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Mirrors > Home > MPE Home > Th. List > Mathboxes > postcposALT | Structured version Visualization version GIF version |
Description: Alternate proof for postcpos 48058. (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 2728 | . . . 4 ⊢ (𝜑 → (Base‘𝐾) = (Base‘𝐾)) | |
4 | 1, 2, 3 | prstcbas 48045 | . . 3 ⊢ (𝜑 → (Base‘𝐾) = (Base‘𝐶)) |
5 | eqidd 2728 | . . . . . . 7 ⊢ (𝜑 → (le‘𝐾) = (le‘𝐾)) | |
6 | 1, 2, 5 | prstcle 48048 | . . . . . 6 ⊢ (𝜑 → (𝑥(le‘𝐾)𝑦 ↔ 𝑥(le‘𝐶)𝑦)) |
7 | 1, 2, 5 | prstcle 48048 | . . . . . 6 ⊢ (𝜑 → (𝑦(le‘𝐾)𝑥 ↔ 𝑦(le‘𝐶)𝑥)) |
8 | 6, 7 | anbi12d 630 | . . . . 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 3324 | . . 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 3324 | . 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 2727 | . . . . 5 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
13 | eqid 2727 | . . . . 5 ⊢ (le‘𝐾) = (le‘𝐾) | |
14 | 12, 13 | ispos2 18300 | . . . 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 48053 | . . 3 ⊢ (𝜑 → 𝐶 ∈ Proset ) |
18 | eqid 2727 | . . . . 5 ⊢ (Base‘𝐶) = (Base‘𝐶) | |
19 | eqid 2727 | . . . . 5 ⊢ (le‘𝐶) = (le‘𝐶) | |
20 | 18, 19 | ispos2 18300 | . . . 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 205 ∧ wa 395 = wceq 1534 ∈ wcel 2099 ∀wral 3056 class class class wbr 5142 ‘cfv 6542 Basecbs 17173 lecple 17233 Proset cproset 18278 Posetcpo 18292 ProsetToCatcprstc 48040 |
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 7734 ax-cnex 11188 ax-resscn 11189 ax-1cn 11190 ax-icn 11191 ax-addcl 11192 ax-addrcl 11193 ax-mulcl 11194 ax-mulrcl 11195 ax-mulcom 11196 ax-addass 11197 ax-mulass 11198 ax-distr 11199 ax-i2m1 11200 ax-1ne0 11201 ax-1rid 11202 ax-rnegex 11203 ax-rrecex 11204 ax-cnre 11205 ax-pre-lttri 11206 ax-pre-lttrn 11207 ax-pre-ltadd 11208 ax-pre-mulgt0 11209 |
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 7865 df-2nd 7988 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-er 8718 df-en 8958 df-dom 8959 df-sdom 8960 df-pnf 11274 df-mnf 11275 df-xr 11276 df-ltxr 11277 df-le 11278 df-sub 11470 df-neg 11471 df-nn 12237 df-2 12299 df-3 12300 df-4 12301 df-5 12302 df-6 12303 df-7 12304 df-8 12305 df-9 12306 df-n0 12497 df-z 12583 df-dec 12702 df-sets 17126 df-slot 17144 df-ndx 17156 df-base 17174 df-ple 17246 df-hom 17250 df-cco 17251 df-proset 18280 df-poset 18298 df-prstc 48041 |
This theorem is referenced by: (None) |
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