mapdpglem29 - Mathbox for Norm Megill

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Theorem mapdpglem29 41701

Description: Lemma for mapdpg 41707. Baer p. 45 line 16: "But Gx' and Gy'' are distinct points and so x' and y'' are independent elements in B. (Contributed by NM, 22-Mar-2015.)

**Hypotheses**
Ref	Expression
mapdpg.h	⊢ 𝐻 = (LHyp‘𝐾)
mapdpg.m	⊢ 𝑀 = ((mapd‘𝐾)‘𝑊)
mapdpg.u	⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
mapdpg.v	⊢ 𝑉 = (Base‘𝑈)
mapdpg.s	⊢ − = (-_g‘𝑈)
mapdpg.z	⊢ 0 = (0_g‘𝑈)
mapdpg.n	⊢ 𝑁 = (LSpan‘𝑈)
mapdpg.c	⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊)
mapdpg.f	⊢ 𝐹 = (Base‘𝐶)
mapdpg.r	⊢ 𝑅 = (-_g‘𝐶)
mapdpg.j	⊢ 𝐽 = (LSpan‘𝐶)
mapdpg.k	⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
mapdpg.x	⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))
mapdpg.y	⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 }))
mapdpg.g	⊢ (𝜑 → 𝐺 ∈ 𝐹)
mapdpg.ne	⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}))
mapdpg.e	⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐺}))
mapdpgem25.h1	⊢ (𝜑 → (ℎ ∈ 𝐹 ∧ ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐺𝑅ℎ)}))))
mapdpgem25.i1	⊢ (𝜑 → (𝑖 ∈ 𝐹 ∧ ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{𝑖}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐺𝑅𝑖)}))))
mapdpglem26.a	⊢ 𝐴 = (Scalar‘𝑈)
mapdpglem26.b	⊢ 𝐵 = (Base‘𝐴)
mapdpglem26.t	⊢ · = ( ·_𝑠 ‘𝐶)
mapdpglem26.o	⊢ 𝑂 = (0_g‘𝐴)
mapdpglem28.ve	⊢ (𝜑 → 𝑣 ∈ 𝐵)
mapdpglem28.u1	⊢ (𝜑 → ℎ = (𝑢 · 𝑖))
mapdpglem28.u2	⊢ (𝜑 → (𝐺𝑅ℎ) = (𝑣 · (𝐺𝑅𝑖)))

**Assertion**
Ref	Expression
mapdpglem29	⊢ (𝜑 → (𝐽‘{𝐺}) ≠ (𝐽‘{𝑖}))

Distinct variable groups: ℎ,𝑖,𝑢,𝑣 𝑢,𝐵,𝑣 𝑢,𝐶,𝑣 𝑢,𝑂,𝑣 𝑢, · ,𝑣 𝑣,𝐺 𝑣,𝑅 Allowed substitution hints: 𝜑(𝑣,𝑢,ℎ,𝑖) 𝐴(𝑣,𝑢,ℎ,𝑖) 𝐵(ℎ,𝑖) 𝐶(ℎ,𝑖) 𝑅(𝑢,ℎ,𝑖) · (ℎ,𝑖) 𝑈(𝑣,𝑢,ℎ,𝑖) 𝐹(𝑣,𝑢,ℎ,𝑖) 𝐺(𝑢,ℎ,𝑖) 𝐻(𝑣,𝑢,ℎ,𝑖) 𝐽(𝑣,𝑢,ℎ,𝑖) 𝐾(𝑣,𝑢,ℎ,𝑖) 𝑀(𝑣,𝑢,ℎ,𝑖) − (𝑣,𝑢,ℎ,𝑖) 𝑁(𝑣,𝑢,ℎ,𝑖) 𝑂(ℎ,𝑖) 𝑉(𝑣,𝑢,ℎ,𝑖) 𝑊(𝑣,𝑢,ℎ,𝑖) 𝑋(𝑣,𝑢,ℎ,𝑖) 𝑌(𝑣,𝑢,ℎ,𝑖) 0 (𝑣,𝑢,ℎ,𝑖)

