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Mirrors > Home > MPE Home > Th. List > Mathboxes > dihcnvord | Structured version Visualization version GIF version |
Description: Ordering property for converse of isomorphism H. (Contributed by NM, 17-Aug-2014.) |
Ref | Expression |
---|---|
dihcnvord.l | ⊢ ≤ = (le‘𝐾) |
dihcnvord.h | ⊢ 𝐻 = (LHyp‘𝐾) |
dihcnvord.i | ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊) |
dihcnvord.k | ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
dihcnvord.x | ⊢ (𝜑 → 𝑋 ∈ ran 𝐼) |
dihcnvord.y | ⊢ (𝜑 → 𝑌 ∈ ran 𝐼) |
Ref | Expression |
---|---|
dihcnvord | ⊢ (𝜑 → ((◡𝐼‘𝑋) ≤ (◡𝐼‘𝑌) ↔ 𝑋 ⊆ 𝑌)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dihcnvord.k | . . 3 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
2 | dihcnvord.x | . . . 4 ⊢ (𝜑 → 𝑋 ∈ ran 𝐼) | |
3 | eqid 2778 | . . . . 5 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
4 | dihcnvord.h | . . . . 5 ⊢ 𝐻 = (LHyp‘𝐾) | |
5 | dihcnvord.i | . . . . 5 ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊) | |
6 | 3, 4, 5 | dihcnvcl 37858 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ ran 𝐼) → (◡𝐼‘𝑋) ∈ (Base‘𝐾)) |
7 | 1, 2, 6 | syl2anc 576 | . . 3 ⊢ (𝜑 → (◡𝐼‘𝑋) ∈ (Base‘𝐾)) |
8 | dihcnvord.y | . . . 4 ⊢ (𝜑 → 𝑌 ∈ ran 𝐼) | |
9 | 3, 4, 5 | dihcnvcl 37858 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑌 ∈ ran 𝐼) → (◡𝐼‘𝑌) ∈ (Base‘𝐾)) |
10 | 1, 8, 9 | syl2anc 576 | . . 3 ⊢ (𝜑 → (◡𝐼‘𝑌) ∈ (Base‘𝐾)) |
11 | dihcnvord.l | . . . 4 ⊢ ≤ = (le‘𝐾) | |
12 | 3, 11, 4, 5 | dihord 37851 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (◡𝐼‘𝑋) ∈ (Base‘𝐾) ∧ (◡𝐼‘𝑌) ∈ (Base‘𝐾)) → ((𝐼‘(◡𝐼‘𝑋)) ⊆ (𝐼‘(◡𝐼‘𝑌)) ↔ (◡𝐼‘𝑋) ≤ (◡𝐼‘𝑌))) |
13 | 1, 7, 10, 12 | syl3anc 1351 | . 2 ⊢ (𝜑 → ((𝐼‘(◡𝐼‘𝑋)) ⊆ (𝐼‘(◡𝐼‘𝑌)) ↔ (◡𝐼‘𝑋) ≤ (◡𝐼‘𝑌))) |
14 | 4, 5 | dihcnvid2 37860 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ ran 𝐼) → (𝐼‘(◡𝐼‘𝑋)) = 𝑋) |
15 | 1, 2, 14 | syl2anc 576 | . . 3 ⊢ (𝜑 → (𝐼‘(◡𝐼‘𝑋)) = 𝑋) |
16 | 4, 5 | dihcnvid2 37860 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑌 ∈ ran 𝐼) → (𝐼‘(◡𝐼‘𝑌)) = 𝑌) |
17 | 1, 8, 16 | syl2anc 576 | . . 3 ⊢ (𝜑 → (𝐼‘(◡𝐼‘𝑌)) = 𝑌) |
18 | 15, 17 | sseq12d 3890 | . 2 ⊢ (𝜑 → ((𝐼‘(◡𝐼‘𝑋)) ⊆ (𝐼‘(◡𝐼‘𝑌)) ↔ 𝑋 ⊆ 𝑌)) |
19 | 13, 18 | bitr3d 273 | 1 ⊢ (𝜑 → ((◡𝐼‘𝑋) ≤ (◡𝐼‘𝑌) ↔ 𝑋 ⊆ 𝑌)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 387 = wceq 1507 ∈ wcel 2050 ⊆ wss 3829 class class class wbr 4929 ◡ccnv 5406 ran crn 5408 ‘cfv 6188 Basecbs 16339 lecple 16428 HLchlt 35937 LHypclh 36571 DIsoHcdih 37815 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-13 2301 ax-ext 2750 ax-rep 5049 ax-sep 5060 ax-nul 5067 ax-pow 5119 ax-pr 5186 ax-un 7279 ax-cnex 10391 ax-resscn 10392 ax-1cn 10393 ax-icn 10394 ax-addcl 10395 ax-addrcl 10396 ax-mulcl 10397 ax-mulrcl 10398 