mapdpglem23 - Mathbox for Norm Megill

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Theorem mapdpglem23 38380

Description: Lemma for mapdpg 38392. 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 2795	. . . 4 ⊢ (LSubSp‘𝑈) = (LSubSp‘𝑈)
5		mapdpglem.c	. . . 4 ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊)
6		eqid 2795	. . . 4 ⊢ (LSubSp‘𝐶) = (LSubSp‘𝐶)
7		mapdpglem.k	. . . 4 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
8	1, 3, 7	dvhlmod 37796	. . . . 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 19439	. . . . 5 ⊢ ((𝑈 ∈ LMod ∧ 𝑌 ∈ 𝑉) → (𝑁‘{𝑌}) ∈ (LSubSp‘𝑈))
13	8, 9, 12	syl2anc 584	. . . 4 ⊢ (𝜑 → (𝑁‘{𝑌}) ∈ (LSubSp‘𝑈))
14	1, 2, 3, 4, 5, 6, 7, 13	mapdcl2 38342	. . 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 38376	. . 3 ⊢ (𝜑 → 𝐸 ∈ (𝑀‘(𝑁‘{𝑌})))
38	19, 6	lssel 19399	. . 3 ⊢ (((𝑀‘(𝑁‘{𝑌})) ∈ (LSubSp‘𝐶) ∧ 𝐸 ∈ (𝑀‘(𝑁‘{𝑌}))) → 𝐸 ∈ 𝐹)
39	14, 37, 38	syl2anc 584	. 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 38377	. 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 38379	. 2 ⊢ (𝜑 → (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐺𝑅𝐸)}))
42		sneq 4482	. . . . . 6 ⊢ (ℎ = 𝐸 → {ℎ} = {𝐸})
43	42	fveq2d 6542	. . . . 5 ⊢ (ℎ = 𝐸 → (𝐽‘{ℎ}) = (𝐽‘{𝐸}))
44	43	eqeq2d 2805	. . . 4 ⊢ (ℎ = 𝐸 → ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{ℎ}) ↔ (𝑀‘(𝑁‘{𝑌})) = (𝐽‘{𝐸})))
45		oveq2 7024	. . . . . . 7 ⊢ (ℎ = 𝐸 → (𝐺𝑅ℎ) = (𝐺𝑅𝐸))
46	45	sneqd 4484	. . . . . 6 ⊢ (ℎ = 𝐸 → {(𝐺𝑅ℎ)} = {(𝐺𝑅𝐸)})
47	46	fveq2d 6542	. . . . 5 ⊢ (ℎ = 𝐸 → (𝐽‘{(𝐺𝑅ℎ)}) = (𝐽‘{(𝐺𝑅𝐸)}))
48	47	eqeq2d 2805	. . . 4 ⊢ (ℎ = 𝐸 → ((𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐺𝑅ℎ)}) ↔ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐺𝑅𝐸)})))
49	44, 48	anbi12d 630	. . 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 3559	. 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 833	1 ⊢ (𝜑 → ∃ℎ ∈ 𝐹 ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐺𝑅ℎ)})))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 396 = wceq 1522 ∈ wcel 2081 ≠ wne 2984 ∃wrex 3106 {csn 4472 ‘cfv 6225 (class class class)co 7016 Basecbs 16312 Scalarcsca 16397 ·_𝑠 cvsca 16398 0_gc0g 16542 -_gcsg 17863 LSSumclsm 18489 inv_rcinvr 19111 LModclmod 19324 LSubSpclss 19393 LSpanclspn 19433 HLchlt 36036 LHypclh 36670 DVecHcdvh 37764 LCDualclcd 38272 mapdcmpd 38310

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1777 ax-4 1791 ax-5 1888 ax-6 1947 ax-7 1992 ax-8 2083 ax-9 2091 ax-10 2112 ax-11 2126 ax-12 2141 ax-13 2344 ax-ext 2769 ax-rep 5081 ax-sep 5094 ax-nul 5101 ax-pow 5157 ax-pr 5221 ax-un 7319 ax-cnex 10439 ax-resscn 10440 ax-1cn 10441 ax-icn 10442 ax-addcl 10443 ax-addrcl 10444 ax-mulcl 10445 ax-mulrcl 10446 ax-mulcom 10447 ax-addass 10448 ax-mulass 10449 ax-distr 10450 ax-i2m1 10451 ax-1ne0 10452 ax-1rid 10453 ax-rnegex 10454 ax-rrecex 10455 ax-cnre 10456 ax-pre-lttri 10457 ax-pre-lttrn 10458 ax-pre-ltadd 10459 ax-pre-mulgt0 10460 ax-riotaBAD 35639

