mapdpglem29 - Mathbox for Norm Megill

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Theorem mapdpglem29 41399

Description: Lemma for mapdpg 41405. 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 2726	. . . . 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 40809	. . . . . 6 ⊢ (𝜑 → 𝑈 ∈ LMod)
8		mapdpg.x	. . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))
9	8	eldifad 3959	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ 𝑉)
10		mapdpg.v	. . . . . . 7 ⊢ 𝑉 = (Base‘𝑈)
11		mapdpg.n	. . . . . . 7 ⊢ 𝑁 = (LSpan‘𝑈)
12	10, 4, 11	lspsncl 20954	. . . . . 6 ⊢ ((𝑈 ∈ LMod ∧ 𝑋 ∈ 𝑉) → (𝑁‘{𝑋}) ∈ (LSubSp‘𝑈))
13	7, 9, 12	syl2anc 582	. . . . 5 ⊢ (𝜑 → (𝑁‘{𝑋}) ∈ (LSubSp‘𝑈))
14		mapdpg.y	. . . . . . 7 ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 }))
15	14	eldifad 3959	. . . . . 6 ⊢ (𝜑 → 𝑌 ∈ 𝑉)
16	10, 4, 11	lspsncl 20954	. . . . . 6 ⊢ ((𝑈 ∈ LMod ∧ 𝑌 ∈ 𝑉) → (𝑁‘{𝑌}) ∈ (LSubSp‘𝑈))
17	7, 15, 16	syl2anc 582	. . . . 5 ⊢ (𝜑 → (𝑁‘{𝑌}) ∈ (LSubSp‘𝑈))
18	2, 3, 4, 5, 6, 13, 17	mapd11 41338	. . . 4 ⊢ (𝜑 → ((𝑀‘(𝑁‘{𝑋})) = (𝑀‘(𝑁‘{𝑌})) ↔ (𝑁‘{𝑋}) = (𝑁‘{𝑌})))
19	18	necon3bid 2975	. . 3 ⊢ (𝜑 → ((𝑀‘(𝑁‘{𝑋})) ≠ (𝑀‘(𝑁‘{𝑌})) ↔ (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌})))
20	1, 19	mpbird 256	. 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 494	. . 3 ⊢ (𝜑 → ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{𝑖}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐺𝑅𝑖)})))
24	23	simpld 493	. 2 ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑌})) = (𝐽‘{𝑖}))
25	20, 21, 24	3netr3d 3007	1 ⊢ (𝜑 → (𝐽‘{𝐺}) ≠ (𝐽‘{𝑖}))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 394 = wceq 1534 ∈ wcel 2099 ≠ wne 2930 ∖ cdif 3944 {csn 4633 ‘cfv 6554 (class class class)co 7424 Basecbs 17213 Scalarcsca 17269 ·_𝑠 cvsca 17270 0_gc0g 17454 -_gcsg 18930 LModclmod 20836 LSubSpclss 20908 LSpanclspn 20948 HLchlt 39048 LHypclh 39683 DVecHcdvh 40777 LCDualclcd 41285 mapdcmpd 41323

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-rep 5290 ax-sep 5304 ax-nul 5311 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-cnex 11214 ax-resscn 11215 ax-1cn 11216 ax-icn 11217 ax-addcl 11218 ax-addrcl 11219 ax-mulcl 11220 ax-mulrcl 11221 ax-mulcom 11222 ax-addass 11223 ax-mulass 11224 ax-distr 11225 ax-i2m1 11226 ax-1ne0 11227 ax-1rid 11228 ax-rnegex 11229 ax-rrecex 11230 ax-cnre 11231 ax-pre-lttri 11232 ax-pre-lttrn 11233 ax-pre-ltadd 11234 ax-pre-mulgt0 11235 ax-riotaBAD 38651

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3967 df-nul 4326 df-if 4534 df-pw 4609 df-sn 4634 df-pr 4636 df-tp 4638 df-op 4640 df-uni 4914 df-int 4955 df-iun 5003 df-iin 5004 df-br 5154 df-opab 5216 df-mpt 5237 df-tr 5271 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6312 df-ord 6379 df-on 6380 df-lim 6381 df-suc 6382 df-iota 6506 df-fun 6556 df-fn 6557 df-f 6558 df-f1 6559 df-fo 6560 df-f1o 6561 df-fv 6562 df-riota 7380 df-ov 7427 df-oprab 7428 df-mpo 7429 df-om 7877 df-1st 8003 df-2nd 8004 df-tpos 8241 df-undef 8288 df-frecs 8296 df-wrecs 8327 df-recs 8401 df-rdg 8440 df-1o 8496 df-er 8734 df-map 8857 df-en 8975 df-dom 8976 df-sdom 8977 df-fin 8978 df-pnf 11300 df-mnf 11301 df-xr 11302 df-ltxr 11303 df-le 11304 df-sub 11496 df-neg 11497 df-nn 12265 df-2 12327 df-3 12328 df-4 12329 df-5 12330 df-6 12331 df-n0 12525 df-z 12611 df-uz 12875 df-fz 13539 df-struct 17149 df-sets 17166 df-slot 17184 df-ndx 17196 df-base 17214 df-ress 17243 df-plusg 17279 df-mulr 17280 df-sca 17282 df-vsca 17283 df-0g 17456 df-proset 18320 df-poset 18338 df-plt 18355 df-lub 18371 df-glb 18372 df-join 18373 df-meet 18374 df-p0 18450 df-p1 18451 df-lat 18457 df-clat 18524 df-mgm 18633 df-sgrp 18712 df-mnd 18728 df-submnd 18774 df-grp 18931 df-minusg 18932 df-sbg 18933 df-subg 19117 df-cntz 19311 df-lsm 19634 df-cmn 19780 df-abl 19781 df-mgp 20118 df-rng 20136 df-ur 20165 df-ring 20218 df-oppr 20316 df-dvdsr 20339 df-unit 20340 df-invr 20370 df-dvr 20383 df-drng 20709 df-lmod 20838 df-lss 20909 df-lsp 20949 df-lvec 21081 df-lsatoms 38674 df-lshyp 38675 df-lfl 38756 df-lkr 38784 df-oposet 38874 df-ol 38876 df-oml 38877 df-covers 38964 df-ats 38965 df-atl 38996 df-cvlat 39020 df-hlat 39049 df-llines 39197 df-lplanes 39198 df-lvols 39199 df-lines 39200 df-psubsp 39202 df-pmap 39203 df-padd 39495 df-lhyp 39687 df-laut 39688 df-ldil 39803 df-ltrn 39804 df-trl 39858 df-tgrp 40442 df-tendo 40454 df-edring 40456 df-dveca 40702 df-disoa 40728 df-dvech 40778 df-dib 40838 df-dic 40872 df-dih 40928 df-doch 41047 df-djh 41094 df-mapd 41324

This theorem is referenced by: mapdpglem30 41401

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