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Mirrors > Home > MPE Home > Th. List > Mathboxes > dvhdimlem | Structured version Visualization version GIF version |
Description: Lemma for dvh2dim 38125 and dvh3dim 38126. TODO: make this obsolete and use dvh4dimlem 38123 directly? (Contributed by NM, 24-Apr-2015.) |
Ref | Expression |
---|---|
dvh3dim.h | ⊢ 𝐻 = (LHyp‘𝐾) |
dvh3dim.u | ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) |
dvh3dim.v | ⊢ 𝑉 = (Base‘𝑈) |
dvh3dim.n | ⊢ 𝑁 = (LSpan‘𝑈) |
dvh3dim.k | ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
dvh3dim.x | ⊢ (𝜑 → 𝑋 ∈ 𝑉) |
dvhdim.y | ⊢ (𝜑 → 𝑌 ∈ 𝑉) |
dvhdim.o | ⊢ 0 = (0g‘𝑈) |
dvhdim.x | ⊢ (𝜑 → 𝑋 ≠ 0 ) |
dvhdimlem.y | ⊢ (𝜑 → 𝑌 ≠ 0 ) |
Ref | Expression |
---|---|
dvhdimlem | ⊢ (𝜑 → ∃𝑧 ∈ 𝑉 ¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑌})) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dvh3dim.h | . . 3 ⊢ 𝐻 = (LHyp‘𝐾) | |
2 | dvh3dim.u | . . 3 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) | |
3 | dvh3dim.v | . . 3 ⊢ 𝑉 = (Base‘𝑈) | |
4 | dvh3dim.n | . . 3 ⊢ 𝑁 = (LSpan‘𝑈) | |
5 | dvh3dim.k | . . 3 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
6 | dvh3dim.x | . . 3 ⊢ (𝜑 → 𝑋 ∈ 𝑉) | |
7 | dvhdim.y | . . 3 ⊢ (𝜑 → 𝑌 ∈ 𝑉) | |
8 | dvhdim.o | . . 3 ⊢ 0 = (0g‘𝑈) | |
9 | dvhdim.x | . . 3 ⊢ (𝜑 → 𝑋 ≠ 0 ) | |
10 | dvhdimlem.y | . . 3 ⊢ (𝜑 → 𝑌 ≠ 0 ) | |
11 | 1, 2, 3, 4, 5, 6, 7, 7, 8, 9, 10, 10 | dvh4dimlem 38123 | . 2 ⊢ (𝜑 → ∃𝑧 ∈ 𝑉 ¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑌, 𝑌})) |
12 | 1, 2, 5 | dvhlmod 37790 | . . . . 5 ⊢ (𝜑 → 𝑈 ∈ LMod) |
13 | df-tp 4479 | . . . . . 6 ⊢ {𝑋, 𝑌, 𝑌} = ({𝑋, 𝑌} ∪ {𝑌}) | |
14 | prssi 4663 | . . . . . . . 8 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → {𝑋, 𝑌} ⊆ 𝑉) | |
15 | 6, 7, 14 | syl2anc 584 | . . . . . . 7 ⊢ (𝜑 → {𝑋, 𝑌} ⊆ 𝑉) |
16 | 7 | snssd 4651 | . . . . . . 7 ⊢ (𝜑 → {𝑌} ⊆ 𝑉) |
17 | 15, 16 | unssd 4085 | . . . . . 6 ⊢ (𝜑 → ({𝑋, 𝑌} ∪ {𝑌}) ⊆ 𝑉) |
18 | 13, 17 | eqsstrid 3938 | . . . . 5 ⊢ (𝜑 → {𝑋, 𝑌, 𝑌} ⊆ 𝑉) |
19 | ssun1 4071 | . . . . . . 7 ⊢ {𝑋, 𝑌} ⊆ ({𝑋, 𝑌} ∪ {𝑌}) | |
20 | 19, 13 | sseqtr4i 3927 | . . . . . 6 ⊢ {𝑋, 𝑌} ⊆ {𝑋, 𝑌, 𝑌} |
21 | 20 | a1i 11 | . . . . 5 ⊢ (𝜑 → {𝑋, 𝑌} ⊆ {𝑋, 𝑌, 𝑌}) |
22 | 3, 4 | lspss 19446 | . . . . 5 ⊢ ((𝑈 ∈ LMod ∧ {𝑋, 𝑌, 𝑌} ⊆ 𝑉 ∧ {𝑋, 𝑌} ⊆ {𝑋, 𝑌, 𝑌}) → (𝑁‘{𝑋, 𝑌}) ⊆ (𝑁‘{𝑋, 𝑌, 𝑌})) |
23 | 12, 18, 21, 22 | syl3anc 1364 | . . . 4 ⊢ (𝜑 → (𝑁‘{𝑋, 𝑌}) ⊆ (𝑁‘{𝑋, 𝑌, 𝑌})) |
24 | 23 | ssneld 3893 | . . 3 ⊢ (𝜑 → (¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑌, 𝑌}) → ¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑌}))) |
25 | 24 | reximdv 3235 | . 2 ⊢ (𝜑 → (∃𝑧 ∈ 𝑉 ¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑌, 𝑌}) → ∃𝑧 ∈ 𝑉 ¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑌}))) |
