hdmaprnlem10N - Mathbox for Norm Megill

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Theorem hdmaprnlem10N 42797

Description: Lemma for hdmaprnN 42802. 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 42796	. 2 ⊢ (𝜑 → ∃𝑡 ∈ ((𝑁‘{𝑣}) ∖ { 0 })(𝐿‘{((𝑆‘𝑢) ✚ 𝑠)}) = (𝑀‘(𝑁‘{(𝑢 + 𝑡)})))
21	9	adantr 486	. . 3 ⊢ ((𝜑 ∧ (𝑡 ∈ ((𝑁‘{𝑣}) ∖ { 0 }) ∧ (𝐿‘{((𝑆‘𝑢) ✚ 𝑠)}) = (𝑀‘(𝑁‘{(𝑢 + 𝑡)})))) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
22	10	adantr 486	. . 3 ⊢ ((𝜑 ∧ (𝑡 ∈ ((𝑁‘{𝑣}) ∖ { 0 }) ∧ (𝐿‘{((𝑆‘𝑢) ✚ 𝑠)}) = (𝑀‘(𝑁‘{(𝑢 + 𝑡)})))) → 𝑠 ∈ (𝐷 ∖ {𝑄}))
23	11	adantr 486	. . 3 ⊢ ((𝜑 ∧ (𝑡 ∈ ((𝑁‘{𝑣}) ∖ { 0 }) ∧ (𝐿‘{((𝑆‘𝑢) ✚ 𝑠)}) = (𝑀‘(𝑁‘{(𝑢 + 𝑡)})))) → 𝑣 ∈ 𝑉)
24	12	adantr 486	. . 3 ⊢ ((𝜑 ∧ (𝑡 ∈ ((𝑁‘{𝑣}) ∖ { 0 }) ∧ (𝐿‘{((𝑆‘𝑢) ✚ 𝑠)}) = (𝑀‘(𝑁‘{(𝑢 + 𝑡)})))) → (𝑀‘(𝑁‘{𝑣})) = (𝐿‘{𝑠}))
25	13	adantr 486	. . 3 ⊢ ((𝜑 ∧ (𝑡 ∈ ((𝑁‘{𝑣}) ∖ { 0 }) ∧ (𝐿‘{((𝑆‘𝑢) ✚ 𝑠)}) = (𝑀‘(𝑁‘{(𝑢 + 𝑡)})))) → 𝑢 ∈ 𝑉)
26	14	adantr 486	. . 3 ⊢ ((𝜑 ∧ (𝑡 ∈ ((𝑁‘{𝑣}) ∖ { 0 }) ∧ (𝐿‘{((𝑆‘𝑢) ✚ 𝑠)}) = (𝑀‘(𝑁‘{(𝑢 + 𝑡)})))) → ¬ 𝑢 ∈ (𝑁‘{𝑣}))
27		simprl 783	. . 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 42790	. 2 ⊢ ((𝜑 ∧ (𝑡 ∈ ((𝑁‘{𝑣}) ∖ { 0 }) ∧ (𝐿‘{((𝑆‘𝑢) ✚ 𝑠)}) = (𝑀‘(𝑁‘{(𝑢 + 𝑡)})))) → 𝑡 ∈ 𝑉)
29		simprr 785	. . . 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 42795	. . 3 ⊢ ((𝜑 ∧ (𝑡 ∈ ((𝑁‘{𝑣}) ∖ { 0 }) ∧ (𝐿‘{((𝑆‘𝑢) ✚ 𝑠)}) = (𝑀‘(𝑁‘{(𝑢 + 𝑡)})))) → 𝑠 = (𝑆‘𝑡))
31	30	eqcomd 2766	. 2 ⊢ ((𝜑 ∧ (𝑡 ∈ ((𝑁‘{𝑣}) ∖ { 0 }) ∧ (𝐿‘{((𝑆‘𝑢) ✚ 𝑠)}) = (𝑀‘(𝑁‘{(𝑢 + 𝑡)})))) → (𝑆‘𝑡) = 𝑠)
32	20, 28, 31	reximssdv 3180	1 ⊢ (𝜑 → ∃𝑡 ∈ 𝑉 (𝑆‘𝑡) = 𝑠)

Colors of variables: wff setvar class

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ∧ wa 401 = wceq 1570 ∈ wcel 2145 ∃wrex 3086 ∖ cdif 3896 {csn 4584 ‘cfv 6535 (class class class)co 7416 Basecbs 17326 +_gcplusg 17367 0_gc0g 17549 LSpanclspn 21185 HLchlt 40288 LHypclh 40922 DVecHcdvh 42016 LCDualclcd 42524 mapdcmpd 42562 HDMapchdma 42730

