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Mirrors > Home > MPE Home > Th. List > Mathboxes > dia2dimlem11 | Structured version Visualization version GIF version |
Description: Lemma for dia2dim 39751. Convert ordering hypothesis on 𝑅‘𝐹 to subspace membership 𝐹 ∈ (𝐼‘(𝑈 ∨ 𝑉)). (Contributed by NM, 8-Sep-2014.) |
Ref | Expression |
---|---|
dia2dimlem11.l | ⊢ ≤ = (le‘𝐾) |
dia2dimlem11.j | ⊢ ∨ = (join‘𝐾) |
dia2dimlem11.m | ⊢ ∧ = (meet‘𝐾) |
dia2dimlem11.a | ⊢ 𝐴 = (Atoms‘𝐾) |
dia2dimlem11.h | ⊢ 𝐻 = (LHyp‘𝐾) |
dia2dimlem11.t | ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) |
dia2dimlem11.r | ⊢ 𝑅 = ((trL‘𝐾)‘𝑊) |
dia2dimlem11.y | ⊢ 𝑌 = ((DVecA‘𝐾)‘𝑊) |
dia2dimlem11.s | ⊢ 𝑆 = (LSubSp‘𝑌) |
dia2dimlem11.pl | ⊢ ⊕ = (LSSum‘𝑌) |
dia2dimlem11.n | ⊢ 𝑁 = (LSpan‘𝑌) |
dia2dimlem11.i | ⊢ 𝐼 = ((DIsoA‘𝐾)‘𝑊) |
dia2dimlem11.k | ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
dia2dimlem11.u | ⊢ (𝜑 → (𝑈 ∈ 𝐴 ∧ 𝑈 ≤ 𝑊)) |
dia2dimlem11.v | ⊢ (𝜑 → (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) |
dia2dimlem11.f | ⊢ (𝜑 → 𝐹 ∈ 𝑇) |
dia2dimlem11.uv | ⊢ (𝜑 → 𝑈 ≠ 𝑉) |
dia2dimlem11.fe | ⊢ (𝜑 → 𝐹 ∈ (𝐼‘(𝑈 ∨ 𝑉))) |
Ref | Expression |
---|---|
dia2dimlem11 | ⊢ (𝜑 → 𝐹 ∈ ((𝐼‘𝑈) ⊕ (𝐼‘𝑉))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dia2dimlem11.l | . 2 ⊢ ≤ = (le‘𝐾) | |
2 | dia2dimlem11.j | . 2 ⊢ ∨ = (join‘𝐾) | |
3 | dia2dimlem11.m | . 2 ⊢ ∧ = (meet‘𝐾) | |
4 | dia2dimlem11.a | . 2 ⊢ 𝐴 = (Atoms‘𝐾) | |
5 | dia2dimlem11.h | . 2 ⊢ 𝐻 = (LHyp‘𝐾) | |
6 | dia2dimlem11.t | . 2 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) | |
7 | dia2dimlem11.r | . 2 ⊢ 𝑅 = ((trL‘𝐾)‘𝑊) | |
8 | dia2dimlem11.y | . 2 ⊢ 𝑌 = ((DVecA‘𝐾)‘𝑊) | |
9 | dia2dimlem11.s | . 2 ⊢ 𝑆 = (LSubSp‘𝑌) | |
10 | dia2dimlem11.pl | . 2 ⊢ ⊕ = (LSSum‘𝑌) | |
11 | dia2dimlem11.n | . 2 ⊢ 𝑁 = (LSpan‘𝑌) | |
12 | dia2dimlem11.i | . 2 ⊢ 𝐼 = ((DIsoA‘𝐾)‘𝑊) | |
13 | dia2dimlem11.k | . 2 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
14 | dia2dimlem11.u | . 2 ⊢ (𝜑 → (𝑈 ∈ 𝐴 ∧ 𝑈 ≤ 𝑊)) | |
15 | dia2dimlem11.v | . 2 ⊢ (𝜑 → (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) | |
16 | dia2dimlem11.f | . 2 ⊢ (𝜑 → 𝐹 ∈ 𝑇) | |
17 | dia2dimlem11.fe | . . 3 ⊢ (𝜑 → 𝐹 ∈ (𝐼‘(𝑈 ∨ 𝑉))) | |
18 | 1, 2, 4, 5, 6, 7, 8, 9, 11, 12, 13, 14, 15, 16, 17 | dia2dimlem10 39747 | . 2 ⊢ (𝜑 → (𝑅‘𝐹) ≤ (𝑈 ∨ 𝑉)) |
19 | dia2dimlem11.uv | . 2 ⊢ (𝜑 → 𝑈 ≠ 𝑉) | |
20 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 18, 19 | dia2dimlem9 39746 | 1 ⊢ (𝜑 → 𝐹 ∈ ((𝐼‘𝑈) ⊕ (𝐼‘𝑉))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ≠ wne 2939 class class class wbr 5141 ‘cfv 6532 (class class class)co 7393 lecple 17186 joincjn 18246 meetcmee 18247 LSSumclsm 19466 LSubSpclss 20491 LSpanclspn 20531 Atomscatm 37936 HLchlt 38023 LHypclh 38658 LTrncltrn 38775 trLctrl 38832 DVecAcdveca 39676 DIsoAcdia 39702 |
