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Mirrors > Home > MPE Home > Th. List > Mathboxes > postcposALT | Structured version Visualization version GIF version |
Description: Alternate proof for postcpos 45975. (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 45964 | . . 3 ⊢ (𝜑 → (Base‘𝐾) = (Base‘𝐶)) |
5 | eqidd 2737 | . . . . . . 7 ⊢ (𝜑 → (le‘𝐾) = (le‘𝐾)) | |
6 | 1, 2, 5 | prstcle 45966 | . . . . . 6 ⊢ (𝜑 → (𝑥(le‘𝐾)𝑦 ↔ 𝑥(le‘𝐶)𝑦)) |
7 | 1, 2, 5 | prstcle 45966 | . . . . . 6 ⊢ (𝜑 → (𝑦(le‘𝐾)𝑥 ↔ 𝑦(le‘𝐶)𝑥)) |
8 | 6, 7 | anbi12d 634 | . . . . 5 ⊢ (𝜑 → ((𝑥(le‘𝐾)𝑦 ∧ 𝑦(le‘𝐾)𝑥) ↔ (𝑥(le‘𝐶)𝑦 ∧ 𝑦(le‘𝐶)𝑥))) |
9 | 8 | imbi1d 345 | . . . 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 3305 | . . 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 3305 | . 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 17776 | . . . 4 ⊢ (𝐾 ∈ Poset ↔ (𝐾 ∈ Proset ∧ ∀𝑥 ∈ (Base‘𝐾)∀𝑦 ∈ (Base‘𝐾)((𝑥(le‘𝐾)𝑦 ∧ 𝑦(le‘𝐾)𝑥) → 𝑥 = 𝑦))) |
15 | 14 | baib 539 | . . 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 45970 | . . 3 ⊢ (𝜑 → 𝐶 ∈ Proset ) |
18 | eqid 2736 | . . . . 5 ⊢ (Base‘𝐶) = (Base‘𝐶) | |
19 | eqid 2736 | . . . . 5 ⊢ (le‘𝐶) = (le‘𝐶) | |
20 | 18, 19 | ispos2 17776 | . . . 4 ⊢ (𝐶 ∈ Poset ↔ (𝐶 ∈ Proset ∧ ∀𝑥 ∈ (Base‘𝐶)∀𝑦 ∈ (Base‘𝐶)((𝑥(le‘𝐶)𝑦 ∧ 𝑦(le‘𝐶)𝑥) → 𝑥 = 𝑦))) |
21 | 20 | baib 539 | . . 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 314 | 1 ⊢ (𝜑 → (𝐾 ∈ Poset ↔ 𝐶 ∈ Poset)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1543 ∈ wcel 2112 ∀wral 3051 class class class wbr 5039 ‘cfv 6358 Basecbs 16666 lecple 16756 Proset cproset 17754 Posetcpo 17768 ProsetToCatcprstc 45959 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501 ax-cnex 10750 ax-resscn 10751 ax-1cn 10752 ax-icn 10753 ax-addcl 10754 ax-addrcl 10755 ax-mulcl 10756 ax-mulrcl 10757 ax-mulcom 10758 ax-addass 10759 ax-mulass 10760 ax-distr 10761 ax-i2m1 10762 ax-1ne0 10763 ax-1rid 10764 ax-rnegex 10765 ax-rrecex 10766 ax-cnre 10767 ax-pre-lttri 10768 ax-pre-lttrn 10769 ax-pre-ltadd 10770 ax-pre-mulgt0 10771 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-nel 3037 df-ral 3056 df-rex 3057 df-reu 3058 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-tp 4532 df-op 4534 df-uni 4806 df-iun 4892 df-br 5040 df-opab 5102 df-mpt 5121 df-tr 5147 df-id 5440 df-eprel 5445 df-po 5453 df-so 5454 df-fr 5494 df-we 5496 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-pred 6140 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7148 df-ov 7194 df-oprab 7195 df-mpo 7196 df-om 7623 df-wrecs 8025 df-recs 8086 df-rdg 8124 df-er 8369 df-en 8605 df-dom 8606 df-sdom 8607 df-pnf 10834 df-mnf 10835 df-xr 10836 df-ltxr 10837 df-le 10838 df-sub 11029 df-neg 11030 df-nn 11796 df-2 11858 df-3 11859 df-4 11860 df-5 11861 df-6 11862 df-7 11863 df-8 11864 df-9 11865 df-n0 12056 df-z 12142 df-dec 12259 df-ndx 16669 df-slot 16670 df-base 16672 df-sets 16673 df-ple 16769 df-hom 16773 df-cco 16774 df-proset 17756 df-poset 17774 df-prstc 45960 |
This theorem is referenced by: (None) |
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