26 | 11, 25 | mpd 15 | 1 ⊢ (𝜑 → ∃𝑧 ∈ 𝑉 ¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑌})) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 396 = wceq 1522 ∈ wcel 2080 ≠ wne 2983 ∃wrex 3105 ∪ cun 3859 ⊆ wss 3861 {csn 4474 {cpr 4476 {ctp 4478 ‘cfv 6228 Basecbs 16312 0gc0g 16542 LModclmod 19324 LSpanclspn 19433 HLchlt 36030 LHypclh 36664 DVecHcdvh 37758 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1778 ax-4 1792 ax-5 1889 ax-6 1948 ax-7 1993 ax-8 2082 ax-9 2090 ax-10 2111 ax-11 2125 ax-12 2140 ax-13 2343 ax-ext 2768 ax-rep 5084 ax-sep 5097 ax-nul 5104 ax-pow 5160 ax-pr 5224 ax-un 7322 ax-cnex 10442 ax-resscn 10443 ax-1cn 10444 ax-icn 10445 ax-addcl 10446 ax-addrcl 10447 ax-mulcl 10448 ax-mulrcl 10449 ax-mulcom 10450 ax-addass 10451 ax-mulass 10452 ax-distr 10453 ax-i2m1 10454 ax-1ne0 10455 ax-1rid 10456 ax-rnegex 10457 ax-rrecex 10458 ax-cnre 10459 ax-pre-lttri 10460 ax-pre-lttrn 10461 ax-pre-ltadd 10462 ax-pre-mulgt0 10463 ax-riotaBAD 35633 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1081 df-3an 1082 df-tru 1525 df-fal 1535 df-ex 1763 df-nf 1767 df-sb 2042 df-mo 2575 df-eu 2611 df-clab 2775 df-cleq 2787 df-clel 2862 df-nfc 2934 df-ne 2984 df-nel 3090 df-ral 3109 df-rex 3110 df-reu 3111 df-rmo 3112 df-rab 3113 df-v 3438 df-sbc 3708 df-csb 3814 df-dif 3864 df-un 3866 df-in 3868 df-ss 3876 df-pss 3878 df-nul 4214 df-if 4384 df-pw 4457 df-sn 4475 df-pr 4477 df-tp 4479 df-op 4481 df-uni 4748 df-int 4785 df-iun 4829 df-iin 4830 df-br 4965 df-opab 5027 df-mpt 5044 df-tr 5067 df-id 5351 df-eprel 5356 df-po 5365 df-so 5366 df-fr 5405 df-we 5407 df-xp 5452 df-rel 5453 df-cnv 5454 df-co 5455 df-dm 5456 df-rn 5457 df-res 5458 df-ima 5459 df-pred 6026 df-ord 6072 df-on 6073 df-lim 6074 df-suc 6075 df-iota 6192 df-fun 6230 df-fn 6231 df-f 6232 df-f1 6233 df-fo 6234 df-f1o 6235 df-fv 6236 df-riota 6980 df-ov 7022 df-oprab 7023 df-mpo 7024 df-om 7440 df-1st 7548 df-2nd 7549 df-tpos 7746 df-undef 7793 df-wrecs 7801 df-recs 7863 df-rdg 7901 df-1o 7956 df-oadd 7960 df-er 8142 df-map 8261 df-en 8361 df-dom 8362 df-sdom 8363 df-fin 8364 df-pnf 10526 df-mnf 10527 df-xr 10528 df-ltxr 10529 df-le 10530 df-sub 10721 df-neg 10722 df-nn 11489 df-2 11550 df-3 11551 df-4 11552 df-5 11553 df-6 11554 df-n0 11748 df-z 11832 df-uz 12094 df-fz 12743 df-struct 16314 df-ndx 16315 df-slot 16316 df-base 16318 df-sets 16319 df-ress 16320 df-plusg 16407 df-mulr 16408 df-sca 16410 df-vsca 16411 df-0g 16544 df-proset 17367 df-poset 17385 df-plt 17397 df-lub 17413 df-glb 17414 df-join 17415 df-meet 17416 df-p0 17478 df-p1 17479 df-lat 17485 df-clat 17547 df-mgm 17681 df-sgrp 17723 df-mnd 17734 df-submnd 17775 df-grp 17864 df-minusg 17865 df-sbg 17866 df-subg 18030 df-cntz 18188 df-lsm 18491 df-cmn 18635 df-abl 18636 df-mgp 18930 df-ur 18942 df-ring 18989 df-oppr 19063 df-dvdsr 19081 df-unit 19082 df-invr 19112 df-dvr 19123 df-drng 19194 df-lmod 19326 df-lss 19394 df-lsp 19434 df-lvec 19565 df-lsatoms 35656 df-oposet 35856 df-ol 35858 df-oml 35859 df-covers 35946 df-ats 35947 df-atl 35978 df-cvlat 36002 df-hlat 36031 df-llines 36178 df-lplanes 36179 df-lvols 36180 df-lines 36181 df-psubsp 36183 df-pmap 36184 df-padd 36476 df-lhyp 36668 df-laut 36669 df-ldil 36784 df-ltrn 36785 df-trl 36839 df-tgrp 37423 df-tendo 37435 df-edring 37437 df-dveca 37683 df-disoa 37709 df-dvech 37759 df-dib 37819 df-dic 37853 df-dih 37909 df-doch 38028 df-djh 38075 |
This theorem is referenced by: dvh2dim 38125 dvh3dim 38126 |
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