mapdpglem27 - Mathbox for Norm Megill

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Theorem mapdpglem27 42580

Description: Lemma for mapdpg 42587. Baer p. 45 line 16: "v(x'-y'') = x'-y'" (with equality swapped). (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)

**Assertion**
Ref	Expression
mapdpglem27	⊢ (𝜑 → ∃𝑣 ∈ (𝐵 ∖ {𝑂})(𝐺𝑅ℎ) = (𝑣 · (𝐺𝑅𝑖)))

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

**Proof of Theorem mapdpglem27**
Step	Hyp	Ref	Expression
1		mapdpg.h	. . . 4 ⊢ 𝐻 = (LHyp‘𝐾)
2		mapdpg.m	. . . 4 ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊)
3		mapdpg.u	. . . 4 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
4		mapdpg.v	. . . 4 ⊢ 𝑉 = (Base‘𝑈)
5		mapdpg.s	. . . 4 ⊢ − = (-_g‘𝑈)
6		mapdpg.z	. . . 4 ⊢ 0 = (0_g‘𝑈)
7		mapdpg.n	. . . 4 ⊢ 𝑁 = (LSpan‘𝑈)
8		mapdpg.c	. . . 4 ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊)
9		mapdpg.f	. . . 4 ⊢ 𝐹 = (Base‘𝐶)
10		mapdpg.r	. . . 4 ⊢ 𝑅 = (-_g‘𝐶)
11		mapdpg.j	. . . 4 ⊢ 𝐽 = (LSpan‘𝐶)
12		mapdpg.k	. . . 4 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
13		mapdpg.x	. . . 4 ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))
14		mapdpg.y	. . . 4 ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 }))
15		mapdpg.g	. . . 4 ⊢ (𝜑 → 𝐺 ∈ 𝐹)
16		mapdpg.ne	. . . 4 ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}))
17		mapdpg.e	. . . 4 ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐺}))
18		mapdpgem25.h1	. . . 4 ⊢ (𝜑 → (ℎ ∈ 𝐹 ∧ ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐺𝑅ℎ)}))))
19		mapdpgem25.i1	. . . 4 ⊢ (𝜑 → (𝑖 ∈ 𝐹 ∧ ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{𝑖}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐺𝑅𝑖)}))))
20	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19	mapdpglem25 42578	. . 3 ⊢ (𝜑 → ((𝐽‘{ℎ}) = (𝐽‘{𝑖}) ∧ (𝐽‘{(𝐺𝑅ℎ)}) = (𝐽‘{(𝐺𝑅𝑖)})))
21	20	simprd 501	. 2 ⊢ (𝜑 → (𝐽‘{(𝐺𝑅ℎ)}) = (𝐽‘{(𝐺𝑅𝑖)}))
22		eqid 2762	. . . 4 ⊢ (Scalar‘𝐶) = (Scalar‘𝐶)
23		eqid 2762	. . . 4 ⊢ (Base‘(Scalar‘𝐶)) = (Base‘(Scalar‘𝐶))
24		eqid 2762	. . . 4 ⊢ (0_g‘(Scalar‘𝐶)) = (0_g‘(Scalar‘𝐶))
25		mapdpglem26.t	. . . 4 ⊢ · = ( ·_𝑠 ‘𝐶)
26	1, 8, 12	lcdlvec 42472	. . . 4 ⊢ (𝜑 → 𝐶 ∈ LVec)
27	1, 8, 12	lcdlmod 42473	. . . . 5 ⊢ (𝜑 → 𝐶 ∈ LMod)
