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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > dihord5a | Structured version Visualization version GIF version |
Description: Part of proof that isomorphism H is order-preserving . (Contributed by NM, 7-Mar-2014.) |
Ref | Expression |
---|---|
dihord.b | ⊢ 𝐵 = (Base‘𝐾) |
dihord.l | ⊢ ≤ = (le‘𝐾) |
dihord.h | ⊢ 𝐻 = (LHyp‘𝐾) |
dihord.i | ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊) |
Ref | Expression |
---|---|
dihord5a | ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊)) ∧ (𝐼‘𝑋) ⊆ (𝐼‘𝑌)) → 𝑋 ≤ 𝑌) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dihord.b | . 2 ⊢ 𝐵 = (Base‘𝐾) | |
2 | dihord.l | . 2 ⊢ ≤ = (le‘𝐾) | |
3 | dihord.h | . 2 ⊢ 𝐻 = (LHyp‘𝐾) | |
4 | eqid 2732 | . 2 ⊢ (join‘𝐾) = (join‘𝐾) | |
5 | eqid 2732 | . 2 ⊢ (meet‘𝐾) = (meet‘𝐾) | |
6 | eqid 2732 | . 2 ⊢ (Atoms‘𝐾) = (Atoms‘𝐾) | |
7 | eqid 2732 | . 2 ⊢ ((DVecH‘𝐾)‘𝑊) = ((DVecH‘𝐾)‘𝑊) | |
8 | eqid 2732 | . 2 ⊢ (LSSum‘((DVecH‘𝐾)‘𝑊)) = (LSSum‘((DVecH‘𝐾)‘𝑊)) | |
9 | dihord.i | . 2 ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊) | |
10 | 1, 2, 3, 4, 5, 6, 7, 8, 9 | dihord5apre 40002 | 1 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊)) ∧ (𝐼‘𝑋) ⊆ (𝐼‘𝑌)) → 𝑋 ≤ 𝑌) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 396 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 ⊆ wss 3945 class class class wbr 5142 ‘cfv 6533 Basecbs 17128 lecple 17188 joincjn 18248 meetcmee 18249 LSSumclsm 19468 Atomscatm 38002 HLchlt 38089 LHypclh 38724 DVecHcdvh 39818 DIsoHcdih 39968 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5357 ax-pr 5421 ax-un 7709 ax-cnex 11150 ax-resscn 11151 ax-1cn 11152 ax-icn 11153 ax-addcl 11154 ax-addrcl 11155 ax-mulcl 11156 ax-mulrcl 11157 ax-mulcom 11158 ax-addass 11159 ax-mulass 11160 ax-distr 11161 ax-i2m1 11162 ax-1ne0 11163 ax-1rid 11164 ax-rnegex 11165 ax-rrecex 11166 ax-cnre 11167 ax-pre-lttri 11168 ax-pre-lttrn 11169 ax-pre-ltadd 11170 ax-pre-mulgt0 11171 ax-riotaBAD 37692 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3775 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-pss 3964 df-nul 4320 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-tp 4628 df-op 4630 df-uni 4903 df-int 4945 df-iun 4993 df-iin 4994 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5568 df-eprel 5574 df-po 5582 df-so 5583 df-fr 5625 df-we 5627 df-xp 5676 df-rel 5677 df-cnv 5678 df-co 5679 df-dm 5680 df-rn 5681 df-res 5682 df-ima 5683 df-pred 6290 df-ord 6357 df-on 6358 df-lim 6359 df-suc 6360 df-iota 6485 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-riota 7350 df-ov 7397 df-oprab 7398 df-mpo 7399 df-om 7840 df-1st 7959 df-2nd 7960 df-tpos 8195 df-undef 8242 df-frecs 8250 df-wrecs 8281 df-recs 8355 df-rdg 8394 df-1o 8450 df-er 8688 df-map 8807 df-en 8925 df-dom 8926 df-sdom 8927 df-fin 8928 df-pnf 11234 df-mnf 11235 df-xr 11236 df-ltxr 11237 df-le 11238 df-sub 11430 df-neg 11431 df-nn 12197 df-2 12259 df-3 12260 df-4 12261 df-5 12262 df-6 12263 df-n0 12457 df-z 12543 df-uz 12807 df-fz 13469 df-struct 17064 df-sets 17081 df-slot 17099 df-ndx 17111 df-base 17129 df-ress 17158 df-plusg 17194 df-mulr 17195 df-sca 17197 df-vsca 17198 df-0g 17371 df-proset 18232 df-poset 18250 df-plt 18267 df-lub 18283 df-glb 18284 df-join 18285 df-meet 18286 df-p0 18362 df-p1 18363 df-lat 18369 df-clat 18436 df-mgm 18545 df-sgrp 18594 df-mnd 18605 df-submnd 18650 df-grp 18799 df-minusg 18800 df-sbg 18801 df-subg 18977 df-cntz 19149 df-lsm 19470 df-cmn 19616 df-abl 19617 df-mgp 19949 df-ur 19966 df-ring 20018 df-oppr 20104 df-dvdsr 20125 df-unit 20126 df-invr 20156 df-dvr 20167 df-drng 20269 df-lmod 20424 df-lss 20494 df-lsp 20534 df-lvec 20665 df-oposet 37915 df-ol 37917 df-oml 37918 df-covers 38005 df-ats 38006 df-atl 38037 df-cvlat 38061 df-hlat 38090 df-llines 38238 df-lplanes 38239 df-lvols 38240 df-lines 38241 df-psubsp 38243 df-pmap 38244 df-padd 38536 df-lhyp 38728 df-laut 38729 df-ldil 38844 df-ltrn 38845 df-trl 38899 df-tendo 39495 df-edring 39497 df-disoa 39769 df-dvech 39819 df-dib 39879 df-dic 39913 df-dih 39969 |
This theorem is referenced by: dihord 40004 |
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