**Proof of Theorem mapdpglem29**
Step	Hyp	Ref	Expression
1		mapdpg.ne	. . 3 ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}))
2		mapdpg.h	. . . . 5 ⊢ 𝐻 = (LHyp‘𝐾)
3		mapdpg.u	. . . . 5 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
4		eqid 2730	. . . . 5 ⊢ (LSubSp‘𝑈) = (LSubSp‘𝑈)
5		mapdpg.m	. . . . 5 ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊)
6		mapdpg.k	. . . . 5 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
7	2, 3, 6	dvhlmod 41111	. . . . . 6 ⊢ (𝜑 → 𝑈 ∈ LMod)
8		mapdpg.x	. . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))
9	8	eldifad 3929	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ 𝑉)
10		mapdpg.v	. . . . . . 7 ⊢ 𝑉 = (Base‘𝑈)
11		mapdpg.n	. . . . . . 7 ⊢ 𝑁 = (LSpan‘𝑈)
12	10, 4, 11	lspsncl 20890	. . . . . 6 ⊢ ((𝑈 ∈ LMod ∧ 𝑋 ∈ 𝑉) → (𝑁‘{𝑋}) ∈ (LSubSp‘𝑈))
13	7, 9, 12	syl2anc 584	. . . . 5 ⊢ (𝜑 → (𝑁‘{𝑋}) ∈ (LSubSp‘𝑈))
14		mapdpg.y	. . . . . . 7 ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 }))
15	14	eldifad 3929	. . . . . 6 ⊢ (𝜑 → 𝑌 ∈ 𝑉)
16	10, 4, 11	lspsncl 20890	. . . . . 6 ⊢ ((𝑈 ∈ LMod ∧ 𝑌 ∈ 𝑉) → (𝑁‘{𝑌}) ∈ (LSubSp‘𝑈))
17	7, 15, 16	syl2anc 584	. . . . 5 ⊢ (𝜑 → (𝑁‘{𝑌}) ∈ (LSubSp‘𝑈))
18	2, 3, 4, 5, 6, 13, 17	mapd11 41640	. . . 4 ⊢ (𝜑 → ((𝑀‘(𝑁‘{𝑋})) = (𝑀‘(𝑁‘{𝑌})) ↔ (𝑁‘{𝑋}) = (𝑁‘{𝑌})))
19	18	necon3bid 2970	. . 3 ⊢ (𝜑 → ((𝑀‘(𝑁‘{𝑋})) ≠ (𝑀‘(𝑁‘{𝑌})) ↔ (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌})))
20	1, 19	mpbird 257	. 2 ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) ≠ (𝑀‘(𝑁‘{𝑌})))
21		mapdpg.e	. 2 ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐺}))
22		mapdpgem25.i1	. . . 4 ⊢ (𝜑 → (𝑖 ∈ 𝐹 ∧ ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{𝑖}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐺𝑅𝑖)}))))
23	22	simprd 495	. . 3 ⊢ (𝜑 → ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{𝑖}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐺𝑅𝑖)})))
24	23	simpld 494	. 2 ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑌})) = (𝐽‘{𝑖}))
25	20, 21, 24	3netr3d 3002	1 ⊢ (𝜑 → (𝐽‘{𝐺}) ≠ (𝐽‘{𝑖}))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2109 ≠ wne 2926 ∖ cdif 3914 {csn 4592 ‘cfv 6514 (class class class)co 7390 Basecbs 17186 Scalarcsca 17230 ·_𝑠 cvsca 17231 0_gc0g 17409 -_gcsg 18874 LModclmod 20773 LSubSpclss 20844 LSpanclspn 20884 HLchlt 39350 LHypclh 39985 DVecHcdvh 41079 LCDualclcd 41587 mapdcmpd 41625

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2702 ax-rep 5237 ax-sep 5254 ax-nul 5264 ax-pow 5323 ax-pr 5390 ax-un 7714 ax-cnex 11131 ax-resscn 11132 ax-1cn 11133 ax-icn 11134 ax-addcl 11135 ax-addrcl 11136 ax-mulcl 11137 ax-mulrcl 11138 ax-mulcom 11139 ax-addass 11140 ax-mulass 11141 ax-distr 11142 ax-i2m1 11143 ax-1ne0 11144 ax-1rid 11145 ax-rnegex 11146 ax-rrecex 11147 ax-cnre 11148 ax-pre-lttri 11149 ax-pre-lttrn 11150 ax-pre-ltadd 11151 ax-pre-mulgt0 11152 ax-riotaBAD 38953

