mapdpglem23 - Mathbox for Norm Megill

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Theorem mapdpglem23 43017

Description: Lemma for mapdpg 43029. Baer p. 45, line 10: "and so y' meets all our requirements." Our ℎ is Baer's y'. (Contributed by NM, 20-Mar-2015.)

**Hypotheses**
Ref	Expression
mapdpglem.h	⊢ 𝐻 = (LHyp‘𝐾)
mapdpglem.m	⊢ 𝑀 = ((mapd‘𝐾)‘𝑊)
mapdpglem.u	⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
mapdpglem.v	⊢ 𝑉 = (Base‘𝑈)
mapdpglem.s	⊢ − = (-_g‘𝑈)
mapdpglem.n	⊢ 𝑁 = (LSpan‘𝑈)
mapdpglem.c	⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊)
mapdpglem.k	⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
mapdpglem.x	⊢ (𝜑 → 𝑋 ∈ 𝑉)
mapdpglem.y	⊢ (𝜑 → 𝑌 ∈ 𝑉)
mapdpglem1.p	⊢ ⊕ = (LSSum‘𝐶)
mapdpglem2.j	⊢ 𝐽 = (LSpan‘𝐶)
mapdpglem3.f	⊢ 𝐹 = (Base‘𝐶)
mapdpglem3.te	⊢ (𝜑 → 𝑡 ∈ ((𝑀‘(𝑁‘{𝑋})) ⊕ (𝑀‘(𝑁‘{𝑌}))))
mapdpglem3.a	⊢ 𝐴 = (Scalar‘𝑈)
mapdpglem3.b	⊢ 𝐵 = (Base‘𝐴)
mapdpglem3.t	⊢ · = ( ·_𝑠 ‘𝐶)
mapdpglem3.r	⊢ 𝑅 = (-_g‘𝐶)
mapdpglem3.g	⊢ (𝜑 → 𝐺 ∈ 𝐹)
mapdpglem3.e	⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐺}))
mapdpglem4.q	⊢ 𝑄 = (0_g‘𝑈)
mapdpglem.ne	⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}))
mapdpglem4.jt	⊢ (𝜑 → (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{𝑡}))
mapdpglem4.z	⊢ 0 = (0_g‘𝐴)
mapdpglem4.g4	⊢ (𝜑 → 𝑔 ∈ 𝐵)
mapdpglem4.z4	⊢ (𝜑 → 𝑧 ∈ (𝑀‘(𝑁‘{𝑌})))
mapdpglem4.t4	⊢ (𝜑 → 𝑡 = ((𝑔 · 𝐺)𝑅𝑧))
mapdpglem4.xn	⊢ (𝜑 → 𝑋 ≠ 𝑄)
mapdpglem12.yn	⊢ (𝜑 → 𝑌 ≠ 𝑄)
mapdpglem17.ep	⊢ 𝐸 = (((inv_r‘𝐴)‘𝑔) · 𝑧)

**Assertion**
Ref	Expression
mapdpglem23	⊢ (𝜑 → ∃ℎ ∈ 𝐹 ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐺𝑅ℎ)})))

Distinct variable groups: 𝑡, − 𝑡,𝐶 𝑡,𝐽 𝑡,𝑀 𝑡,𝑁 𝑡,𝑋 𝑡,𝑌 𝐵,𝑔 𝑧,𝑔,𝐶 𝑔,𝐹 𝑔,𝐺,𝑧 𝑔,𝐽,𝑧 𝑔,𝑀,𝑧 𝑔,𝑁,𝑧 𝑅,𝑔,𝑧 · ,𝑔,𝑧 𝑔,𝑌,𝑧,𝑡 ℎ,𝐸 ℎ,𝐹 ℎ,𝐺 ℎ,𝐽 ℎ,𝑀 ℎ,𝑁 𝑅,ℎ − ,ℎ ℎ,𝑋 ℎ,𝑌 Allowed substitution hints: 𝜑(𝑧, 𝑡, 𝑔, ℎ) 𝐴(𝑧, 𝑡, 𝑔, ℎ) 𝐵(𝑧, 𝑡, ℎ) 𝐶(ℎ) ⊕ (𝑧, 𝑡, 𝑔, ℎ) 𝑄(𝑧, 𝑡, 𝑔, ℎ) 𝑅(𝑡) · (𝑡, ℎ) 𝑈(𝑧, 𝑡, 𝑔, ℎ) 𝐸(𝑧, 𝑡, 𝑔) 𝐹(𝑧, 𝑡) 𝐺(𝑡) 𝐻(𝑧, 𝑡, 𝑔, ℎ) 𝐾(𝑧, 𝑡, 𝑔, ℎ) − (𝑧, 𝑔) 𝑉(𝑧, 𝑡, 𝑔, ℎ) 𝑊(𝑧, 𝑡, 𝑔, ℎ) 𝑋(𝑧, 𝑔) 0 (𝑧, 𝑡, 𝑔, ℎ)

