hdmaprnlem16N - Mathbox for Norm Megill

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Theorem hdmaprnlem16N 37930

Description: Lemma for hdmaprnN 37932. 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 37178	. . . 4 ⊢ (𝜑 → 𝑈 ∈ LMod)
5		hdmaprnlem15.m	. . . . 5 ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊)
6		eqid 2825	. . . . 5 ⊢ (LSAtoms‘𝑈) = (LSAtoms‘𝑈)
7		hdmaprnlem15.c	. . . . 5 ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊)
8		eqid 2825	. . . . 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 37660	. . . . . 6 ⊢ (𝜑 → 𝐶 ∈ LMod)
13		hdmaprnlem16.se	. . . . . 6 ⊢ (𝜑 → 𝑠 ∈ (𝐷 ∖ { 0 }))
14	9, 10, 11, 8, 12, 13	lsatlspsn 35061	. . . . 5 ⊢ (𝜑 → (𝐿‘{𝑠}) ∈ (LSAtoms‘𝐶))
15	1, 5, 2, 6, 7, 8, 3, 14	mapdcnvatN 37734	. . . 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 35062	. . . 4 ⊢ ((𝑈 ∈ LMod ∧ (^◡𝑀‘(𝐿‘{𝑠})) ∈ (LSAtoms‘𝑈)) → ∃𝑣 ∈ 𝑉 (^◡𝑀‘(𝐿‘{𝑠})) = (𝑁‘{𝑣}))
19	4, 15, 18	syl2anc 579	. . 3 ⊢ (𝜑 → ∃𝑣 ∈ 𝑉 (^◡𝑀‘(𝐿‘{𝑠})) = (𝑁‘{𝑣}))
20		simpr 479	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑣 ∈ 𝑉) ∧ (^◡𝑀‘(𝐿‘{𝑠})) = (𝑁‘{𝑣})) → (^◡𝑀‘(𝐿‘{𝑠})) = (𝑁‘{𝑣}))
21	20	fveq2d 6437	. . . . . 6 ⊢ (((𝜑 ∧ 𝑣 ∈ 𝑉) ∧ (^◡𝑀‘(𝐿‘{𝑠})) = (𝑁‘{𝑣})) → (𝑀‘(^◡𝑀‘(𝐿‘{𝑠}))) = (𝑀‘(𝑁‘{𝑣})))
22	3	ad2antrr 717	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑣 ∈ 𝑉) ∧ (^◡𝑀‘(𝐿‘{𝑠})) = (𝑁‘{𝑣})) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
23	13	eldifad 3810	. . . . . . . . . 10 ⊢ (𝜑 → 𝑠 ∈ 𝐷)
24		eqid 2825	. . . . . . . . . . 11 ⊢ (LSubSp‘𝐶) = (LSubSp‘𝐶)
25	9, 24, 10	lspsncl 19336	. . . . . . . . . 10 ⊢ ((𝐶 ∈ LMod ∧ 𝑠 ∈ 𝐷) → (𝐿‘{𝑠}) ∈ (LSubSp‘𝐶))
26	12, 23, 25	syl2anc 579	. . . . . . . . 9 ⊢ (𝜑 → (𝐿‘{𝑠}) ∈ (LSubSp‘𝐶))
27	1, 5, 7, 24, 3	mapdrn2 37719	. . . . . . . . 9 ⊢ (𝜑 → ran 𝑀 = (LSubSp‘𝐶))
28	26, 27	eleqtrrd 2909	. . . . . . . 8 ⊢ (𝜑 → (𝐿‘{𝑠}) ∈ ran 𝑀)
29	28	ad2antrr 717	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑣 ∈ 𝑉) ∧ (^◡𝑀‘(𝐿‘{𝑠})) = (𝑁‘{𝑣})) → (𝐿‘{𝑠}) ∈ ran 𝑀)
30	1, 5, 22, 29	mapdcnvid2 37725	. . . . . 6 ⊢ (((𝜑 ∧ 𝑣 ∈ 𝑉) ∧ (^◡𝑀‘(𝐿‘{𝑠})) = (𝑁‘{𝑣})) → (𝑀‘(^◡𝑀‘(𝐿‘{𝑠}))) = (𝐿‘{𝑠}))
31	21, 30	eqtr3d 2863	. . . . 5 ⊢ (((𝜑 ∧ 𝑣 ∈ 𝑉) ∧ (^◡𝑀‘(𝐿‘{𝑠})) = (𝑁‘{𝑣})) → (𝑀‘(𝑁‘{𝑣})) = (𝐿‘{𝑠}))
32	31	ex 403	. . . 4 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑉) → ((^◡𝑀‘(𝐿‘{𝑠})) = (𝑁‘{𝑣}) → (𝑀‘(𝑁‘{𝑣})) = (𝐿‘{𝑠})))
33	32	reximdva 3225	. . 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 1167	. . . 4 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑉 ∧ (𝑀‘(𝑁‘{𝑣})) = (𝐿‘{𝑠})) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
37	13	3ad2ant1 1167	. . . 4 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑉 ∧ (𝑀‘(𝑁‘{𝑣})) = (𝐿‘{𝑠})) → 𝑠 ∈ (𝐷 ∖ { 0 }))
38		simp2 1171	. . . 4 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑉 ∧ (𝑀‘(𝑁‘{𝑣})) = (𝐿‘{𝑠})) → 𝑣 ∈ 𝑉)
39		simp3 1172	. . . 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 37929	. . 3 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑉 ∧ (𝑀‘(𝑁‘{𝑣})) = (𝐿‘{𝑠})) → 𝑠 ∈ ran 𝑆)
41	40	rexlimdv3a 3242	. 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 386 ∧ w3a 1111 = wceq 1656 ∈ wcel 2164 ∃wrex 3118 ∖ cdif 3795 {csn 4397 ^◡ccnv 5341 ran crn 5343 ‘cfv 6123 Basecbs 16222 0_gc0g 16453 LModclmod 19219 LSubSpclss 19288 LSpanclspn 19330 LSAtomsclsa 35042 HLchlt 35418 LHypclh 36052 DVecHcdvh 37146 LCDualclcd 37654 mapdcmpd 37692 HDMapchdma 37860

