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Mirrors > Home > MPE Home > Th. List > Mathboxes > postcposALT | Structured version Visualization version GIF version |
Description: Alternate proof of postcpos 49144. (Contributed by Zhi Wang, 25-Sep-2024.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
postc.c | ⊢ (𝜑 → 𝐶 = (ProsetToCat‘𝐾)) |
postc.k | ⊢ (𝜑 → 𝐾 ∈ Proset ) |
Ref | Expression |
---|---|
postcposALT | ⊢ (𝜑 → (𝐾 ∈ Poset ↔ 𝐶 ∈ Poset)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | postc.c | . . . 4 ⊢ (𝜑 → 𝐶 = (ProsetToCat‘𝐾)) | |
2 | postc.k | . . . 4 ⊢ (𝜑 → 𝐾 ∈ Proset ) | |
3 | eqidd 2737 | . . . 4 ⊢ (𝜑 → (Base‘𝐾) = (Base‘𝐾)) | |
4 | 1, 2, 3 | prstcbas 49129 | . . 3 ⊢ (𝜑 → (Base‘𝐾) = (Base‘𝐶)) |
5 | eqidd 2737 | . . . . . . 7 ⊢ (𝜑 → (le‘𝐾) = (le‘𝐾)) | |
6 | 1, 2, 5 | prstcle 49132 | . . . . . 6 ⊢ (𝜑 → (𝑥(le‘𝐾)𝑦 ↔ 𝑥(le‘𝐶)𝑦)) |
7 | 1, 2, 5 | prstcle 49132 | . . . . . 6 ⊢ (𝜑 → (𝑦(le‘𝐾)𝑥 ↔ 𝑦(le‘𝐶)𝑥)) |
8 | 6, 7 | anbi12d 632 | . . . . 5 ⊢ (𝜑 → ((𝑥(le‘𝐾)𝑦 ∧ 𝑦(le‘𝐾)𝑥) ↔ (𝑥(le‘𝐶)𝑦 ∧ 𝑦(le‘𝐶)𝑥))) |
9 | 8 | imbi1d 341 | . . . 4 ⊢ (𝜑 → (((𝑥(le‘𝐾)𝑦 ∧ 𝑦(le‘𝐾)𝑥) → 𝑥 = 𝑦) ↔ ((𝑥(le‘𝐶)𝑦 ∧ 𝑦(le‘𝐶)𝑥) → 𝑥 = 𝑦))) |
10 | 4, 9 | raleqbidvv 3333 | . . 3 ⊢ (𝜑 → (∀𝑦 ∈ (Base‘𝐾)((𝑥(le‘𝐾)𝑦 ∧ 𝑦(le‘𝐾)𝑥) → 𝑥 = 𝑦) ↔ ∀𝑦 ∈ (Base‘𝐶)((𝑥(le‘𝐶)𝑦 ∧ 𝑦(le‘𝐶)𝑥) → 𝑥 = 𝑦))) |
11 | 4, 10 | raleqbidvv 3333 | . 2 ⊢ (𝜑 → (∀𝑥 ∈ (Base‘𝐾)∀𝑦 ∈ (Base‘𝐾)((𝑥(le‘𝐾)𝑦 ∧ 𝑦(le‘𝐾)𝑥) → 𝑥 = 𝑦) ↔ ∀𝑥 ∈ (Base‘𝐶)∀𝑦 ∈ (Base‘𝐶)((𝑥(le‘𝐶)𝑦 ∧ 𝑦(le‘𝐶)𝑥) → 𝑥 = 𝑦))) |
12 | eqid 2736 | . . . . 5 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
13 | eqid 2736 | . . . . 5 ⊢ (le‘𝐾) = (le‘𝐾) | |
14 | 12, 13 | ispos2 18357 | . . . 4 ⊢ (𝐾 ∈ Poset ↔ (𝐾 ∈ Proset ∧ ∀𝑥 ∈ (Base‘𝐾)∀𝑦 ∈ (Base‘𝐾)((𝑥(le‘𝐾)𝑦 ∧ 𝑦(le‘𝐾)𝑥) → 𝑥 = 𝑦))) |
15 | 14 | baib 535 | . . 3 ⊢ (𝐾 ∈ Proset → (𝐾 ∈ Poset ↔ ∀𝑥 ∈ (Base‘𝐾)∀𝑦 ∈ (Base‘𝐾)((𝑥(le‘𝐾)𝑦 ∧ 𝑦(le‘𝐾)𝑥) → 𝑥 = 𝑦))) |
16 | 2, 15 | syl 17 | . 2 ⊢ (𝜑 → (𝐾 ∈ Poset ↔ ∀𝑥 ∈ (Base‘𝐾)∀𝑦 ∈ (Base‘𝐾)((𝑥(le‘𝐾)𝑦 ∧ 𝑦(le‘𝐾)𝑥) → 𝑥 = 𝑦))) |
17 | 1, 2 | prstcprs 49137 | . . 3 ⊢ (𝜑 → 𝐶 ∈ Proset ) |
18 | eqid 2736 | . . . . 5 ⊢ (Base‘𝐶) = (Base‘𝐶) | |
19 | eqid 2736 | . . . . 5 ⊢ (le‘𝐶) = (le‘𝐶) | |
