lspsnne1 - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > lspsnne1		Structured version Visualization version GIF version

Theorem lspsnne1 20379

Description: Two ways to express that vectors have different spans. (Contributed by NM, 28-May-2015.)

**Hypotheses**
Ref	Expression
lspsnne1.v	⊢ 𝑉 = (Base‘𝑊)
lspsnne1.o	⊢ 0 = (0_g‘𝑊)
lspsnne1.n	⊢ 𝑁 = (LSpan‘𝑊)
lspsnne1.w	⊢ (𝜑 → 𝑊 ∈ LVec)
lspsnne1.x	⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))
lspsnne1.y	⊢ (𝜑 → 𝑌 ∈ 𝑉)
lspsnne1.e	⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}))

**Assertion**
Ref	Expression
lspsnne1	⊢ (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌}))

**Proof of Theorem lspsnne1**
Step	Hyp	Ref	Expression
1		lspsnne1.e	. 2 ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}))
2		lspsnne1.v	. . . . 5 ⊢ 𝑉 = (Base‘𝑊)
3		eqid 2738	. . . . 5 ⊢ (LSubSp‘𝑊) = (LSubSp‘𝑊)
4		lspsnne1.n	. . . . 5 ⊢ 𝑁 = (LSpan‘𝑊)
5		lspsnne1.w	. . . . . 6 ⊢ (𝜑 → 𝑊 ∈ LVec)
6		lveclmod 20368	. . . . . 6 ⊢ (𝑊 ∈ LVec → 𝑊 ∈ LMod)
7	5, 6	syl 17	. . . . 5 ⊢ (𝜑 → 𝑊 ∈ LMod)
8		lspsnne1.y	. . . . . 6 ⊢ (𝜑 → 𝑌 ∈ 𝑉)
9	2, 3, 4	lspsncl 20239	. . . . . 6 ⊢ ((𝑊 ∈ LMod ∧ 𝑌 ∈ 𝑉) → (𝑁‘{𝑌}) ∈ (LSubSp‘𝑊))
10	7, 8, 9	syl2anc 584	. . . . 5 ⊢ (𝜑 → (𝑁‘{𝑌}) ∈ (LSubSp‘𝑊))
11		lspsnne1.x	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))
12	11	eldifad 3899	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝑉)
13	2, 3, 4, 7, 10, 12	lspsnel5 20257	. . . 4 ⊢ (𝜑 → (𝑋 ∈ (𝑁‘{𝑌}) ↔ (𝑁‘{𝑋}) ⊆ (𝑁‘{𝑌})))
14	13	notbid 318	. . 3 ⊢ (𝜑 → (¬ 𝑋 ∈ (𝑁‘{𝑌}) ↔ ¬ (𝑁‘{𝑋}) ⊆ (𝑁‘{𝑌})))
15		lspsnne1.o	. . . . 5 ⊢ 0 = (0_g‘𝑊)
16	2, 15, 4, 5, 11, 8	lspsncmp 20378	. . . 4 ⊢ (𝜑 → ((𝑁‘{𝑋}) ⊆ (𝑁‘{𝑌}) ↔ (𝑁‘{𝑋}) = (𝑁‘{𝑌})))
17	16	necon3bbid 2981	. . 3 ⊢ (𝜑 → (¬ (𝑁‘{𝑋}) ⊆ (𝑁‘{𝑌}) ↔ (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌})))
18	14, 17	bitrd 278	. 2 ⊢ (𝜑 → (¬ 𝑋 ∈ (𝑁‘{𝑌}) ↔ (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌})))
19	1, 18	mpbird 256	1 ⊢ (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌}))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 = wceq 1539 ∈ wcel 2106 ≠ wne 2943 ∖ cdif 3884 ⊆ wss 3887 {csn 4561 ‘cfv 6433 Basecbs 16912 0_gc0g 17150 LModclmod 20123 LSubSpclss 20193 LSpanclspn 20233 LVecclvec 20364

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 5209 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-cnex 10927 ax-resscn 10928 ax-1cn 10929 ax-icn 10930 ax-addcl 10931 ax-addrcl 10932 ax-mulcl 10933 ax-mulrcl 10934 ax-mulcom 10935 ax-addass 10936 ax-mulass 10937 ax-distr 10938 ax-i2m1 10939 ax-1ne0 10940 ax-1rid 10941 ax-rnegex 10942 ax-rrecex 10943 ax-cnre 10944 ax-pre-lttri 10945 ax-pre-lttrn 10946 ax-pre-ltadd 10947 ax-pre-mulgt0 10948

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-rmo 3071 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-int 4880 df-iun 4926 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6202 df-ord 6269 df-on 6270 df-lim 6271 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-riota 7232 df-ov 7278 df-oprab 7279 df-mpo 7280 df-om 7713 df-1st 7831 df-2nd 7832 df-tpos 8042 df-frecs 8097 df-wrecs 8128 df-recs 8202 df-rdg 8241 df-er 8498 df-en 8734 df-dom 8735 df-sdom 8736 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014 df-le 11015 df-sub 11207 df-neg 11208 df-nn 11974 df-2 12036 df-3 12037 df-sets 16865 df-slot 16883 df-ndx 16895 df-base 16913 df-ress 16942 df-plusg 16975 df-mulr 16976 df-0g 17152 df-mgm 18326 df-sgrp 18375 df-mnd 18386 df-grp 18580 df-minusg 18581 df-sbg 18582 df-mgp 19721 df-ur 19738 df-ring 19785 df-oppr 19862 df-dvdsr 19883 df-unit 19884 df-invr 19914 df-drng 19993 df-lmod 20125 df-lss 20194 df-lsp 20234 df-lvec 20365

This theorem is referenced by: lspsnnecom 20381 lsatfixedN 37023 baerlem5amN 39730 baerlem5bmN 39731 baerlem5abmN 39732 mapdh6dN 39753 hdmaplem4 39788 hdmap1l6d 39827 hdmaprnlem3N 39864

W3C validator