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Mathbox for Zhi Wang |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > postcposALT | Structured version Visualization version GIF version |
Description: Alternate proof for postcpos 47999. (Contributed by Zhi Wang, 25-Sep-2024.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
postc.c | β’ (π β πΆ = (ProsetToCatβπΎ)) |
postc.k | β’ (π β πΎ β Proset ) |
Ref | Expression |
---|---|
postcposALT | β’ (π β (πΎ β Poset β πΆ β Poset)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | postc.c | . . . 4 β’ (π β πΆ = (ProsetToCatβπΎ)) | |
2 | postc.k | . . . 4 β’ (π β πΎ β Proset ) | |
3 | eqidd 2728 | . . . 4 β’ (π β (BaseβπΎ) = (BaseβπΎ)) | |
4 | 1, 2, 3 | prstcbas 47986 | . . 3 β’ (π β (BaseβπΎ) = (BaseβπΆ)) |
5 | eqidd 2728 | . . . . . . 7 β’ (π β (leβπΎ) = (leβπΎ)) | |
6 | 1, 2, 5 | prstcle 47989 | . . . . . 6 β’ (π β (π₯(leβπΎ)π¦ β π₯(leβπΆ)π¦)) |
7 | 1, 2, 5 | prstcle 47989 | . . . . . 6 β’ (π β (π¦(leβπΎ)π₯ β π¦(leβπΆ)π₯)) |
8 | 6, 7 | anbi12d 630 | . . . . 5 β’ (π β ((π₯(leβπΎ)π¦ β§ π¦(leβπΎ)π₯) β (π₯(leβπΆ)π¦ β§ π¦(leβπΆ)π₯))) |
9 | 8 | imbi1d 341 | . . . 4 β’ (π β (((π₯(leβπΎ)π¦ β§ π¦(leβπΎ)π₯) β π₯ = π¦) β ((π₯(leβπΆ)π¦ β§ π¦(leβπΆ)π₯) β π₯ = π¦))) |
10 | 4, 9 | raleqbidvv 3324 | . . 3 β’ (π β (βπ¦ β (BaseβπΎ)((π₯(leβπΎ)π¦ β§ π¦(leβπΎ)π₯) β π₯ = π¦) β βπ¦ β (BaseβπΆ)((π₯(leβπΆ)π¦ β§ π¦(leβπΆ)π₯) β π₯ = π¦))) |
11 | 4, 10 | raleqbidvv 3324 | . 2 β’ (π β (βπ₯ β (BaseβπΎ)βπ¦ β (BaseβπΎ)((π₯(leβπΎ)π¦ β§ π¦(leβπΎ)π₯) β π₯ = π¦) β βπ₯ β (BaseβπΆ)βπ¦ β (BaseβπΆ)((π₯(leβπΆ)π¦ β§ π¦(leβπΆ)π₯) β π₯ = π¦))) |
12 | eqid 2727 | . . . . 5 β’ (BaseβπΎ) = (BaseβπΎ) | |
13 | eqid 2727 | . . . . 5 β’ (leβπΎ) = (leβπΎ) | |
14 | 12, 13 | ispos2 18292 | . . . 4 β’ (πΎ β Poset β (πΎ β Proset β§ βπ₯ β (BaseβπΎ)βπ¦ β (BaseβπΎ)((π₯(leβπΎ)π¦ β§ π¦(leβπΎ)π₯) β π₯ = π¦))) |
15 | 14 | baib 535 | . . 3 β’ (πΎ β Proset β (πΎ β Poset β βπ₯ β (BaseβπΎ)βπ¦ β (BaseβπΎ)((π₯(leβπΎ)π¦ β§ π¦(leβπΎ)π₯) β π₯ = π¦))) |
16 | 2, 15 | syl 17 | . 2 β’ (π β (πΎ β Poset β βπ₯ β (BaseβπΎ)βπ¦ β (BaseβπΎ)((π₯(leβπΎ)π¦ β§ π¦(leβπΎ)π₯) β π₯ = π¦))) |
17 | 1, 2 | prstcprs 47994 | . . 3 β’ (π β πΆ β Proset ) |
18 | eqid 2727 | . . . . 5 β’ (BaseβπΆ) = (BaseβπΆ) | |
19 | eqid 2727 | . . . . 5 β’ (leβπΆ) = (leβπΆ) | |
20 | 18, 19 | ispos2 18292 | . . . 4 β’ (πΆ β Poset β (πΆ β Proset β§ βπ₯ β (BaseβπΆ)βπ¦ β (BaseβπΆ)((π₯(leβπΆ)π¦ β§ π¦(leβπΆ)π₯) β π₯ = π¦))) |
21 | 20 | baib 535 | . . 3 β’ (πΆ β Proset β (πΆ β Poset β βπ₯ β (BaseβπΆ)βπ¦ β (BaseβπΆ)((π₯(leβπΆ)π¦ β§ π¦(leβπΆ)π₯) β π₯ = π¦))) |
22 | 17, 21 | syl 17 | . 2 β’ (π β (πΆ β Poset β βπ₯ β (BaseβπΆ)βπ¦ β (BaseβπΆ)((π₯(leβπΆ)π¦ β§ π¦(leβπΆ)π₯) β π₯ = π¦))) |
23 | 11, 16, 22 | 3bitr4d 311 | 1 β’ (π β (πΎ β Poset β πΆ β Poset)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β§ wa 395 = wceq 1534 β wcel 2099 βwral 3056 class class class wbr 5142 βcfv 6542 Basecbs 17165 lecple 17225 Proset cproset 18270 Posetcpo 18284 ProsetToCatcprstc 47981 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7732 ax-cnex 11180 ax-resscn 11181 ax-1cn 11182 ax-icn 11183 ax-addcl 11184 ax-addrcl 11185 ax-mulcl 11186 ax-mulrcl 11187 ax-mulcom 11188 ax-addass 11189 ax-mulass 11190 ax-distr 11191 ax-i2m1 11192 ax-1ne0 11193 ax-1rid 11194 ax-rnegex 11195 ax-rrecex 11196 ax-cnre 11197 ax-pre-lttri 11198 ax-pre-lttrn 11199 ax-pre-ltadd 11200 ax-pre-mulgt0 11201 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-nel 3042 df-ral 3057 df-rex 3066 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 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 5143 df-opab 5205 df-mpt 5226 df-tr 5260 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 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7863 df-2nd 7986 df-frecs 8278 df-wrecs 8309 df-recs 8383 df-rdg 8422 df-er 8716 df-en 8954 df-dom 8955 df-sdom 8956 df-pnf 11266 df-mnf 11267 df-xr 11268 df-ltxr 11269 df-le 11270 df-sub 11462 df-neg 11463 df-nn 12229 df-2 12291 df-3 12292 df-4 12293 df-5 12294 df-6 12295 df-7 12296 df-8 12297 df-9 12298 df-n0 12489 df-z 12575 df-dec 12694 df-sets 17118 df-slot 17136 df-ndx 17148 df-base 17166 df-ple 17238 df-hom 17242 df-cco 17243 df-proset 18272 df-poset 18290 df-prstc 47982 |
This theorem is referenced by: (None) |
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