hgmaprnlem4N - Mathbox for Norm Megill

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Theorem hgmaprnlem4N 42391

Description: Lemma for hgmaprnN 42393. Eliminate 𝑠. (Contributed by NM, 7-Jun-2015.) (New usage is discouraged.)

**Hypotheses**
Ref	Expression
hgmaprnlem1.h	⊢ 𝐻 = (LHyp‘𝐾)
hgmaprnlem1.u	⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
hgmaprnlem1.v	⊢ 𝑉 = (Base‘𝑈)
hgmaprnlem1.r	⊢ 𝑅 = (Scalar‘𝑈)
hgmaprnlem1.b	⊢ 𝐵 = (Base‘𝑅)
hgmaprnlem1.t	⊢ · = ( ·_𝑠 ‘𝑈)
hgmaprnlem1.o	⊢ 0 = (0_g‘𝑈)
hgmaprnlem1.c	⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊)
hgmaprnlem1.d	⊢ 𝐷 = (Base‘𝐶)
hgmaprnlem1.p	⊢ 𝑃 = (Scalar‘𝐶)
hgmaprnlem1.a	⊢ 𝐴 = (Base‘𝑃)
hgmaprnlem1.e	⊢ ∙ = ( ·_𝑠 ‘𝐶)
hgmaprnlem1.q	⊢ 𝑄 = (0_g‘𝐶)
hgmaprnlem1.s	⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊)
hgmaprnlem1.g	⊢ 𝐺 = ((HGMap‘𝐾)‘𝑊)
hgmaprnlem1.k	⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
hgmaprnlem1.z	⊢ (𝜑 → 𝑧 ∈ 𝐴)
hgmaprnlem1.t2	⊢ (𝜑 → 𝑡 ∈ (𝑉 ∖ { 0 }))

**Assertion**
Ref	Expression
hgmaprnlem4N	⊢ (𝜑 → 𝑧 ∈ ran 𝐺)

Distinct variable group: 𝑧,𝑡 Allowed substitution hints: 𝜑(𝑧,𝑡) 𝐴(𝑧,𝑡) 𝐵(𝑧,𝑡) 𝐶(𝑧,𝑡) 𝐷(𝑧,𝑡) 𝑃(𝑧,𝑡) 𝑄(𝑧,𝑡) 𝑅(𝑧,𝑡) 𝑆(𝑧,𝑡) ∙ (𝑧,𝑡) · (𝑧,𝑡) 𝑈(𝑧,𝑡) 𝐺(𝑧,𝑡) 𝐻(𝑧,𝑡) 𝐾(𝑧,𝑡) 𝑉(𝑧,𝑡) 𝑊(𝑧,𝑡) 0 (𝑧,𝑡)

