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Mirrors > Home > MPE Home > Th. List > evlf2val | Structured version Visualization version GIF version |
Description: Value of the evaluation natural transformation at an object. (Contributed by Mario Carneiro, 12-Jan-2017.) |
Ref | Expression |
---|---|
evlfval.e | โข ๐ธ = (๐ถ evalF ๐ท) |
evlfval.c | โข (๐ โ ๐ถ โ Cat) |
evlfval.d | โข (๐ โ ๐ท โ Cat) |
evlfval.b | โข ๐ต = (Baseโ๐ถ) |
evlfval.h | โข ๐ป = (Hom โ๐ถ) |
evlfval.o | โข ยท = (compโ๐ท) |
evlfval.n | โข ๐ = (๐ถ Nat ๐ท) |
evlf2.f | โข (๐ โ ๐น โ (๐ถ Func ๐ท)) |
evlf2.g | โข (๐ โ ๐บ โ (๐ถ Func ๐ท)) |
evlf2.x | โข (๐ โ ๐ โ ๐ต) |
evlf2.y | โข (๐ โ ๐ โ ๐ต) |
evlf2.l | โข ๐ฟ = (โจ๐น, ๐โฉ(2nd โ๐ธ)โจ๐บ, ๐โฉ) |
evlf2val.a | โข (๐ โ ๐ด โ (๐น๐๐บ)) |
evlf2val.k | โข (๐ โ ๐พ โ (๐๐ป๐)) |
Ref | Expression |
---|---|
evlf2val | โข (๐ โ (๐ด๐ฟ๐พ) = ((๐ดโ๐)(โจ((1st โ๐น)โ๐), ((1st โ๐น)โ๐)โฉ ยท ((1st โ๐บ)โ๐))((๐(2nd โ๐น)๐)โ๐พ))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | evlfval.e | . . 3 โข ๐ธ = (๐ถ evalF ๐ท) | |
2 | evlfval.c | . . 3 โข (๐ โ ๐ถ โ Cat) | |
3 | evlfval.d | . . 3 โข (๐ โ ๐ท โ Cat) | |
4 | evlfval.b | . . 3 โข ๐ต = (Baseโ๐ถ) | |
5 | evlfval.h | . . 3 โข ๐ป = (Hom โ๐ถ) | |
6 | evlfval.o | . . 3 โข ยท = (compโ๐ท) | |
7 | evlfval.n | . . 3 โข ๐ = (๐ถ Nat ๐ท) | |
8 | evlf2.f | . . 3 โข (๐ โ ๐น โ (๐ถ Func ๐ท)) | |
9 | evlf2.g | . . 3 โข (๐ โ ๐บ โ (๐ถ Func ๐ท)) | |
10 | evlf2.x | . . 3 โข (๐ โ ๐ โ ๐ต) | |
11 | evlf2.y | . . 3 โข (๐ โ ๐ โ ๐ต) | |
12 | evlf2.l | . . 3 โข ๐ฟ = (โจ๐น, ๐โฉ(2nd โ๐ธ)โจ๐บ, ๐โฉ) | |
13 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 | evlf2 18176 | . 2 โข (๐ โ ๐ฟ = (๐ โ (๐น๐๐บ), ๐ โ (๐๐ป๐) โฆ ((๐โ๐)(โจ((1st โ๐น)โ๐), ((1st โ๐น)โ๐)โฉ ยท ((1st โ๐บ)โ๐))((๐(2nd โ๐น)๐)โ๐)))) |
14 | simprl 768 | . . . 4 โข ((๐ โง (๐ = ๐ด โง ๐ = ๐พ)) โ ๐ = ๐ด) | |
15 | 14 | fveq1d 6893 | . . 3 โข ((๐ โง (๐ = ๐ด โง ๐ = ๐พ)) โ (๐โ๐) = (๐ดโ๐)) |
16 | simprr 770 | . . . 4 โข ((๐ โง (๐ = ๐ด โง ๐ = ๐พ)) โ ๐ = ๐พ) | |
17 | 16 | fveq2d 6895 | . . 3 โข ((๐ โง (๐ = ๐ด โง ๐ = ๐พ)) โ ((๐(2nd โ๐น)๐)โ๐) = ((๐(2nd โ๐น)๐)โ๐พ)) |
18 | 15, 17 | oveq12d 7430 | . 2 โข ((๐ โง (๐ = ๐ด โง ๐ = ๐พ)) โ ((๐โ๐)(โจ((1st โ๐น)โ๐), ((1st โ๐น)โ๐)โฉ ยท ((1st โ๐บ)โ๐))((๐(2nd โ๐น)๐)โ๐)) = ((๐ดโ๐)(โจ((1st โ๐น)โ๐), ((1st โ๐น)โ๐)โฉ ยท ((1st โ๐บ)โ๐))((๐(2nd โ๐น)๐)โ๐พ))) |
19 | evlf2val.a | . 2 โข (๐ โ ๐ด โ (๐น๐๐บ)) | |
20 | evlf2val.k | . 2 โข (๐ โ ๐พ โ (๐๐ป๐)) | |
21 | ovexd 7447 | . 2 โข (๐ โ ((๐ดโ๐)(โจ((1st โ๐น)โ๐), ((1st โ๐น)โ๐)โฉ ยท ((1st โ๐บ)โ๐))((๐(2nd โ๐น)๐)โ๐พ)) โ V) | |
22 | 13, 18, 19, 20, 21 | ovmpod 7563 | 1 โข (๐ โ (๐ด๐ฟ๐พ) = ((๐ดโ๐)(โจ((1st โ๐น)โ๐), ((1st โ๐น)โ๐)โฉ ยท ((1st โ๐บ)โ๐))((๐(2nd โ๐น)๐)โ๐พ))) |
Colors of variables: wff setvar class |
Syntax hints: โ wi 4 โง wa 395 = wceq 1540 โ wcel 2105 Vcvv 3473 โจcop 4634 โcfv 6543 (class class class)co 7412 1st c1st 7976 2nd c2nd 7977 Basecbs 17149 Hom chom 17213 compcco 17214 Catccat 17613 Func cfunc 17809 Nat cnat 17897 evalF cevlf 18167 |
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 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7728 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-ral 3061 df-rex 3070 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-ov 7415 df-oprab 7416 df-mpo 7417 df-1st 7978 df-2nd 7979 df-evlf 18171 |
This theorem is referenced by: evlfcllem 18179 evlfcl 18180 uncf2 18195 yonedalem3b 18237 |
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