mapdpglem11 - Mathbox for Norm Megill

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Theorem mapdpglem11 38820

Description: Lemma for mapdpg 38844. (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)

**Assertion**
Ref	Expression
mapdpglem11	⊢ (𝜑 → 𝑔 ≠ 0 )

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 mapdpglem11**
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 483	. . . . 5 ⊢ ((𝜑 ∧ 𝑔 = 0 ) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
11		mapdpglem.x	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ 𝑉)
12	11	adantr 483	. . . . 5 ⊢ ((𝜑 ∧ 𝑔 = 0 ) → 𝑋 ∈ 𝑉)
13		mapdpglem.y	. . . . . 6 ⊢ (𝜑 → 𝑌 ∈ 𝑉)
14	13	adantr 483	. . . . 5 ⊢ ((𝜑 ∧ 𝑔 = 0 ) → 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 483	. . . . 5 ⊢ ((𝜑 ∧ 𝑔 = 0 ) → 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 483	. . . . 5 ⊢ ((𝜑 ∧ 𝑔 = 0 ) → 𝐺 ∈ 𝐹)
26		mapdpglem3.e	. . . . . 6 ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐺}))
27	26	adantr 483	. . . . 5 ⊢ ((𝜑 ∧ 𝑔 = 0 ) → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐺}))
28		mapdpglem4.q	. . . . 5 ⊢ 𝑄 = (0_g‘𝑈)
29	1	adantr 483	. . . . 5 ⊢ ((𝜑 ∧ 𝑔 = 0 ) → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}))
30		mapdpglem4.jt	. . . . . 6 ⊢ (𝜑 → (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{𝑡}))
31	30	adantr 483	. . . . 5 ⊢ ((𝜑 ∧ 𝑔 = 0 ) → (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{𝑡}))
32		mapdpglem4.z	. . . . 5 ⊢ 0 = (0_g‘𝐴)
33		mapdpglem4.g4	. . . . . 6 ⊢ (𝜑 → 𝑔 ∈ 𝐵)
34	33	adantr 483	. . . . 5 ⊢ ((𝜑 ∧ 𝑔 = 0 ) → 𝑔 ∈ 𝐵)
35		mapdpglem4.z4	. . . . . 6 ⊢ (𝜑 → 𝑧 ∈ (𝑀‘(𝑁‘{𝑌})))
36	35	adantr 483	. . . . 5 ⊢ ((𝜑 ∧ 𝑔 = 0 ) → 𝑧 ∈ (𝑀‘(𝑁‘{𝑌})))
37		mapdpglem4.t4	. . . . . 6 ⊢ (𝜑 → 𝑡 = ((𝑔 · 𝐺)𝑅𝑧))
38	37	adantr 483	. . . . 5 ⊢ ((𝜑 ∧ 𝑔 = 0 ) → 𝑡 = ((𝑔 · 𝐺)𝑅𝑧))
39		mapdpglem4.xn	. . . . . 6 ⊢ (𝜑 → 𝑋 ≠ 𝑄)
40	39	adantr 483	. . . . 5 ⊢ ((𝜑 ∧ 𝑔 = 0 ) → 𝑋 ≠ 𝑄)
41		simpr 487	. . . . 5 ⊢ ((𝜑 ∧ 𝑔 = 0 ) → 𝑔 = 0 )
42	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, 41	mapdpglem10 38819	. . . 4 ⊢ ((𝜑 ∧ 𝑔 = 0 ) → (𝑁‘{𝑋}) = (𝑁‘{𝑌}))
43	42	ex 415	. . 3 ⊢ (𝜑 → (𝑔 = 0 → (𝑁‘{𝑋}) = (𝑁‘{𝑌})))
44	43	necon3d 3039	. 2 ⊢ (𝜑 → ((𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}) → 𝑔 ≠ 0 ))
45	1, 44	mpd 15	1 ⊢ (𝜑 → 𝑔 ≠ 0 )

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ≠ wne 3018 {csn 4569 ‘cfv 6357 (class class class)co 7158 Basecbs 16485 Scalarcsca 16570 ·_𝑠 cvsca 16571 0_gc0g 16715 -_gcsg 18107 LSSumclsm 18761 LSpanclspn 19745 HLchlt 36488 LHypclh 37122 DVecHcdvh 38216 LCDualclcd 38724 mapdcmpd 38762

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 ax-riotaBAD 36091

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-fal 1550 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-iin 4924 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-of 7411 df-om 7583 df-1st 7691 df-2nd 7692 df-tpos 7894 df-undef 7941 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-1o 8104 df-oadd 8108 df-er 8291 df-map 8410 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-nn 11641 df-2 11703 df-3 11704 df-4 11705 df-5 11706 df-6 11707 df-n0 11901 df-z 11985 df-uz 12247 df-fz 12896 df-struct 16487 df-ndx 16488 df-slot 16489 df-base 16491 df-sets 16492 df-ress 16493 df-plusg 16580 df-mulr 16581 df-sca 16583 df-vsca 16584 df-0g 16717 df-mre 16859 df-mrc 16860 df-acs 16862 df-proset 17540 df-poset 17558 df-plt 17570 df-lub 17586 df-glb 17587 df-join 17588 df-meet 17589 df-p0 17651 df-p1 17652 df-lat 17658 df-clat 17720 df-mgm 17854 df-sgrp 17903 df-mnd 17914 df-submnd 17959 df-grp 18108 df-minusg 18109 df-sbg 18110 df-subg 18278 df-cntz 18449 df-oppg 18476 df-lsm 18763 df-cmn 18910 df-abl 18911 df-mgp 19242 df-ur 19254 df-ring 19301 df-oppr 19375 df-dvdsr 19393 df-unit 19394 df-invr 19424 df-dvr 19435 df-drng 19506 df-lmod 19638 df-lss 19706 df-lsp 19746 df-lvec 19877 df-lsatoms 36114 df-lshyp 36115 df-lcv 36157 df-lfl 36196 df-lkr 36224 df-ldual 36262 df-oposet 36314 df-ol 36316 df-oml 36317 df-covers 36404 df-ats 36405 df-atl 36436 df-cvlat 36460 df-hlat 36489 df-llines 36636 df-lplanes 36637 df-lvols 36638 df-lines 36639 df-psubsp 36641 df-pmap 36642 df-padd 36934 df-lhyp 37126 df-laut 37127 df-ldil 37242 df-ltrn 37243 df-trl 37297 df-tgrp 37881 df-tendo 37893 df-edring 37895 df-dveca 38141 df-disoa 38167 df-dvech 38217 df-dib 38277 df-dic 38311 df-dih 38367 df-doch 38486 df-djh 38533 df-lcdual 38725 df-mapd 38763

This theorem is referenced by: mapdpglem17N 38826 mapdpglem18 38827 mapdpglem19 38828 mapdpglem21 38830 mapdpglem22 38831

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