**Proof of Theorem mapdpglem23**
Step	Hyp	Ref	Expression
1		mapdpglem.h	. . . 4 ⊢ 𝐻 = (LHyp‘𝐾)
2		mapdpglem.m	. . . 4 ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊)
3		mapdpglem.u	. . . 4 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
4		eqid 2763	. . . 4 ⊢ (LSubSp‘𝑈) = (LSubSp‘𝑈)
5		mapdpglem.c	. . . 4 ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊)
6		eqid 2763	. . . 4 ⊢ (LSubSp‘𝐶) = (LSubSp‘𝐶)
7		mapdpglem.k	. . . 4 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
8	1, 3, 7	dvhlmod 42433	. . . . 5 ⊢ (𝜑 → 𝑈 ∈ LMod)
9		mapdpglem.y	. . . . 5 ⊢ (𝜑 → 𝑌 ∈ 𝑉)
10		mapdpglem.v	. . . . . 6 ⊢ 𝑉 = (Base‘𝑈)
11		mapdpglem.n	. . . . . 6 ⊢ 𝑁 = (LSpan‘𝑈)
12	10, 4, 11	lspsncl 21303	. . . . 5 ⊢ ((𝑈 ∈ LMod ∧ 𝑌 ∈ 𝑉) → (𝑁‘{𝑌}) ∈ (LSubSp‘𝑈))
13	8, 9, 12	syl2anc 596	. . . 4 ⊢ (𝜑 → (𝑁‘{𝑌}) ∈ (LSubSp‘𝑈))
14	1, 2, 3, 4, 5, 6, 7, 13	mapdcl2 42979	. . 3 ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑌})) ∈ (LSubSp‘𝐶))
15		mapdpglem.s	. . . 4 ⊢ − = (-_g‘𝑈)
16		mapdpglem.x	. . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝑉)
17		mapdpglem1.p	. . . 4 ⊢ ⊕ = (LSSum‘𝐶)
18		mapdpglem2.j	. . . 4 ⊢ 𝐽 = (LSpan‘𝐶)
19		mapdpglem3.f	. . . 4 ⊢ 𝐹 = (Base‘𝐶)
20		mapdpglem3.te	. . . 4 ⊢ (𝜑 → 𝑡 ∈ ((𝑀‘(𝑁‘{𝑋})) ⊕ (𝑀‘(𝑁‘{𝑌}))))
21		mapdpglem3.a	. . . 4 ⊢ 𝐴 = (Scalar‘𝑈)
22		mapdpglem3.b	. . . 4 ⊢ 𝐵 = (Base‘𝐴)
23		mapdpglem3.t	. . . 4 ⊢ · = ( ·_𝑠 ‘𝐶)
24		mapdpglem3.r	. . . 4 ⊢ 𝑅 = (-_g‘𝐶)
25		mapdpglem3.g	. . . 4 ⊢ (𝜑 → 𝐺 ∈ 𝐹)
26		mapdpglem3.e	. . . 4 ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐺}))
27		mapdpglem4.q	. . . 4 ⊢ 𝑄 = (0_g‘𝑈)
28		mapdpglem.ne	. . . 4 ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}))
29		mapdpglem4.jt	. . . 4 ⊢ (𝜑 → (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{𝑡}))
30		mapdpglem4.z	. . . 4 ⊢ 0 = (0_g‘𝐴)
31		mapdpglem4.g4	. . . 4 ⊢ (𝜑 → 𝑔 ∈ 𝐵)
32		mapdpglem4.z4	. . . 4 ⊢ (𝜑 → 𝑧 ∈ (𝑀‘(𝑁‘{𝑌})))
33		mapdpglem4.t4	. . . 4 ⊢ (𝜑 → 𝑡 = ((𝑔 · 𝐺)𝑅𝑧))
34		mapdpglem4.xn	. . . 4 ⊢ (𝜑 → 𝑋 ≠ 𝑄)
35		mapdpglem12.yn	. . . 4 ⊢ (𝜑 → 𝑌 ≠ 𝑄)
36		mapdpglem17.ep	. . . 4 ⊢ 𝐸 = (((inv_r‘𝐴)‘𝑔) · 𝑧)