20 | 18, 19 | ispos2 18357 | . . . 4 ⊢ (𝐶 ∈ Poset ↔ (𝐶 ∈ Proset ∧ ∀𝑥 ∈ (Base‘𝐶)∀𝑦 ∈ (Base‘𝐶)((𝑥(le‘𝐶)𝑦 ∧ 𝑦(le‘𝐶)𝑥) → 𝑥 = 𝑦))) |
21 | 20 | baib 535 | . . 3 ⊢ (𝐶 ∈ Proset → (𝐶 ∈ Poset ↔ ∀𝑥 ∈ (Base‘𝐶)∀𝑦 ∈ (Base‘𝐶)((𝑥(le‘𝐶)𝑦 ∧ 𝑦(le‘𝐶)𝑥) → 𝑥 = 𝑦))) |
22 | 17, 21 | syl 17 | . 2 ⊢ (𝜑 → (𝐶 ∈ Poset ↔ ∀𝑥 ∈ (Base‘𝐶)∀𝑦 ∈ (Base‘𝐶)((𝑥(le‘𝐶)𝑦 ∧ 𝑦(le‘𝐶)𝑥) → 𝑥 = 𝑦))) |
23 | 11, 16, 22 | 3bitr4d 311 | 1 ⊢ (𝜑 → (𝐾 ∈ Poset ↔ 𝐶 ∈ Poset)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ∀wral 3060 class class class wbr 5141 ‘cfv 6559 Basecbs 17243 lecple 17300 Proset cproset 18334 Posetcpo 18349 ProsetToCatcprstc 49124 |
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 2707 ax-sep 5294 ax-nul 5304 ax-pow 5363 ax-pr 5430 ax-un 7751 ax-cnex 11207 ax-resscn 11208 ax-1cn 11209 ax-icn 11210 ax-addcl 11211 ax-addrcl 11212 ax-mulcl 11213 ax-mulrcl 11214 ax-mulcom 11215 ax-addass 11216 ax-mulass 11217 ax-distr 11218 ax-i2m1 11219 ax-1ne0 11220 ax-1rid 11221 ax-rnegex 11222 ax-rrecex 11223 ax-cnre 11224 ax-pre-lttri 11225 ax-pre-lttrn 11226 ax-pre-ltadd 11227 ax-pre-mulgt0 11228 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-nul 4333 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4906 df-iun 4991 df-br 5142 df-opab 5204 df-mpt 5224 df-tr 5258 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5635 df-we 5637 df-xp 5689 df-rel 5690 df-cnv 5691 df-co 5692 df-dm 5693 df-rn 5694 df-res 5695 df-ima 5696 df-pred 6319 df-ord 6385 df-on 6386 df-lim 6387 df-suc 6388 df-iota 6512 df-fun 6561 df-fn 6562 df-f 6563 df-f1 6564 df-fo 6565 df-f1o 6566 df-fv 6567 df-riota 7386 df-ov 7432 df-oprab 7433 df-mpo 7434 df-om 7884 df-2nd 8011 df-frecs 8302 df-wrecs 8333 df-recs 8407 df-rdg 8446 df-er 8741 df-en 8982 df-dom 8983 df-sdom 8984 df-pnf 11293 df-mnf 11294 df-xr 11295 df-ltxr 11296 df-le 11297 df-sub 11490 df-neg 11491 df-nn 12263 df-2 12325 df-3 12326 df-4 12327 df-5 12328 df-6 12329 df-7 12330 df-8 12331 df-9 12332 df-n0 12523 df-z 12610 df-dec 12730 df-sets 17197 df-slot 17215 df-ndx 17227 df-base 17244 df-ple 17313 df-hom 17317 df-cco 17318 df-proset 18336 df-poset 18355 df-prstc 49125 |
This theorem is referenced by: (None) |
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