hdmaprnlem10N - Mathbox for Norm Megill

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Theorem hdmaprnlem10N 42661

Description: Lemma for hdmaprnN 42666. Show 𝑠 is in the range of 𝑆. (Contributed by NM, 29-May-2015.) (New usage is discouraged.)

**Hypotheses**
Ref	Expression
hdmaprnlem1.h	⊢ 𝐻 = (LHyp‘𝐾)
hdmaprnlem1.u	⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
hdmaprnlem1.v	⊢ 𝑉 = (Base‘𝑈)
hdmaprnlem1.n	⊢ 𝑁 = (LSpan‘𝑈)
hdmaprnlem1.c	⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊)
hdmaprnlem1.l	⊢ 𝐿 = (LSpan‘𝐶)
hdmaprnlem1.m	⊢ 𝑀 = ((mapd‘𝐾)‘𝑊)
hdmaprnlem1.s	⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊)
hdmaprnlem1.k	⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
hdmaprnlem1.se	⊢ (𝜑 → 𝑠 ∈ (𝐷 ∖ {𝑄}))
hdmaprnlem1.ve	⊢ (𝜑 → 𝑣 ∈ 𝑉)
hdmaprnlem1.e	⊢ (𝜑 → (𝑀‘(𝑁‘{𝑣})) = (𝐿‘{𝑠}))
hdmaprnlem1.ue	⊢ (𝜑 → 𝑢 ∈ 𝑉)
hdmaprnlem1.un	⊢ (𝜑 → ¬ 𝑢 ∈ (𝑁‘{𝑣}))
hdmaprnlem1.d	⊢ 𝐷 = (Base‘𝐶)
hdmaprnlem1.q	⊢ 𝑄 = (0_g‘𝐶)
hdmaprnlem1.o	⊢ 0 = (0_g‘𝑈)
hdmaprnlem1.a	⊢ ✚ = (+_g‘𝐶)
hdmaprnlem3e.p	⊢ + = (+_g‘𝑈)

**Assertion**
Ref	Expression
hdmaprnlem10N	⊢ (𝜑 → ∃𝑡 ∈ 𝑉 (𝑆‘𝑡) = 𝑠)

Distinct variable groups: 𝑡, ✚ 𝑡,𝐿 𝑡,𝑀 𝑡,𝑁 𝑡, 0 𝑡, + 𝑡,𝑆 𝑡,𝑈 𝑡,𝑉 𝜑,𝑡 𝑡,𝑠,𝑢,𝑣 Allowed substitution hints: 𝜑(𝑣, 𝑢, 𝑠) 𝐶(𝑣, 𝑢, 𝑡, 𝑠) 𝐷(𝑣, 𝑢, 𝑡, 𝑠) + (𝑣, 𝑢, 𝑠) ✚ (𝑣, 𝑢, 𝑠) 𝑄(𝑣, 𝑢, 𝑡, 𝑠) 𝑆(𝑣, 𝑢, 𝑠) 𝑈(𝑣, 𝑢, 𝑠) 𝐻(𝑣, 𝑢, 𝑡, 𝑠) 𝐾(𝑣, 𝑢, 𝑡, 𝑠) 𝐿(𝑣, 𝑢, 𝑠) 𝑀(𝑣, 𝑢, 𝑠) 𝑁(𝑣, 𝑢, 𝑠) 𝑉(𝑣, 𝑢, 𝑠) 𝑊(𝑣, 𝑢, 𝑡, 𝑠) 0 (𝑣, 𝑢, 𝑠)

