hdmaprnlem16N - Mathbox for Norm Megill

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Theorem hdmaprnlem16N 40123

Description: Lemma for hdmaprnN 40125. Eliminate 𝑣. (Contributed by NM, 30-May-2015.) (New usage is discouraged.)

**Hypotheses**
Ref	Expression
hdmaprnlem15.h	⊢ 𝐻 = (LHyp‘𝐾)
hdmaprnlem15.u	⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
hdmaprnlem15.v	⊢ 𝑉 = (Base‘𝑈)
hdmaprnlem15.n	⊢ 𝑁 = (LSpan‘𝑈)
hdmaprnlem15.c	⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊)
hdmaprnlem15.d	⊢ 𝐷 = (Base‘𝐶)
hdmaprnlem15.q	⊢ 0 = (0_g‘𝐶)
hdmaprnlem15.l	⊢ 𝐿 = (LSpan‘𝐶)
hdmaprnlem15.m	⊢ 𝑀 = ((mapd‘𝐾)‘𝑊)
hdmaprnlem15.s	⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊)
hdmaprnlem15.k	⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
hdmaprnlem16.se	⊢ (𝜑 → 𝑠 ∈ (𝐷 ∖ { 0 }))

**Assertion**
Ref	Expression
hdmaprnlem16N	⊢ (𝜑 → 𝑠 ∈ ran 𝑆)

Proof of Theorem hdmaprnlem16N Dummy variable 𝑣 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		hdmaprnlem15.h	. . . . 5 ⊢ 𝐻 = (LHyp‘𝐾)
2		hdmaprnlem15.u	. . . . 5 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
3		hdmaprnlem15.k	. . . . 5 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
4	1, 2, 3	dvhlmod 39371	. . . 4 ⊢ (𝜑 → 𝑈 ∈ LMod)
5		hdmaprnlem15.m	. . . . 5 ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊)
6		eqid 2736	. . . . 5 ⊢ (LSAtoms‘𝑈) = (LSAtoms‘𝑈)
7		hdmaprnlem15.c	. . . . 5 ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊)
8		eqid 2736	. . . . 5 ⊢ (LSAtoms‘𝐶) = (LSAtoms‘𝐶)
9		hdmaprnlem15.d	. . . . . 6 ⊢ 𝐷 = (Base‘𝐶)
10		hdmaprnlem15.l	. . . . . 6 ⊢ 𝐿 = (LSpan‘𝐶)
11		hdmaprnlem15.q	. . . . . 6 ⊢ 0 = (0_g‘𝐶)
12	1, 7, 3	lcdlmod 39853	. . . . . 6 ⊢ (𝜑 → 𝐶 ∈ LMod)
13		hdmaprnlem16.se	. . . . . 6 ⊢ (𝜑 → 𝑠 ∈ (𝐷 ∖ { 0 }))
14	9, 10, 11, 8, 12, 13	lsatlspsn 37253	. . . . 5 ⊢ (𝜑 → (𝐿‘{𝑠}) ∈ (LSAtoms‘𝐶))
15	1, 5, 2, 6, 7, 8, 3, 14	mapdcnvatN 39927	. . . 4 ⊢ (𝜑 → (^◡𝑀‘(𝐿‘{𝑠})) ∈ (LSAtoms‘𝑈))
16		hdmaprnlem15.v	. . . . 5 ⊢ 𝑉 = (Base‘𝑈)
17		hdmaprnlem15.n	. . . . 5 ⊢ 𝑁 = (LSpan‘𝑈)
18	16, 17, 6	islsati 37254	. . . 4 ⊢ ((𝑈 ∈ LMod ∧ (^◡𝑀‘(𝐿‘{𝑠})) ∈ (LSAtoms‘𝑈)) → ∃𝑣 ∈ 𝑉 (^◡𝑀‘(𝐿‘{𝑠})) = (𝑁‘{𝑣}))
19	4, 15, 18	syl2anc 584	. . 3 ⊢ (𝜑 → ∃𝑣 ∈ 𝑉 (^◡𝑀‘(𝐿‘{𝑠})) = (𝑁‘{𝑣}))
20		simpr 485	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑣 ∈ 𝑉) ∧ (^◡𝑀‘(𝐿‘{𝑠})) = (𝑁‘{𝑣})) → (^◡𝑀‘(𝐿‘{𝑠})) = (𝑁‘{𝑣}))
21	20	fveq2d 6823	. . . . . 6 ⊢ (((𝜑 ∧ 𝑣 ∈ 𝑉) ∧ (^◡𝑀‘(𝐿‘{𝑠})) = (𝑁‘{𝑣})) → (𝑀‘(^◡𝑀‘(𝐿‘{𝑠}))) = (𝑀‘(𝑁‘{𝑣})))
