mapdpglem19 - Mathbox for Norm Megill

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Theorem mapdpglem19 42749

Description: Lemma for mapdpg 42765. Baer p. 45, line 8: "...is in (Fy)*..." (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
mapdpglem19	⊢ (𝜑 → 𝐸 ∈ (𝑀‘(𝑁‘{𝑌})))

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 Allowed substitution hints: 𝜑(𝑧, 𝑡, 𝑔) 𝐴(𝑧, 𝑡, 𝑔) 𝐵(𝑧, 𝑡) ⊕ (𝑧, 𝑡, 𝑔) 𝑄(𝑧, 𝑡, 𝑔) 𝑅(𝑡) · (𝑡) 𝑈(𝑧, 𝑡, 𝑔) 𝐸(𝑧, 𝑡, 𝑔) 𝐹(𝑧, 𝑡) 𝐺(𝑡) 𝐻(𝑧, 𝑡, 𝑔) 𝐾(𝑧, 𝑡, 𝑔) − (𝑧, 𝑔) 𝑉(𝑧, 𝑡, 𝑔) 𝑊(𝑧, 𝑡, 𝑔) 𝑋(𝑧, 𝑔) 0 (𝑧, 𝑡, 𝑔)

**Proof of Theorem mapdpglem19**
Step	Hyp	Ref	Expression
1		mapdpglem17.ep	. 2 ⊢ 𝐸 = (((inv_r‘𝐴)‘𝑔) · 𝑧)
2		mapdpglem.h	. . 3 ⊢ 𝐻 = (LHyp‘𝐾)
3		mapdpglem.u	. . 3 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
4		mapdpglem3.a	. . 3 ⊢ 𝐴 = (Scalar‘𝑈)
5		mapdpglem3.b	. . 3 ⊢ 𝐵 = (Base‘𝐴)
6		mapdpglem.c	. . 3 ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊)
7		mapdpglem3.f	. . 3 ⊢ 𝐹 = (Base‘𝐶)
8		mapdpglem3.t	. . 3 ⊢ · = ( ·_𝑠 ‘𝐶)
9		eqid 2761	. . 3 ⊢ (LSubSp‘𝐶) = (LSubSp‘𝐶)
10		mapdpglem.k	. . 3 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
11		mapdpglem.m	. . . 4 ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊)
12		eqid 2761	. . . 4 ⊢ (LSubSp‘𝑈) = (LSubSp‘𝑈)
13	2, 3, 10	dvhlmod 42169	. . . . 5 ⊢ (𝜑 → 𝑈 ∈ LMod)
14		mapdpglem.y	. . . . 5 ⊢ (𝜑 → 𝑌 ∈ 𝑉)
15		mapdpglem.v	. . . . . 6 ⊢ 𝑉 = (Base‘𝑈)
16		mapdpglem.n	. . . . . 6 ⊢ 𝑁 = (LSpan‘𝑈)
17	15, 12, 16	lspsncl 21253	. . . . 5 ⊢ ((𝑈 ∈ LMod ∧ 𝑌 ∈ 𝑉) → (𝑁‘{𝑌}) ∈ (LSubSp‘𝑈))
18	13, 14, 17	syl2anc 596	. . . 4 ⊢ (𝜑 → (𝑁‘{𝑌}) ∈ (LSubSp‘𝑈))
19	2, 11, 3, 12, 6, 9, 10, 18	mapdcl2 42715	. . 3 ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑌})) ∈ (LSubSp‘𝐶))
20	2, 3, 10	dvhlvec 42168	. . . . 5 ⊢ (𝜑 → 𝑈 ∈ LVec)
21	4	lvecdrng 21381	. . . . 5 ⊢ (𝑈 ∈ LVec → 𝐴 ∈ DivRing)
22	20, 21	syl 18	. . . 4 ⊢ (𝜑 → 𝐴 ∈ DivRing)
