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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 6906 | . . . . . . 7 ⊢ (𝑟 = 𝑅 → (Base‘𝑟) = (Base‘𝑅)) | |
| 4 | evlval.b | . . . . . . 7 ⊢ 𝐵 = (Base‘𝑅) | |
| 5 | 3, 4 | eqtr4di 2795 | . . . . . 6 ⊢ (𝑟 = 𝑅 → (Base‘𝑟) = 𝐵) |
| 6 | 5 | adantl 481 | . . . . 5 ⊢ ((𝑖 = 𝐼 ∧ 𝑟 = 𝑅) → (Base‘𝑟) = 𝐵) |
| 7 | 2, 6 | fveq12d 6913 | . . . 4 ⊢ ((𝑖 = 𝐼 ∧ 𝑟 = 𝑅) → ((𝑖 evalSub 𝑟)‘(Base‘𝑟)) = ((𝐼 evalSub 𝑅)‘𝐵)) |
| 8 | df-evl 22099 | . . . 4 ⊢ eval = (𝑖 ∈ V, 𝑟 ∈ V ↦ ((𝑖 evalSub 𝑟)‘(Base‘𝑟))) | |
| 9 | fvex 6919 | . . . 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 6950 | . . . . 5 ⊢ (∅‘𝐵) = ∅ | |
| 13 | 11, 12 | eqtr4di 2795 | . . . 4 ⊢ (¬ (𝐼 ∈ V ∧ 𝑅 ∈ V) → (𝐼 eval 𝑅) = (∅‘𝐵)) |
| 14 | reldmevls 22108 | . . . . . 6 ⊢ Rel dom evalSub | |
| 15 | 14 | ovprc 7469 | . . . . 5 ⊢ (¬ (𝐼 ∈ V ∧ 𝑅 ∈ V) → (𝐼 evalSub 𝑅) = ∅) |
| 16 | 15 | fveq1d 6908 | . . . 4 ⊢ (¬ (𝐼 ∈ V ∧ 𝑅 ∈ V) → ((𝐼 evalSub 𝑅)‘𝐵) = (∅‘𝐵)) |
| 17 | 13, 16 | eqtr4d 2780 | . . 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 2765 | 1 ⊢ 𝑄 = ((𝐼 evalSub 𝑅)‘𝐵) |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 ∧ wa 395 = wceq 1540 ∈ wcel 2108 Vcvv 3480 ∅c0 4333 ‘cfv 6561 (class class class)co 7431 Basecbs 17247 evalSub ces 22096 eval cevl 22097 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-iota 6514 df-fun 6563 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-evls 22098 df-evl 22099 |
| This theorem is referenced by: evlrhm 22120 evlsscasrng 22121 evlsvarsrng 22123 evl1fval1lem 22334 evl1sca 22338 evl1var 22340 pf1rcl 22353 mpfpf1 22355 pf1ind 22359 evlsevl 42581 mhphf4 42610 mzpmfp 42758 |
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