37	1, 2, 3, 10, 15, 11, 5, 7, 16, 9, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36	mapdpglem19 43013	. . 3 ⊢ (𝜑 → 𝐸 ∈ (𝑀‘(𝑁‘{𝑌})))
38	19, 6	lssel 21263	. . 3 ⊢ (((𝑀‘(𝑁‘{𝑌})) ∈ (LSubSp‘𝐶) ∧ 𝐸 ∈ (𝑀‘(𝑁‘{𝑌}))) → 𝐸 ∈ 𝐹)
39	14, 37, 38	syl2anc 596	. 2 ⊢ (𝜑 → 𝐸 ∈ 𝐹)
40	1, 2, 3, 10, 15, 11, 5, 7, 16, 9, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36	mapdpglem20 43014	. 2 ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑌})) = (𝐽‘{𝐸}))
41	1, 2, 3, 10, 15, 11, 5, 7, 16, 9, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36	mapdpglem22 43016	. 2 ⊢ (𝜑 → (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐺𝑅𝐸)}))
42		sneq 4594	. . . . . 6 ⊢ (ℎ = 𝐸 → {ℎ} = {𝐸})
43	42	fveq2d 6920	. . . . 5 ⊢ (ℎ = 𝐸 → (𝐽‘{ℎ}) = (𝐽‘{𝐸}))
44	43	eqeq2d 2774	. . . 4 ⊢ (ℎ = 𝐸 → ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{ℎ}) ↔ (𝑀‘(𝑁‘{𝑌})) = (𝐽‘{𝐸})))
45		oveq2 7459	. . . . . . 7 ⊢ (ℎ = 𝐸 → (𝐺𝑅ℎ) = (𝐺𝑅𝐸))
46	45	sneqd 4596	. . . . . 6 ⊢ (ℎ = 𝐸 → {(𝐺𝑅ℎ)} = {(𝐺𝑅𝐸)})
47	46	fveq2d 6920	. . . . 5 ⊢ (ℎ = 𝐸 → (𝐽‘{(𝐺𝑅ℎ)}) = (𝐽‘{(𝐺𝑅𝐸)}))
48	47	eqeq2d 2774	. . . 4 ⊢ (ℎ = 𝐸 → ((𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐺𝑅ℎ)}) ↔ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐺𝑅𝐸)})))
49	44, 48	anbi12d 644	. . 3 ⊢ (ℎ = 𝐸 → (((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐺𝑅ℎ)})) ↔ ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{𝐸}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐺𝑅𝐸)}))))
50	49	rspcev 3580	. 2 ⊢ ((𝐸 ∈ 𝐹 ∧ ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{𝐸}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐺𝑅𝐸)}))) → ∃ℎ ∈ 𝐹 ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐺𝑅ℎ)})))
51	39, 40, 41, 50	syl12anc 850	1 ⊢ (𝜑 → ∃ℎ ∈ 𝐹 ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐺𝑅ℎ)})))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∧ wa 401 = wceq 1570 ∈ wcel 2147 ≠ wne 2959 ∃wrex 3090 {csn 4588 ‘cfv 6566 (class class class)co 7451 Basecbs 17431 Scalarcsca 17475 ·_𝑠 cvsca 17476 0_gc0g 17660 -_gcsg 19196 LSSumclsm 19899 inv_rcinvr 20668 LModclmod 21186 LSubSpclss 21257 LSpanclspn 21297 HLchlt 40673 LHypclh 41307 DVecHcdvh 42401 LCDualclcd 42909 mapdcmpd 42947

