mapdpglem27 - Mathbox for Norm Megill

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Theorem mapdpglem27 41693

Description: Lemma for mapdpg 41700. 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 41691	. . 3 ⊢ (𝜑 → ((𝐽‘{ℎ}) = (𝐽‘{𝑖}) ∧ (𝐽‘{(𝐺𝑅ℎ)}) = (𝐽‘{(𝐺𝑅𝑖)})))
21	20	simprd 495	. 2 ⊢ (𝜑 → (𝐽‘{(𝐺𝑅ℎ)}) = (𝐽‘{(𝐺𝑅𝑖)}))
22		eqid 2729	. . . 4 ⊢ (Scalar‘𝐶) = (Scalar‘𝐶)
23		eqid 2729	. . . 4 ⊢ (Base‘(Scalar‘𝐶)) = (Base‘(Scalar‘𝐶))
24		eqid 2729	. . . 4 ⊢ (0_g‘(Scalar‘𝐶)) = (0_g‘(Scalar‘𝐶))
25		mapdpglem26.t	. . . 4 ⊢ · = ( ·_𝑠 ‘𝐶)
26	1, 8, 12	lcdlvec 41585	. . . 4 ⊢ (𝜑 → 𝐶 ∈ LVec)
27	1, 8, 12	lcdlmod 41586	. . . . 5 ⊢ (𝜑 → 𝐶 ∈ LMod)
28	18	simpld 494	. . . . 5 ⊢ (𝜑 → ℎ ∈ 𝐹)
29	9, 10	lmodvsubcl 20813	. . . . 5 ⊢ ((𝐶 ∈ LMod ∧ 𝐺 ∈ 𝐹 ∧ ℎ ∈ 𝐹) → (𝐺𝑅ℎ) ∈ 𝐹)
30	27, 15, 28, 29	syl3anc 1373	. . . 4 ⊢ (𝜑 → (𝐺𝑅ℎ) ∈ 𝐹)
31	19	simpld 494	. . . . 5 ⊢ (𝜑 → 𝑖 ∈ 𝐹)
32	9, 10	lmodvsubcl 20813	. . . . 5 ⊢ ((𝐶 ∈ LMod ∧ 𝐺 ∈ 𝐹 ∧ 𝑖 ∈ 𝐹) → (𝐺𝑅𝑖) ∈ 𝐹)
33	27, 15, 31, 32	syl3anc 1373	. . . 4 ⊢ (𝜑 → (𝐺𝑅𝑖) ∈ 𝐹)
34	9, 22, 23, 24, 25, 11, 26, 30, 33	lspsneq 21032	. . 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 41594	. . . . 5 ⊢ (𝜑 → (Base‘(Scalar‘𝐶)) = 𝐵)
38		mapdpglem26.o	. . . . . . 7 ⊢ 𝑂 = (0_g‘𝐴)
39	1, 3, 35, 38, 8, 22, 24, 12	lcd0 41602	. . . . . 6 ⊢ (𝜑 → (0_g‘(Scalar‘𝐶)) = 𝑂)
40	39	sneqd 4601	. . . . 5 ⊢ (𝜑 → {(0_g‘(Scalar‘𝐶))} = {𝑂})
41	37, 40	difeq12d 4090	. . . 4 ⊢ (𝜑 → ((Base‘(Scalar‘𝐶)) ∖ {(0_g‘(Scalar‘𝐶))}) = (𝐵 ∖ {𝑂}))
42	41	rexeqdv 3300	. . 3 ⊢ (𝜑 → (∃𝑣 ∈ ((Base‘(Scalar‘𝐶)) ∖ {(0_g‘(Scalar‘𝐶))})(𝐺𝑅ℎ) = (𝑣 · (𝐺𝑅𝑖)) ↔ ∃𝑣 ∈ (𝐵 ∖ {𝑂})(𝐺𝑅ℎ) = (𝑣 · (𝐺𝑅𝑖))))
43	34, 42	bitrd 279	. 2 ⊢ (𝜑 → ((𝐽‘{(𝐺𝑅ℎ)}) = (𝐽‘{(𝐺𝑅𝑖)}) ↔ ∃𝑣 ∈ (𝐵 ∖ {𝑂})(𝐺𝑅ℎ) = (𝑣 · (𝐺𝑅𝑖))))
44	21, 43	mpbid 232	1 ⊢ (𝜑 → ∃𝑣 ∈ (𝐵 ∖ {𝑂})(𝐺𝑅ℎ) = (𝑣 · (𝐺𝑅𝑖)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2109 ≠ wne 2925 ∃wrex 3053 ∖ cdif 3911 {csn 4589 ‘cfv 6511 (class class class)co 7387 Basecbs 17179 Scalarcsca 17223 ·_𝑠 cvsca 17224 0_gc0g 17402 -_gcsg 18867 LModclmod 20766 LSpanclspn 20877 HLchlt 39343 LHypclh 39978 DVecHcdvh 41072 LCDualclcd 41580 mapdcmpd 41618