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-8 2166 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-rep 4994 ax-sep 5005 ax-nul 5013 ax-pow 5065 ax-pr 5127 ax-un 7209 ax-cnex 10308 ax-resscn 10309 ax-1cn 10310 ax-icn 10311 ax-addcl 10312 ax-addrcl 10313 ax-mulcl 10314 ax-mulrcl 10315 ax-mulcom 10316 ax-addass 10317 ax-mulass 10318 ax-distr 10319 ax-i2m1 10320 ax-1ne0 10321 ax-1rid 10322 ax-rnegex 10323 ax-rrecex 10324 ax-cnre 10325 ax-pre-lttri 10326 ax-pre-lttrn 10327 ax-pre-ltadd 10328 ax-pre-mulgt0 10329 ax-riotaBAD 35021

This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3or 1112 df-3an 1113 df-tru 1660 df-fal 1670 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-nel 3103 df-ral 3122 df-rex 3123 df-reu 3124 df-rmo 3125 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-pss 3814 df-nul 4145 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-tp 4402 df-op 4404 df-ot 4406 df-uni 4659 df-int 4698 df-iun 4742 df-iin 4743 df-br 4874 df-opab 4936 df-mpt 4953 df-tr 4976 df-id 5250 df-eprel 5255 df-po 5263 df-so 5264 df-fr 5301 df-we 5303 df-xp 5348 df-rel 5349 df-cnv 5350 df-co 5351 df-dm 5352 df-rn 5353 df-res 5354 df-ima 5355 df-pred 5920 df-ord 5966 df-on 5967 df-lim 5968 df-suc 5969 df-iota 6086 df-fun 6125 df-fn 6126 df-f 6127 df-f1 6128 df-fo 6129 df-f1o 6130 df-fv 6131 df-riota 6866 df-ov 6908 df-oprab 6909 df-mpt2 6910 df-of 7157 df-om 7327 df-1st 7428 df-2nd 7429 df-tpos 7617 df-undef 7664 df-wrecs 7672 df-recs 7734 df-rdg 7772 df-1o 7826 df-oadd 7830 df-er 8009 df-map 8124 df-en 8223 df-dom 8224 df-sdom 8225 df-fin 8226 df-pnf 10393 df-mnf 10394 df-xr 10395 df-ltxr 10396 df-le 10397 df-sub 10587 df-neg 10588 df-nn 11351 df-2 11414 df-3 11415 df-4 11416 df-5 11417 df-6 11418 df-n0 11619 df-z 11705 df-uz 11969 df-fz 12620 df-struct 16224 df-ndx 16225 df-slot 16226 df-base 16228 df-sets 16229 df-ress 16230 df-plusg 16318 df-mulr 16319 df-sca 16321 df-vsca 16322 df-0g 16455 df-mre 16599 df-mrc 16600 df-acs 16602 df-proset 17281 df-poset 17299 df-plt 17311 df-lub 17327 df-glb 17328 df-join 17329 df-meet 17330 df-p0 17392 df-p1 17393 df-lat 17399 df-clat 17461 df-mgm 17595 df-sgrp 17637 df-mnd 17648 df-submnd 17689 df-grp 17779 df-minusg 17780 df-sbg 17781 df-subg 17942 df-cntz 18100 df-oppg 18126 df-lsm 18402 df-cmn 18548 df-abl 18549 df-mgp 18844 df-ur 18856 df-ring 18903 df-oppr 18977 df-dvdsr 18995 df-unit 18996 df-invr 19026 df-dvr 19037 df-drng 19105 df-lmod 19221 df-lss 19289 df-lsp 19331 df-lvec 19462 df-lsatoms 35044 df-lshyp 35045 df-lcv 35087 df-lfl 35126 df-lkr 35154 df-ldual 35192 df-oposet 35244 df-ol 35246 df-oml 35247 df-covers 35334 df-ats 35335 df-atl 35366 df-cvlat 35390 df-hlat 35419 df-llines 35566 df-lplanes 35567 df-lvols 35568 df-lines 35569 df-psubsp 35571 df-pmap 35572 df-padd 35864 df-lhyp 36056 df-laut 36057 df-ldil 36172 df-ltrn 36173 df-trl 36227 df-tgrp 36811 df-tendo 36823 df-edring 36825 df-dveca 37071 df-disoa 37097 df-dvech 37147 df-dib 37207 df-dic 37241 df-dih 37297 df-doch 37416 df-djh 37463 df-lcdual 37655 df-mapd 37693 df-hvmap 37825 df-hdmap1 37861 df-hdmap 37862

This theorem is referenced by: hdmaprnlem17N 37931

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