**Proof of Theorem hdmaprnlem10N**
Step	Hyp	Ref	Expression
1		hdmaprnlem1.h	. . 3 ⊢ 𝐻 = (LHyp‘𝐾)
2		hdmaprnlem1.u	. . 3 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
3		hdmaprnlem1.v	. . 3 ⊢ 𝑉 = (Base‘𝑈)
4		hdmaprnlem1.n	. . 3 ⊢ 𝑁 = (LSpan‘𝑈)
5		hdmaprnlem1.c	. . 3 ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊)
6		hdmaprnlem1.l	. . 3 ⊢ 𝐿 = (LSpan‘𝐶)
7		hdmaprnlem1.m	. . 3 ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊)
8		hdmaprnlem1.s	. . 3 ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊)
9		hdmaprnlem1.k	. . 3 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
10		hdmaprnlem1.se	. . 3 ⊢ (𝜑 → 𝑠 ∈ (𝐷 ∖ {𝑄}))
11		hdmaprnlem1.ve	. . 3 ⊢ (𝜑 → 𝑣 ∈ 𝑉)
12		hdmaprnlem1.e	. . 3 ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑣})) = (𝐿‘{𝑠}))
13		hdmaprnlem1.ue	. . 3 ⊢ (𝜑 → 𝑢 ∈ 𝑉)
14		hdmaprnlem1.un	. . 3 ⊢ (𝜑 → ¬ 𝑢 ∈ (𝑁‘{𝑣}))
15		hdmaprnlem1.d	. . 3 ⊢ 𝐷 = (Base‘𝐶)
16		hdmaprnlem1.q	. . 3 ⊢ 𝑄 = (0_g‘𝐶)
17		hdmaprnlem1.o	. . 3 ⊢ 0 = (0_g‘𝑈)
18		hdmaprnlem1.a	. . 3 ⊢ ✚ = (+_g‘𝐶)
19		hdmaprnlem3e.p	. . 3 ⊢ + = (+_g‘𝑈)
20	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19	hdmaprnlem3eN 42660	. 2 ⊢ (𝜑 → ∃𝑡 ∈ ((𝑁‘{𝑣}) ∖ { 0 })(𝐿‘{((𝑆‘𝑢) ✚ 𝑠)}) = (𝑀‘(𝑁‘{(𝑢 + 𝑡)})))
21	9	adantr 485	. . 3 ⊢ ((𝜑 ∧ (𝑡 ∈ ((𝑁‘{𝑣}) ∖ { 0 }) ∧ (𝐿‘{((𝑆‘𝑢) ✚ 𝑠)}) = (𝑀‘(𝑁‘{(𝑢 + 𝑡)})))) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
22	10	adantr 485	. . 3 ⊢ ((𝜑 ∧ (𝑡 ∈ ((𝑁‘{𝑣}) ∖ { 0 }) ∧ (𝐿‘{((𝑆‘𝑢) ✚ 𝑠)}) = (𝑀‘(𝑁‘{(𝑢 + 𝑡)})))) → 𝑠 ∈ (𝐷 ∖ {𝑄}))
23	11	adantr 485	. . 3 ⊢ ((𝜑 ∧ (𝑡 ∈ ((𝑁‘{𝑣}) ∖ { 0 }) ∧ (𝐿‘{((𝑆‘𝑢) ✚ 𝑠)}) = (𝑀‘(𝑁‘{(𝑢 + 𝑡)})))) → 𝑣 ∈ 𝑉)
24	12	adantr 485	. . 3 ⊢ ((𝜑 ∧ (𝑡 ∈ ((𝑁‘{𝑣}) ∖ { 0 }) ∧ (𝐿‘{((𝑆‘𝑢) ✚ 𝑠)}) = (𝑀‘(𝑁‘{(𝑢 + 𝑡)})))) → (𝑀‘(𝑁‘{𝑣})) = (𝐿‘{𝑠}))
25	13	adantr 485	. . 3 ⊢ ((𝜑 ∧ (𝑡 ∈ ((𝑁‘{𝑣}) ∖ { 0 }) ∧ (𝐿‘{((𝑆‘𝑢) ✚ 𝑠)}) = (𝑀‘(𝑁‘{(𝑢 + 𝑡)})))) → 𝑢 ∈ 𝑉)
26	14	adantr 485	. . 3 ⊢ ((𝜑 ∧ (𝑡 ∈ ((𝑁‘{𝑣}) ∖ { 0 }) ∧ (𝐿‘{((𝑆‘𝑢) ✚ 𝑠)}) = (𝑀‘(𝑁‘{(𝑢 + 𝑡)})))) → ¬ 𝑢 ∈ (𝑁‘{𝑣}))
27		simprl 782	. . 3 ⊢ ((𝜑 ∧ (𝑡 ∈ ((𝑁‘{𝑣}) ∖ { 0 }) ∧ (𝐿‘{((𝑆‘𝑢) ✚ 𝑠)}) = (𝑀‘(𝑁‘{(𝑢 + 𝑡)})))) → 𝑡 ∈ ((𝑁‘{𝑣}) ∖ { 0 }))
28	1, 2, 3, 4, 5, 6, 7, 8, 21, 22, 23, 24, 25, 26, 15, 16, 17, 18, 27	hdmaprnlem4tN 42654	. 2 ⊢ ((𝜑 ∧ (𝑡 ∈ ((𝑁‘{𝑣}) ∖ { 0 }) ∧ (𝐿‘{((𝑆‘𝑢) ✚ 𝑠)}) = (𝑀‘(𝑁‘{(𝑢 + 𝑡)})))) → 𝑡 ∈ 𝑉)
29		simprr 784	. . . 4 ⊢ ((𝜑 ∧ (𝑡 ∈ ((𝑁‘{𝑣}) ∖ { 0 }) ∧ (𝐿‘{((𝑆‘𝑢) ✚ 𝑠)}) = (𝑀‘(𝑁‘{(𝑢 + 𝑡)})))) → (𝐿‘{((𝑆‘𝑢) ✚ 𝑠)}) = (𝑀‘(𝑁‘{(𝑢 + 𝑡)})))
30	1, 2, 3, 4, 5, 6, 7, 8, 21, 22, 23, 24, 25, 26, 15, 16, 17, 18, 27, 19, 29	hdmaprnlem9N 42659	. . 3 ⊢ ((𝜑 ∧ (𝑡 ∈ ((𝑁‘{𝑣}) ∖ { 0 }) ∧ (𝐿‘{((𝑆‘𝑢) ✚ 𝑠)}) = (𝑀‘(𝑁‘{(𝑢 + 𝑡)})))) → 𝑠 = (𝑆‘𝑡))
31	30	eqcomd 2769	. 2 ⊢ ((𝜑 ∧ (𝑡 ∈ ((𝑁‘{𝑣}) ∖ { 0 }) ∧ (𝐿‘{((𝑆‘𝑢) ✚ 𝑠)}) = (𝑀‘(𝑁‘{(𝑢 + 𝑡)})))) → (𝑆‘𝑡) = 𝑠)
32	20, 28, 31	reximssdv 3183	1 ⊢ (𝜑 → ∃𝑡 ∈ 𝑉 (𝑆‘𝑡) = 𝑠)

