hdmaprnlem16N - Mathbox for Norm Megill

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Theorem hdmaprnlem16N 41856

Description: Lemma for hdmaprnN 41858. 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 41104	. . . 4 ⊢ (𝜑 → 𝑈 ∈ LMod)
5		hdmaprnlem15.m	. . . . 5 ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊)
6		eqid 2729	. . . . 5 ⊢ (LSAtoms‘𝑈) = (LSAtoms‘𝑈)
7		hdmaprnlem15.c	. . . . 5 ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊)
8		eqid 2729	. . . . 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 41586	. . . . . 6 ⊢ (𝜑 → 𝐶 ∈ LMod)
13		hdmaprnlem16.se	. . . . . 6 ⊢ (𝜑 → 𝑠 ∈ (𝐷 ∖ { 0 }))
14	9, 10, 11, 8, 12, 13	lsatlspsn 38986	. . . . 5 ⊢ (𝜑 → (𝐿‘{𝑠}) ∈ (LSAtoms‘𝐶))
15	1, 5, 2, 6, 7, 8, 3, 14	mapdcnvatN 41660	. . . 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 38987	. . . 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 484	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑣 ∈ 𝑉) ∧ (^◡𝑀‘(𝐿‘{𝑠})) = (𝑁‘{𝑣})) → (^◡𝑀‘(𝐿‘{𝑠})) = (𝑁‘{𝑣}))
21	20	fveq2d 6862	. . . . . 6 ⊢ (((𝜑 ∧ 𝑣 ∈ 𝑉) ∧ (^◡𝑀‘(𝐿‘{𝑠})) = (𝑁‘{𝑣})) → (𝑀‘(^◡𝑀‘(𝐿‘{𝑠}))) = (𝑀‘(𝑁‘{𝑣})))
22	3	ad2antrr 726	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑣 ∈ 𝑉) ∧ (^◡𝑀‘(𝐿‘{𝑠})) = (𝑁‘{𝑣})) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
23	13	eldifad 3926	. . . . . . . . . 10 ⊢ (𝜑 → 𝑠 ∈ 𝐷)
24		eqid 2729	. . . . . . . . . . 11 ⊢ (LSubSp‘𝐶) = (LSubSp‘𝐶)
25	9, 24, 10	lspsncl 20883	. . . . . . . . . 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 41645	. . . . . . . . 9 ⊢ (𝜑 → ran 𝑀 = (LSubSp‘𝐶))
28	26, 27	eleqtrrd 2831	. . . . . . . 8 ⊢ (𝜑 → (𝐿‘{𝑠}) ∈ ran 𝑀)
29	28	ad2antrr 726	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑣 ∈ 𝑉) ∧ (^◡𝑀‘(𝐿‘{𝑠})) = (𝑁‘{𝑣})) → (𝐿‘{𝑠}) ∈ ran 𝑀)
30	1, 5, 22, 29	mapdcnvid2 41651	. . . . . 6 ⊢ (((𝜑 ∧ 𝑣 ∈ 𝑉) ∧ (^◡𝑀‘(𝐿‘{𝑠})) = (𝑁‘{𝑣})) → (𝑀‘(^◡𝑀‘(𝐿‘{𝑠}))) = (𝐿‘{𝑠}))
31	21, 30	eqtr3d 2766	. . . . 5 ⊢ (((𝜑 ∧ 𝑣 ∈ 𝑉) ∧ (^◡𝑀‘(𝐿‘{𝑠})) = (𝑁‘{𝑣})) → (𝑀‘(𝑁‘{𝑣})) = (𝐿‘{𝑠}))
32	31	ex 412	. . . 4 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑉) → ((^◡𝑀‘(𝐿‘{𝑠})) = (𝑁‘{𝑣}) → (𝑀‘(𝑁‘{𝑣})) = (𝐿‘{𝑠})))
33	32	reximdva 3146	. . 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 1133	. . . 4 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑉 ∧ (𝑀‘(𝑁‘{𝑣})) = (𝐿‘{𝑠})) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
37	13	3ad2ant1 1133	. . . 4 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑉 ∧ (𝑀‘(𝑁‘{𝑣})) = (𝐿‘{𝑠})) → 𝑠 ∈ (𝐷 ∖ { 0 }))
38		simp2 1137	. . . 4 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑉 ∧ (𝑀‘(𝑁‘{𝑣})) = (𝐿‘{𝑠})) → 𝑣 ∈ 𝑉)
39		simp3 1138	. . . 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 41855	. . 3 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑉 ∧ (𝑀‘(𝑁‘{𝑣})) = (𝐿‘{𝑠})) → 𝑠 ∈ ran 𝑆)
41	40	rexlimdv3a 3138	. 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 395 ∧ w3a 1086 = wceq 1540 ∈ wcel 2109 ∃wrex 3053 ∖ cdif 3911 {csn 4589 ^◡ccnv 5637 ran crn 5639 ‘cfv 6511 Basecbs 17179 0_gc0g 17402 LModclmod 20766 LSubSpclss 20837 LSpanclspn 20877 LSAtomsclsa 38967 HLchlt 39343 LHypclh 39978 DVecHcdvh 41072 LCDualclcd 41580 mapdcmpd 41618 HDMapchdma 41786

