hdmaprnlem16N - Mathbox for Norm Megill

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Theorem hdmaprnlem16N 43062

Description: Lemma for hdmaprnN 43064. 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 42310	. . . 4 ⊢ (𝜑 → 𝑈 ∈ LMod)
5		hdmaprnlem15.m	. . . . 5 ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊)
6		eqid 2763	. . . . 5 ⊢ (LSAtoms‘𝑈) = (LSAtoms‘𝑈)
7		hdmaprnlem15.c	. . . . 5 ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊)
8		eqid 2763	. . . . 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 42792	. . . . . 6 ⊢ (𝜑 → 𝐶 ∈ LMod)
13		hdmaprnlem16.se	. . . . . 6 ⊢ (𝜑 → 𝑠 ∈ (𝐷 ∖ { 0 }))
14	9, 10, 11, 8, 12, 13	lsatlspsn 40193	. . . . 5 ⊢ (𝜑 → (𝐿‘{𝑠}) ∈ (LSAtoms‘𝐶))
15	1, 5, 2, 6, 7, 8, 3, 14	mapdcnvatN 42866	. . . 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 40194	. . . 4 ⊢ ((𝑈 ∈ LMod ∧ (^◡𝑀‘(𝐿‘{𝑠})) ∈ (LSAtoms‘𝑈)) → ∃𝑣 ∈ 𝑉 (^◡𝑀‘(𝐿‘{𝑠})) = (𝑁‘{𝑣}))
19	4, 15, 18	syl2anc 596	. . 3 ⊢ (𝜑 → ∃𝑣 ∈ 𝑉 (^◡𝑀‘(𝐿‘{𝑠})) = (𝑁‘{𝑣}))
20		simpr 490	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑣 ∈ 𝑉) ∧ (^◡𝑀‘(𝐿‘{𝑠})) = (𝑁‘{𝑣})) → (^◡𝑀‘(𝐿‘{𝑠})) = (𝑁‘{𝑣}))
21	20	fveq2d 6900	. . . . . 6 ⊢ (((𝜑 ∧ 𝑣 ∈ 𝑉) ∧ (^◡𝑀‘(𝐿‘{𝑠})) = (𝑁‘{𝑣})) → (𝑀‘(^◡𝑀‘(𝐿‘{𝑠}))) = (𝑀‘(𝑁‘{𝑣})))
22	3	ad2antrr 739	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑣 ∈ 𝑉) ∧ (^◡𝑀‘(𝐿‘{𝑠})) = (𝑁‘{𝑣})) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
23	13	eldifad 4054	. . . . . . . . . 10 ⊢ (𝜑 → 𝑠 ∈ 𝐷)
24		eqid 2763	. . . . . . . . . . 11 ⊢ (LSubSp‘𝐶) = (LSubSp‘𝐶)
25	9, 24, 10	lspsncl 21267	. . . . . . . . . 10 ⊢ ((𝐶 ∈ LMod ∧ 𝑠 ∈ 𝐷) → (𝐿‘{𝑠}) ∈ (LSubSp‘𝐶))
26	12, 23, 25	syl2anc 596	. . . . . . . . 9 ⊢ (𝜑 → (𝐿‘{𝑠}) ∈ (LSubSp‘𝐶))
27	1, 5, 7, 24, 3	mapdrn2 42851	. . . . . . . . 9 ⊢ (𝜑 → ran 𝑀 = (LSubSp‘𝐶))
28	26, 27	eleqtrrd 2867	. . . . . . . 8 ⊢ (𝜑 → (𝐿‘{𝑠}) ∈ ran 𝑀)
29	28	ad2antrr 739	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑣 ∈ 𝑉) ∧ (^◡𝑀‘(𝐿‘{𝑠})) = (𝑁‘{𝑣})) → (𝐿‘{𝑠}) ∈ ran 𝑀)
30	1, 5, 22, 29	mapdcnvid2 42857	. . . . . 6 ⊢ (((𝜑 ∧ 𝑣 ∈ 𝑉) ∧ (^◡𝑀‘(𝐿‘{𝑠})) = (𝑁‘{𝑣})) → (𝑀‘(^◡𝑀‘(𝐿‘{𝑠}))) = (𝐿‘{𝑠}))
31	21, 30	eqtr3d 2800	. . . . 5 ⊢ (((𝜑 ∧ 𝑣 ∈ 𝑉) ∧ (^◡𝑀‘(𝐿‘{𝑠})) = (𝑁‘{𝑣})) → (𝑀‘(𝑁‘{𝑣})) = (𝐿‘{𝑠}))
32	31	ex 418	. . . 4 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑉) → ((^◡𝑀‘(𝐿‘{𝑠})) = (𝑁‘{𝑣}) → (𝑀‘(𝑁‘{𝑣})) = (𝐿‘{𝑠})))
33	32	reximdva 3179	. . 3 ⊢ (𝜑 → (∃𝑣 ∈ 𝑉 (^◡𝑀‘(𝐿‘{𝑠})) = (𝑁‘{𝑣}) → ∃𝑣 ∈ 𝑉 (𝑀‘(𝑁‘{𝑣})) = (𝐿‘{𝑠})))
34	19, 33	mpd 16	. 2 ⊢ (𝜑 → ∃𝑣 ∈ 𝑉 (𝑀‘(𝑁‘{𝑣})) = (𝐿‘{𝑠}))
35		hdmaprnlem15.s	. . . 4 ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊)
36	3	3ad2ant1 1151	. . . 4 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑉 ∧ (𝑀‘(𝑁‘{𝑣})) = (𝐿‘{𝑠})) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
37	13	3ad2ant1 1151	. . . 4 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑉 ∧ (𝑀‘(𝑁‘{𝑣})) = (𝐿‘{𝑠})) → 𝑠 ∈ (𝐷 ∖ { 0 }))
38		simp2 1155	. . . 4 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑉 ∧ (𝑀‘(𝑁‘{𝑣})) = (𝐿‘{𝑠})) → 𝑣 ∈ 𝑉)
39		simp3 1156	. . . 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 43061	. . 3 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑉 ∧ (𝑀‘(𝑁‘{𝑣})) = (𝐿‘{𝑠})) → 𝑠 ∈ ran 𝑆)
41	40	rexlimdv3a 3171	. 2 ⊢ (𝜑 → (∃𝑣 ∈ 𝑉 (𝑀‘(𝑁‘{𝑣})) = (𝐿‘{𝑠}) → 𝑠 ∈ ran 𝑆))
42	34, 41	mpd 16	1 ⊢ (𝜑 → 𝑠 ∈ ran 𝑆)

