mapdpglem16 - Mathbox for Norm Megill

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Theorem mapdpglem16 42884

Description: Lemma for mapdpg 42903. 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 486	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 = (0_g‘𝐶)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
11		mapdpglem.x	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ 𝑉)
12	11	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 = (0_g‘𝐶)) → 𝑋 ∈ 𝑉)
13		mapdpglem.y	. . . . . 6 ⊢ (𝜑 → 𝑌 ∈ 𝑉)
14	13	adantr 486	. . . . 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 486	. . . . 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 486	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 = (0_g‘𝐶)) → 𝐺 ∈ 𝐹)
26		mapdpglem3.e	. . . . . 6 ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐺}))
27	26	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 = (0_g‘𝐶)) → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐺}))
28		mapdpglem4.q	. . . . 5 ⊢ 𝑄 = (0_g‘𝑈)
29	1	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 = (0_g‘𝐶)) → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}))
30		mapdpglem4.jt	. . . . . 6 ⊢ (𝜑 → (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{𝑡}))
31	30	adantr 486	. . . . 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 486	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 = (0_g‘𝐶)) → 𝑔 ∈ 𝐵)
35		mapdpglem4.z4	. . . . . 6 ⊢ (𝜑 → 𝑧 ∈ (𝑀‘(𝑁‘{𝑌})))
36	35	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 = (0_g‘𝐶)) → 𝑧 ∈ (𝑀‘(𝑁‘{𝑌})))
37		mapdpglem4.t4	. . . . . 6 ⊢ (𝜑 → 𝑡 = ((𝑔 · 𝐺)𝑅𝑧))
38	37	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 = (0_g‘𝐶)) → 𝑡 = ((𝑔 · 𝐺)𝑅𝑧))
39		mapdpglem4.xn	. . . . . 6 ⊢ (𝜑 → 𝑋 ≠ 𝑄)
40	39	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 = (0_g‘𝐶)) → 𝑋 ≠ 𝑄)
41		mapdpglem12.yn	. . . . . 6 ⊢ (𝜑 → 𝑌 ≠ 𝑄)
42	41	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 = (0_g‘𝐶)) → 𝑌 ≠ 𝑄)
43		simpr 490	. . . . 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 42883	. . . 4 ⊢ ((𝜑 ∧ 𝑧 = (0_g‘𝐶)) → (𝑁‘{𝑋}) = (𝑁‘{𝑌}))
45	44	ex 418	. . 3 ⊢ (𝜑 → (𝑧 = (0_g‘𝐶) → (𝑁‘{𝑋}) = (𝑁‘{𝑌})))
46	45	necon3d 2979	. 2 ⊢ (𝜑 → ((𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}) → 𝑧 ≠ (0_g‘𝐶)))
47	1, 46	mpd 16	1 ⊢ (𝜑 → 𝑧 ≠ (0_g‘𝐶))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∧ wa 401 = wceq 1569 ∈ wcel 2146 ≠ wne 2958 {csn 4587 ‘cfv 6546 (class class class)co 7427 Basecbs 17399 Scalarcsca 17443 ·_𝑠 cvsca 17444 0_gc0g 17622 -_gcsg 19158 LSSumclsm 19860 LSpanclspn 21258 HLchlt 40547 LHypclh 41181 DVecHcdvh 42275 LCDualclcd 42783 mapdcmpd 42821

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1827 ax-4 1841 ax-5 1944 ax-6 2001 ax-7 2042 ax-8 2148 ax-9 2156 ax-10 2179 ax-11 2196 ax-12 2215 ax-ext 2735 ax-rep 5234 ax-sep 5251 ax-nul 5262 ax-pow 5330 ax-pr 5394 ax-un 7758 ax-cnex 11266 ax-resscn 11267 ax-1cn 11268 ax-icn 11269 ax-addcl 11270 ax-addrcl 11271 ax-mulcl 11272 ax-mulrcl 11273 ax-mulcom 11274 ax-addass 11275 ax-mulass 11276 ax-distr 11277 ax-i2m1 11278 ax-1ne0 11279 ax-1rid 11280 ax-rnegex 11281 ax-rrecex 11282 ax-cnre 11283 ax-pre-lttri 11284 ax-pre-lttrn 11285 ax-pre-ltadd 11286 ax-pre-mulgt0 11287 ax-riotaBAD 40150