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5234 ax-sep 5251 ax-nul 5261 ax-pow 5320 ax-pr 5387 ax-un 7711 ax-cnex 11124 ax-resscn 11125 ax-1cn 11126 ax-icn 11127 ax-addcl 11128 ax-addrcl 11129 ax-mulcl 11130 ax-mulrcl 11131 ax-mulcom 11132 ax-addass 11133 ax-mulass 11134 ax-distr 11135 ax-i2m1 11136 ax-1ne0 11137 ax-1rid 11138 ax-rnegex 11139 ax-rrecex 11140 ax-cnre 11141 ax-pre-lttri 11142 ax-pre-lttrn 11143 ax-pre-ltadd 11144 ax-pre-mulgt0 11145 ax-riotaBAD 38946

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3354 df-reu 3355 df-rab 3406 df-v 3449 df-sbc 3754 df-csb 3863 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3934 df-nul 4297 df-if 4489 df-pw 4565 df-sn 4590 df-pr 4592 df-tp 4594 df-op 4596 df-ot 4598 df-uni 4872 df-int 4911 df-iun 4957 df-iin 4958 df-br 5108 df-opab 5170 df-mpt 5189 df-tr 5215 df-id 5533 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5591 df-we 5593 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6274 df-ord 6335 df-on 6336 df-lim 6337 df-suc 6338 df-iota 6464 df-fun 6513 df-fn 6514 df-f 6515 df-f1 6516 df-fo 6517 df-f1o 6518 df-fv 6519 df-riota 7344 df-ov 7390 df-oprab 7391 df-mpo 7392 df-of 7653 df-om 7843 df-1st 7968 df-2nd 7969 df-tpos 8205 df-undef 8252 df-frecs 8260 df-wrecs 8291 df-recs 8340 df-rdg 8378 df-1o 8434 df-2o 8435 df-er 8671 df-map 8801 df-en 8919 df-dom 8920 df-sdom 8921 df-fin 8922 df-pnf 11210 df-mnf 11211 df-xr 11212 df-ltxr 11213 df-le 11214 df-sub 11407 df-neg 11408 df-nn 12187 df-2 12249 df-3 12250 df-4 12251 df-5 12252 df-6 12253 df-n0 12443 df-z 12530 df-uz 12794 df-fz 13469 df-struct 17117 df-sets 17134 df-slot 17152 df-ndx 17164 df-base 17180 df-ress 17201 df-plusg 17233 df-mulr 17234 df-sca 17236 df-vsca 17237 df-0g 17404 df-mre 17547 df-mrc 17548 df-acs 17550 df-proset 18255 df-poset 18274 df-plt 18289 df-lub 18305 df-glb 18306 df-join 18307 df-meet 18308 df-p0 18384 df-p1 18385 df-lat 18391 df-clat 18458 df-mgm 18567 df-sgrp 18646 df-mnd 18662 df-submnd 18711 df-grp 18868 df-minusg 18869 df-sbg 18870 df-subg 19055 df-cntz 19249 df-oppg 19278 df-lsm 19566 df-cmn 19712 df-abl 19713 df-mgp 20050 df-rng 20062 df-ur 20091 df-ring 20144 df-oppr 20246 df-dvdsr 20266 df-unit 20267 df-invr 20297 df-dvr 20310 df-nzr 20422 df-rlreg 20603 df-domn 20604 df-drng 20640 df-lmod 20768 df-lss 20838 df-lsp 20878 df-lvec 21010 df-lsatoms 38969 df-lshyp 38970 df-lcv 39012 df-lfl 39051 df-lkr 39079 df-ldual 39117 df-oposet 39169 df-ol 39171 df-oml 39172 df-covers 39259 df-ats 39260 df-atl 39291 df-cvlat 39315 df-hlat 39344 df-llines 39492 df-lplanes 39493 df-lvols 39494 df-lines 39495 df-psubsp 39497 df-pmap 39498 df-padd 39790 df-lhyp 39982 df-laut 39983 df-ldil 40098 df-ltrn 40099 df-trl 40153 df-tgrp 40737 df-tendo 40749 df-edring 40751 df-dveca 40997 df-disoa 41023 df-dvech 41073 df-dib 41133 df-dic 41167 df-dih 41223 df-doch 41342 df-djh 41389 df-lcdual 41581 df-mapd 41619 df-hvmap 41751 df-hdmap1 41787 df-hdmap 41788

This theorem is referenced by: hdmaprnlem17N 41857

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