ax-mulcom 10399 ax-addass 10400 ax-mulass 10401 ax-distr 10402 ax-i2m1 10403 ax-1ne0 10404 ax-1rid 10405 ax-rnegex 10406 ax-rrecex 10407 ax-cnre 10408 ax-pre-lttri 10409 ax-pre-lttrn 10410 ax-pre-ltadd 10411 ax-pre-mulgt0 10412 ax-riotaBAD 35540 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3or 1069 df-3an 1070 df-tru 1510 df-fal 1520 df-ex 1743 df-nf 1747 df-sb 2016 df-mo 2547 df-eu 2584 df-clab 2759 df-cleq 2771 df-clel 2846 df-nfc 2918 df-ne 2968 df-nel 3074 df-ral 3093 df-rex 3094 df-reu 3095 df-rmo 3096 df-rab 3097 df-v 3417 df-sbc 3682 df-csb 3787 df-dif 3832 df-un 3834 df-in 3836 df-ss 3843 df-pss 3845 df-nul 4179 df-if 4351 df-pw 4424 df-sn 4442 df-pr 4444 df-tp 4446 df-op 4448 df-uni 4713 df-int 4750 df-iun 4794 df-iin 4795 df-br 4930 df-opab 4992 df-mpt 5009 df-tr 5031 df-id 5312 df-eprel 5317 df-po 5326 df-so 5327 df-fr 5366 df-we 5368 df-xp 5413 df-rel 5414 df-cnv 5415 df-co 5416 df-dm 5417 df-rn 5418 df-res 5419 df-ima 5420 df-pred 5986 df-ord 6032 df-on 6033 df-lim 6034 df-suc 6035 df-iota 6152 df-fun 6190 df-fn 6191 df-f 6192 df-f1 6193 df-fo 6194 df-f1o 6195 df-fv 6196 df-riota 6937 df-ov 6979 df-oprab 6980 df-mpo 6981 df-om 7397 df-1st 7501 df-2nd 7502 df-tpos 7695 df-undef 7742 df-wrecs 7750 df-recs 7812 df-rdg 7850 df-1o 7905 df-oadd 7909 df-er 8089 df-map 8208 df-en 8307 df-dom 8308 df-sdom 8309 df-fin 8310 df-pnf 10476 df-mnf 10477 df-xr 10478 df-ltxr 10479 df-le 10480 df-sub 10672 df-neg 10673 df-nn 11440 df-2 11503 df-3 11504 df-4 11505 df-5 11506 df-6 11507 df-n0 11708 df-z 11794 df-uz 12059 df-fz 12709 df-struct 16341 df-ndx 16342 df-slot 16343 df-base 16345 df-sets 16346 df-ress 16347 df-plusg 16434 df-mulr 16435 df-sca 16437 df-vsca 16438 df-0g 16571 df-proset 17396 df-poset 17414 df-plt 17426 df-lub 17442 df-glb 17443 df-join 17444 df-meet 17445 df-p0 17507 df-p1 17508 df-lat 17514 df-clat 17576 df-mgm 17710 df-sgrp 17752 df-mnd 17763 df-submnd 17804 df-grp 17894 df-minusg 17895 df-sbg 17896 df-subg 18060 df-cntz 18218 df-lsm 18522 df-cmn 18668 df-abl 18669 df-mgp 18963 df-ur 18975 df-ring 19022 df-oppr 19096 df-dvdsr 19114 df-unit 19115 df-invr 19145 df-dvr 19156 df-drng 19227 df-lmod 19358 df-lss 19426 df-lsp 19466 df-lvec 19597 df-oposet 35763 df-ol 35765 df-oml 35766 df-covers 35853 df-ats 35854 df-atl 35885 df-cvlat 35909 df-hlat 35938 df-llines 36085 df-lplanes 36086 df-lvols 36087 df-lines 36088 df-psubsp 36090 df-pmap 36091 df-padd 36383 df-lhyp 36575 df-laut 36576 df-ldil 36691 df-ltrn 36692 df-trl 36746 df-tendo 37342 df-edring 37344 df-disoa 37616 df-dvech 37666 df-dib 37726 df-dic 37760 df-dih 37816 |
This theorem is referenced by: dihoml4c 37963 djhcvat42 38002 |
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