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Mirrors > Home > MPE Home > Th. List > oppchomfvalOLD | Structured version Visualization version GIF version |
Description: Obsolete proof of oppchomfval 17601 as of 14-Oct-2024. (Contributed by Mario Carneiro, 2-Jan-2017.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
oppchom.h | β’ π» = (Hom βπΆ) |
oppchom.o | β’ π = (oppCatβπΆ) |
Ref | Expression |
---|---|
oppchomfvalOLD | β’ tpos π» = (Hom βπ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | homid 17300 | . . . 4 β’ Hom = Slot (Hom βndx) | |
2 | 1nn0 12436 | . . . . . . . 8 β’ 1 β β0 | |
3 | 4nn 12243 | . . . . . . . 8 β’ 4 β β | |
4 | 2, 3 | decnncl 12645 | . . . . . . 7 β’ ;14 β β |
5 | 4 | nnrei 12169 | . . . . . 6 β’ ;14 β β |
6 | 4nn0 12439 | . . . . . . 7 β’ 4 β β0 | |
7 | 5nn 12246 | . . . . . . 7 β’ 5 β β | |
8 | 4lt5 12337 | . . . . . . 7 β’ 4 < 5 | |
9 | 2, 6, 7, 8 | declt 12653 | . . . . . 6 β’ ;14 < ;15 |
10 | 5, 9 | ltneii 11275 | . . . . 5 β’ ;14 β ;15 |
11 | homndx 17299 | . . . . . 6 β’ (Hom βndx) = ;14 | |
12 | ccondx 17301 | . . . . . 6 β’ (compβndx) = ;15 | |
13 | 11, 12 | neeq12i 3011 | . . . . 5 β’ ((Hom βndx) β (compβndx) β ;14 β ;15) |
14 | 10, 13 | mpbir 230 | . . . 4 β’ (Hom βndx) β (compβndx) |
15 | 1, 14 | setsnid 17088 | . . 3 β’ (Hom β(πΆ sSet β¨(Hom βndx), tpos π»β©)) = (Hom β((πΆ sSet β¨(Hom βndx), tpos π»β©) sSet β¨(compβndx), (π’ β ((BaseβπΆ) Γ (BaseβπΆ)), π§ β (BaseβπΆ) β¦ tpos (β¨π§, (2nd βπ’)β©(compβπΆ)(1st βπ’)))β©)) |
16 | oppchom.h | . . . . . 6 β’ π» = (Hom βπΆ) | |
17 | 16 | fvexi 6861 | . . . . 5 β’ π» β V |
18 | 17 | tposex 8196 | . . . 4 β’ tpos π» β V |
19 | 1 | setsid 17087 | . . . 4 β’ ((πΆ β V β§ tpos π» β V) β tpos π» = (Hom β(πΆ sSet β¨(Hom βndx), tpos π»β©))) |
20 | 18, 19 | mpan2 690 | . . 3 β’ (πΆ β V β tpos π» = (Hom β(πΆ sSet β¨(Hom βndx), tpos π»β©))) |
21 | eqid 2737 | . . . . 5 β’ (BaseβπΆ) = (BaseβπΆ) | |
22 | eqid 2737 | . . . . 5 β’ (compβπΆ) = (compβπΆ) | |
23 | oppchom.o | . . . . 5 β’ π = (oppCatβπΆ) | |
24 | 21, 16, 22, 23 | oppcval 17600 | . . . 4 β’ (πΆ β V β π = ((πΆ sSet β¨(Hom βndx), tpos π»β©) sSet β¨(compβndx), (π’ β ((BaseβπΆ) Γ (BaseβπΆ)), π§ β (BaseβπΆ) β¦ tpos (β¨π§, (2nd βπ’)β©(compβπΆ)(1st βπ’)))β©)) |
25 | 24 | fveq2d 6851 | . . 3 β’ (πΆ β V β (Hom βπ) = (Hom β((πΆ sSet β¨(Hom βndx), tpos π»β©) sSet β¨(compβndx), (π’ β ((BaseβπΆ) Γ (BaseβπΆ)), π§ β (BaseβπΆ) β¦ tpos (β¨π§, (2nd βπ’)β©(compβπΆ)(1st βπ’)))β©))) |
26 | 15, 20, 25 | 3eqtr4a 2803 | . 2 β’ (πΆ β V β tpos π» = (Hom βπ)) |
27 | tpos0 8192 | . . 3 β’ tpos β = β | |
28 | fvprc 6839 | . . . . 5 β’ (Β¬ πΆ β V β (Hom βπΆ) = β ) | |
