mapdpglem29 - Mathbox for Norm Megill

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Theorem mapdpglem29 37863

Description: Lemma for mapdpg 37869. 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 2778	. . . . 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 37273	. . . . . 6 ⊢ (𝜑 → 𝑈 ∈ LMod)
8		mapdpg.x	. . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))
9	8	eldifad 3804	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ 𝑉)
10		mapdpg.v	. . . . . . 7 ⊢ 𝑉 = (Base‘𝑈)
11		mapdpg.n	. . . . . . 7 ⊢ 𝑁 = (LSpan‘𝑈)
12	10, 4, 11	lspsncl 19383	. . . . . 6 ⊢ ((𝑈 ∈ LMod ∧ 𝑋 ∈ 𝑉) → (𝑁‘{𝑋}) ∈ (LSubSp‘𝑈))
13	7, 9, 12	syl2anc 579	. . . . 5 ⊢ (𝜑 → (𝑁‘{𝑋}) ∈ (LSubSp‘𝑈))
14		mapdpg.y	. . . . . . 7 ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 }))
15	14	eldifad 3804	. . . . . 6 ⊢ (𝜑 → 𝑌 ∈ 𝑉)
16	10, 4, 11	lspsncl 19383	. . . . . 6 ⊢ ((𝑈 ∈ LMod ∧ 𝑌 ∈ 𝑉) → (𝑁‘{𝑌}) ∈ (LSubSp‘𝑈))
17	7, 15, 16	syl2anc 579	. . . . 5 ⊢ (𝜑 → (𝑁‘{𝑌}) ∈ (LSubSp‘𝑈))
18	2, 3, 4, 5, 6, 13, 17	mapd11 37802	. . . 4 ⊢ (𝜑 → ((𝑀‘(𝑁‘{𝑋})) = (𝑀‘(𝑁‘{𝑌})) ↔ (𝑁‘{𝑋}) = (𝑁‘{𝑌})))
19	18	necon3bid 3013	. . 3 ⊢ (𝜑 → ((𝑀‘(𝑁‘{𝑋})) ≠ (𝑀‘(𝑁‘{𝑌})) ↔ (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌})))
20	1, 19	mpbird 249	. 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 491	. . 3 ⊢ (𝜑 → ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{𝑖}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐺𝑅𝑖)})))
24	23	simpld 490	. 2 ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑌})) = (𝐽‘{𝑖}))
25	20, 21, 24	3netr3d 3045	1 ⊢ (𝜑 → (𝐽‘{𝐺}) ≠ (𝐽‘{𝑖}))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 386 = wceq 1601 ∈ wcel 2107 ≠ wne 2969 ∖ cdif 3789 {csn 4398 ‘cfv 6137 (class class class)co 6924 Basecbs 16266 Scalarcsca 16352 ·_𝑠 cvsca 16353 0_gc0g 16497 -_gcsg 17822 LModclmod 19266 LSubSpclss 19335 LSpanclspn 19377 HLchlt 35513 LHypclh 36147 DVecHcdvh 37241 LCDualclcd 37749 mapdcmpd 37787

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-rep 5008 ax-sep 5019 ax-nul 5027 ax-pow 5079 ax-pr 5140 ax-un 7228 ax-cnex 10330 ax-resscn 10331 ax-1cn 10332 ax-icn 10333 ax-addcl 10334 ax-addrcl 10335 ax-mulcl 10336 ax-mulrcl 10337 ax-mulcom 10338 ax-addass 10339 ax-mulass 10340 ax-distr 10341 ax-i2m1 10342 ax-1ne0 10343 ax-1rid 10344 ax-rnegex 10345 ax-rrecex 10346 ax-cnre 10347 ax-pre-lttri 10348 ax-pre-lttrn 10349 ax-pre-ltadd 10350 ax-pre-mulgt0 10351 ax-riotaBAD 35116

This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-fal 1615 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-nel 3076 df-ral 3095 df-rex 3096 df-reu 3097 df-rmo 3098 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-pss 3808 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4674 df-int 4713 df-iun 4757 df-iin 4758 df-br 4889 df-opab 4951 df-mpt 4968 df-tr 4990 df-id 5263 df-eprel 5268 df-po 5276 df-so 5277 df-fr 5316 df-we 5318 df-xp 5363 df-rel 5364 df-cnv 5365 df-co 5366 df-dm 5367 df-rn 5368 df-res 5369 df-ima 5370 df-pred 5935 df-ord 5981 df-on 5982 df-lim 5983 df-suc 5984 df-iota 6101 df-fun 6139 df-fn 6140 df-f 6141 df-f1 6142 df-fo 6143 df-f1o 6144 df-fv 6145 df-riota 6885 df-ov 6927 df-oprab 6928 df-mpt2 6929 df-om 7346 df-1st 7447 df-2nd 7448 df-tpos 7636 df-undef 7683 df-wrecs 7691 df-recs 7753 df-rdg 7791 df-1o 7845 df-oadd 7849 df-er 8028 df-map 8144 df-en 8244 df-dom 8245 df-sdom 8246 df-fin 8247 df-pnf 10415 df-mnf 10416 df-xr 10417 df-ltxr 10418 df-le 10419 df-sub 10610 df-neg 10611 df-nn 11380 df-2 11443 df-3 11444 df-4 11445 df-5 11446 df-6 11447 df-n0 11648 df-z 11734 df-uz 11998 df-fz 12649 df-struct 16268 df-ndx 16269 df-slot 16270 df-base 16272 df-sets 16273 df-ress 16274 df-plusg 16362 df-mulr 16363 df-sca 16365 df-vsca 16366 df-0g 16499 df-proset 17325 df-poset 17343 df-plt 17355 df-lub 17371 df-glb 17372 df-join 17373 df-meet 17374 df-p0 17436 df-p1 17437 df-lat 17443 df-clat 17505 df-mgm 17639 df-sgrp 17681 df-mnd 17692 df-submnd 17733 df-grp 17823 df-minusg 17824 df-sbg 17825 df-subg 17986 df-cntz 18144 df-lsm 18446 df-cmn 18592 df-abl 18593 df-mgp 18888 df-ur 18900 df-ring 18947 df-oppr 19021 df-dvdsr 19039 df-unit 19040 df-invr 19070 df-dvr 19081 df-drng 19152 df-lmod 19268 df-lss 19336 df-lsp 19378 df-lvec 19509 df-lsatoms 35139 df-lshyp 35140 df-lfl 35221 df-lkr 35249 df-oposet 35339 df-ol 35341 df-oml 35342 df-covers 35429 df-ats 35430 df-atl 35461 df-cvlat 35485 df-hlat 35514 df-llines 35661 df-lplanes 35662 df-lvols 35663 df-lines 35664 df-psubsp 35666 df-pmap 35667 df-padd 35959 df-lhyp 36151 df-laut 36152 df-ldil 36267 df-ltrn 36268 df-trl 36322 df-tgrp 36906 df-tendo 36918 df-edring 36920 df-dveca 37166 df-disoa 37192 df-dvech 37242 df-dib 37302 df-dic 37336 df-dih 37392 df-doch 37511 df-djh 37558 df-mapd 37788

This theorem is referenced by: mapdpglem30 37865

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