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Mirrors > Home > MPE Home > Th. List > Mathboxes > prstclevalOLD | Structured version Visualization version GIF version |
Description: Obsolete proof of prstcleval 48868 as of 12-Nov-2024. Value of the less-than-or-equal-to relation 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 ) |
prstcle.l | ⊢ (𝜑 → ≤ = (le‘𝐾)) |
Ref | Expression |
---|---|
prstclevalOLD | ⊢ (𝜑 → ≤ = (le‘𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | prstcle.l | . 2 ⊢ (𝜑 → ≤ = (le‘𝐾)) | |
2 | prstcnid.c | . . 3 ⊢ (𝜑 → 𝐶 = (ProsetToCat‘𝐾)) | |
3 | prstcnid.k | . . 3 ⊢ (𝜑 → 𝐾 ∈ Proset ) | |
4 | pleid 17412 | . . 3 ⊢ le = Slot (le‘ndx) | |
5 | 10re 12749 | . . . . 5 ⊢ ;10 ∈ ℝ | |
6 | 1nn0 12539 | . . . . . 6 ⊢ 1 ∈ ℕ0 | |
7 | 0nn0 12538 | . . . . . 6 ⊢ 0 ∈ ℕ0 | |
8 | 5nn 12349 | . . . . . 6 ⊢ 5 ∈ ℕ | |
9 | 8 | nngt0i 12302 | . . . . . 6 ⊢ 0 < 5 |
10 | 6, 7, 8, 9 | declt 12758 | . . . . 5 ⊢ ;10 < ;15 |
11 | 5, 10 | ltneii 11371 | . . . 4 ⊢ ;10 ≠ ;15 |
12 | plendx 17411 | . . . . 5 ⊢ (le‘ndx) = ;10 | |
13 | ccondx 17458 | . . . . 5 ⊢ (comp‘ndx) = ;15 | |
14 | 12, 13 | neeq12i 3004 | . . . 4 ⊢ ((le‘ndx) ≠ (comp‘ndx) ↔ ;10 ≠ ;15) |
15 | 11, 14 | mpbir 231 | . . 3 ⊢ (le‘ndx) ≠ (comp‘ndx) |
16 | 4nn 12346 | . . . . . 6 ⊢ 4 ∈ ℕ | |
17 | 16 | nngt0i 12302 | . . . . . 6 ⊢ 0 < 4 |
18 | 6, 7, 16, 17 | declt 12758 | . . . . 5 ⊢ ;10 < ;14 |
19 | 5, 18 | ltneii 11371 | . . . 4 ⊢ ;10 ≠ ;14 |
20 | homndx 17456 | . . . . 5 ⊢ (Hom ‘ndx) = ;14 | |
21 | 12, 20 | neeq12i 3004 | . . . 4 ⊢ ((le‘ndx) ≠ (Hom ‘ndx) ↔ ;10 ≠ ;14) |
22 | 19, 21 | mpbir 231 | . . 3 ⊢ (le‘ndx) ≠ (Hom ‘ndx) |
23 | 2, 3, 4, 15, 22 | prstcnid 48866 | . 2 ⊢ (𝜑 → (le‘𝐾) = (le‘𝐶)) |
24 | 1, 23 | eqtrd 2774 | 1 ⊢ (𝜑 → ≤ = (le‘𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1536 ∈ wcel 2105 ≠ wne 2937 ‘cfv 6562 0cc0 11152 1c1 11153 4c4 12320 5c5 12321 ;cdc 12730 ndxcnx 17226 lecple 17304 Hom chom 17308 compcco 17309 Proset cproset 18349 ProsetToCatcprstc 48862 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 ax-cnex 11208 ax-resscn 11209 ax-1cn 11210 ax-icn 11211 ax-addcl 11212 ax-addrcl 11213 ax-mulcl 11214 ax-mulrcl 11215 ax-mulcom 11216 ax-addass 11217 ax-mulass 11218 ax-distr 11219 ax-i2m1 11220 ax-1ne0 11221 ax-1rid 11222 ax-rnegex 11223 ax-rrecex 11224 ax-cnre 11225 ax-pre-lttri 11226 ax-pre-lttrn 11227 ax-pre-ltadd 11228 ax-pre-mulgt0 11229 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-pss 3982 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-iun 4997 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5640 df-we 5642 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-pred 6322 df-ord 6388 df-on 6389 df-lim 6390 df-suc 6391 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-riota 7387 df-ov 7433 df-oprab 7434 df-mpo 7435 df-om 7887 df-2nd 8013 df-frecs 8304 df-wrecs 8335 df-recs 8409 df-rdg 8448 df-er 8743 df-en 8984 df-dom 8985 df-sdom 8986 df-pnf 11294 df-mnf 11295 df-xr 11296 df-ltxr 11297 df-le 11298 df-sub 11491 df-neg 11492 df-nn 12264 df-2 12326 df-3 12327 df-4 12328 df-5 12329 df-6 12330 df-7 12331 df-8 12332 df-9 12333 df-n0 12524 df-dec 12731 df-sets 17197 df-slot 17215 df-ndx 17227 df-ple 17317 df-hom 17321 df-cco 17322 df-prstc 48863 |
This theorem is referenced by: (None) |
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