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Mirrors > Home > MPE Home > Th. List > Mathboxes > mapdsord | Structured version Visualization version GIF version |
Description: Strong ordering property of themap defined by df-mapd 38154. (Contributed by NM, 13-Mar-2015.) |
Ref | Expression |
---|---|
mapdcnvcl.h | ⊢ 𝐻 = (LHyp‘𝐾) |
mapdcnvcl.m | ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) |
mapdcnvcl.u | ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) |
mapdcnvcl.s | ⊢ 𝑆 = (LSubSp‘𝑈) |
mapdcnvcl.k | ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
mapdcl.x | ⊢ (𝜑 → 𝑋 ∈ 𝑆) |
mapdsord.x | ⊢ (𝜑 → 𝑌 ∈ 𝑆) |
Ref | Expression |
---|---|
mapdsord | ⊢ (𝜑 → ((𝑀‘𝑋) ⊊ (𝑀‘𝑌) ↔ 𝑋 ⊊ 𝑌)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mapdcnvcl.h | . . . 4 ⊢ 𝐻 = (LHyp‘𝐾) | |
2 | mapdcnvcl.u | . . . 4 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) | |
3 | mapdcnvcl.s | . . . 4 ⊢ 𝑆 = (LSubSp‘𝑈) | |
4 | mapdcnvcl.m | . . . 4 ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) | |
5 | mapdcnvcl.k | . . . 4 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
6 | mapdcl.x | . . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝑆) | |
7 | mapdsord.x | . . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝑆) | |
8 | 1, 2, 3, 4, 5, 6, 7 | mapdord 38167 | . . 3 ⊢ (𝜑 → ((𝑀‘𝑋) ⊆ (𝑀‘𝑌) ↔ 𝑋 ⊆ 𝑌)) |
9 | 1, 2, 3, 4, 5, 6, 7 | mapd11 38168 | . . . 4 ⊢ (𝜑 → ((𝑀‘𝑋) = (𝑀‘𝑌) ↔ 𝑋 = 𝑌)) |
10 | 9 | necon3bid 3005 | . . 3 ⊢ (𝜑 → ((𝑀‘𝑋) ≠ (𝑀‘𝑌) ↔ 𝑋 ≠ 𝑌)) |
11 | 8, 10 | anbi12d 621 | . 2 ⊢ (𝜑 → (((𝑀‘𝑋) ⊆ (𝑀‘𝑌) ∧ (𝑀‘𝑋) ≠ (𝑀‘𝑌)) ↔ (𝑋 ⊆ 𝑌 ∧ 𝑋 ≠ 𝑌))) |
12 | df-pss 3841 | . 2 ⊢ ((𝑀‘𝑋) ⊊ (𝑀‘𝑌) ↔ ((𝑀‘𝑋) ⊆ (𝑀‘𝑌) ∧ (𝑀‘𝑋) ≠ (𝑀‘𝑌))) | |
13 | df-pss 3841 | . 2 ⊢ (𝑋 ⊊ 𝑌 ↔ (𝑋 ⊆ 𝑌 ∧ 𝑋 ≠ 𝑌)) | |
14 | 11, 12, 13 | 3bitr4g 306 | 1 ⊢ (𝜑 → ((𝑀‘𝑋) ⊊ (𝑀‘𝑌) ↔ 𝑋 ⊊ 𝑌)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 387 = wceq 1507 ∈ wcel 2048 ≠ wne 2961 ⊆ wss 3825 ⊊ wpss 3826 ‘cfv 6182 LSubSpclss 19415 HLchlt 35879 LHypclh 36513 DVecHcdvh 37607 mapdcmpd 38153 |
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 1964 ax-8 2050 ax-9 2057 ax-10 2077 ax-11 2091 ax-12 2104 ax-13 2299 ax-ext 2745 ax-rep 5043 ax-sep 5054 ax-nul 5061 ax-pow 5113 ax-pr 5180 ax-un 7273 ax-cnex 10383 ax-resscn 10384 ax-1cn 10385 ax-icn 10386 ax-addcl 10387 ax-addrcl 10388 ax-mulcl 10389 ax-mulrcl 10390 ax-mulcom 10391 ax-addass 10392 ax-mulass 10393 ax-distr 10394 ax-i2m1 10395 ax-1ne0 10396 ax-1rid 10397 ax-rnegex 10398 ax-rrecex 10399 ax-cnre 10400 ax-pre-lttri 10401 ax-pre-lttrn 10402 ax-pre-ltadd 10403 ax-pre-mulgt0 10404 ax-riotaBAD 35482 |
