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Theorem lspsnne1 19889

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 2821	. . . . 5 ⊢ (LSubSp‘𝑊) = (LSubSp‘𝑊)
4		lspsnne1.n	. . . . 5 ⊢ 𝑁 = (LSpan‘𝑊)
5		lspsnne1.w	. . . . . 6 ⊢ (𝜑 → 𝑊 ∈ LVec)
6		lveclmod 19878	. . . . . 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 19749	. . . . . 6 ⊢ ((𝑊 ∈ LMod ∧ 𝑌 ∈ 𝑉) → (𝑁‘{𝑌}) ∈ (LSubSp‘𝑊))
10	7, 8, 9	syl2anc 586	. . . . 5 ⊢ (𝜑 → (𝑁‘{𝑌}) ∈ (LSubSp‘𝑊))
11		lspsnne1.x	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))
12	11	eldifad 3948	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝑉)
13	2, 3, 4, 7, 10, 12	lspsnel5 19767	. . . 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 19888	. . . 4 ⊢ (𝜑 → ((𝑁‘{𝑋}) ⊆ (𝑁‘{𝑌}) ↔ (𝑁‘{𝑋}) = (𝑁‘{𝑌})))
17	16	necon3bbid 3053	. . 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 1537 ∈ wcel 2114 ≠ wne 3016 ∖ cdif 3933 ⊆ wss 3936 {csn 4567 ‘cfv 6355 Basecbs 16483 0_gc0g 16713 LModclmod 19634 LSubSpclss 19703 LSpanclspn 19743 LVecclvec 19874

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-1st 7689 df-2nd 7690 df-tpos 7892 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-er 8289 df-en 8510 df-dom 8511 df-sdom 8512 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-nn 11639 df-2 11701 df-3 11702 df-ndx 16486 df-slot 16487 df-base 16489 df-sets 16490 df-ress 16491 df-plusg 16578 df-mulr 16579 df-0g 16715 df-mgm 17852 df-sgrp 17901 df-mnd 17912 df-grp 18106 df-minusg 18107 df-sbg 18108 df-mgp 19240 df-ur 19252 df-ring 19299 df-oppr 19373 df-dvdsr 19391 df-unit 19392 df-invr 19422 df-drng 19504 df-lmod 19636 df-lss 19704 df-lsp 19744 df-lvec 19875

This theorem is referenced by: lspsnnecom 19891 lsatfixedN 36160 baerlem5amN 38867 baerlem5bmN 38868 baerlem5abmN 38869 mapdh6dN 38890 hdmaplem4 38925 hdmap1l6d 38964 hdmaprnlem3N 39001

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