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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 7440 | . . . . 5 ⊢ ((𝑖 = 𝐼 ∧ 𝑟 = 𝑅) → (𝑖 evalSub 𝑟) = (𝐼 evalSub 𝑅)) | |
3 | fveq2 6907 | . . . . . . 7 ⊢ (𝑟 = 𝑅 → (Base‘𝑟) = (Base‘𝑅)) | |
4 | evlval.b | . . . . . . 7 ⊢ 𝐵 = (Base‘𝑅) | |
5 | 3, 4 | eqtr4di 2793 | . . . . . 6 ⊢ (𝑟 = 𝑅 → (Base‘𝑟) = 𝐵) |
6 | 5 | adantl 481 | . . . . 5 ⊢ ((𝑖 = 𝐼 ∧ 𝑟 = 𝑅) → (Base‘𝑟) = 𝐵) |
7 | 2, 6 | fveq12d 6914 | . . . 4 ⊢ ((𝑖 = 𝐼 ∧ 𝑟 = 𝑅) → ((𝑖 evalSub 𝑟)‘(Base‘𝑟)) = ((𝐼 evalSub 𝑅)‘𝐵)) |
8 | df-evl 22117 | . . . 4 ⊢ eval = (𝑖 ∈ V, 𝑟 ∈ V ↦ ((𝑖 evalSub 𝑟)‘(Base‘𝑟))) | |
9 | fvex 6920 | . . . 4 ⊢ ((𝐼 evalSub 𝑅)‘𝐵) ∈ V | |
10 | 7, 8, 9 | ovmpoa 7588 | . . 3 ⊢ ((𝐼 ∈ V ∧ 𝑅 ∈ V) → (𝐼 eval 𝑅) = ((𝐼 evalSub 𝑅)‘𝐵)) |
11 | 8 | mpondm0 7673 | . . . . 5 ⊢ (¬ (𝐼 ∈ V ∧ 𝑅 ∈ V) → (𝐼 eval 𝑅) = ∅) |
12 | 0fv 6951 | . . . . 5 ⊢ (∅‘𝐵) = ∅ | |
13 | 11, 12 | eqtr4di 2793 | . . . 4 ⊢ (¬ (𝐼 ∈ V ∧ 𝑅 ∈ V) → (𝐼 eval 𝑅) = (∅‘𝐵)) |
14 | reldmevls 22126 | . . . . . 6 ⊢ Rel dom evalSub | |
15 | 14 | ovprc 7469 | . . . . 5 ⊢ (¬ (𝐼 ∈ V ∧ 𝑅 ∈ V) → (𝐼 evalSub 𝑅) = ∅) |
16 | 15 | fveq1d 6909 | . . . 4 ⊢ (¬ (𝐼 ∈ V ∧ 𝑅 ∈ V) → ((𝐼 evalSub 𝑅)‘𝐵) = (∅‘𝐵)) |
17 | 13, 16 | eqtr4d 2778 | . . 3 ⊢ (¬ (𝐼 ∈ V ∧ 𝑅 ∈ V) → (𝐼 eval 𝑅) = ((𝐼 evalSub 𝑅)‘𝐵)) |
18 | 10, 17 | pm2.61i 182 | . 2 ⊢ (𝐼 eval 𝑅) = ((𝐼 evalSub 𝑅)‘𝐵) |
19 | 1, 18 | eqtri 2763 | 1 ⊢ 𝑄 = ((𝐼 evalSub 𝑅)‘𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∧ wa 395 = wceq 1537 ∈ wcel 2106 Vcvv 3478 ∅c0 4339 ‘cfv 6563 (class class class)co 7431 Basecbs 17245 evalSub ces 22114 eval cevl 22115 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-sbc 3792 df-dif 3966 df-un 3968 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-iota 6516 df-fun 6565 df-fv 6571 df-ov 7434 df-oprab 7435 df-mpo 7436 df-evls 22116 df-evl 22117 |
This theorem is referenced by: evlrhm 22138 evlsscasrng 22139 evlsvarsrng 22141 evl1fval1lem 22350 evl1sca 22354 evl1var 22356 pf1rcl 22369 mpfpf1 22371 pf1ind 22375 evlsevl 42558 mhphf4 42587 mzpmfp 42735 |
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