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 2014 df-mo 2544 df-eu 2580 df-clab 2754 df-cleq 2765 df-clel 2840 df-nfc 2912 df-ne 2962 df-nel 3068 df-ral 3087 df-rex 3088 df-reu 3089 df-rmo 3090 df-rab 3091 df-v 3411 df-sbc 3678 df-csb 3783 df-dif 3828 df-un 3830 df-in 3832 df-ss 3839 df-pss 3841 df-nul 4174 df-if 4345 df-pw 4418 df-sn 4436 df-pr 4438 df-tp 4440 df-op 4442 df-uni 4707 df-int 4744 df-iun 4788 df-iin 4789 df-br 4924 df-opab 4986 df-mpt 5003 df-tr 5025 df-id 5305 df-eprel 5310 df-po 5319 df-so 5320 df-fr 5359 df-we 5361 df-xp 5406 df-rel 5407 df-cnv 5408 df-co 5409 df-dm 5410 df-rn 5411 df-res 5412 df-ima 5413 df-pred 5980 df-ord 6026 df-on 6027 df-lim 6028 df-suc 6029 df-iota 6146 df-fun 6184 df-fn 6185 df-f 6186 df-f1 6187 df-fo 6188 df-f1o 6189 df-fv 6190 df-riota 6931 df-ov 6973 df-oprab 6974 df-mpo 6975 df-om 7391 df-1st 7494 df-2nd 7495 df-tpos 7688 df-undef 7735 df-wrecs 7743 df-recs 7805 df-rdg 7843 df-1o 7897 df-oadd 7901 df-er 8081 df-map 8200 df-en 8299 df-dom 8300 df-sdom 8301 df-fin 8302 df-pnf 10468 df-mnf 10469 df-xr 10470 df-ltxr 10471 df-le 10472 df-sub 10664 df-neg 10665 df-nn 11432 df-2 11496 df-3 11497 df-4 11498 df-5 11499 df-6 11500 df-n0 11701 df-z 11787 df-uz 12052 df-fz 12702 df-struct 16331 df-ndx 16332 df-slot 16333 df-base 16335 df-sets 16336 df-ress 16337 df-plusg 16424 df-mulr 16425 df-sca 16427 df-vsca 16428 df-0g 16561 df-proset 17386 df-poset 17404 df-plt 17416 df-lub 17432 df-glb 17433 df-join 17434 df-meet 17435 df-p0 17497 df-p1 17498 df-lat 17504 df-clat 17566 df-mgm 17700 df-sgrp 17742 df-mnd 17753 df-submnd 17794 df-grp 17884 df-minusg 17885 df-sbg 17886 df-subg 18050 df-cntz 18208 df-lsm 18512 df-cmn 18658 df-abl 18659 df-mgp 18953 df-ur 18965 df-ring 19012 df-oppr 19086 df-dvdsr 19104 df-unit 19105 df-invr 19135 df-dvr 19146 df-drng 19217 df-lmod 19348 df-lss 19416 df-lsp 19456 df-lvec 19587 df-lsatoms 35505 df-lshyp 35506 df-lfl 35587 df-lkr 35615 df-oposet 35705 df-ol 35707 df-oml 35708 df-covers 35795 df-ats 35796 df-atl 35827 df-cvlat 35851 df-hlat 35880 df-llines 36027 df-lplanes 36028 df-lvols 36029 df-lines 36030 df-psubsp 36032 df-pmap 36033 df-padd 36325 df-lhyp 36517 df-laut 36518 df-ldil 36633 df-ltrn 36634 df-trl 36688 df-tgrp 37272 df-tendo 37284 df-edring 37286 df-dveca 37532 df-disoa 37558 df-dvech 37608 df-dib 37668 df-dic 37702 df-dih 37758 df-doch 37877 df-djh 37924 df-mapd 38154 |
This theorem is referenced by: mapdcv 38189 |
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