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Mirrors > Home > MPE Home > Th. List > Mathboxes > prstcocvalOLD | Structured version Visualization version GIF version |
Description: Obsolete proof of prstcocval 48185 as of 12-Nov-2024. Orthocomplementation is unchanged. (Contributed by Zhi Wang, 20-Sep-2024.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
prstcnid.c | ⊢ (𝜑 → 𝐶 = (ProsetToCat‘𝐾)) |
prstcnid.k | ⊢ (𝜑 → 𝐾 ∈ Proset ) |
prstcoc.oc | ⊢ (𝜑 → ⊥ = (oc‘𝐾)) |
Ref | Expression |
---|---|
prstcocvalOLD | ⊢ (𝜑 → ⊥ = (oc‘𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | prstcoc.oc | . 2 ⊢ (𝜑 → ⊥ = (oc‘𝐾)) | |
2 | prstcnid.c | . . 3 ⊢ (𝜑 → 𝐶 = (ProsetToCat‘𝐾)) | |
3 | prstcnid.k | . . 3 ⊢ (𝜑 → 𝐾 ∈ Proset ) | |
4 | ocid 17357 | . . 3 ⊢ oc = Slot (oc‘ndx) | |
5 | 1nn0 12513 | . . . . . . 7 ⊢ 1 ∈ ℕ0 | |
6 | 5, 5 | deccl 12717 | . . . . . 6 ⊢ ;11 ∈ ℕ0 |
7 | 6 | nn0rei 12508 | . . . . 5 ⊢ ;11 ∈ ℝ |
8 | 5nn 12323 | . . . . . 6 ⊢ 5 ∈ ℕ | |
9 | 1lt5 12417 | . . . . . 6 ⊢ 1 < 5 | |
10 | 5, 5, 8, 9 | declt 12730 | . . . . 5 ⊢ ;11 < ;15 |
11 | 7, 10 | ltneii 11352 | . . . 4 ⊢ ;11 ≠ ;15 |
12 | ocndx 17356 | . . . . 5 ⊢ (oc‘ndx) = ;11 | |
13 | ccondx 17388 | . . . . 5 ⊢ (comp‘ndx) = ;15 | |
14 | 12, 13 | neeq12i 2997 | . . . 4 ⊢ ((oc‘ndx) ≠ (comp‘ndx) ↔ ;11 ≠ ;15) |
15 | 11, 14 | mpbir 230 | . . 3 ⊢ (oc‘ndx) ≠ (comp‘ndx) |
16 | 4nn 12320 | . . . . . 6 ⊢ 4 ∈ ℕ | |
17 | 1lt4 12413 | . . . . . 6 ⊢ 1 < 4 | |
18 | 5, 5, 16, 17 | declt 12730 | . . . . 5 ⊢ ;11 < ;14 |
19 | 7, 18 | ltneii 11352 | . . . 4 ⊢ ;11 ≠ ;14 |
20 | homndx 17386 | . . . . 5 ⊢ (Hom ‘ndx) = ;14 | |
21 | 12, 20 | neeq12i 2997 | . . . 4 ⊢ ((oc‘ndx) ≠ (Hom ‘ndx) ↔ ;11 ≠ ;14) |
22 | 19, 21 | mpbir 230 | . . 3 ⊢ (oc‘ndx) ≠ (Hom ‘ndx) |
23 | 2, 3, 4, 15, 22 | prstcnid 48180 | . 2 ⊢ (𝜑 → (oc‘𝐾) = (oc‘𝐶)) |
24 | 1, 23 | eqtrd 2765 | 1 ⊢ (𝜑 → ⊥ = (oc‘𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2098 ≠ wne 2930 ‘cfv 6543 1c1 11134 4c4 12294 5c5 12295 ;cdc 12702 ndxcnx 17156 occoc 17235 Hom chom 17238 compcco 17239 Proset cproset 18279 ProsetToCatcprstc 48176 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5295 ax-nul 5302 ax-pow 5360 ax-pr 5424 ax-un 7735 ax-cnex 11189 ax-resscn 11190 ax-1cn 11191 ax-icn 11192 ax-addcl 11193 ax-addrcl 11194 ax-mulcl 11195 ax-mulrcl 11196 ax-mulcom 11197 ax-addass 11198 ax-mulass 11199 ax-distr 11200 ax-i2m1 11201 ax-1ne0 11202 ax-1rid 11203 ax-rnegex 11204 ax-rrecex 11205 ax-cnre 11206 ax-pre-lttri 11207 ax-pre-lttrn 11208 ax-pre-ltadd 11209 ax-pre-mulgt0 11210 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-pss 3961 df-nul 4320 df-if 4526 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4905 df-iun 4994 df-br 5145 df-opab 5207 df-mpt 5228 df-tr 5262 df-id 5571 df-eprel 5577 df-po 5585 df-so 5586 df-fr 5628 df-we 5630 df-xp 5679 df-rel 5680 df-cnv 5681 df-co 5682 df-dm 5683 df-rn 5684 df-res 5685 df-ima 5686 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7369 df-ov 7416 df-oprab 7417 df-mpo 7418 df-om 7866 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 11275 df-mnf 11276 df-xr 11277 df-ltxr 11278 df-le 11279 df-sub 11471 df-neg 11472 df-nn 12238 df-2 12300 df-3 12301 df-4 12302 df-5 12303 df-6 12304 df-7 12305 df-8 12306 df-9 12307 df-n0 12498 df-dec 12703 df-sets 17127 df-slot 17145 df-ndx 17157 df-ocomp 17248 df-hom 17251 df-cco 17252 df-prstc 48177 |
This theorem is referenced by: (None) |
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