This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1081 df-3an 1082 df-tru 1525 df-fal 1535 df-ex 1762 df-nf 1766 df-sb 2043 df-mo 2576 df-eu 2612 df-clab 2776 df-cleq 2788 df-clel 2863 df-nfc 2935 df-ne 2985 df-nel 3091 df-ral 3110 df-rex 3111 df-reu 3112 df-rmo 3113 df-rab 3114 df-v 3439 df-sbc 3707 df-csb 3812 df-dif 3862 df-un 3864 df-in 3866 df-ss 3874 df-pss 3876 df-nul 4212 df-if 4382 df-pw 4455 df-sn 4473 df-pr 4475 df-tp 4477 df-op 4479 df-uni 4746 df-int 4783 df-iun 4827 df-iin 4828 df-br 4963 df-opab 5025 df-mpt 5042 df-tr 5064 df-id 5348 df-eprel 5353 df-po 5362 df-so 5363 df-fr 5402 df-we 5404 df-xp 5449 df-rel 5450 df-cnv 5451 df-co 5452 df-dm 5453 df-rn 5454 df-res 5455 df-ima 5456 df-pred 6023 df-ord 6069 df-on 6070 df-lim 6071 df-suc 6072 df-iota 6189 df-fun 6227 df-fn 6228 df-f 6229 df-f1 6230 df-fo 6231 df-f1o 6232 df-fv 6233 df-riota 6977 df-ov 7019 df-oprab 7020 df-mpo 7021 df-of 7267 df-om 7437 df-1st 7545 df-2nd 7546 df-tpos 7743 df-undef 7790 df-wrecs 7798 df-recs 7860 df-rdg 7898 df-1o 7953 df-oadd 7957 df-er 8139 df-map 8258 df-en 8358 df-dom 8359 df-sdom 8360 df-fin 8361 df-pnf 10523 df-mnf 10524 df-xr 10525 df-ltxr 10526 df-le 10527 df-sub 10719 df-neg 10720 df-nn 11487 df-2 11548 df-3 11549 df-4 11550 df-5 11551 df-6 11552 df-n0 11746 df-z 11830 df-uz 12094 df-fz 12743 df-struct 16314 df-ndx 16315 df-slot 16316 df-base 16318 df-sets 16319 df-ress 16320 df-plusg 16407 df-mulr 16408 df-sca 16410 df-vsca 16411 df-0g 16544 df-mre 16686 df-mrc 16687 df-acs 16689 df-proset 17367 df-poset 17385 df-plt 17397 df-lub 17413 df-glb 17414 df-join 17415 df-meet 17416 df-p0 17478 df-p1 17479 df-lat 17485 df-clat 17547 df-mgm 17681 df-sgrp 17723 df-mnd 17734 df-submnd 17775 df-grp 17864 df-minusg 17865 df-sbg 17866 df-subg 18030 df-cntz 18188 df-oppg 18215 df-lsm 18491 df-cmn 18635 df-abl 18636 df-mgp 18930 df-ur 18942 df-ring 18989 df-oppr 19063 df-dvdsr 19081 df-unit 19082 df-invr 19112 df-dvr 19123 df-drng 19194 df-lmod 19326 df-lss 19394 df-lsp 19434 df-lvec 19565 df-lsatoms 35662 df-lshyp 35663 df-lcv 35705 df-lfl 35744 df-lkr 35772 df-ldual 35810 df-oposet 35862 df-ol 35864 df-oml 35865 df-covers 35952 df-ats 35953 df-atl 35984 df-cvlat 36008 df-hlat 36037 df-llines 36184 df-lplanes 36185 df-lvols 36186 df-lines 36187 df-psubsp 36189 df-pmap 36190 df-padd 36482 df-lhyp 36674 df-laut 36675 df-ldil 36790 df-ltrn 36791 df-trl 36845 df-tgrp 37429 df-tendo 37441 df-edring 37443 df-dveca 37689 df-disoa 37715 df-dvech 37765 df-dib 37825 df-dic 37859 df-dih 37915 df-doch 38034 df-djh 38081 df-lcdual 38273 df-mapd 38311

This theorem is referenced by: mapdpglem24 38390

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