mapdpglem29 - Mathbox for Norm Megill

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Theorem mapdpglem29 42199

Description: Lemma for mapdpg 42205. 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 2740	. . . . 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 41609	. . . . . 6 ⊢ (𝜑 → 𝑈 ∈ LMod)
8		mapdpg.x	. . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))
9	8	eldifad 3902	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ 𝑉)
10		mapdpg.v	. . . . . . 7 ⊢ 𝑉 = (Base‘𝑈)
11		mapdpg.n	. . . . . . 7 ⊢ 𝑁 = (LSpan‘𝑈)
12	10, 4, 11	lspsncl 20974	. . . . . 6 ⊢ ((𝑈 ∈ LMod ∧ 𝑋 ∈ 𝑉) → (𝑁‘{𝑋}) ∈ (LSubSp‘𝑈))
13	7, 9, 12	syl2anc 590	. . . . 5 ⊢ (𝜑 → (𝑁‘{𝑋}) ∈ (LSubSp‘𝑈))
14		mapdpg.y	. . . . . . 7 ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 }))
15	14	eldifad 3902	. . . . . 6 ⊢ (𝜑 → 𝑌 ∈ 𝑉)
16	10, 4, 11	lspsncl 20974	. . . . . 6 ⊢ ((𝑈 ∈ LMod ∧ 𝑌 ∈ 𝑉) → (𝑁‘{𝑌}) ∈ (LSubSp‘𝑈))
17	7, 15, 16	syl2anc 590	. . . . 5 ⊢ (𝜑 → (𝑁‘{𝑌}) ∈ (LSubSp‘𝑈))
18	2, 3, 4, 5, 6, 13, 17	mapd11 42138	. . . 4 ⊢ (𝜑 → ((𝑀‘(𝑁‘{𝑋})) = (𝑀‘(𝑁‘{𝑌})) ↔ (𝑁‘{𝑋}) = (𝑁‘{𝑌})))
19	18	necon3bid 2979	. . 3 ⊢ (𝜑 → ((𝑀‘(𝑁‘{𝑋})) ≠ (𝑀‘(𝑁‘{𝑌})) ↔ (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌})))
20	1, 19	mpbird 258	. 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 496	. . 3 ⊢ (𝜑 → ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{𝑖}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐺𝑅𝑖)})))
24	23	simpld 495	. 2 ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑌})) = (𝐽‘{𝑖}))
25	20, 21, 24	3netr3d 3011	1 ⊢ (𝜑 → (𝐽‘{𝐺}) ≠ (𝐽‘{𝑖}))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 396 = wceq 1547 ∈ wcel 2119 ≠ wne 2935 ∖ cdif 3887 {csn 4562 ‘cfv 6492 (class class class)co 7363 Basecbs 17177 Scalarcsca 17221 ·_𝑠 cvsca 17222 0_gc0g 17400 -_gcsg 18909 LModclmod 20857 LSubSpclss 20928 LSpanclspn 20968 HLchlt 39849 LHypclh 40483 DVecHcdvh 41577 LCDualclcd 42085 mapdcmpd 42123

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1974 ax-7 2015 ax-8 2121 ax-9 2129 ax-10 2152 ax-11 2168 ax-12 2189 ax-ext 2712 ax-rep 5206 ax-sep 5225 ax-nul 5235 ax-pow 5301 ax-pr 5369 ax-un 7685 ax-cnex 11092 ax-resscn 11093 ax-1cn 11094 ax-icn 11095 ax-addcl 11096 ax-addrcl 11097 ax-mulcl 11098 ax-mulrcl 11099 ax-mulcom 11100 ax-addass 11101 ax-mulass 11102 ax-distr 11103 ax-i2m1 11104 ax-1ne0 11105 ax-1rid 11106 ax-rnegex 11107 ax-rrecex 11108 ax-cnre 11109 ax-pre-lttri 11110 ax-pre-lttrn 11111 ax-pre-ltadd 11112 ax-pre-mulgt0 11113 ax-riotaBAD 39452

This theorem depends on definitions: df-bi 208 df-an 397 df-or 854 df-3or 1093 df-3an 1094 df-tru 1550 df-fal 1560 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2543 df-eu 2573 df-clab 2719 df-cleq 2732 df-clel 2815 df-nfc 2889 df-ne 2936 df-nel 3040 df-ral 3055 df-rex 3065 df-rmo 3345 df-reu 3346 df-rab 3393 df-v 3434 df-sbc 3731 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-pss 3910 df-nul 4269 df-if 4462 df-pw 4538 df-sn 4563 df-pr 4565 df-tp 4567 df-op 4569 df-uni 4846 df-int 4885 df-iun 4930 df-iin 4931 df-br 5080 df-opab 5142 df-mpt 5161 df-tr 5187 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 6259 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-riota 7320 df-ov 7366 df-oprab 7367 df-mpo 7368 df-om 7814 df-1st 7938 df-2nd 7939 df-tpos 8173 df-undef 8220 df-frecs 8228 df-wrecs 8259 df-recs 8308 df-rdg 8346 df-1o 8402 df-er 8640 df-map 8772 df-en 8891 df-dom 8892 df-sdom 8893 df-fin 8894 df-pnf 11179 df-mnf 11180 df-xr 11181 df-ltxr 11182 df-le 11183 df-sub 11377 df-neg 11378 df-nn 12173 df-2 12242 df-3 12243 df-4 12244 df-5 12245 df-6 12246 df-n0 12436 df-z 12523 df-uz 12787 df-fz 13460 df-struct 17115 df-sets 17132 df-slot 17150 df-ndx 17162 df-base 17178 df-ress 17199 df-plusg 17231 df-mulr 17232 df-sca 17234 df-vsca 17235 df-0g 17402 df-proset 18258 df-poset 18277 df-plt 18292 df-lub 18308 df-glb 18309 df-join 18310 df-meet 18311 df-p0 18387 df-p1 18388 df-lat 18396 df-clat 18463 df-mgm 18606 df-sgrp 18685 df-mnd 18701 df-submnd 18750 df-grp 18910 df-minusg 18911 df-sbg 18912 df-subg 19097 df-cntz 19290 df-lsm 19609 df-cmn 19755 df-abl 19756 df-mgp 20120 df-rng 20132 df-ur 20161 df-ring 20214 df-oppr 20315 df-dvdsr 20335 df-unit 20336 df-invr 20366 df-dvr 20379 df-drng 20710 df-lmod 20859 df-lss 20929 df-lsp 20969 df-lvec 21100 df-lsatoms 39475 df-lshyp 39476 df-lfl 39557 df-lkr 39585 df-oposet 39675 df-ol 39677 df-oml 39678 df-covers 39765 df-ats 39766 df-atl 39797 df-cvlat 39821 df-hlat 39850 df-llines 39997 df-lplanes 39998 df-lvols 39999 df-lines 40000 df-psubsp 40002 df-pmap 40003 df-padd 40295 df-lhyp 40487 df-laut 40488 df-ldil 40603 df-ltrn 40604 df-trl 40658 df-tgrp 41242 df-tendo 41254 df-edring 41256 df-dveca 41502 df-disoa 41528 df-dvech 41578 df-dib 41638 df-dic 41672 df-dih 41728 df-doch 41847 df-djh 41894 df-mapd 42124

This theorem is referenced by: mapdpglem30 42201

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