hgmaprnlem4N - Mathbox for Norm Megill

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Theorem hgmaprnlem4N 41502

Description: Lemma for hgmaprnN 41504. 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 41195	. . . . 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 3956	. . . . . 6 ⊢ (𝜑 → 𝑡 ∈ 𝑉)
12	1, 6, 7, 2, 8, 9, 3, 11	hdmapcl 41433	. . . . 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 20773	. . . . 5 ⊢ ((𝐶 ∈ LMod ∧ 𝑧 ∈ 𝐴 ∧ (𝑆‘𝑡) ∈ 𝐷) → (𝑧 ∙ (𝑆‘𝑡)) ∈ 𝐷)
17	4, 5, 12, 16	syl3anc 1368	. . . 4 ⊢ (𝜑 → (𝑧 ∙ (𝑆‘𝑡)) ∈ 𝐷)
18	1, 2, 8, 9, 3	hdmaprnN 41467	. . . 4 ⊢ (𝜑 → ran 𝑆 = 𝐷)
19	17, 18	eleqtrrd 2828	. . 3 ⊢ (𝜑 → (𝑧 ∙ (𝑆‘𝑡)) ∈ ran 𝑆)
20	1, 6, 7, 9, 3	hdmapfnN 41432	. . . 4 ⊢ (𝜑 → 𝑆 Fn 𝑉)
21		fvelrnb 6958	. . . 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 231	. 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 1130	. . . 4 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑉 ∧ (𝑆‘𝑠) = (𝑧 ∙ (𝑆‘𝑡))) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
31	5	3ad2ant1 1130	. . . 4 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑉 ∧ (𝑆‘𝑠) = (𝑧 ∙ (𝑆‘𝑡))) → 𝑧 ∈ 𝐴)
32	10	3ad2ant1 1130	. . . 4 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑉 ∧ (𝑆‘𝑠) = (𝑧 ∙ (𝑆‘𝑡))) → 𝑡 ∈ (𝑉 ∖ { 0 }))
33		simp2 1134	. . . 4 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑉 ∧ (𝑆‘𝑠) = (𝑧 ∙ (𝑆‘𝑡))) → 𝑠 ∈ 𝑉)
34		simp3 1135	. . . 4 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑉 ∧ (𝑆‘𝑠) = (𝑧 ∙ (𝑆‘𝑡))) → (𝑆‘𝑠) = (𝑧 ∙ (𝑆‘𝑡)))
35		eqid 2725	. . . 4 ⊢ ((mapd‘𝐾)‘𝑊) = ((mapd‘𝐾)‘𝑊)
36		eqid 2725	. . . 4 ⊢ (LSpan‘𝑈) = (LSpan‘𝑈)
37		eqid 2725	. . . 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 41501	. . 3 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑉 ∧ (𝑆‘𝑠) = (𝑧 ∙ (𝑆‘𝑡))) → 𝑧 ∈ ran 𝐺)
39	38	rexlimdv3a 3148	. 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 205 ∧ wa 394 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 ∃wrex 3059 ∖ cdif 3941 {csn 4630 ran crn 5679 Fn wfn 6544 ‘cfv 6549 (class class class)co 7419 Basecbs 17183 Scalarcsca 17239 ·_𝑠 cvsca 17240 0_gc0g 17424 LModclmod 20755 LSpanclspn 20867 HLchlt 38952 LHypclh 39587 DVecHcdvh 40681 LCDualclcd 41189 mapdcmpd 41227 HDMapchdma 41395 HGMapchg 41486

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 ax-cnex 11196 ax-resscn 11197 ax-1cn 11198 ax-icn 11199 ax-addcl 11200 ax-addrcl 11201 ax-mulcl 11202 ax-mulrcl 11203 ax-mulcom 11204 ax-addass 11205 ax-mulass 11206 ax-distr 11207 ax-i2m1 11208 ax-1ne0 11209 ax-1rid 11210 ax-rnegex 11211 ax-rrecex 11212 ax-cnre 11213 ax-pre-lttri 11214 ax-pre-lttrn 11215 ax-pre-ltadd 11216 ax-pre-mulgt0 11217 ax-riotaBAD 38555

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-nel 3036 df-ral 3051 df-rex 3060 df-rmo 3363 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3964 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-tp 4635 df-op 4637 df-ot 4639 df-uni 4910 df-int 4951 df-iun 4999 df-iin 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6307 df-ord 6374 df-on 6375 df-lim 6376 df-suc 6377 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-riota 7375 df-ov 7422 df-oprab 7423 df-mpo 7424 df-of 7685 df-om 7872 df-1st 7994 df-2nd 7995 df-tpos 8232 df-undef 8279 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-1o 8487 df-er 8725 df-map 8847 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-pnf 11282 df-mnf 11283 df-xr 11284 df-ltxr 11285 df-le 11286 df-sub 11478 df-neg 11479 df-nn 12246 df-2 12308 df-3 12309 df-4 12310 df-5 12311 df-6 12312 df-n0 12506 df-z 12592 df-uz 12856 df-fz 13520 df-struct 17119 df-sets 17136 df-slot 17154 df-ndx 17166 df-base 17184 df-ress 17213 df-plusg 17249 df-mulr 17250 df-sca 17252 df-vsca 17253 df-0g 17426 df-mre 17569 df-mrc 17570 df-acs 17572 df-proset 18290 df-poset 18308 df-plt 18325 df-lub 18341 df-glb 18342 df-join 18343 df-meet 18344 df-p0 18420 df-p1 18421 df-lat 18427 df-clat 18494 df-mgm 18603 df-sgrp 18682 df-mnd 18698 df-submnd 18744 df-grp 18901 df-minusg 18902 df-sbg 18903 df-subg 19086 df-cntz 19280 df-oppg 19309 df-lsm 19603 df-cmn 19749 df-abl 19750 df-mgp 20087 df-rng 20105 df-ur 20134 df-ring 20187 df-oppr 20285 df-dvdsr 20308 df-unit 20309 df-invr 20339 df-dvr 20352 df-drng 20638 df-lmod 20757 df-lss 20828 df-lsp 20868 df-lvec 21000 df-lsatoms 38578 df-lshyp 38579 df-lcv 38621 df-lfl 38660 df-lkr 38688 df-ldual 38726 df-oposet 38778 df-ol 38780 df-oml 38781 df-covers 38868 df-ats 38869 df-atl 38900 df-cvlat 38924 df-hlat 38953 df-llines 39101 df-lplanes 39102 df-lvols 39103 df-lines 39104 df-psubsp 39106 df-pmap 39107 df-padd 39399 df-lhyp 39591 df-laut 39592 df-ldil 39707 df-ltrn 39708 df-trl 39762 df-tgrp 40346 df-tendo 40358 df-edring 40360 df-dveca 40606 df-disoa 40632 df-dvech 40682 df-dib 40742 df-dic 40776 df-dih 40832 df-doch 40951 df-djh 40998 df-lcdual 41190 df-mapd 41228 df-hvmap 41360 df-hdmap1 41396 df-hdmap 41397 df-hgmap 41487

This theorem is referenced by: hgmaprnlem5N 41503

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