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Mirrors > Home > MPE Home > Th. List > Mathboxes > prstcocvalOLD | Structured version Visualization version GIF version |
Description: Obsolete proof of prstcocval 48792 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 17417 | . . 3 ⊢ oc = Slot (oc‘ndx) | |
5 | 1nn0 12533 | . . . . . . 7 ⊢ 1 ∈ ℕ0 | |
6 | 5, 5 | deccl 12739 | . . . . . 6 ⊢ ;11 ∈ ℕ0 |
7 | 6 | nn0rei 12528 | . . . . 5 ⊢ ;11 ∈ ℝ |
8 | 5nn 12343 | . . . . . 6 ⊢ 5 ∈ ℕ | |
9 | 1lt5 12437 | . . . . . 6 ⊢ 1 < 5 | |
10 | 5, 5, 8, 9 | declt 12752 | . . . . 5 ⊢ ;11 < ;15 |
11 | 7, 10 | ltneii 11365 | . . . 4 ⊢ ;11 ≠ ;15 |
12 | ocndx 17416 | . . . . 5 ⊢ (oc‘ndx) = ;11 | |
13 | ccondx 17448 | . . . . 5 ⊢ (comp‘ndx) = ;15 | |
14 | 12, 13 | neeq12i 3003 | . . . 4 ⊢ ((oc‘ndx) ≠ (comp‘ndx) ↔ ;11 ≠ ;15) |
15 | 11, 14 | mpbir 231 | . . 3 ⊢ (oc‘ndx) ≠ (comp‘ndx) |
16 | 4nn 12340 | . . . . . 6 ⊢ 4 ∈ ℕ | |
17 | 1lt4 12433 | . . . . . 6 ⊢ 1 < 4 | |
18 | 5, 5, 16, 17 | declt 12752 | . . . . 5 ⊢ ;11 < ;14 |
19 | 7, 18 | ltneii 11365 | . . . 4 ⊢ ;11 ≠ ;14 |
20 | homndx 17446 | . . . . 5 ⊢ (Hom ‘ndx) = ;14 | |
21 | 12, 20 | neeq12i 3003 | . . . 4 ⊢ ((oc‘ndx) ≠ (Hom ‘ndx) ↔ ;11 ≠ ;14) |
22 | 19, 21 | mpbir 231 | . . 3 ⊢ (oc‘ndx) ≠ (Hom ‘ndx) |
23 | 2, 3, 4, 15, 22 | prstcnid 48787 | . 2 ⊢ (𝜑 → (oc‘𝐾) = (oc‘𝐶)) |
24 | 1, 23 | eqtrd 2773 | 1 ⊢ (𝜑 → ⊥ = (oc‘𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1535 ∈ wcel 2104 ≠ wne 2936 ‘cfv 6558 1c1 11147 4c4 12314 5c5 12315 ;cdc 12724 ndxcnx 17216 occoc 17295 Hom chom 17298 compcco 17299 Proset cproset 18339 ProsetToCatcprstc 48783 |
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 1963 ax-7 2003 ax-8 2106 ax-9 2114 ax-10 2137 ax-11 2153 ax-12 2173 ax-ext 2704 ax-sep 5300 ax-nul 5307 ax-pow 5366 ax-pr 5430 ax-un 7747 ax-cnex 11202 ax-resscn 11203 ax-1cn 11204 ax-icn 11205 ax-addcl 11206 ax-addrcl 11207 ax-mulcl 11208 ax-mulrcl 11209 ax-mulcom 11210 ax-addass 11211 ax-mulass 11212 ax-distr 11213 ax-i2m1 11214 ax-1ne0 11215 ax-1rid 11216 ax-rnegex 11217 ax-rrecex 11218 ax-cnre 11219 ax-pre-lttri 11220 ax-pre-lttrn 11221 ax-pre-ltadd 11222 ax-pre-mulgt0 11223 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1538 df-fal 1548 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2536 df-eu 2565 df-clab 2711 df-cleq 2725 df-clel 2812 df-nfc 2888 df-ne 2937 df-nel 3043 df-ral 3058 df-rex 3067 df-reu 3377 df-rab 3433 df-v 3479 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4915 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 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 6317 df-ord 6383 df-on 6384 df-lim 6385 df-suc 6386 df-iota 6510 df-fun 6560 df-fn 6561 df-f 6562 df-f1 6563 df-fo 6564 df-f1o 6565 df-fv 6566 df-riota 7381 df-ov 7428 df-oprab 7429 df-mpo 7430 df-om 7881 df-2nd 8008 df-frecs 8299 df-wrecs 8330 df-recs 8404 df-rdg 8443 df-er 8738 df-en 8979 df-dom 8980 df-sdom 8981 df-pnf 11288 df-mnf 11289 df-xr 11290 df-ltxr 11291 df-le 11292 df-sub 11485 df-neg 11486 df-nn 12258 df-2 12320 df-3 12321 df-4 12322 df-5 12323 df-6 12324 df-7 12325 df-8 12326 df-9 12327 df-n0 12518 df-dec 12725 df-sets 17187 df-slot 17205 df-ndx 17217 df-ocomp 17308 df-hom 17311 df-cco 17312 df-prstc 48784 |
This theorem is referenced by: (None) |
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