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2534 df-eu 2563 df-clab 2709 df-cleq 2722 df-clel 2804 df-nfc 2879 df-ne 2927 df-nel 3031 df-ral 3046 df-rex 3055 df-rmo 3356 df-reu 3357 df-rab 3409 df-v 3452 df-sbc 3757 df-csb 3866 df-dif 3920 df-un 3922 df-in 3924 df-ss 3934 df-pss 3937 df-nul 4300 df-if 4492 df-pw 4568 df-sn 4593 df-pr 4595 df-tp 4597 df-op 4599 df-uni 4875 df-int 4914 df-iun 4960 df-iin 4961 df-br 5111 df-opab 5173 df-mpt 5192 df-tr 5218 df-id 5536 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5594 df-we 5596 df-xp 5647 df-rel 5648 df-cnv 5649 df-co 5650 df-dm 5651 df-rn 5652 df-res 5653 df-ima 5654 df-pred 6277 df-ord 6338 df-on 6339 df-lim 6340 df-suc 6341 df-iota 6467 df-fun 6516 df-fn 6517 df-f 6518 df-f1 6519 df-fo 6520 df-f1o 6521 df-fv 6522 df-riota 7347 df-ov 7393 df-oprab 7394 df-mpo 7395 df-om 7846 df-1st 7971 df-2nd 7972 df-tpos 8208 df-undef 8255 df-frecs 8263 df-wrecs 8294 df-recs 8343 df-rdg 8381 df-1o 8437 df-er 8674 df-map 8804 df-en 8922 df-dom 8923 df-sdom 8924 df-fin 8925 df-pnf 11217 df-mnf 11218 df-xr 11219 df-ltxr 11220 df-le 11221 df-sub 11414 df-neg 11415 df-nn 12194 df-2 12256 df-3 12257 df-4 12258 df-5 12259 df-6 12260 df-n0 12450 df-z 12537 df-uz 12801 df-fz 13476 df-struct 17124 df-sets 17141 df-slot 17159 df-ndx 17171 df-base 17187 df-ress 17208 df-plusg 17240 df-mulr 17241 df-sca 17243 df-vsca 17244 df-0g 17411 df-proset 18262 df-poset 18281 df-plt 18296 df-lub 18312 df-glb 18313 df-join 18314 df-meet 18315 df-p0 18391 df-p1 18392 df-lat 18398 df-clat 18465 df-mgm 18574 df-sgrp 18653 df-mnd 18669 df-submnd 18718 df-grp 18875 df-minusg 18876 df-sbg 18877 df-subg 19062 df-cntz 19256 df-lsm 19573 df-cmn 19719 df-abl 19720 df-mgp 20057 df-rng 20069 df-ur 20098 df-ring 20151 df-oppr 20253 df-dvdsr 20273 df-unit 20274 df-invr 20304 df-dvr 20317 df-drng 20647 df-lmod 20775 df-lss 20845 df-lsp 20885 df-lvec 21017 df-lsatoms 38976 df-lshyp 38977 df-lfl 39058 df-lkr 39086 df-oposet 39176 df-ol 39178 df-oml 39179 df-covers 39266 df-ats 39267 df-atl 39298 df-cvlat 39322 df-hlat 39351 df-llines 39499 df-lplanes 39500 df-lvols 39501 df-lines 39502 df-psubsp 39504 df-pmap 39505 df-padd 39797 df-lhyp 39989 df-laut 39990 df-ldil 40105 df-ltrn 40106 df-trl 40160 df-tgrp 40744 df-tendo 40756 df-edring 40758 df-dveca 41004 df-disoa 41030 df-dvech 41080 df-dib 41140 df-dic 41174 df-dih 41230 df-doch 41349 df-djh 41396 df-mapd 41626

This theorem is referenced by: mapdpglem30 41703

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