = (∅‘𝐵)) |
17 | 13, 16 | eqtr4d 2647 | . . 3 ⊢ (¬ (𝐼 ∈ V ∧ 𝑅 ∈ V) → (𝐼 eval 𝑅) = ((𝐼 evalSub 𝑅)‘𝐵)) |
18 | 10, 17 | pm2.61i 175 | . 2 ⊢ (𝐼 eval 𝑅) = ((𝐼 evalSub 𝑅)‘𝐵) |
19 | 1, 18 | eqtri 2632 | 1 ⊢ 𝑄 = ((𝐼 evalSub 𝑅)‘𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∧ wa 383 = wceq 1475 ∈ wcel 1977 Vcvv 3173 ∅c0 3874 ‘cfv 5804 (class class class)co 6549 Basecbs 15695 evalSub ces 19325 eval cevl 19326 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-iota 5768 df-fun 5806 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-evls 19327 df-evl 19328 |
This theorem is referenced by: evlrhm 19346 evlsscasrng 19347 evlsvarsrng 19349 evl1fval1lem 19515 evl1sca 19519 evl1var 19521 pf1rcl 19534 mpfpf1 19536 pf1ind 19540 mzpmfp 36328 |
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