29 | 16, 28 | eqtrid 2789 | . . . 4 β’ (Β¬ πΆ β V β π» = β ) |
30 | 29 | tposeqd 8165 | . . 3 β’ (Β¬ πΆ β V β tpos π» = tpos β ) |
31 | fvprc 6839 | . . . . . 6 β’ (Β¬ πΆ β V β (oppCatβπΆ) = β ) | |
32 | 23, 31 | eqtrid 2789 | . . . . 5 β’ (Β¬ πΆ β V β π = β ) |
33 | 32 | fveq2d 6851 | . . . 4 β’ (Β¬ πΆ β V β (Hom βπ) = (Hom ββ )) |
34 | df-hom 17164 | . . . . 5 β’ Hom = Slot ;14 | |
35 | 34 | str0 17068 | . . . 4 β’ β = (Hom ββ ) |
36 | 33, 35 | eqtr4di 2795 | . . 3 β’ (Β¬ πΆ β V β (Hom βπ) = β ) |
37 | 27, 30, 36 | 3eqtr4a 2803 | . 2 β’ (Β¬ πΆ β V β tpos π» = (Hom βπ)) |
38 | 26, 37 | pm2.61i 182 | 1 β’ tpos π» = (Hom βπ) |
Colors of variables: wff setvar class |
Syntax hints: Β¬ wn 3 = wceq 1542 β wcel 2107 β wne 2944 Vcvv 3448 β c0 4287 β¨cop 4597 Γ cxp 5636 βcfv 6501 (class class class)co 7362 β cmpo 7364 1st c1st 7924 2nd c2nd 7925 tpos ctpos 8161 1c1 11059 4c4 12217 5c5 12218 ;cdc 12625 sSet csts 17042 ndxcnx 17072 Basecbs 17090 Hom chom 17151 compcco 17152 oppCatcoppc 17598 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-sep 5261 ax-nul 5268 ax-pow 5325 ax-pr 5389 ax-un 7677 ax-cnex 11114 ax-resscn 11115 ax-1cn 11116 ax-icn 11117 ax-addcl 11118 ax-addrcl 11119 ax-mulcl 11120 ax-mulrcl 11121 ax-mulcom 11122 ax-addass 11123 ax-mulass 11124 ax-distr 11125 ax-i2m1 11126 ax-1ne0 11127 ax-1rid 11128 ax-rnegex 11129 ax-rrecex 11130 ax-cnre 11131 ax-pre-lttri 11132 ax-pre-lttrn 11133 ax-pre-ltadd 11134 ax-pre-mulgt0 11135 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3066 df-rex 3075 df-reu 3357 df-rab 3411 df-v 3450 df-sbc 3745 df-csb 3861 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-pss 3934 df-nul 4288 df-if 4492 df-pw 4567 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4871 df-iun 4961 df-br 5111 df-opab 5173 df-mpt 5194 df-tr 5228 df-id 5536 df-eprel 5542 df-po 5550 df-so 5551 df-fr 5593 df-we 5595 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6258 df-ord 6325 df-on 6326 df-lim 6327 df-suc 6328 df-iota 6453 df-fun 6503 df-fn 6504 df-f 6505 df-f1 6506 df-fo 6507 df-f1o 6508 df-fv 6509 df-riota 7318 df-ov 7365 df-oprab 7366 df-mpo 7367 df-om 7808 df-2nd 7927 df-tpos 8162 df-frecs 8217 df-wrecs 8248 df-recs 8322 df-rdg 8361 df-er 8655 df-en 8891 df-dom 8892 df-sdom 8893 df-pnf 11198 df-mnf 11199 df-xr 11200 df-ltxr 11201 df-le 11202 df-sub 11394 df-neg 11395 df-nn 12161 df-2 12223 df-3 12224 df-4 12225 df-5 12226 df-6 12227 df-7 12228 df-8 12229 df-9 12230 df-n0 12421 df-dec 12626 df-sets 17043 df-slot 17061 df-ndx 17073 df-hom 17164 df-cco 17165 df-oppc 17599 |
This theorem is referenced by: (None) |
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