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2732 ax-rep 5232 ax-sep 5251 ax-nul 5263 ax-pow 5330 ax-pr 5398 ax-un 7742 ax-cnex 11205 ax-resscn 11206 ax-1cn 11207 ax-icn 11208 ax-addcl 11209 ax-addrcl 11210 ax-mulcl 11211 ax-mulrcl 11212 ax-mulcom 11213 ax-addass 11214 ax-mulass 11215 ax-distr 11216 ax-i2m1 11217 ax-1ne0 11218 ax-1rid 11219 ax-rnegex 11220 ax-rrecex 11221 ax-cnre 11222 ax-pre-lttri 11223 ax-pre-lttrn 11224 ax-pre-ltadd 11225 ax-pre-mulgt0 11226 ax-riotaBAD 39891

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3365 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4591 df-ot 4593 df-uni 4868 df-int 4908 df-iun 4953 df-iin 4954 df-br 5104 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5550 df-eprel 5555 df-po 5563 df-so 5564 df-fr 5608 df-we 5610 df-xp 5661 df-rel 5662 df-cnv 5663 df-co 5664 df-dm 5665 df-rn 5666 df-res 5667 df-ima 5668 df-pred 6301 df-ord 6362 df-on 6363 df-lim 6364 df-suc 6365 df-iota 6491 df-fun 6537 df-fn 6538 df-f 6539 df-f1 6540 df-fo 6541 df-f1o 6542 df-fv 6543 df-riota 7373 df-ov 7419 df-oprab 7420 df-mpo 7421 df-of 7684 df-om 7869 df-1st 7992 df-2nd 7993 df-tpos 8229 df-undef 8276 df-frecs 8285 df-wrecs 8316 df-recs 8365 df-rdg 8404 df-1o 8462 df-2o 8463 df-er 8703 df-map 8835 df-en 8960 df-dom 8961 df-sdom 8962 df-fin 8963 df-pnf 11294 df-mnf 11295 df-xr 11296 df-ltxr 11297 df-le 11298 df-sub 11492 df-neg 11493 df-nn 12283 df-2 12352 df-3 12353 df-4 12354 df-5 12355 df-6 12356 df-n0 12554 df-z 12641 df-uz 12913 df-fz 13587 df-struct 17264 df-sets 17281 df-slot 17299 df-ndx 17311 df-base 17327 df-ress 17348 df-plusg 17380 df-mulr 17381 df-sca 17383 df-vsca 17384 df-0g 17551 df-mre 17695 df-mrc 17696 df-acs 17698 df-proset 18407 df-poset 18426 df-plt 18441 df-lub 18457 df-glb 18458 df-join 18459 df-meet 18460 df-p0 18536 df-p1 18537 df-lat 18545 df-clat 18612 df-mgm 18755 df-sgrp 18847 df-mnd 18863 df-submnd 18918 df-grp 19086 df-minusg 19087 df-sbg 19088 df-subg 19272 df-cntz 19470 df-oppg 19499 df-lsm 19789 df-cmn 19935 df-abl 19936 df-mgp 20300 df-rng 20314 df-ur 20347 df-ring 20400 df-oppr 20506 df-dvdsr 20526 df-unit 20527 df-invr 20557 df-dvr 20570 df-nzr 20702 df-rlreg 20885 df-domn 20886 df-drng 20921 df-lmod 21076 df-lss 21146 df-lsp 21186 df-lvec 21317 df-lsatoms 39914 df-lshyp 39915 df-lcv 39957 df-lfl 39996 df-lkr 40024 df-ldual 40062 df-oposet 40114 df-ol 40116 df-oml 40117 df-covers 40204 df-ats 40205 df-atl 40236 df-cvlat 40260 df-hlat 40289 df-llines 40436 df-lplanes 40437 df-lvols 40438 df-lines 40439 df-psubsp 40441 df-pmap 40442 df-padd 40734 df-lhyp 40926 df-laut 40927 df-ldil 41042 df-ltrn 41043 df-trl 41097 df-tgrp 41681 df-tendo 41693 df-edring 41695 df-dveca 41941 df-disoa 41967 df-dvech 42017 df-dib 42077 df-dic 42111 df-dih 42167 df-doch 42286 df-djh 42333 df-lcdual 42525 df-mapd 42563 df-hvmap 42695 df-hdmap1 42731 df-hdmap 42732

This theorem is used by: hdmaprnlem11N 42798

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