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Mirrors > Home > MPE Home > Th. List > Mathboxes > dvh2dimatN | Structured version Visualization version GIF version |
Description: Given an atom, there exists another. (Contributed by NM, 25-Apr-2015.) (New usage is discouraged.) |
Ref | Expression |
---|---|
dvh4dimat.h | ⊢ 𝐻 = (LHyp‘𝐾) |
dvh4dimat.u | ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) |
dvh2dimat.a | ⊢ 𝐴 = (LSAtoms‘𝑈) |
dvh2dimat.k | ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
dvh2dimat.p | ⊢ (𝜑 → 𝑃 ∈ 𝐴) |
Ref | Expression |
---|---|
dvh2dimatN | ⊢ (𝜑 → ∃𝑠 ∈ 𝐴 𝑠 ≠ 𝑃) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dvh4dimat.h | . . 3 ⊢ 𝐻 = (LHyp‘𝐾) | |
2 | dvh4dimat.u | . . 3 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) | |
3 | eqid 2738 | . . 3 ⊢ (LSSum‘𝑈) = (LSSum‘𝑈) | |
4 | dvh2dimat.a | . . 3 ⊢ 𝐴 = (LSAtoms‘𝑈) | |
5 | dvh2dimat.k | . . 3 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
6 | dvh2dimat.p | . . 3 ⊢ (𝜑 → 𝑃 ∈ 𝐴) | |
7 | 1, 2, 3, 4, 5, 6, 6 | dvh3dimatN 39461 | . 2 ⊢ (𝜑 → ∃𝑠 ∈ 𝐴 ¬ 𝑠 ⊆ (𝑃(LSSum‘𝑈)𝑃)) |
8 | 1, 2, 5 | dvhlmod 39132 | . . . . . . . . 9 ⊢ (𝜑 → 𝑈 ∈ LMod) |
9 | eqid 2738 | . . . . . . . . . 10 ⊢ (LSubSp‘𝑈) = (LSubSp‘𝑈) | |
10 | 9, 4, 8, 6 | lsatlssel 37019 | . . . . . . . . 9 ⊢ (𝜑 → 𝑃 ∈ (LSubSp‘𝑈)) |
11 | 9 | lsssubg 20229 | . . . . . . . . 9 ⊢ ((𝑈 ∈ LMod ∧ 𝑃 ∈ (LSubSp‘𝑈)) → 𝑃 ∈ (SubGrp‘𝑈)) |
12 | 8, 10, 11 | syl2anc 584 | . . . . . . . 8 ⊢ (𝜑 → 𝑃 ∈ (SubGrp‘𝑈)) |
13 | 3 | lsmidm 19278 | . . . . . . . 8 ⊢ (𝑃 ∈ (SubGrp‘𝑈) → (𝑃(LSSum‘𝑈)𝑃) = 𝑃) |
14 | 12, 13 | syl 17 | . . . . . . 7 ⊢ (𝜑 → (𝑃(LSSum‘𝑈)𝑃) = 𝑃) |
15 | 14 | sseq2d 3952 | . . . . . 6 ⊢ (𝜑 → (𝑠 ⊆ (𝑃(LSSum‘𝑈)𝑃) ↔ 𝑠 ⊆ 𝑃)) |
16 | 15 | adantr 481 | . . . . 5 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐴) → (𝑠 ⊆ (𝑃(LSSum‘𝑈)𝑃) ↔ 𝑠 ⊆ 𝑃)) |
17 | 1, 2, 5 | dvhlvec 39131 | . . . . . . 7 ⊢ (𝜑 → 𝑈 ∈ LVec) |
18 | 17 | adantr 481 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐴) → 𝑈 ∈ LVec) |
19 | simpr 485 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐴) → 𝑠 ∈ 𝐴) | |
20 | 6 | adantr 481 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐴) → 𝑃 ∈ 𝐴) |
21 | 4, 18, 19, 20 | lsatcmp 37025 | . . . . 5 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐴) → (𝑠 ⊆ 𝑃 ↔ 𝑠 = 𝑃)) |
22 | 16, 21 | bitrd 278 | . . . 4 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐴) → (𝑠 ⊆ (𝑃(LSSum‘𝑈)𝑃) ↔ 𝑠 = 𝑃)) |
23 | 22 | necon3bbid 2981 | . . 3 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐴) → (¬ 𝑠 ⊆ (𝑃(LSSum‘𝑈)𝑃) ↔ 𝑠 ≠ 𝑃)) |
24 | 23 | rexbidva 3223 | . 2 ⊢ (𝜑 → (∃𝑠 ∈ 𝐴 ¬ 𝑠 ⊆ (𝑃(LSSum‘𝑈)𝑃) ↔ ∃𝑠 ∈ 𝐴 𝑠 ≠ 𝑃)) |