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 2702 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7708 ax-cnex 11148 ax-resscn 11149 ax-1cn 11150 ax-icn 11151 ax-addcl 11152 ax-addrcl 11153 ax-mulcl 11154 ax-mulrcl 11155 ax-mulcom 11156 ax-addass 11157 ax-mulass 11158 ax-distr 11159 ax-i2m1 11160 ax-1ne0 11161 ax-1rid 11162 ax-rnegex 11163 ax-rrecex 11164 ax-cnre 11165 ax-pre-lttri 11166 ax-pre-lttrn 11167 ax-pre-ltadd 11168 ax-pre-mulgt0 11169 ax-riotaBAD 37626 |
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 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4523 df-pw 4598 df-sn 4623 df-pr 4625 df-tp 4627 df-op 4629 df-uni 4902 df-int 4944 df-iun 4992 df-iin 4993 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6289 df-ord 6356 df-on 6357 df-lim 6358 df-suc 6359 df-iota 6484 df-fun 6534 df-fn 6535 df-f 6536 df-f1 6537 df-fo 6538 df-f1o 6539 df-fv 6540 df-riota 7349 df-ov 7396 df-oprab 7397 df-mpo 7398 df-om 7839 df-1st 7957 df-2nd 7958 df-tpos 8193 df-undef 8240 df-frecs 8248 df-wrecs 8279 df-recs 8353 df-rdg 8392 df-1o 8448 df-er 8686 df-map 8805 df-en 8923 df-dom 8924 df-sdom 8925 df-fin 8926 df-pnf 11232 df-mnf 11233 df-xr 11234 df-ltxr 11235 df-le 11236 df-sub 11428 df-neg 11429 df-nn 12195 df-2 12257 df-3 12258 df-4 12259 df-5 12260 df-6 12261 df-n0 12455 df-z 12541 df-uz 12805 df-fz 13467 df-struct 17062 df-sets 17079 df-slot 17097 df-ndx 17109 df-base 17127 df-ress 17156 df-plusg 17192 df-mulr 17193 df-sca 17195 df-vsca 17196 df-0g 17369 df-proset 18230 df-poset 18248 df-plt 18265 df-lub 18281 df-glb 18282 df-join 18283 df-meet 18284 df-p0 18360 df-p1 18361 df-lat 18367 df-clat 18434 df-mgm 18543 df-sgrp 18592 df-mnd 18603 df-submnd 18648 df-grp 18797 df-minusg 18798 df-sbg 18799 df-subg 18975 df-cntz 19147 df-lsm 19468 df-cmn 19614 df-abl 19615 df-mgp 19947 df-ur 19964 df-ring 20016 df-oppr 20102 df-dvdsr 20123 df-unit 20124 df-invr 20154 df-dvr 20165 df-drng 20267 df-lmod 20422 df-lss 20492 df-lsp 20532 df-lvec 20663 df-oposet 37849 df-ol 37851 df-oml 37852 df-covers 37939 df-ats 37940 df-atl 37971 df-cvlat 37995 df-hlat 38024 df-llines 38172 df-lplanes 38173 df-lvols 38174 df-lines 38175 df-psubsp 38177 df-pmap 38178 df-padd 38470 df-lhyp 38662 df-laut 38663 df-ldil 38778 df-ltrn 38779 df-trl 38833 df-tgrp 39417 df-tendo 39429 df-edring 39431 df-dveca 39677 df-disoa 39703 |
This theorem is referenced by: dia2dimlem12 39749 |
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