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Theorem lspindp2 21128

Description: Alternate way to say 3 vectors are mutually independent (rotate right). (Contributed by NM, 12-Apr-2015.)

**Hypotheses**
Ref	Expression
lspindp1.v	⊢ 𝑉 = (Base‘𝑊)
lspindp1.o	⊢ 0 = (0_g‘𝑊)
lspindp1.n	⊢ 𝑁 = (LSpan‘𝑊)
lspindp1.w	⊢ (𝜑 → 𝑊 ∈ LVec)
lspindp2.x	⊢ (𝜑 → 𝑋 ∈ 𝑉)
lspindp2.y	⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 }))
lspindp2.z	⊢ (𝜑 → 𝑍 ∈ 𝑉)
lspindp2.q	⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}))
lspindp2.e	⊢ (𝜑 → ¬ 𝑍 ∈ (𝑁‘{𝑋, 𝑌}))

**Assertion**
Ref	Expression
lspindp2	⊢ (𝜑 → ((𝑁‘{𝑍}) ≠ (𝑁‘{𝑋}) ∧ ¬ 𝑌 ∈ (𝑁‘{𝑍, 𝑋})))

**Proof of Theorem lspindp2**
Step	Hyp	Ref	Expression
1		lspindp1.v	. 2 ⊢ 𝑉 = (Base‘𝑊)
2		lspindp1.o	. 2 ⊢ 0 = (0_g‘𝑊)
3		lspindp1.n	. 2 ⊢ 𝑁 = (LSpan‘𝑊)
4		lspindp1.w	. 2 ⊢ (𝜑 → 𝑊 ∈ LVec)
5		lspindp2.y	. 2 ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 }))
6		lspindp2.x	. 2 ⊢ (𝜑 → 𝑋 ∈ 𝑉)
7		lspindp2.z	. 2 ⊢ (𝜑 → 𝑍 ∈ 𝑉)
8		lspindp2.q	. . 3 ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}))
9	8	necomd 2988	. 2 ⊢ (𝜑 → (𝑁‘{𝑌}) ≠ (𝑁‘{𝑋}))
10		lspindp2.e	. . 3 ⊢ (𝜑 → ¬ 𝑍 ∈ (𝑁‘{𝑋, 𝑌}))
11		prcom 4677	. . . . 5 ⊢ {𝑋, 𝑌} = {𝑌, 𝑋}
12	11	fveq2i 6838	. . . 4 ⊢ (𝑁‘{𝑋, 𝑌}) = (𝑁‘{𝑌, 𝑋})
13	12	eleq2i 2829	. . 3 ⊢ (𝑍 ∈ (𝑁‘{𝑋, 𝑌}) ↔ 𝑍 ∈ (𝑁‘{𝑌, 𝑋}))
14	10, 13	sylnib 328	. 2 ⊢ (𝜑 → ¬ 𝑍 ∈ (𝑁‘{𝑌, 𝑋}))
15	1, 2, 3, 4, 5, 6, 7, 9, 14	lspindp1 21126	1 ⊢ (𝜑 → ((𝑁‘{𝑍}) ≠ (𝑁‘{𝑋}) ∧ ¬ 𝑌 ∈ (𝑁‘{𝑍, 𝑋})))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 = wceq 1542 ∈ wcel 2114 ≠ wne 2933 ∖ cdif 3887 {csn 4568 {cpr 4570 ‘cfv 6493 Basecbs 17173 0_gc0g 17396 LSpanclspn 20960 LVecclvec 21092

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5213 ax-sep 5232 ax-nul 5242 ax-pow 5303 ax-pr 5371 ax-un 7683 ax-cnex 11088 ax-resscn 11089 ax-1cn 11090 ax-icn 11091 ax-addcl 11092 ax-addrcl 11093 ax-mulcl 11094 ax-mulrcl 11095 ax-mulcom 11096 ax-addass 11097 ax-mulass 11098 ax-distr 11099 ax-i2m1 11100 ax-1ne0 11101 ax-1rid 11102 ax-rnegex 11103 ax-rrecex 11104 ax-cnre 11105 ax-pre-lttri 11106 ax-pre-lttrn 11107 ax-pre-ltadd 11108 ax-pre-mulgt0 11109

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rmo 3343 df-reu 3344 df-rab 3391 df-v 3432 df-sbc 3730 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-pss 3910 df-nul 4275 df-if 4468 df-pw 4544 df-sn 4569 df-pr 4571 df-op 4575 df-uni 4852 df-int 4891 df-iun 4936 df-br 5087 df-opab 5149 df-mpt 5168 df-tr 5194 df-id 5520 df-eprel 5525 df-po 5533 df-so 5534 df-fr 5578 df-we 5580 df-xp 5631 df-rel 5632 df-cnv 5633 df-co 5634 df-dm 5635 df-rn 5636 df-res 5637 df-ima 5638 df-pred 6260 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-riota 7318 df-ov 7364 df-oprab 7365 df-mpo 7366 df-om 7812 df-1st 7936 df-2nd 7937 df-tpos 8170 df-frecs 8225 df-wrecs 8256 df-recs 8305 df-rdg 8343 df-er 8637 df-en 8888 df-dom 8889 df-sdom 8890 df-pnf 11175 df-mnf 11176 df-xr 11177 df-ltxr 11178 df-le 11179 df-sub 11373 df-neg 11374 df-nn 12169 df-2 12238 df-3 12239 df-sets 17128 df-slot 17146 df-ndx 17158 df-base 17174 df-ress 17195 df-plusg 17227 df-mulr 17228 df-0g 17398 df-mgm 18602 df-sgrp 18681 df-mnd 18697 df-submnd 18746 df-grp 18906 df-minusg 18907 df-sbg 18908 df-subg 19093 df-cntz 19286 df-lsm 19605 df-cmn 19751 df-abl 19752 df-mgp 20116 df-rng 20128 df-ur 20157 df-ring 20210 df-oppr 20311 df-dvdsr 20331 df-unit 20332 df-invr 20362 df-drng 20702 df-lmod 20851 df-lss 20921 df-lsp 20961 df-lvec 21093

This theorem is referenced by: mapdheq4lem 42194 mapdheq4 42195 mapdh6lem1N 42196 mapdh6lem2N 42197 mapdh6aN 42198 hdmap1l6lem1 42270 hdmap1l6lem2 42271 hdmap1l6a 42272

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