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| Mirrors > Home > MPE Home > Th. List > Mathboxes > prstclevalOLD | Structured version Visualization version GIF version | ||
| Description: Obsolete proof of prstcleval 49157 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 17411 | . . 3 ⊢ le = Slot (le‘ndx) | |
| 5 | 10re 12752 | . . . . 5 ⊢ ;10 ∈ ℝ | |
| 6 | 1nn0 12542 | . . . . . 6 ⊢ 1 ∈ ℕ0 | |
| 7 | 0nn0 12541 | . . . . . 6 ⊢ 0 ∈ ℕ0 | |
| 8 | 5nn 12352 | . . . . . 6 ⊢ 5 ∈ ℕ | |
| 9 | 8 | nngt0i 12305 | . . . . . 6 ⊢ 0 < 5 |
| 10 | 6, 7, 8, 9 | declt 12761 | . . . . 5 ⊢ ;10 < ;15 |
| 11 | 5, 10 | ltneii 11374 | . . . 4 ⊢ ;10 ≠ ;15 |
| 12 | plendx 17410 | . . . . 5 ⊢ (le‘ndx) = ;10 | |
| 13 | ccondx 17457 | . . . . 5 ⊢ (comp‘ndx) = ;15 | |
| 14 | 12, 13 | neeq12i 3007 | . . . 4 ⊢ ((le‘ndx) ≠ (comp‘ndx) ↔ ;10 ≠ ;15) |
| 15 | 11, 14 | mpbir 231 | . . 3 ⊢ (le‘ndx) ≠ (comp‘ndx) |
| 16 | 4nn 12349 | . . . . . 6 ⊢ 4 ∈ ℕ | |
| 17 | 16 | nngt0i 12305 | . . . . . 6 ⊢ 0 < 4 |
| 18 | 6, 7, 16, 17 | declt 12761 | . . . . 5 ⊢ ;10 < ;14 |
| 19 | 5, 18 | ltneii 11374 | . . . 4 ⊢ ;10 ≠ ;14 |
| 20 | homndx 17455 | . . . . 5 ⊢ (Hom ‘ndx) = ;14 | |
| 21 | 12, 20 | neeq12i 3007 | . . . 4 ⊢ ((le‘ndx) ≠ (Hom ‘ndx) ↔ ;10 ≠ ;14) |
| 22 | 19, 21 | mpbir 231 | . . 3 ⊢ (le‘ndx) ≠ (Hom ‘ndx) |
| 23 | 2, 3, 4, 15, 22 | prstcnid 49155 | . 2 ⊢ (𝜑 → (le‘𝐾) = (le‘𝐶)) |
| 24 | 1, 23 | eqtrd 2777 | 1 ⊢ (𝜑 → ≤ = (le‘𝐶)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2108 ≠ wne 2940 ‘cfv 6561 0cc0 11155 1c1 11156 4c4 12323 5c5 12324 ;cdc 12733 ndxcnx 17230 lecple 17304 Hom chom 17308 compcco 17309 Proset cproset 18338 ProsetToCatcprstc 49151 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-5 12332 df-6 12333 df-7 12334 df-8 12335 df-9 12336 df-n0 12527 df-dec 12734 df-sets 17201 df-slot 17219 df-ndx 17231 df-ple 17317 df-hom 17321 df-cco 17322 df-prstc 49152 |
| This theorem is referenced by: (None) |
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