Proof of Theorem hgmaprnlem4N Dummy variable 𝑠 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		hgmaprnlem1.h	. . . . . 6 ⊢ 𝐻 = (LHyp‘𝐾)
2		hgmaprnlem1.c	. . . . . 6 ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊)
3		hgmaprnlem1.k	. . . . . 6 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
4	1, 2, 3	lcdlmod 42084	. . . . 5 ⊢ (𝜑 → 𝐶 ∈ LMod)
5		hgmaprnlem1.z	. . . . 5 ⊢ (𝜑 → 𝑧 ∈ 𝐴)
6		hgmaprnlem1.u	. . . . . 6 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
7		hgmaprnlem1.v	. . . . . 6 ⊢ 𝑉 = (Base‘𝑈)
8		hgmaprnlem1.d	. . . . . 6 ⊢ 𝐷 = (Base‘𝐶)
9		hgmaprnlem1.s	. . . . . 6 ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊)
10		hgmaprnlem1.t2	. . . . . . 7 ⊢ (𝜑 → 𝑡 ∈ (𝑉 ∖ { 0 }))
11	10	eldifad 3895	. . . . . 6 ⊢ (𝜑 → 𝑡 ∈ 𝑉)
12	1, 6, 7, 2, 8, 9, 3, 11	hdmapcl 42322	. . . . 5 ⊢ (𝜑 → (𝑆‘𝑡) ∈ 𝐷)
13		hgmaprnlem1.p	. . . . . 6 ⊢ 𝑃 = (Scalar‘𝐶)
14		hgmaprnlem1.e	. . . . . 6 ⊢ ∙ = ( ·_𝑠 ‘𝐶)
15		hgmaprnlem1.a	. . . . . 6 ⊢ 𝐴 = (Base‘𝑃)
16	8, 13, 14, 15	lmodvscl 20868	. . . . 5 ⊢ ((𝐶 ∈ LMod ∧ 𝑧 ∈ 𝐴 ∧ (𝑆‘𝑡) ∈ 𝐷) → (𝑧 ∙ (𝑆‘𝑡)) ∈ 𝐷)
17	4, 5, 12, 16	syl3anc 1379	. . . 4 ⊢ (𝜑 → (𝑧 ∙ (𝑆‘𝑡)) ∈ 𝐷)
18	1, 2, 8, 9, 3	hdmaprnN 42356	. . . 4 ⊢ (𝜑 → ran 𝑆 = 𝐷)
19	17, 18	eleqtrrd 2842	. . 3 ⊢ (𝜑 → (𝑧 ∙ (𝑆‘𝑡)) ∈ ran 𝑆)
20	1, 6, 7, 9, 3	hdmapfnN 42321	. . . 4 ⊢ (𝜑 → 𝑆 Fn 𝑉)
21		fvelrnb 6887	. . . 4 ⊢ (𝑆 Fn 𝑉 → ((𝑧 ∙ (𝑆‘𝑡)) ∈ ran 𝑆 ↔ ∃𝑠 ∈ 𝑉 (𝑆‘𝑠) = (𝑧 ∙ (𝑆‘𝑡))))
22	20, 21	syl 17	. . 3 ⊢ (𝜑 → ((𝑧 ∙ (𝑆‘𝑡)) ∈ ran 𝑆 ↔ ∃𝑠 ∈ 𝑉 (𝑆‘𝑠) = (𝑧 ∙ (𝑆‘𝑡))))
23	19, 22	mpbid 233	. 2 ⊢ (𝜑 → ∃𝑠 ∈ 𝑉 (𝑆‘𝑠) = (𝑧 ∙ (𝑆‘𝑡)))
24		hgmaprnlem1.r	. . . 4 ⊢ 𝑅 = (Scalar‘𝑈)
25		hgmaprnlem1.b	. . . 4 ⊢ 𝐵 = (Base‘𝑅)
26		hgmaprnlem1.t	. . . 4 ⊢ · = ( ·_𝑠 ‘𝑈)
27		hgmaprnlem1.o	. . . 4 ⊢ 0 = (0_g‘𝑈)
28		hgmaprnlem1.q	. . . 4 ⊢ 𝑄 = (0_g‘𝐶)
29		hgmaprnlem1.g	. . . 4 ⊢ 𝐺 = ((HGMap‘𝐾)‘𝑊)
30	3	3ad2ant1 1139	. . . 4 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑉 ∧ (𝑆‘𝑠) = (𝑧 ∙ (𝑆‘𝑡))) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
31	5	3ad2ant1 1139	. . . 4 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑉 ∧ (𝑆‘𝑠) = (𝑧 ∙ (𝑆‘𝑡))) → 𝑧 ∈ 𝐴)
32	10	3ad2ant1 1139	. . . 4 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑉 ∧ (𝑆‘𝑠) = (𝑧 ∙ (𝑆‘𝑡))) → 𝑡 ∈ (𝑉 ∖ { 0 }))
33		simp2 1143	. . . 4 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑉 ∧ (𝑆‘𝑠) = (𝑧 ∙ (𝑆‘𝑡))) → 𝑠 ∈ 𝑉)
34		simp3 1144	. . . 4 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑉 ∧ (𝑆‘𝑠) = (𝑧 ∙ (𝑆‘𝑡))) → (𝑆‘𝑠) = (𝑧 ∙ (𝑆‘𝑡)))
35		eqid 2739	. . . 4 ⊢ ((mapd‘𝐾)‘𝑊) = ((mapd‘𝐾)‘𝑊)
36		eqid 2739	. . . 4 ⊢ (LSpan‘𝑈) = (LSpan‘𝑈)
37		eqid 2739	. . . 4 ⊢ (LSpan‘𝐶) = (LSpan‘𝐶)
38	1, 6, 7, 24, 25, 26, 27, 2, 8, 13, 15, 14, 28, 9, 29, 30, 31, 32, 33, 34, 35, 36, 37	hgmaprnlem3N 42390	. . 3 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑉 ∧ (𝑆‘𝑠) = (𝑧 ∙ (𝑆‘𝑡))) → 𝑧 ∈ ran 𝐺)
39	38	rexlimdv3a 3144	. 2 ⊢ (𝜑 → (∃𝑠 ∈ 𝑉 (𝑆‘𝑠) = (𝑧 ∙ (𝑆‘𝑡)) → 𝑧 ∈ ran 𝐺))
40	23, 39	mpd 15	1 ⊢ (𝜑 → 𝑧 ∈ ran 𝐺)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 ∧ w3a 1092 = wceq 1547 ∈ wcel 2119 ∃wrex 3063 ∖ cdif 3880 {csn 4555 ran crn 5619 Fn wfn 6480 ‘cfv 6485 (class class class)co 7356 Basecbs 17170 Scalarcsca 17214 ·_𝑠 cvsca 17215 0_gc0g 17393 LModclmod 20850 LSpanclspn 20961 HLchlt 39842 LHypclh 40476 DVecHcdvh 41570 LCDualclcd 42078 mapdcmpd 42116 HDMapchdma 42284 HGMapchg 42375