28	18	simpld 500	. . . . 5 ⊢ (𝜑 → ℎ ∈ 𝐹)
29	9, 10	lmodvsubcl 21097	. . . . 5 ⊢ ((𝐶 ∈ LMod ∧ 𝐺 ∈ 𝐹 ∧ ℎ ∈ 𝐹) → (𝐺𝑅ℎ) ∈ 𝐹)
30	27, 15, 28, 29	syl3anc 1398	. . . 4 ⊢ (𝜑 → (𝐺𝑅ℎ) ∈ 𝐹)
31	19	simpld 500	. . . . 5 ⊢ (𝜑 → 𝑖 ∈ 𝐹)
32	9, 10	lmodvsubcl 21097	. . . . 5 ⊢ ((𝐶 ∈ LMod ∧ 𝐺 ∈ 𝐹 ∧ 𝑖 ∈ 𝐹) → (𝐺𝑅𝑖) ∈ 𝐹)
33	27, 15, 31, 32	syl3anc 1398	. . . 4 ⊢ (𝜑 → (𝐺𝑅𝑖) ∈ 𝐹)
34	9, 22, 23, 24, 25, 11, 26, 30, 33	lspsneq 21315	. . 3 ⊢ (𝜑 → ((𝐽‘{(𝐺𝑅ℎ)}) = (𝐽‘{(𝐺𝑅𝑖)}) ↔ ∃𝑣 ∈ ((Base‘(Scalar‘𝐶)) ∖ {(0_g‘(Scalar‘𝐶))})(𝐺𝑅ℎ) = (𝑣 · (𝐺𝑅𝑖))))
35		mapdpglem26.a	. . . . . 6 ⊢ 𝐴 = (Scalar‘𝑈)
36		mapdpglem26.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝐴)
37	1, 3, 35, 36, 8, 22, 23, 12	lcdsbase 42481	. . . . 5 ⊢ (𝜑 → (Base‘(Scalar‘𝐶)) = 𝐵)
38		mapdpglem26.o	. . . . . . 7 ⊢ 𝑂 = (0_g‘𝐴)
39	1, 3, 35, 38, 8, 22, 24, 12	lcd0 42489	. . . . . 6 ⊢ (𝜑 → (0_g‘(Scalar‘𝐶)) = 𝑂)
40	39	sneqd 4599	. . . . 5 ⊢ (𝜑 → {(0_g‘(Scalar‘𝐶))} = {𝑂})
41	37, 40	difeq12d 4078	. . . 4 ⊢ (𝜑 → ((Base‘(Scalar‘𝐶)) ∖ {(0_g‘(Scalar‘𝐶))}) = (𝐵 ∖ {𝑂}))
42	41	rexeqdv 3322	. . 3 ⊢ (𝜑 → (∃𝑣 ∈ ((Base‘(Scalar‘𝐶)) ∖ {(0_g‘(Scalar‘𝐶))})(𝐺𝑅ℎ) = (𝑣 · (𝐺𝑅𝑖)) ↔ ∃𝑣 ∈ (𝐵 ∖ {𝑂})(𝐺𝑅ℎ) = (𝑣 · (𝐺𝑅𝑖))))
43	34, 42	bitrd 282	. 2 ⊢ (𝜑 → ((𝐽‘{(𝐺𝑅ℎ)}) = (𝐽‘{(𝐺𝑅𝑖)}) ↔ ∃𝑣 ∈ (𝐵 ∖ {𝑂})(𝐺𝑅ℎ) = (𝑣 · (𝐺𝑅𝑖))))
44	21, 43	mpbid 235	1 ⊢ (𝜑 → ∃𝑣 ∈ (𝐵 ∖ {𝑂})(𝐺𝑅ℎ) = (𝑣 · (𝐺𝑅𝑖)))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∧ wa 401 = wceq 1570 ∈ wcel 2145 ≠ wne 2957 ∃wrex 3088 ∖ cdif 3899 {csn 4587 ‘cfv 6537 (class class class)co 7417 Basecbs 17307 Scalarcsca 17351 ·_𝑠 cvsca 17352 0_gc0g 17530 -_gcsg 19065 LModclmod 21050 LSpanclspn 21161 HLchlt 40231 LHypclh 40865 DVecHcdvh 41959 LCDualclcd 42467 mapdcmpd 42505

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2215 ax-ext 2734 ax-rep 5236 ax-sep 5255 ax-nul 5267 ax-pow 5334 ax-pr 5402 ax-un 7740 ax-cnex 11184 ax-resscn 11185 ax-1cn 11186 ax-icn 11187 ax-addcl 11188 ax-addrcl 11189 ax-mulcl 11190 ax-mulrcl 11191 ax-mulcom 11192 ax-addass 11193 ax-mulass 11194 ax-distr 11195 ax-i2m1 11196 ax-1ne0 11197 ax-1rid 11198 ax-rnegex 11199 ax-rrecex 11200 ax-cnre 11201 ax-pre-lttri 11202 ax-pre-lttrn 11203 ax-pre-ltadd 11204 ax-pre-mulgt0 11205 ax-riotaBAD 39834