22	3	ad2antrr 723	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑣 ∈ 𝑉) ∧ (^◡𝑀‘(𝐿‘{𝑠})) = (𝑁‘{𝑣})) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
23	13	eldifad 3909	. . . . . . . . . 10 ⊢ (𝜑 → 𝑠 ∈ 𝐷)
24		eqid 2736	. . . . . . . . . . 11 ⊢ (LSubSp‘𝐶) = (LSubSp‘𝐶)
25	9, 24, 10	lspsncl 20337	. . . . . . . . . 10 ⊢ ((𝐶 ∈ LMod ∧ 𝑠 ∈ 𝐷) → (𝐿‘{𝑠}) ∈ (LSubSp‘𝐶))
26	12, 23, 25	syl2anc 584	. . . . . . . . 9 ⊢ (𝜑 → (𝐿‘{𝑠}) ∈ (LSubSp‘𝐶))
27	1, 5, 7, 24, 3	mapdrn2 39912	. . . . . . . . 9 ⊢ (𝜑 → ran 𝑀 = (LSubSp‘𝐶))
28	26, 27	eleqtrrd 2840	. . . . . . . 8 ⊢ (𝜑 → (𝐿‘{𝑠}) ∈ ran 𝑀)
29	28	ad2antrr 723	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑣 ∈ 𝑉) ∧ (^◡𝑀‘(𝐿‘{𝑠})) = (𝑁‘{𝑣})) → (𝐿‘{𝑠}) ∈ ran 𝑀)
30	1, 5, 22, 29	mapdcnvid2 39918	. . . . . 6 ⊢ (((𝜑 ∧ 𝑣 ∈ 𝑉) ∧ (^◡𝑀‘(𝐿‘{𝑠})) = (𝑁‘{𝑣})) → (𝑀‘(^◡𝑀‘(𝐿‘{𝑠}))) = (𝐿‘{𝑠}))
31	21, 30	eqtr3d 2778	. . . . 5 ⊢ (((𝜑 ∧ 𝑣 ∈ 𝑉) ∧ (^◡𝑀‘(𝐿‘{𝑠})) = (𝑁‘{𝑣})) → (𝑀‘(𝑁‘{𝑣})) = (𝐿‘{𝑠}))
32	31	ex 413	. . . 4 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑉) → ((^◡𝑀‘(𝐿‘{𝑠})) = (𝑁‘{𝑣}) → (𝑀‘(𝑁‘{𝑣})) = (𝐿‘{𝑠})))
33	32	reximdva 3161	. . 3 ⊢ (𝜑 → (∃𝑣 ∈ 𝑉 (^◡𝑀‘(𝐿‘{𝑠})) = (𝑁‘{𝑣}) → ∃𝑣 ∈ 𝑉 (𝑀‘(𝑁‘{𝑣})) = (𝐿‘{𝑠})))
34	19, 33	mpd 15	. 2 ⊢ (𝜑 → ∃𝑣 ∈ 𝑉 (𝑀‘(𝑁‘{𝑣})) = (𝐿‘{𝑠}))
35		hdmaprnlem15.s	. . . 4 ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊)
36	3	3ad2ant1 1132	. . . 4 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑉 ∧ (𝑀‘(𝑁‘{𝑣})) = (𝐿‘{𝑠})) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
37	13	3ad2ant1 1132	. . . 4 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑉 ∧ (𝑀‘(𝑁‘{𝑣})) = (𝐿‘{𝑠})) → 𝑠 ∈ (𝐷 ∖ { 0 }))
38		simp2 1136	. . . 4 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑉 ∧ (𝑀‘(𝑁‘{𝑣})) = (𝐿‘{𝑠})) → 𝑣 ∈ 𝑉)
39		simp3 1137	. . . 4 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑉 ∧ (𝑀‘(𝑁‘{𝑣})) = (𝐿‘{𝑠})) → (𝑀‘(𝑁‘{𝑣})) = (𝐿‘{𝑠}))
40	1, 2, 16, 17, 7, 9, 11, 10, 5, 35, 36, 37, 38, 39	hdmaprnlem15N 40122	. . 3 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑉 ∧ (𝑀‘(𝑁‘{𝑣})) = (𝐿‘{𝑠})) → 𝑠 ∈ ran 𝑆)
41	40	rexlimdv3a 3152	. 2 ⊢ (𝜑 → (∃𝑣 ∈ 𝑉 (𝑀‘(𝑁‘{𝑣})) = (𝐿‘{𝑠}) → 𝑠 ∈ ran 𝑆))
42	34, 41	mpd 15	1 ⊢ (𝜑 → 𝑠 ∈ ran 𝑆)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 396 ∧ w3a 1086 = wceq 1540 ∈ wcel 2105 ∃wrex 3070 ∖ cdif 3894 {csn 4572 ^◡ccnv 5613 ran crn 5615 ‘cfv 6473 Basecbs 17001 0_gc0g 17239 LModclmod 20221 LSubSpclss 20291 LSpanclspn 20331 LSAtomsclsa 37234 HLchlt 37610 LHypclh 38245 DVecHcdvh 39339 LCDualclcd 39847 mapdcmpd 39885 HDMapchdma 40053

