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 21175

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 2761	. . . . 5 ⊢ (LSubSp‘𝑊) = (LSubSp‘𝑊)
4		lspsnne1.n	. . . . 5 ⊢ 𝑁 = (LSpan‘𝑊)
5		lspsnne1.w	. . . . . 6 ⊢ (𝜑 → 𝑊 ∈ LVec)
6		lveclmod 21161	. . . . . 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 21032	. . . . . 6 ⊢ ((𝑊 ∈ LMod ∧ 𝑌 ∈ 𝑉) → (𝑁‘{𝑌}) ∈ (LSubSp‘𝑊))
10	7, 8, 9	syl2anc 593	. . . . 5 ⊢ (𝜑 → (𝑁‘{𝑌}) ∈ (LSubSp‘𝑊))
11		lspsnne1.x	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))
12	11	eldifad 3914	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝑉)
13	2, 3, 4, 7, 10, 12	ellspsn5b 21050	. . . 4 ⊢ (𝜑 → (𝑋 ∈ (𝑁‘{𝑌}) ↔ (𝑁‘{𝑋}) ⊆ (𝑁‘{𝑌})))
14	13	notbid 320	. . 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 21174	. . . 4 ⊢ (𝜑 → ((𝑁‘{𝑋}) ⊆ (𝑁‘{𝑌}) ↔ (𝑁‘{𝑋}) = (𝑁‘{𝑌})))
17	16	necon3bbid 2993	. . 3 ⊢ (𝜑 → (¬ (𝑁‘{𝑋}) ⊆ (𝑁‘{𝑌}) ↔ (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌})))
18	14, 17	bitrd 281	. 2 ⊢ (𝜑 → (¬ 𝑋 ∈ (𝑁‘{𝑌}) ↔ (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌})))
19	1, 18	mpbird 259	1 ⊢ (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌}))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 = wceq 1559 ∈ wcel 2141 ≠ wne 2956 ∖ cdif 3899 ⊆ wss 3902 {csn 4579 ‘cfv 6516 Basecbs 17236 0_gc0g 17459 LModclmod 20915 LSubSpclss 20986 LSpanclspn 21026 LVecclvec 21157

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1814 ax-4 1828 ax-5 1929 ax-6 1986 ax-7 2027 ax-8 2143 ax-9 2151 ax-10 2174 ax-11 2190 ax-12 2211 ax-ext 2733 ax-rep 5224 ax-sep 5243 ax-nul 5253 ax-pow 5319 ax-pr 5387 ax-un 7713 ax-cnex 11123 ax-resscn 11124 ax-1cn 11125 ax-icn 11126 ax-addcl 11127 ax-addrcl 11128 ax-mulcl 11129 ax-mulrcl 11130 ax-mulcom 11131 ax-addass 11132 ax-mulass 11133 ax-distr 11134 ax-i2m1 11135 ax-1ne0 11136 ax-1rid 11137 ax-rnegex 11138 ax-rrecex 11139 ax-cnre 11140 ax-pre-lttri 11141 ax-pre-lttrn 11142 ax-pre-ltadd 11143 ax-pre-mulgt0 11144

This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-3or 1098 df-3an 1099 df-tru 1562 df-fal 1572 df-ex 1799 df-nf 1803 df-sb 2090 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-nel 3061 df-ral 3076 df-rex 3086 df-rmo 3366 df-reu 3367 df-rab 3414 df-v 3455 df-sbc 3743 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-pss 3922 df-nul 4284 df-if 4478 df-pw 4554 df-sn 4580 df-pr 4582 df-op 4586 df-uni 4863 df-int 4903 df-iun 4948 df-br 5098 df-opab 5160 df-mpt 5179 df-tr 5205 df-id 5538 df-eprel 5543 df-po 5551 df-so 5552 df-fr 5596 df-we 5598 df-xp 5649 df-rel 5650 df-cnv 5651 df-co 5652 df-dm 5653 df-rn 5654 df-res 5655 df-ima 5656 df-pred 6283 df-ord 6344 df-on 6345 df-lim 6346 df-suc 6347 df-iota 6472 df-fun 6518 df-fn 6519 df-f 6520 df-f1 6521 df-fo 6522 df-f1o 6523 df-fv 6524 df-riota 7348 df-ov 7394 df-oprab 7395 df-mpo 7396 df-om 7842 df-1st 7965 df-2nd 7966 df-tpos 8200 df-frecs 8256 df-wrecs 8287 df-recs 8336 df-rdg 8375 df-er 8672 df-en 8922 df-dom 8923 df-sdom 8924 df-pnf 11212 df-mnf 11213 df-xr 11214 df-ltxr 11215 df-le 11216 df-sub 11410 df-neg 11411 df-nn 12205 df-2 12274 df-3 12275 df-sets 17191 df-slot 17209 df-ndx 17221 df-base 17237 df-ress 17258 df-plusg 17290 df-mulr 17291 df-0g 17461 df-mgm 18665 df-sgrp 18744 df-mnd 18760 df-grp 18969 df-minusg 18970 df-sbg 18971 df-cmn 19813 df-abl 19814 df-mgp 20178 df-rng 20190 df-ur 20219 df-ring 20272 df-oppr 20373 df-dvdsr 20393 df-unit 20394 df-invr 20424 df-drng 20768 df-lmod 20917 df-lss 20987 df-lsp 21027 df-lvec 21158

This theorem is referenced by: lspsnnecom 21177 lsatfixedN 39594 baerlem5amN 42301 baerlem5bmN 42302 baerlem5abmN 42303 mapdh6dN 42324 hdmaplem4 42359 hdmap1l6d 42398 hdmaprnlem3N 42435

W3C validator