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1974 ax-7 2015 ax-8 2121 ax-9 2129 ax-10 2152 ax-11 2168 ax-12 2189 ax-ext 2711 ax-rep 5199 ax-sep 5218 ax-nul 5228 ax-pow 5294 ax-pr 5362 ax-un 7678 ax-cnex 11085 ax-resscn 11086 ax-1cn 11087 ax-icn 11088 ax-addcl 11089 ax-addrcl 11090 ax-mulcl 11091 ax-mulrcl 11092 ax-mulcom 11093 ax-addass 11094 ax-mulass 11095 ax-distr 11096 ax-i2m1 11097 ax-1ne0 11098 ax-1rid 11099 ax-rnegex 11100 ax-rrecex 11101 ax-cnre 11102 ax-pre-lttri 11103 ax-pre-lttrn 11104 ax-pre-ltadd 11105 ax-pre-mulgt0 11106 ax-riotaBAD 39445

This theorem depends on definitions: df-bi 208 df-an 397 df-or 854 df-3or 1093 df-3an 1094 df-tru 1550 df-fal 1560 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2543 df-eu 2573 df-clab 2718 df-cleq 2731 df-clel 2814 df-nfc 2888 df-ne 2935 df-nel 3039 df-ral 3054 df-rex 3064 df-rmo 3344 df-reu 3345 df-rab 3392 df-v 3433 df-sbc 3724 df-csb 3832 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3903 df-nul 4262 df-if 4455 df-pw 4531 df-sn 4556 df-pr 4558 df-tp 4560 df-op 4562 df-ot 4564 df-uni 4839 df-int 4878 df-iun 4923 df-iin 4924 df-br 5073 df-opab 5135 df-mpt 5154 df-tr 5180 df-id 5513 df-eprel 5518 df-po 5526 df-so 5527 df-fr 5571 df-we 5573 df-xp 5624 df-rel 5625 df-cnv 5626 df-co 5627 df-dm 5628 df-rn 5629 df-res 5630 df-ima 5631 df-pred 6252 df-ord 6313 df-on 6314 df-lim 6315 df-suc 6316 df-iota 6441 df-fun 6487 df-fn 6488 df-f 6489 df-f1 6490 df-fo 6491 df-f1o 6492 df-fv 6493 df-riota 7313 df-ov 7359 df-oprab 7360 df-mpo 7361 df-of 7620 df-om 7807 df-1st 7931 df-2nd 7932 df-tpos 8166 df-undef 8213 df-frecs 8221 df-wrecs 8252 df-recs 8301 df-rdg 8339 df-1o 8395 df-2o 8396 df-er 8633 df-map 8765 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-pnf 11172 df-mnf 11173 df-xr 11174 df-ltxr 11175 df-le 11176 df-sub 11370 df-neg 11371 df-nn 12166 df-2 12235 df-3 12236 df-4 12237 df-5 12238 df-6 12239 df-n0 12429 df-z 12516 df-uz 12780 df-fz 13453 df-struct 17108 df-sets 17125 df-slot 17143 df-ndx 17155 df-base 17171 df-ress 17192 df-plusg 17224 df-mulr 17225 df-sca 17227 df-vsca 17228 df-0g 17395 df-mre 17539 df-mrc 17540 df-acs 17542 df-proset 18251 df-poset 18270 df-plt 18285 df-lub 18301 df-glb 18302 df-join 18303 df-meet 18304 df-p0 18380 df-p1 18381 df-lat 18389 df-clat 18456 df-mgm 18599 df-sgrp 18678 df-mnd 18694 df-submnd 18743 df-grp 18903 df-minusg 18904 df-sbg 18905 df-subg 19090 df-cntz 19283 df-oppg 19312 df-lsm 19602 df-cmn 19748 df-abl 19749 df-mgp 20113 df-rng 20125 df-ur 20154 df-ring 20207 df-oppr 20308 df-dvdsr 20328 df-unit 20329 df-invr 20359 df-dvr 20372 df-nzr 20485 df-rlreg 20666 df-domn 20667 df-drng 20703 df-lmod 20852 df-lss 20922 df-lsp 20962 df-lvec 21093 df-lsatoms 39468 df-lshyp 39469 df-lcv 39511 df-lfl 39550 df-lkr 39578 df-ldual 39616 df-oposet 39668 df-ol 39670 df-oml 39671 df-covers 39758 df-ats 39759 df-atl 39790 df-cvlat 39814 df-hlat 39843 df-llines 39990 df-lplanes 39991 df-lvols 39992 df-lines 39993 df-psubsp 39995 df-pmap 39996 df-padd 40288 df-lhyp 40480 df-laut 40481 df-ldil 40596 df-ltrn 40597 df-trl 40651 df-tgrp 41235 df-tendo 41247 df-edring 41249 df-dveca 41495 df-disoa 41521 df-dvech 41571 df-dib 41631 df-dic 41665 df-dih 41721 df-doch 41840 df-djh 41887 df-lcdual 42079 df-mapd 42117 df-hvmap 42249 df-hdmap1 42285 df-hdmap 42286 df-hgmap 42376

This theorem is referenced by: hgmaprnlem5N 42392

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