Colors of variables: wff setvar class

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ∧ wa 400 = wceq 1570 ∈ wcel 2143 ∃wrex 3089 ∖ cdif 3902 {csn 4589 ‘cfv 6536 (class class class)co 7410 Basecbs 17273 +_gcplusg 17314 0_gc0g 17496 LSpanclspn 21101 HLchlt 40152 LHypclh 40786 DVecHcdvh 41880 LCDualclcd 42388 mapdcmpd 42426 HDMapchdma 42594

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1825 ax-4 1839 ax-5 1940 ax-6 1997 ax-7 2038 ax-8 2145 ax-9 2153 ax-10 2176 ax-11 2192 ax-12 2213 ax-ext 2735 ax-rep 5238 ax-sep 5257 ax-nul 5269 ax-pow 5336 ax-pr 5404 ax-un 7732 ax-cnex 11160 ax-resscn 11161 ax-1cn 11162 ax-icn 11163 ax-addcl 11164 ax-addrcl 11165 ax-mulcl 11166 ax-mulrcl 11167 ax-mulcom 11168 ax-addass 11169 ax-mulass 11170 ax-distr 11171 ax-i2m1 11172 ax-1ne0 11173 ax-1rid 11174 ax-rnegex 11175 ax-rrecex 11176 ax-cnre 11177 ax-pre-lttri 11178 ax-pre-lttrn 11179 ax-pre-ltadd 11180 ax-pre-mulgt0 11181 ax-riotaBAD 39755

