mapdpglem16 - Mathbox for Norm Megill

	Mathbox for Norm Megill		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > mapdpglem16		Structured version Visualization version GIF version

Theorem mapdpglem16 41711

Description: Lemma for mapdpg 41730. Baer p. 45, line 7: "Likewise we see that z =/= 0." (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)

**Assertion**
Ref	Expression
mapdpglem16	⊢ (𝜑 → 𝑧 ≠ (0_g‘𝐶))

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) 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 mapdpglem16**
Step	Hyp	Ref	Expression
1		mapdpglem.ne	. 2 ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}))
2		mapdpglem.h	. . . . 5 ⊢ 𝐻 = (LHyp‘𝐾)
3		mapdpglem.m	. . . . 5 ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊)
4		mapdpglem.u	. . . . 5 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
5		mapdpglem.v	. . . . 5 ⊢ 𝑉 = (Base‘𝑈)
6		mapdpglem.s	. . . . 5 ⊢ − = (-_g‘𝑈)
7		mapdpglem.n	. . . . 5 ⊢ 𝑁 = (LSpan‘𝑈)
8		mapdpglem.c	. . . . 5 ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊)
9		mapdpglem.k	. . . . . 6 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
10	9	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 = (0_g‘𝐶)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
11		mapdpglem.x	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ 𝑉)
12	11	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 = (0_g‘𝐶)) → 𝑋 ∈ 𝑉)
13		mapdpglem.y	. . . . . 6 ⊢ (𝜑 → 𝑌 ∈ 𝑉)
14	13	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 = (0_g‘𝐶)) → 𝑌 ∈ 𝑉)
15		mapdpglem1.p	. . . . 5 ⊢ ⊕ = (LSSum‘𝐶)
16		mapdpglem2.j	. . . . 5 ⊢ 𝐽 = (LSpan‘𝐶)
17		mapdpglem3.f	. . . . 5 ⊢ 𝐹 = (Base‘𝐶)
18		mapdpglem3.te	. . . . . 6 ⊢ (𝜑 → 𝑡 ∈ ((𝑀‘(𝑁‘{𝑋})) ⊕ (𝑀‘(𝑁‘{𝑌}))))
19	18	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 = (0_g‘𝐶)) → 𝑡 ∈ ((𝑀‘(𝑁‘{𝑋})) ⊕ (𝑀‘(𝑁‘{𝑌}))))
20		mapdpglem3.a	. . . . 5 ⊢ 𝐴 = (Scalar‘𝑈)
21		mapdpglem3.b	. . . . 5 ⊢ 𝐵 = (Base‘𝐴)
22		mapdpglem3.t	. . . . 5 ⊢ · = ( ·_𝑠 ‘𝐶)
23		mapdpglem3.r	. . . . 5 ⊢ 𝑅 = (-_g‘𝐶)
24		mapdpglem3.g	. . . . . 6 ⊢ (𝜑 → 𝐺 ∈ 𝐹)
25	24	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 = (0_g‘𝐶)) → 𝐺 ∈ 𝐹)
26		mapdpglem3.e	. . . . . 6 ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐺}))
27	26	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 = (0_g‘𝐶)) → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐺}))
28		mapdpglem4.q	. . . . 5 ⊢ 𝑄 = (0_g‘𝑈)
29	1	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 = (0_g‘𝐶)) → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}))
30		mapdpglem4.jt	. . . . . 6 ⊢ (𝜑 → (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{𝑡}))
31	30	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 = (0_g‘𝐶)) → (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{𝑡}))
32		mapdpglem4.z	. . . . 5 ⊢ 0 = (0_g‘𝐴)
33		mapdpglem4.g4	. . . . . 6 ⊢ (𝜑 → 𝑔 ∈ 𝐵)
34	33	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 = (0_g‘𝐶)) → 𝑔 ∈ 𝐵)
35		mapdpglem4.z4	. . . . . 6 ⊢ (𝜑 → 𝑧 ∈ (𝑀‘(𝑁‘{𝑌})))
36	35	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 = (0_g‘𝐶)) → 𝑧 ∈ (𝑀‘(𝑁‘{𝑌})))
37		mapdpglem4.t4	. . . . . 6 ⊢ (𝜑 → 𝑡 = ((𝑔 · 𝐺)𝑅𝑧))
38	37	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 = (0_g‘𝐶)) → 𝑡 = ((𝑔 · 𝐺)𝑅𝑧))
39		mapdpglem4.xn	. . . . . 6 ⊢ (𝜑 → 𝑋 ≠ 𝑄)
40	39	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 = (0_g‘𝐶)) → 𝑋 ≠ 𝑄)
41		mapdpglem12.yn	. . . . . 6 ⊢ (𝜑 → 𝑌 ≠ 𝑄)
42	41	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 = (0_g‘𝐶)) → 𝑌 ≠ 𝑄)
43		simpr 484	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 = (0_g‘𝐶)) → 𝑧 = (0_g‘𝐶))
44	2, 3, 4, 5, 6, 7, 8, 10, 12, 14, 15, 16, 17, 19, 20, 21, 22, 23, 25, 27, 28, 29, 31, 32, 34, 36, 38, 40, 42, 43	mapdpglem15 41710	. . . 4 ⊢ ((𝜑 ∧ 𝑧 = (0_g‘𝐶)) → (𝑁‘{𝑋}) = (𝑁‘{𝑌}))
45	44	ex 412	. . 3 ⊢ (𝜑 → (𝑧 = (0_g‘𝐶) → (𝑁‘{𝑋}) = (𝑁‘{𝑌})))
46	45	necon3d 2954	. 2 ⊢ (𝜑 → ((𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}) → 𝑧 ≠ (0_g‘𝐶)))
47	1, 46	mpd 15	1 ⊢ (𝜑 → 𝑧 ≠ (0_g‘𝐶))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2109 ≠ wne 2933 {csn 4606 ‘cfv 6536 (class class class)co 7410 Basecbs 17233 Scalarcsca 17279 ·_𝑠 cvsca 17280 0_gc0g 17458 -_gcsg 18923 LSSumclsm 19620 LSpanclspn 20933 HLchlt 39373 LHypclh 40008 DVecHcdvh 41102 LCDualclcd 41610 mapdcmpd 41648