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 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2707 ax-rep 5226 ax-sep 5240 ax-nul 5247 ax-pow 5305 ax-pr 5369 ax-un 7642 ax-cnex 11020 ax-resscn 11021 ax-1cn 11022 ax-icn 11023 ax-addcl 11024 ax-addrcl 11025 ax-mulcl 11026 ax-mulrcl 11027 ax-mulcom 11028 ax-addass 11029 ax-mulass 11030 ax-distr 11031 ax-i2m1 11032 ax-1ne0 11033 ax-1rid 11034 ax-rnegex 11035 ax-rrecex 11036 ax-cnre 11037 ax-pre-lttri 11038 ax-pre-lttrn 11039 ax-pre-ltadd 11040 ax-pre-mulgt0 11041 ax-riotaBAD 37213

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3349 df-reu 3350 df-rab 3404 df-v 3443 df-sbc 3727 df-csb 3843 df-dif 3900 df-un 3902 df-in 3904 df-ss 3914 df-pss 3916 df-nul 4269 df-if 4473 df-pw 4548 df-sn 4573 df-pr 4575 df-tp 4577 df-op 4579 df-ot 4581 df-uni 4852 df-int 4894 df-iun 4940 df-iin 4941 df-br 5090 df-opab 5152 df-mpt 5173 df-tr 5207 df-id 5512 df-eprel 5518 df-po 5526 df-so 5527 df-fr 5569 df-we 5571 df-xp 5620 df-rel 5621 df-cnv 5622 df-co 5623 df-dm 5624 df-rn 5625 df-res 5626 df-ima 5627 df-pred 6232 df-ord 6299 df-on 6300 df-lim 6301 df-suc 6302 df-iota 6425 df-fun 6475 df-fn 6476 df-f 6477 df-f1 6478 df-fo 6479 df-f1o 6480 df-fv 6481 df-riota 7286 df-ov 7332 df-oprab 7333 df-mpo 7334 df-of 7587 df-om 7773 df-1st 7891 df-2nd 7892 df-tpos 8104 df-undef 8151 df-frecs 8159 df-wrecs 8190 df-recs 8264 df-rdg 8303 df-1o 8359 df-er 8561 df-map 8680 df-en 8797 df-dom 8798 df-sdom 8799 df-fin 8800 df-pnf 11104 df-mnf 11105 df-xr 11106 df-ltxr 11107 df-le 11108 df-sub 11300 df-neg 11301 df-nn 12067 df-2 12129 df-3 12130 df-4 12131 df-5 12132 df-6 12133 df-n0 12327 df-z 12413 df-uz 12676 df-fz 13333 df-struct 16937 df-sets 16954 df-slot 16972 df-ndx 16984 df-base 17002 df-ress 17031 df-plusg 17064 df-mulr 17065 df-sca 17067 df-vsca 17068 df-0g 17241 df-mre 17384 df-mrc 17385 df-acs 17387 df-proset 18102 df-poset 18120 df-plt 18137 df-lub 18153 df-glb 18154 df-join 18155 df-meet 18156 df-p0 18232 df-p1 18233 df-lat 18239 df-clat 18306 df-mgm 18415 df-sgrp 18464 df-mnd 18475 df-submnd 18520 df-grp 18668 df-minusg 18669 df-sbg 18670 df-subg 18840 df-cntz 19011 df-oppg 19038 df-lsm 19329 df-cmn 19475 df-abl 19476 df-mgp 19808 df-ur 19825 df-ring 19872 df-oppr 19949 df-dvdsr 19970 df-unit 19971 df-invr 20001 df-dvr 20012 df-drng 20087 df-lmod 20223 df-lss 20292 df-lsp 20332 df-lvec 20463 df-lsatoms 37236 df-lshyp 37237 df-lcv 37279 df-lfl 37318 df-lkr 37346 df-ldual 37384 df-oposet 37436 df-ol 37438 df-oml 37439 df-covers 37526 df-ats 37527 df-atl 37558 df-cvlat 37582 df-hlat 37611 df-llines 37759 df-lplanes 37760 df-lvols 37761 df-lines 37762 df-psubsp 37764 df-pmap 37765 df-padd 38057 df-lhyp 38249 df-laut 38250 df-ldil 38365 df-ltrn 38366 df-trl 38420 df-tgrp 39004 df-tendo 39016 df-edring 39018 df-dveca 39264 df-disoa 39290 df-dvech 39340 df-dib 39400 df-dic 39434 df-dih 39490 df-doch 39609 df-djh 39656 df-lcdual 39848 df-mapd 39886 df-hvmap 40018 df-hdmap1 40054 df-hdmap 40055

This theorem is referenced by: hdmaprnlem17N 40124

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