This proof depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1810 df-nf 1814 df-sb 2097 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 3369 df-reu 3370 df-rab 3417 df-v 3457 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 4488 df-pw 4564 df-sn 4590 df-pr 4592 df-tp 4594 df-op 4596 df-ot 4598 df-uni 4873 df-int 4913 df-iun 4958 df-iin 4959 df-br 5110 df-opab 5174 df-mpt 5193 df-tr 5219 df-id 5556 df-eprel 5561 df-po 5569 df-so 5570 df-fr 5614 df-we 5616 df-xp 5667 df-rel 5668 df-cnv 5669 df-co 5670 df-dm 5671 df-rn 5672 df-res 5673 df-ima 5674 df-pred 6302 df-ord 6363 df-on 6364 df-lim 6365 df-suc 6366 df-iota 6492 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 7674 df-om 7859 df-1st 7982 df-2nd 7983 df-tpos 8218 df-undef 8265 df-frecs 8274 df-wrecs 8305 df-recs 8354 df-rdg 8393 df-1o 8449 df-2o 8450 df-er 8690 df-map 8822 df-en 8940 df-dom 8941 df-sdom 8942 df-fin 8943 df-pnf 11249 df-mnf 11250 df-xr 11251 df-ltxr 11252 df-le 11253 df-sub 11447 df-neg 11448 df-nn 12238 df-2 12307 df-3 12308 df-4 12309 df-5 12310 df-6 12311 df-n0 12509 df-z 12596 df-uz 12867 df-fz 13540 df-struct 17211 df-sets 17228 df-slot 17246 df-ndx 17258 df-base 17274 df-ress 17295 df-plusg 17327 df-mulr 17328 df-sca 17330 df-vsca 17331 df-0g 17498 df-mre 17642 df-mrc 17643 df-acs 17645 df-proset 18354 df-poset 18373 df-plt 18388 df-lub 18404 df-glb 18405 df-join 18406 df-meet 18407 df-p0 18483 df-p1 18484 df-lat 18492 df-clat 18559 df-mgm 18702 df-sgrp 18781 df-mnd 18797 df-submnd 18846 df-grp 19007 df-minusg 19008 df-sbg 19009 df-subg 19193 df-cntz 19391 df-oppg 19420 df-lsm 19710 df-cmn 19856 df-abl 19857 df-mgp 20221 df-rng 20235 df-ur 20268 df-ring 20321 df-oppr 20424 df-dvdsr 20444 df-unit 20445 df-invr 20475 df-dvr 20488 df-nzr 20619 df-rlreg 20802 df-domn 20803 df-drng 20838 df-lmod 20992 df-lss 21062 df-lsp 21102 df-lvec 21233 df-lsatoms 39778 df-lshyp 39779 df-lcv 39821 df-lfl 39860 df-lkr 39888 df-ldual 39926 df-oposet 39978 df-ol 39980 df-oml 39981 df-covers 40068 df-ats 40069 df-atl 40100 df-cvlat 40124 df-hlat 40153 df-llines 40300 df-lplanes 40301 df-lvols 40302 df-lines 40303 df-psubsp 40305 df-pmap 40306 df-padd 40598 df-lhyp 40790 df-laut 40791 df-ldil 40906 df-ltrn 40907 df-trl 40961 df-tgrp 41545 df-tendo 41557 df-edring 41559 df-dveca 41805 df-disoa 41831 df-dvech 41881 df-dib 41941 df-dic 41975 df-dih 42031 df-doch 42150 df-djh 42197 df-lcdual 42389 df-mapd 42427 df-hvmap 42559 df-hdmap1 42595 df-hdmap 42596

This theorem is used by: hdmaprnlem11N 42662

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