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∧ wa 401 ∧ w3a 1103 = wceq 1569 ∈ wcel 2146 ∃wrex 3090 ∖ cdif 4048 {csn 4588 ^◡ccnv 5654 ran crn 5656 ‘cfv 6547 Basecbs 17402 0_gc0g 17625 LModclmod 21150 LSubSpclss 21221 LSpanclspn 21261 LSAtomsclsa 40174 HLchlt 40550 LHypclh 41184 DVecHcdvh 42278 LCDualclcd 42786 mapdcmpd 42824 HDMapchdma 42992

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1827 ax-4 1841 ax-5 1944 ax-6 2001 ax-7 2042 ax-8 2148 ax-9 2156 ax-10 2179 ax-11 2196 ax-12 2215 ax-ext 2735 ax-rep 5235 ax-sep 5252 ax-nul 5263 ax-pow 5331 ax-pr 5395 ax-un 7761 ax-cnex 11269 ax-resscn 11270 ax-1cn 11271 ax-icn 11272 ax-addcl 11273 ax-addrcl 11274 ax-mulcl 11275 ax-mulrcl 11276 ax-mulcom 11277 ax-addass 11278 ax-mulass 11279 ax-distr 11280 ax-i2m1 11281 ax-1ne0 11282 ax-1rid 11283 ax-rnegex 11284 ax-rrecex 11285 ax-cnre 11286 ax-pre-lttri 11287 ax-pre-lttrn 11288 ax-pre-ltadd 11289 ax-pre-mulgt0 11290 ax-riotaBAD 40153