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 2701 ax-rep 5234 ax-sep 5251 ax-nul 5261 ax-pow 5320 ax-pr 5387 ax-un 7711 ax-cnex 11124 ax-resscn 11125 ax-1cn 11126 ax-icn 11127 ax-addcl 11128 ax-addrcl 11129 ax-mulcl 11130 ax-mulrcl 11131 ax-mulcom 11132 ax-addass 11133 ax-mulass 11134 ax-distr 11135 ax-i2m1 11136 ax-1ne0 11137 ax-1rid 11138 ax-rnegex 11139 ax-rrecex 11140 ax-cnre 11141 ax-pre-lttri 11142 ax-pre-lttrn 11143 ax-pre-ltadd 11144 ax-pre-mulgt0 11145 ax-riotaBAD 38946

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 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3354 df-reu 3355 df-rab 3406 df-v 3449 df-sbc 3754 df-csb 3863 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3934 df-nul 4297 df-if 4489 df-pw 4565 df-sn 4590 df-pr 4592 df-tp 4594 df-op 4596 df-uni 4872 df-int 4911 df-iun 4957 df-iin 4958 df-br 5108 df-opab 5170 df-mpt 5189 df-tr 5215 df-id 5533 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5591 df-we 5593 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6274 df-ord 6335 df-on 6336 df-lim 6337 df-suc 6338 df-iota 6464 df-fun 6513 df-fn 6514 df-f 6515 df-f1 6516 df-fo 6517 df-f1o 6518 df-fv 6519 df-riota 7344 df-ov 7390 df-oprab 7391 df-mpo 7392 df-of 7653 df-om 7843 df-1st 7968 df-2nd 7969 df-tpos 8205 df-undef 8252 df-frecs 8260 df-wrecs 8291 df-recs 8340 df-rdg 8378 df-1o 8434 df-2o 8435 df-er 8671 df-map 8801 df-en 8919 df-dom 8920 df-sdom 8921 df-fin 8922 df-pnf 11210 df-mnf 11211 df-xr 11212 df-ltxr 11213 df-le 11214 df-sub 11407 df-neg 11408 df-nn 12187 df-2 12249 df-3 12250 df-4 12251 df-5 12252 df-6 12253 df-n0 12443 df-z 12530 df-uz 12794 df-fz 13469 df-struct 17117 df-sets 17134 df-slot 17152 df-ndx 17164 df-base 17180 df-ress 17201 df-plusg 17233 df-mulr 17234 df-sca 17236 df-vsca 17237 df-0g 17404 df-mre 17547 df-mrc 17548 df-acs 17550 df-proset 18255 df-poset 18274 df-plt 18289 df-lub 18305 df-glb 18306 df-join 18307 df-meet 18308 df-p0 18384 df-p1 18385 df-lat 18391 df-clat 18458 df-mgm 18567 df-sgrp 18646 df-mnd 18662 df-submnd 18711 df-grp 18868 df-minusg 18869 df-sbg 18870 df-subg 19055 df-cntz 19249 df-oppg 19278 df-lsm 19566 df-cmn 19712 df-abl 19713 df-mgp 20050 df-rng 20062 df-ur 20091 df-ring 20144 df-oppr 20246 df-dvdsr 20266 df-unit 20267 df-invr 20297 df-dvr 20310 df-nzr 20422 df-rlreg 20603 df-domn 20604 df-drng 20640 df-lmod 20768 df-lss 20838 df-lsp 20878 df-lvec 21010 df-lsatoms 38969 df-lshyp 38970 df-lcv 39012 df-lfl 39051 df-lkr 39079 df-ldual 39117 df-oposet 39169 df-ol 39171 df-oml 39172 df-covers 39259 df-ats 39260 df-atl 39291 df-cvlat 39315 df-hlat 39344 df-llines 39492 df-lplanes 39493 df-lvols 39494 df-lines 39495 df-psubsp 39497 df-pmap 39498 df-padd 39790 df-lhyp 39982 df-laut 39983 df-ldil 40098 df-ltrn 40099 df-trl 40153 df-tgrp 40737 df-tendo 40749 df-edring 40751 df-dveca 40997 df-disoa 41023 df-dvech 41073 df-dib 41133 df-dic 41167 df-dih 41223 df-doch 41342 df-djh 41389 df-lcdual 41581

This theorem is referenced by: mapdpglem32 41699

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