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2566 df-eu 2596 df-clab 2741 df-cleq 2754 df-clel 2837 df-nfc 2911 df-ne 2958 df-nel 3064 df-ral 3079 df-rex 3089 df-rmo 3367 df-reu 3368 df-rab 3415 df-v 3455 df-sbc 3743 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-pss 3922 df-nul 4283 df-if 4486 df-pw 4562 df-sn 4588 df-pr 4590 df-tp 4592 df-op 4594 df-uni 4871 df-int 4911 df-iun 4956 df-iin 4957 df-br 5108 df-opab 5172 df-mpt 5191 df-tr 5217 df-id 5554 df-eprel 5559 df-po 5567 df-so 5568 df-fr 5612 df-we 5614 df-xp 5665 df-rel 5666 df-cnv 5667 df-co 5668 df-dm 5669 df-rn 5670 df-res 5671 df-ima 5672 df-pred 6303 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6493 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-riota 7374 df-ov 7420 df-oprab 7421 df-mpo 7422 df-of 7682 df-om 7867 df-1st 7990 df-2nd 7991 df-tpos 8228 df-undef 8275 df-frecs 8284 df-wrecs 8315 df-recs 8364 df-rdg 8403 df-1o 8459 df-2o 8460 df-er 8700 df-map 8832 df-en 8957 df-dom 8958 df-sdom 8959 df-fin 8960 df-pnf 11273 df-mnf 11274 df-xr 11275 df-ltxr 11276 df-le 11277 df-sub 11471 df-neg 11472 df-nn 12262 df-2 12331 df-3 12332 df-4 12333 df-5 12334 df-6 12335 df-n0 12533 df-z 12620 df-uz 12892 df-fz 13566 df-struct 17245 df-sets 17262 df-slot 17280 df-ndx 17292 df-base 17308 df-ress 17329 df-plusg 17361 df-mulr 17362 df-sca 17364 df-vsca 17365 df-0g 17532 df-mre 17676 df-mrc 17677 df-acs 17679 df-proset 18388 df-poset 18407 df-plt 18422 df-lub 18438 df-glb 18439 df-join 18440 df-meet 18441 df-p0 18517 df-p1 18518 df-lat 18526 df-clat 18593 df-mgm 18736 df-sgrp 18827 df-mnd 18843 df-submnd 18898 df-grp 19066 df-minusg 19067 df-sbg 19068 df-subg 19252 df-cntz 19450 df-oppg 19479 df-lsm 19769 df-cmn 19915 df-abl 19916 df-mgp 20280 df-rng 20294 df-ur 20327 df-ring 20380 df-oppr 20484 df-dvdsr 20504 df-unit 20505 df-invr 20535 df-dvr 20548 df-nzr 20679 df-rlreg 20862 df-domn 20863 df-drng 20898 df-lmod 21052 df-lss 21122 df-lsp 21162 df-lvec 21293 df-lsatoms 39857 df-lshyp 39858 df-lcv 39900 df-lfl 39939 df-lkr 39967 df-ldual 40005 df-oposet 40057 df-ol 40059 df-oml 40060 df-covers 40147 df-ats 40148 df-atl 40179 df-cvlat 40203 df-hlat 40232 df-llines 40379 df-lplanes 40380 df-lvols 40381 df-lines 40382 df-psubsp 40384 df-pmap 40385 df-padd 40677 df-lhyp 40869 df-laut 40870 df-ldil 40985 df-ltrn 40986 df-trl 41040 df-tgrp 41624 df-tendo 41636 df-edring 41638 df-dveca 41884 df-disoa 41910 df-dvech 41960 df-dib 42020 df-dic 42054 df-dih 42110 df-doch 42229 df-djh 42276 df-lcdual 42468

This theorem is used by: mapdpglem32 42586

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