23		mapdpglem4.g4	. . . 4 ⊢ (𝜑 → 𝑔 ∈ 𝐵)
24		mapdpglem.s	. . . . 5 ⊢ − = (-_g‘𝑈)
25		mapdpglem.x	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝑉)
26		mapdpglem1.p	. . . . 5 ⊢ ⊕ = (LSSum‘𝐶)
27		mapdpglem2.j	. . . . 5 ⊢ 𝐽 = (LSpan‘𝐶)
28		mapdpglem3.te	. . . . 5 ⊢ (𝜑 → 𝑡 ∈ ((𝑀‘(𝑁‘{𝑋})) ⊕ (𝑀‘(𝑁‘{𝑌}))))
29		mapdpglem3.r	. . . . 5 ⊢ 𝑅 = (-_g‘𝐶)
30		mapdpglem3.g	. . . . 5 ⊢ (𝜑 → 𝐺 ∈ 𝐹)
31		mapdpglem3.e	. . . . 5 ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐺}))
32		mapdpglem4.q	. . . . 5 ⊢ 𝑄 = (0_g‘𝑈)
33		mapdpglem.ne	. . . . 5 ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}))
34		mapdpglem4.jt	. . . . 5 ⊢ (𝜑 → (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{𝑡}))
35		mapdpglem4.z	. . . . 5 ⊢ 0 = (0_g‘𝐴)
36		mapdpglem4.z4	. . . . 5 ⊢ (𝜑 → 𝑧 ∈ (𝑀‘(𝑁‘{𝑌})))
37		mapdpglem4.t4	. . . . 5 ⊢ (𝜑 → 𝑡 = ((𝑔 · 𝐺)𝑅𝑧))
38		mapdpglem4.xn	. . . . 5 ⊢ (𝜑 → 𝑋 ≠ 𝑄)
39	2, 11, 3, 15, 24, 16, 6, 10, 25, 14, 26, 27, 7, 28, 4, 5, 8, 29, 30, 31, 32, 33, 34, 35, 23, 36, 37, 38	mapdpglem11 42741	. . . 4 ⊢ (𝜑 → 𝑔 ≠ 0 )
40		eqid 2761	. . . . 5 ⊢ (inv_r‘𝐴) = (inv_r‘𝐴)
41	5, 35, 40	drnginvrcl 21012	. . . 4 ⊢ ((𝐴 ∈ DivRing ∧ 𝑔 ∈ 𝐵 ∧ 𝑔 ≠ 0 ) → ((inv_r‘𝐴)‘𝑔) ∈ 𝐵)
42	22, 23, 39, 41	syl3anc 1398	. . 3 ⊢ (𝜑 → ((inv_r‘𝐴)‘𝑔) ∈ 𝐵)
43	2, 3, 4, 5, 6, 7, 8, 9, 10, 19, 42, 36	lcdlssvscl 42665	. 2 ⊢ (𝜑 → (((inv_r‘𝐴)‘𝑔) · 𝑧) ∈ (𝑀‘(𝑁‘{𝑌})))
44	1, 43	eqeltrid 2865	1 ⊢ (𝜑 → 𝐸 ∈ (𝑀‘(𝑁‘{𝑌})))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∧ wa 401 = wceq 1570 ∈ wcel 2145 ≠ wne 2956 {csn 4584 ‘cfv 6539 (class class class)co 7421 Basecbs 17388 Scalarcsca 17432 ·_𝑠 cvsca 17433 0_gc0g 17611 -_gcsg 19147 LSSumclsm 19849 inv_rcinvr 20618 DivRingcdr 20981 LModclmod 21136 LSubSpclss 21207 LSpanclspn 21247 LVecclvec 21378 HLchlt 40409 LHypclh 41043 DVecHcdvh 42137 LCDualclcd 42645 mapdcmpd 42683

