mapdpglem16 - Mathbox for Norm Megill

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Theorem mapdpglem16 41686

Description: Lemma for mapdpg 41705. 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 41685	. . . 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 2946	. 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 2925 {csn 4579 ‘cfv 6486 (class class class)co 7353 Basecbs 17139 Scalarcsca 17183 ·_𝑠 cvsca 17184 0_gc0g 17362 -_gcsg 18833 LSSumclsm 19532 LSpanclspn 20893 HLchlt 39348 LHypclh 39983 DVecHcdvh 41077 LCDualclcd 41585 mapdcmpd 41623

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 2701 ax-rep 5221 ax-sep 5238 ax-nul 5248 ax-pow 5307 ax-pr 5374 ax-un 7675 ax-cnex 11084 ax-resscn 11085 ax-1cn 11086 ax-icn 11087 ax-addcl 11088 ax-addrcl 11089 ax-mulcl 11090 ax-mulrcl 11091 ax-mulcom 11092 ax-addass 11093 ax-mulass 11094 ax-distr 11095 ax-i2m1 11096 ax-1ne0 11097 ax-1rid 11098 ax-rnegex 11099 ax-rrecex 11100 ax-cnre 11101 ax-pre-lttri 11102 ax-pre-lttrn 11103 ax-pre-ltadd 11104 ax-pre-mulgt0 11105 ax-riotaBAD 38951

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 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3345 df-reu 3346 df-rab 3397 df-v 3440 df-sbc 3745 df-csb 3854 df-dif 3908 df-un 3910 df-in 3912 df-ss 3922 df-pss 3925 df-nul 4287 df-if 4479 df-pw 4555 df-sn 4580 df-pr 4582 df-tp 4584 df-op 4586 df-uni 4862 df-int 4900 df-iun 4946 df-iin 4947 df-br 5096 df-opab 5158 df-mpt 5177 df-tr 5203 df-id 5518 df-eprel 5523 df-po 5531 df-so 5532 df-fr 5576 df-we 5578 df-xp 5629 df-rel 5630 df-cnv 5631 df-co 5632 df-dm 5633 df-rn 5634 df-res 5635 df-ima 5636 df-pred 6253 df-ord 6314 df-on 6315 df-lim 6316 df-suc 6317 df-iota 6442 df-fun 6488 df-fn 6489 df-f 6490 df-f1 6491 df-fo 6492 df-f1o 6493 df-fv 6494 df-riota 7310 df-ov 7356 df-oprab 7357 df-mpo 7358 df-of 7617 df-om 7807 df-1st 7931 df-2nd 7932 df-tpos 8166 df-undef 8213 df-frecs 8221 df-wrecs 8252 df-recs 8301 df-rdg 8339 df-1o 8395 df-2o 8396 df-er 8632 df-map 8762 df-en 8880 df-dom 8881 df-sdom 8882 df-fin 8883 df-pnf 11170 df-mnf 11171 df-xr 11172 df-ltxr 11173 df-le 11174 df-sub 11368 df-neg 11369 df-nn 12148 df-2 12210 df-3 12211 df-4 12212 df-5 12213 df-6 12214 df-n0 12404 df-z 12491 df-uz 12755 df-fz 13430 df-struct 17077 df-sets 17094 df-slot 17112 df-ndx 17124 df-base 17140 df-ress 17161 df-plusg 17193 df-mulr 17194 df-sca 17196 df-vsca 17197 df-0g 17364 df-mre 17507 df-mrc 17508 df-acs 17510 df-proset 18219 df-poset 18238 df-plt 18253 df-lub 18269 df-glb 18270 df-join 18271 df-meet 18272 df-p0 18348 df-p1 18349 df-lat 18357 df-clat 18424 df-mgm 18533 df-sgrp 18612 df-mnd 18628 df-submnd 18677 df-grp 18834 df-minusg 18835 df-sbg 18836 df-subg 19021 df-cntz 19215 df-oppg 19244 df-lsm 19534 df-cmn 19680 df-abl 19681 df-mgp 20045 df-rng 20057 df-ur 20086 df-ring 20139 df-oppr 20241 df-dvdsr 20261 df-unit 20262 df-invr 20292 df-dvr 20305 df-nzr 20417 df-rlreg 20598 df-domn 20599 df-drng 20635 df-lmod 20784 df-lss 20854 df-lsp 20894 df-lvec 21026 df-lsatoms 38974 df-lshyp 38975 df-lcv 39017 df-lfl 39056 df-lkr 39084 df-ldual 39122 df-oposet 39174 df-ol 39176 df-oml 39177 df-covers 39264 df-ats 39265 df-atl 39296 df-cvlat 39320 df-hlat 39349 df-llines 39497 df-lplanes 39498 df-lvols 39499 df-lines 39500 df-psubsp 39502 df-pmap 39503 df-padd 39795 df-lhyp 39987 df-laut 39988 df-ldil 40103 df-ltrn 40104 df-trl 40158 df-tgrp 40742 df-tendo 40754 df-edring 40756 df-dveca 41002 df-disoa 41028 df-dvech 41078 df-dib 41138 df-dic 41172 df-dih 41228 df-doch 41347 df-djh 41394 df-lcdual 41586 df-mapd 41624

This theorem is referenced by: mapdpglem18 41688

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