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 1945 ax-6 2002 ax-7 2043 ax-8 2149 ax-9 2157 ax-10 2180 ax-11 2197 ax-12 2216 ax-ext 2735 ax-rep 5238 ax-sep 5255 ax-nul 5266 ax-pow 5334 ax-pr 5398 ax-un 7783 ax-cnex 11295 ax-resscn 11296 ax-1cn 11297 ax-icn 11298 ax-addcl 11299 ax-addrcl 11300 ax-mulcl 11301 ax-mulrcl 11302 ax-mulcom 11303 ax-addass 11304 ax-mulass 11305 ax-distr 11306 ax-i2m1 11307 ax-1ne0 11308 ax-1rid 11309 ax-rnegex 11310 ax-rrecex 11311 ax-cnre 11312 ax-pre-lttri 11313 ax-pre-lttrn 11314 ax-pre-ltadd 11315 ax-pre-mulgt0 11316 ax-riotaBAD 40276

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 2102 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2839 df-nfc 2913 df-ne 2960 df-nel 3066 df-ral 3081 df-rex 3091 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3456 df-sbc 3743 df-csb 3852 df-ss 3901 df-pss 4017 df-dif 4050 df-in 4094 df-un 4154 df-nul 4282 df-if 4486 df-pw 4562 df-sn 4589 df-pr 4591 df-tp 4593 df-op 4841 df-uni 4873 df-int 4913 df-iun 4958 df-iin 4959 df-br 5109 df-opab 5174 df-mpt 5193 df-tr 5219 df-id 5553 df-eprel 5558 df-po 5566 df-so 5567 df-fr 5611 df-we 5613 df-xp 5664 df-rel 5665 df-cnv 5666 df-co 5667 df-dm 5668 df-rn 5669 df-res 5670 df-ima 5671 df-pred 6330 df-ord 6391 df-on 6392 df-lim 6393 df-suc 6394 df-iota 6520 df-fun 6568 df-fn 6569 df-f 6570 df-f1 6571 df-fo 6572 df-f1o 6573 df-fv 6574 df-riota 7408 df-ov 7454 df-oprab 7455 df-mpo 7456 df-of 7725 df-om 7912 df-1st 8035 df-2nd 8036 df-tpos 8281 df-undef 8328 df-frecs 8337 df-wrecs 8368 df-recs 8417 df-rdg 8456 df-1o 8514 df-2o 8515 df-er 8755 df-map 8887 df-en 9012 df-dom 9013 df-sdom 9014 df-fin 9015 df-pnf 11384 df-mnf 11385 df-xr 11386 df-ltxr 11387 df-le 11388 df-sub 11582 df-neg 11583 df-nn 12376 df-2 12445 df-3 12446 df-4 12447 df-5 12448 df-6 12449 df-n0 12647 df-z 12734 df-uz 13006 df-fz 13680 df-struct 17369 df-sets 17386 df-slot 17404 df-ndx 17416 df-base 17432 df-ress 17453 df-plusg 17486 df-mulr 17487 df-sca 17489 df-vsca 17490 df-0g 17662 df-mre 17806 df-mrc 17807 df-acs 17809 df-proset 18518 df-poset 18537 df-plt 18552 df-lub 18568 df-glb 18569 df-join 18570 df-meet 18571 df-p0 18647 df-p1 18648 df-lat 18656 df-clat 18723 df-mgm 18866 df-sgrp 18958 df-mnd 18974 df-submnd 19029 df-grp 19197 df-minusg 19198 df-sbg 19199 df-subg 19383 df-cntz 19582 df-oppg 19611 df-lsm 19901 df-cmn 20047 df-abl 20048 df-mgp 20412 df-rng 20426 df-ur 20459 df-ring 20512 df-oppr 20618 df-dvdsr 20638 df-unit 20639 df-invr 20669 df-dvr 20682 df-nzr 20814 df-rlreg 20997 df-domn 20998 df-drng 21033 df-lmod 21188 df-lss 21258 df-lsp 21298 df-lvec 21429 df-lsatoms 40299 df-lshyp 40300 df-lcv 40342 df-lfl 40381 df-lkr 40409 df-ldual 40447 df-oposet 40499 df-ol 40501 df-oml 40502 df-covers 40589 df-ats 40590 df-atl 40621 df-cvlat 40645 df-hlat 40674 df-llines 40821 df-lplanes 40822 df-lvols 40823 df-lines 40824 df-psubsp 40826 df-pmap 40827 df-padd 41119 df-lhyp 41311 df-laut 41312 df-ldil 41427 df-ltrn 41428 df-trl 41482 df-tgrp 42066 df-tendo 42078 df-edring 42080 df-dveca 42326 df-disoa 42352 df-dvech 42402 df-dib 42462 df-dic 42496 df-dih 42552 df-doch 42671 df-djh 42718 df-lcdual 42910 df-mapd 42948

This theorem is used by: mapdpglem24 43027

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