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1572 df-fal 1582 df-ex 1812 df-nf 1816 df-sb 2101 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2839 df-nfc 2913 df-ne 2960 df-nel 3066 df-ral 3081 df-rex 3091 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3456 df-sbc 3743 df-csb 3852 df-ss 3901 df-pss 4017 df-dif 4050 df-in 4094 df-un 4154 df-nul 4282 df-if 4486 df-pw 4562 df-sn 4589 df-pr 4591 df-tp 4593 df-op 4839 df-ot 4862 df-uni 4871 df-int 4911 df-iun 4956 df-iin 4957 df-br 5107 df-opab 5171 df-mpt 5190 df-tr 5216 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 6311 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6501 df-fun 6549 df-fn 6550 df-f 6551 df-f1 6552 df-fo 6553 df-f1o 6554 df-fv 6555 df-riota 7387 df-ov 7433 df-oprab 7434 df-mpo 7435 df-of 7703 df-om 7888 df-1st 8011 df-2nd 8012 df-tpos 8257 df-undef 8304 df-frecs 8313 df-wrecs 8344 df-recs 8393 df-rdg 8432 df-1o 8490 df-2o 8491 df-er 8731 df-map 8863 df-en 8988 df-dom 8989 df-sdom 8990 df-fin 8991 df-pnf 11358 df-mnf 11359 df-xr 11360 df-ltxr 11361 df-le 11362 df-sub 11556 df-neg 11557 df-nn 12349 df-2 12418 df-3 12419 df-4 12420 df-5 12421 df-6 12422 df-n0 12620 df-z 12707 df-uz 12979 df-fz 13653 df-struct 17340 df-sets 17357 df-slot 17375 df-ndx 17387 df-base 17403 df-ress 17424 df-plusg 17456 df-mulr 17457 df-sca 17459 df-vsca 17460 df-0g 17627 df-mre 17771 df-mrc 17772 df-acs 17774 df-proset 18483 df-poset 18502 df-plt 18517 df-lub 18533 df-glb 18534 df-join 18535 df-meet 18536 df-p0 18612 df-p1 18613 df-lat 18621 df-clat 18688 df-mgm 18831 df-sgrp 18923 df-mnd 18939 df-submnd 18994 df-grp 19162 df-minusg 19163 df-sbg 19164 df-subg 19348 df-cntz 19546 df-oppg 19575 df-lsm 19865 df-cmn 20011 df-abl 20012 df-mgp 20376 df-rng 20390 df-ur 20423 df-ring 20476 df-oppr 20582 df-dvdsr 20602 df-unit 20603 df-invr 20633 df-dvr 20646 df-nzr 20778 df-rlreg 20961 df-domn 20962 df-drng 20997 df-lmod 21152 df-lss 21222 df-lsp 21262 df-lvec 21393 df-lsatoms 40176 df-lshyp 40177 df-lcv 40219 df-lfl 40258 df-lkr 40286 df-ldual 40324 df-oposet 40376 df-ol 40378 df-oml 40379 df-covers 40466 df-ats 40467 df-atl 40498 df-cvlat 40522 df-hlat 40551 df-llines 40698 df-lplanes 40699 df-lvols 40700 df-lines 40701 df-psubsp 40703 df-pmap 40704 df-padd 40996 df-lhyp 41188 df-laut 41189 df-ldil 41304 df-ltrn 41305 df-trl 41359 df-tgrp 41943 df-tendo 41955 df-edring 41957 df-dveca 42203 df-disoa 42229 df-dvech 42279 df-dib 42339 df-dic 42373 df-dih 42429 df-doch 42548 df-djh 42595 df-lcdual 42787 df-mapd 42825 df-hvmap 42957 df-hdmap1 42993 df-hdmap 42994

This theorem is used by: hdmaprnlem17N 43063

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