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 2733 ax-rep 5232 ax-sep 5249 ax-nul 5260 ax-pow 5327 ax-pr 5391 ax-un 7752 ax-cnex 11257 ax-resscn 11258 ax-1cn 11259 ax-icn 11260 ax-addcl 11261 ax-addrcl 11262 ax-mulcl 11263 ax-mulrcl 11264 ax-mulcom 11265 ax-addass 11266 ax-mulass 11267 ax-distr 11268 ax-i2m1 11269 ax-1ne0 11270 ax-1rid 11271 ax-rnegex 11272 ax-rrecex 11273 ax-cnre 11274 ax-pre-lttri 11275 ax-pre-lttrn 11276 ax-pre-ltadd 11277 ax-pre-mulgt0 11278 ax-riotaBAD 40012

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 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-nel 3063 df-ral 3078 df-rex 3088 df-rmo 3366 df-reu 3367 df-rab 3414 df-v 3453 df-sbc 3740 df-csb 3848 df-dif 3900 df-un 3902 df-in 3904 df-ss 3915 df-pss 4035 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 5546 df-eprel 5551 df-po 5559 df-so 5560 df-fr 5604 df-we 5606 df-xp 5657 df-rel 5658 df-cnv 5659 df-co 5660 df-dm 5661 df-rn 5662 df-res 5663 df-ima 5664 df-pred 6305 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6495 df-fun 6541 df-fn 6542 df-f 6543 df-f1 6544 df-fo 6545 df-f1o 6546 df-fv 6547 df-riota 7378 df-ov 7424 df-oprab 7425 df-mpo 7426 df-of 7694 df-om 7879 df-1st 8002 df-2nd 8003 df-tpos 8244 df-undef 8291 df-frecs 8300 df-wrecs 8331 df-recs 8380 df-rdg 8419 df-1o 8477 df-2o 8478 df-er 8718 df-map 8850 df-en 8975 df-dom 8976 df-sdom 8977 df-fin 8978 df-pnf 11346 df-mnf 11347 df-xr 11348 df-ltxr 11349 df-le 11350 df-sub 11544 df-neg 11545 df-nn 12337 df-2 12406 df-3 12407 df-4 12408 df-5 12409 df-6 12410 df-n0 12608 df-z 12695 df-uz 12967 df-fz 13641 df-struct 17326 df-sets 17343 df-slot 17361 df-ndx 17373 df-base 17389 df-ress 17410 df-plusg 17442 df-mulr 17443 df-sca 17445 df-vsca 17446 df-0g 17613 df-mre 17757 df-mrc 17758 df-acs 17760 df-proset 18469 df-poset 18488 df-plt 18503 df-lub 18519 df-glb 18520 df-join 18521 df-meet 18522 df-p0 18598 df-p1 18599 df-lat 18607 df-clat 18674 df-mgm 18817 df-sgrp 18909 df-mnd 18925 df-submnd 18980 df-grp 19148 df-minusg 19149 df-sbg 19150 df-subg 19334 df-cntz 19532 df-oppg 19561 df-lsm 19851 df-cmn 19997 df-abl 19998 df-mgp 20362 df-rng 20376 df-ur 20409 df-ring 20462 df-oppr 20568 df-dvdsr 20588 df-unit 20589 df-invr 20619 df-dvr 20632 df-nzr 20764 df-rlreg 20947 df-domn 20948 df-drng 20983 df-lmod 21138 df-lss 21208 df-lsp 21248 df-lvec 21379 df-lsatoms 40035 df-lshyp 40036 df-lcv 40078 df-lfl 40117 df-lkr 40145 df-ldual 40183 df-oposet 40235 df-ol 40237 df-oml 40238 df-covers 40325 df-ats 40326 df-atl 40357 df-cvlat 40381 df-hlat 40410 df-llines 40557 df-lplanes 40558 df-lvols 40559 df-lines 40560 df-psubsp 40562 df-pmap 40563 df-padd 40855 df-lhyp 41047 df-laut 41048 df-ldil 41163 df-ltrn 41164 df-trl 41218 df-tgrp 41802 df-tendo 41814 df-edring 41816 df-dveca 42062 df-disoa 42088 df-dvech 42138 df-dib 42198 df-dic 42232 df-dih 42288 df-doch 42407 df-djh 42454 df-lcdual 42646 df-mapd 42684

This theorem is used by: mapdpglem20 42750 mapdpglem23 42753

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