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1572 df-fal 1582 df-ex 1812 df-nf 1816 df-sb 2101 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2838 df-nfc 2912 df-ne 2959 df-nel 3065 df-ral 3080 df-rex 3090 df-rmo 3368 df-reu 3369 df-rab 3416 df-v 3455 df-sbc 3742 df-csb 3851 df-ss 3900 df-pss 4016 df-dif 4049 df-in 4093 df-un 4153 df-nul 4281 df-if 4485 df-pw 4561 df-sn 4588 df-pr 4590 df-tp 4592 df-op 4838 df-uni 4870 df-int 4910 df-iun 4955 df-iin 4956 df-br 5106 df-opab 5170 df-mpt 5189 df-tr 5215 df-id 5549 df-eprel 5554 df-po 5562 df-so 5563 df-fr 5607 df-we 5609 df-xp 5660 df-rel 5661 df-cnv 5662 df-co 5663 df-dm 5664 df-rn 5665 df-res 5666 df-ima 5667 df-pred 6310 df-ord 6371 df-on 6372 df-lim 6373 df-suc 6374 df-iota 6500 df-fun 6548 df-fn 6549 df-f 6550 df-f1 6551 df-fo 6552 df-f1o 6553 df-fv 6554 df-riota 7384 df-ov 7430 df-oprab 7431 df-mpo 7432 df-of 7700 df-om 7885 df-1st 8008 df-2nd 8009 df-tpos 8254 df-undef 8301 df-frecs 8310 df-wrecs 8341 df-recs 8390 df-rdg 8429 df-1o 8487 df-2o 8488 df-er 8728 df-map 8860 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-pnf 11355 df-mnf 11356 df-xr 11357 df-ltxr 11358 df-le 11359 df-sub 11553 df-neg 11554 df-nn 12346 df-2 12415 df-3 12416 df-4 12417 df-5 12418 df-6 12419 df-n0 12617 df-z 12704 df-uz 12976 df-fz 13650 df-struct 17337 df-sets 17354 df-slot 17372 df-ndx 17384 df-base 17400 df-ress 17421 df-plusg 17453 df-mulr 17454 df-sca 17456 df-vsca 17457 df-0g 17624 df-mre 17768 df-mrc 17769 df-acs 17771 df-proset 18480 df-poset 18499 df-plt 18514 df-lub 18530 df-glb 18531 df-join 18532 df-meet 18533 df-p0 18609 df-p1 18610 df-lat 18618 df-clat 18685 df-mgm 18828 df-sgrp 18920 df-mnd 18936 df-submnd 18991 df-grp 19159 df-minusg 19160 df-sbg 19161 df-subg 19345 df-cntz 19543 df-oppg 19572 df-lsm 19862 df-cmn 20008 df-abl 20009 df-mgp 20373 df-rng 20387 df-ur 20420 df-ring 20473 df-oppr 20579 df-dvdsr 20599 df-unit 20600 df-invr 20630 df-dvr 20643 df-nzr 20775 df-rlreg 20958 df-domn 20959 df-drng 20994 df-lmod 21149 df-lss 21219 df-lsp 21259 df-lvec 21390 df-lsatoms 40173 df-lshyp 40174 df-lcv 40216 df-lfl 40255 df-lkr 40283 df-ldual 40321 df-oposet 40373 df-ol 40375 df-oml 40376 df-covers 40463 df-ats 40464 df-atl 40495 df-cvlat 40519 df-hlat 40548 df-llines 40695 df-lplanes 40696 df-lvols 40697 df-lines 40698 df-psubsp 40700 df-pmap 40701 df-padd 40993 df-lhyp 41185 df-laut 41186 df-ldil 41301 df-ltrn 41302 df-trl 41356 df-tgrp 41940 df-tendo 41952 df-edring 41954 df-dveca 42200 df-disoa 42226 df-dvech 42276 df-dib 42336 df-dic 42370 df-dih 42426 df-doch 42545 df-djh 42592 df-lcdual 42784 df-mapd 42822

This theorem is used by: mapdpglem18 42886

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