25 | 7, 24 | mpbid 231 | 1 ⊢ (𝜑 → ∃𝑠 ∈ 𝐴 𝑠 ≠ 𝑃) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1539 ∈ wcel 2106 ≠ wne 2943 ∃wrex 3065 ⊆ wss 3886 ‘cfv 6426 (class class class)co 7267 SubGrpcsubg 18759 LSSumclsm 19249 LModclmod 20133 LSubSpclss 20203 LVecclvec 20374 LSAtomsclsa 36996 HLchlt 37372 LHypclh 38006 DVecHcdvh 39100 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 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 2709 ax-rep 5208 ax-sep 5221 ax-nul 5228 ax-pow 5286 ax-pr 5350 ax-un 7578 ax-cnex 10937 ax-resscn 10938 ax-1cn 10939 ax-icn 10940 ax-addcl 10941 ax-addrcl 10942 ax-mulcl 10943 ax-mulrcl 10944 ax-mulcom 10945 ax-addass 10946 ax-mulass 10947 ax-distr 10948 ax-i2m1 10949 ax-1ne0 10950 ax-1rid 10951 ax-rnegex 10952 ax-rrecex 10953 ax-cnre 10954 ax-pre-lttri 10955 ax-pre-lttrn 10956 ax-pre-ltadd 10957 ax-pre-mulgt0 10958 ax-riotaBAD 36975 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-reu 3071 df-rmo 3072 df-rab 3073 df-v 3431 df-sbc 3716 df-csb 3832 df-dif 3889 df-un 3891 df-in 3893 df-ss 3903 df-pss 3905 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4840 df-int 4880 df-iun 4926 df-iin 4927 df-br 5074 df-opab 5136 df-mpt 5157 df-tr 5191 df-id 5484 df-eprel 5490 df-po 5498 df-so 5499 df-fr 5539 df-we 5541 df-xp 5590 df-rel 5591 df-cnv 5592 df-co 5593 df-dm 5594 df-rn 5595 df-res 5596 df-ima 5597 df-pred 6195 df-ord 6262 df-on 6263 df-lim 6264 df-suc 6265 df-iota 6384 df-fun 6428 df-fn 6429 df-f 6430 df-f1 6431 df-fo 6432 df-f1o 6433 df-fv 6434 df-riota 7224 df-ov 7270 df-oprab 7271 df-mpo 7272 df-om 7703 df-1st 7820 df-2nd 7821 df-tpos 8029 df-undef 8076 df-frecs 8084 df-wrecs 8115 df-recs 8189 df-rdg 8228 df-1o 8284 df-er 8485 df-map 8604 df-en 8721 df-dom 8722 df-sdom 8723 df-fin 8724 df-pnf 11021 df-mnf 11022 df-xr 11023 df-ltxr 11024 df-le 11025 df-sub 11217 df-neg 11218 df-nn 11984 df-2 12046 df-3 12047 df-4 12048 df-5 12049 df-6 12050 df-n0 12244 df-z 12330 df-uz 12593 df-fz 13250 df-struct 16858 df-sets 16875 df-slot 16893 df-ndx 16905 df-base 16923 df-ress 16952 df-plusg 16985 df-mulr 16986 df-sca 16988 df-vsca 16989 df-0g 17162 df-proset 18023 df-poset 18041 df-plt 18058 df-lub 18074 df-glb 18075 df-join 18076 df-meet 18077 df-p0 18153 df-p1 18154 df-lat 18160 df-clat 18227 df-mgm 18336 df-sgrp 18385 df-mnd 18396 df-submnd 18441 df-grp 18590 df-minusg 18591 df-sbg 18592 df-subg 18762 df-cntz 18933 df-lsm 19251 df-cmn 19398 df-abl 19399 df-mgp 19731 df-ur 19748 df-ring 19795 df-oppr 19872 df-dvdsr 19893 df-unit 19894 df-invr 19924 df-dvr 19935 df-drng 20003 df-lmod 20135 df-lss 20204 df-lsp 20244 df-lvec 20375 df-lsatoms 36998 df-oposet 37198 df-ol 37200 df-oml 37201 df-covers 37288 df-ats 37289 df-atl 37320 df-cvlat 37344 df-hlat 37373 df-llines 37520 df-lplanes 37521 df-lvols 37522 df-lines 37523 df-psubsp 37525 df-pmap 37526 df-padd 37818 df-lhyp 38010 df-laut 38011 df-ldil 38126 df-ltrn 38127 df-trl 38181 df-tgrp 38765 df-tendo 38777 df-edring 38779 df-dveca 39025 df-disoa 39051 df-dvech 39101 df-dib 39161 df-dic 39195 df-dih 39251 df-doch 39370 df-djh 39417 |
This theorem is referenced by: (None) |
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