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 2708 ax-rep 5254 ax-sep 5271 ax-nul 5281 ax-pow 5340 ax-pr 5407 ax-un 7734 ax-cnex 11190 ax-resscn 11191 ax-1cn 11192 ax-icn 11193 ax-addcl 11194 ax-addrcl 11195 ax-mulcl 11196 ax-mulrcl 11197 ax-mulcom 11198 ax-addass 11199 ax-mulass 11200 ax-distr 11201 ax-i2m1 11202 ax-1ne0 11203 ax-1rid 11204 ax-rnegex 11205 ax-rrecex 11206 ax-cnre 11207 ax-pre-lttri 11208 ax-pre-lttrn 11209 ax-pre-ltadd 11210 ax-pre-mulgt0 11211 ax-riotaBAD 38976

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 2540 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2810 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3062 df-rmo 3364 df-reu 3365 df-rab 3421 df-v 3466 df-sbc 3771 df-csb 3880 df-dif 3934 df-un 3936 df-in 3938 df-ss 3948 df-pss 3951 df-nul 4314 df-if 4506 df-pw 4582 df-sn 4607 df-pr 4609 df-tp 4611 df-op 4613 df-uni 4889 df-int 4928 df-iun 4974 df-iin 4975 df-br 5125 df-opab 5187 df-mpt 5207 df-tr 5235 df-id 5553 df-eprel 5558 df-po 5566 df-so 5567 df-fr 5611 df-we 5613 df-xp 5665 df-rel 5666 df-cnv 5667 df-co 5668 df-dm 5669 df-rn 5670 df-res 5671 df-ima 5672 df-pred 6295 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6489 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-riota 7367 df-ov 7413 df-oprab 7414 df-mpo 7415 df-of 7676 df-om 7867 df-1st 7993 df-2nd 7994 df-tpos 8230 df-undef 8277 df-frecs 8285 df-wrecs 8316 df-recs 8390 df-rdg 8429 df-1o 8485 df-2o 8486 df-er 8724 df-map 8847 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-pnf 11276 df-mnf 11277 df-xr 11278 df-ltxr 11279 df-le 11280 df-sub 11473 df-neg 11474 df-nn 12246 df-2 12308 df-3 12309 df-4 12310 df-5 12311 df-6 12312 df-n0 12507 df-z 12594 df-uz 12858 df-fz 13530 df-struct 17171 df-sets 17188 df-slot 17206 df-ndx 17218 df-base 17234 df-ress 17257 df-plusg 17289 df-mulr 17290 df-sca 17292 df-vsca 17293 df-0g 17460 df-mre 17603 df-mrc 17604 df-acs 17606 df-proset 18311 df-poset 18330 df-plt 18345 df-lub 18361 df-glb 18362 df-join 18363 df-meet 18364 df-p0 18440 df-p1 18441 df-lat 18447 df-clat 18514 df-mgm 18623 df-sgrp 18702 df-mnd 18718 df-submnd 18767 df-grp 18924 df-minusg 18925 df-sbg 18926 df-subg 19111 df-cntz 19305 df-oppg 19334 df-lsm 19622 df-cmn 19768 df-abl 19769 df-mgp 20106 df-rng 20118 df-ur 20147 df-ring 20200 df-oppr 20302 df-dvdsr 20322 df-unit 20323 df-invr 20353 df-dvr 20366 df-nzr 20478 df-rlreg 20659 df-domn 20660 df-drng 20696 df-lmod 20824 df-lss 20894 df-lsp 20934 df-lvec 21066 df-lsatoms 38999 df-lshyp 39000 df-lcv 39042 df-lfl 39081 df-lkr 39109 df-ldual 39147 df-oposet 39199 df-ol 39201 df-oml 39202 df-covers 39289 df-ats 39290 df-atl 39321 df-cvlat 39345 df-hlat 39374 df-llines 39522 df-lplanes 39523 df-lvols 39524 df-lines 39525 df-psubsp 39527 df-pmap 39528 df-padd 39820 df-lhyp 40012 df-laut 40013 df-ldil 40128 df-ltrn 40129 df-trl 40183 df-tgrp 40767 df-tendo 40779 df-edring 40781 df-dveca 41027 df-disoa 41053 df-dvech 41103 df-dib 41163 df-dic 41197 df-dih 41253 df-doch 41372 df-djh 41419 df-lcdual 41611 df-mapd 41649

This theorem is referenced by: mapdpglem18 41713

W3C validator