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Mirrors > Home > MPE Home > Th. List > resccoOLD | Structured version Visualization version GIF version |
Description: Obsolete proof of rescco 17815 as of 14-Oct-2024. (Contributed by Mario Carneiro, 4-Jan-2017.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
rescbas.d | β’ π· = (πΆ βΎcat π») |
rescbas.b | β’ π΅ = (BaseβπΆ) |
rescbas.c | β’ (π β πΆ β π) |
rescbas.h | β’ (π β π» Fn (π Γ π)) |
rescbas.s | β’ (π β π β π΅) |
rescco.o | β’ Β· = (compβπΆ) |
Ref | Expression |
---|---|
resccoOLD | β’ (π β Β· = (compβπ·)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ccoid 17394 | . . 3 β’ comp = Slot (compβndx) | |
2 | 1nn0 12518 | . . . . . . 7 β’ 1 β β0 | |
3 | 4nn 12325 | . . . . . . 7 β’ 4 β β | |
4 | 2, 3 | decnncl 12727 | . . . . . 6 β’ ;14 β β |
5 | 4 | nnrei 12251 | . . . . 5 β’ ;14 β β |
6 | 4nn0 12521 | . . . . . 6 β’ 4 β β0 | |
7 | 5nn 12328 | . . . . . 6 β’ 5 β β | |
8 | 4lt5 12419 | . . . . . 6 β’ 4 < 5 | |
9 | 2, 6, 7, 8 | declt 12735 | . . . . 5 β’ ;14 < ;15 |
10 | 5, 9 | gtneii 11356 | . . . 4 β’ ;15 β ;14 |
11 | ccondx 17393 | . . . . 5 β’ (compβndx) = ;15 | |
12 | homndx 17391 | . . . . 5 β’ (Hom βndx) = ;14 | |
13 | 11, 12 | neeq12i 2997 | . . . 4 β’ ((compβndx) β (Hom βndx) β ;15 β ;14) |
14 | 10, 13 | mpbir 230 | . . 3 β’ (compβndx) β (Hom βndx) |
15 | 1, 14 | setsnid 17177 | . 2 β’ (compβ(πΆ βΎs π)) = (compβ((πΆ βΎs π) sSet β¨(Hom βndx), π»β©)) |
16 | rescbas.s | . . . 4 β’ (π β π β π΅) | |
17 | rescbas.b | . . . . . 6 β’ π΅ = (BaseβπΆ) | |
18 | 17 | fvexi 6906 | . . . . 5 β’ π΅ β V |
19 | 18 | ssex 5316 | . . . 4 β’ (π β π΅ β π β V) |
20 | 16, 19 | syl 17 | . . 3 β’ (π β π β V) |
21 | eqid 2725 | . . . 4 β’ (πΆ βΎs π) = (πΆ βΎs π) | |
22 | rescco.o | . . . 4 β’ Β· = (compβπΆ) | |
23 | 21, 22 | ressco 17400 | . . 3 β’ (π β V β Β· = (compβ(πΆ βΎs π))) |
24 | 20, 23 | syl 17 | . 2 β’ (π β Β· = (compβ(πΆ βΎs π))) |
25 | rescbas.d | . . . 4 β’ π· = (πΆ βΎcat π») | |
26 | rescbas.c | . . . 4 β’ (π β πΆ β π) | |
27 | rescbas.h | . . . 4 β’ (π β π» Fn (π Γ π)) | |
28 | 25, 26, 20, 27 | rescval2 17810 | . . 3 β’ (π β π· = ((πΆ βΎs π) sSet β¨(Hom βndx), π»β©)) |
29 | 28 | fveq2d 6896 | . 2 β’ (π β (compβπ·) = (compβ((πΆ βΎs π) sSet β¨(Hom βndx), π»β©))) |
30 | 15, 24, 29 | 3eqtr4a 2791 | 1 β’ (π β Β· = (compβπ·)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1533 β wcel 2098 β wne 2930 Vcvv 3463 β wss 3939 β¨cop 4630 Γ cxp 5670 Fn wfn 6538 βcfv 6543 (class class class)co 7416 1c1 11139 4c4 12299 5c5 12300 ;cdc 12707 sSet csts 17131 ndxcnx 17161 Basecbs 17179 βΎs cress 17208 Hom chom 17243 compcco 17244 βΎcat cresc 17790 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5280 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7738 ax-cnex 11194 ax-resscn 11195 ax-1cn 11196 ax-icn 11197 ax-addcl 11198 ax-addrcl 11199 ax-mulcl 11200 ax-mulrcl 11201 ax-mulcom 11202 ax-addass 11203 ax-mulass 11204 ax-distr 11205 ax-i2m1 11206 ax-1ne0 11207 ax-1rid 11208 ax-rnegex 11209 ax-rrecex 11210 ax-cnre 11211 ax-pre-lttri 11212 ax-pre-lttrn 11213 ax-pre-ltadd 11214 ax-pre-mulgt0 11215 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3769 df-csb 3885 df-dif 3942 df-un 3944 df-in 3946 df-ss 3956 df-pss 3959 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 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-riota 7372 df-ov 7419 df-oprab 7420 df-mpo 7421 df-om 7869 df-2nd 7992 df-frecs 8285 df-wrecs 8316 df-recs 8390 df-rdg 8429 df-er 8723 df-en 8963 df-dom 8964 df-sdom 8965 df-pnf 11280 df-mnf 11281 df-xr 11282 df-ltxr 11283 df-le 11284 df-sub 11476 df-neg 11477 df-nn 12243 df-2 12305 df-3 12306 df-4 12307 df-5 12308 df-6 12309 df-7 12310 df-8 12311 df-9 12312 df-n0 12503 df-z 12589 df-dec 12708 df-sets 17132 df-slot 17150 df-ndx 17162 df-base 17180 df-ress 17209 df-hom 17256 df-cco 17257 df-resc 17793 |
This theorem is referenced by: (None) |
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