mapdpglem23 - Mathbox for Norm Megill

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Theorem mapdpglem23 42567

Description: Lemma for mapdpg 42579. 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 2760	. . . 4 ⊢ (LSubSp‘𝑈) = (LSubSp‘𝑈)
5		mapdpglem.c	. . . 4 ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊)
6		eqid 2760	. . . 4 ⊢ (LSubSp‘𝐶) = (LSubSp‘𝐶)
7		mapdpglem.k	. . . 4 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
8	1, 3, 7	dvhlmod 41983	. . . . 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 21161	. . . . 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 42529	. . 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 42563	. . 3 ⊢ (𝜑 → 𝐸 ∈ (𝑀‘(𝑁‘{𝑌})))
38	19, 6	lssel 21121	. . 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 42564	. 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 42566	. 2 ⊢ (𝜑 → (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐺𝑅𝐸)}))
42		sneq 4594	. . . . . 6 ⊢ (ℎ = 𝐸 → {ℎ} = {𝐸})
43	42	fveq2d 6882	. . . . 5 ⊢ (ℎ = 𝐸 → (𝐽‘{ℎ}) = (𝐽‘{𝐸}))
44	43	eqeq2d 2771	. . . 4 ⊢ (ℎ = 𝐸 → ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{ℎ}) ↔ (𝑀‘(𝑁‘{𝑌})) = (𝐽‘{𝐸})))
45		oveq2 7421	. . . . . . 7 ⊢ (ℎ = 𝐸 → (𝐺𝑅ℎ) = (𝐺𝑅𝐸))
46	45	sneqd 4596	. . . . . 6 ⊢ (ℎ = 𝐸 → {(𝐺𝑅ℎ)} = {(𝐺𝑅𝐸)})
47	46	fveq2d 6882	. . . . 5 ⊢ (ℎ = 𝐸 → (𝐽‘{(𝐺𝑅ℎ)}) = (𝐽‘{(𝐺𝑅𝐸)}))
48	47	eqeq2d 2771	. . . 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 3576	. 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 2145 ≠ wne 2955 ∃wrex 3086 {csn 4584 ‘cfv 6533 (class class class)co 7413 Basecbs 17301 Scalarcsca 17345 ·_𝑠 cvsca 17346 0_gc0g 17524 -_gcsg 19059 LSSumclsm 19761 inv_rcinvr 20528 LModclmod 21044 LSubSpclss 21115 LSpanclspn 21155 HLchlt 40223 LHypclh 40857 DVecHcdvh 41951 LCDualclcd 42459 mapdcmpd 42497

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 2213 ax-ext 2732 ax-rep 5232 ax-sep 5251 ax-nul 5263 ax-pow 5330 ax-pr 5398 ax-un 7736 ax-cnex 11180 ax-resscn 11181 ax-1cn 11182 ax-icn 11183 ax-addcl 11184 ax-addrcl 11185 ax-mulcl 11186 ax-mulrcl 11187 ax-mulcom 11188 ax-addass 11189 ax-mulass 11190 ax-distr 11191 ax-i2m1 11192 ax-1ne0 11193 ax-1rid 11194 ax-rnegex 11195 ax-rrecex 11196 ax-cnre 11197 ax-pre-lttri 11198 ax-pre-lttrn 11199 ax-pre-ltadd 11200 ax-pre-mulgt0 11201 ax-riotaBAD 39826

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 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3365 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4591 df-uni 4868 df-int 4908 df-iun 4953 df-iin 4954 df-br 5104 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5550 df-eprel 5555 df-po 5563 df-so 5564 df-fr 5608 df-we 5610 df-xp 5661 df-rel 5662 df-cnv 5663 df-co 5664 df-dm 5665 df-rn 5666 df-res 5667 df-ima 5668 df-pred 6299 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6489 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-riota 7370 df-ov 7416 df-oprab 7417 df-mpo 7418 df-of 7678 df-om 7863 df-1st 7986 df-2nd 7987 df-tpos 8224 df-undef 8271 df-frecs 8280 df-wrecs 8311 df-recs 8360 df-rdg 8399 df-1o 8455 df-2o 8456 df-er 8696 df-map 8828 df-en 8953 df-dom 8954 df-sdom 8955 df-fin 8956 df-pnf 11269 df-mnf 11270 df-xr 11271 df-ltxr 11272 df-le 11273 df-sub 11467 df-neg 11468 df-nn 12258 df-2 12327 df-3 12328 df-4 12329 df-5 12330 df-6 12331 df-n0 12529 df-z 12616 df-uz 12888 df-fz 13562 df-struct 17239 df-sets 17256 df-slot 17274 df-ndx 17286 df-base 17302 df-ress 17323 df-plusg 17355 df-mulr 17356 df-sca 17358 df-vsca 17359 df-0g 17526 df-mre 17670 df-mrc 17671 df-acs 17673 df-proset 18382 df-poset 18401 df-plt 18416 df-lub 18432 df-glb 18433 df-join 18434 df-meet 18435 df-p0 18511 df-p1 18512 df-lat 18520 df-clat 18587 df-mgm 18730 df-sgrp 18821 df-mnd 18837 df-submnd 18892 df-grp 19060 df-minusg 19061 df-sbg 19062 df-subg 19246 df-cntz 19444 df-oppg 19473 df-lsm 19763 df-cmn 19909 df-abl 19910 df-mgp 20274 df-rng 20288 df-ur 20321 df-ring 20374 df-oppr 20478 df-dvdsr 20498 df-unit 20499 df-invr 20529 df-dvr 20542 df-nzr 20673 df-rlreg 20856 df-domn 20857 df-drng 20892 df-lmod 21046 df-lss 21116 df-lsp 21156 df-lvec 21287 df-lsatoms 39849 df-lshyp 39850 df-lcv 39892 df-lfl 39931 df-lkr 39959 df-ldual 39997 df-oposet 40049 df-ol 40051 df-oml 40052 df-covers 40139 df-ats 40140 df-atl 40171 df-cvlat 40195 df-hlat 40224 df-llines 40371 df-lplanes 40372 df-lvols 40373 df-lines 40374 df-psubsp 40376 df-pmap 40377 df-padd 40669 df-lhyp 40861 df-laut 40862 df-ldil 40977 df-ltrn 40978 df-trl 41032 df-tgrp 41616 df-tendo 41628 df-edring 41630 df-dveca 41876 df-disoa 41902 df-dvech 41952 df-dib 42012 df-dic 42046 df-dih 42102 df-doch 42221 df-djh 42268 df-lcdual 42460 df-mapd 42498

This theorem is used by: mapdpglem24 42577

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