hgmaprnlem4N - Mathbox for Norm Megill

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Theorem hgmaprnlem4N 43065

Description: Lemma for hgmaprnN 43067. 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 42758	. . . . 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 4050	. . . . . 6 ⊢ (𝜑 → 𝑡 ∈ 𝑉)
12	1, 6, 7, 2, 8, 9, 3, 11	hdmapcl 42996	. . . . 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 21162	. . . . 5 ⊢ ((𝐶 ∈ LMod ∧ 𝑧 ∈ 𝐴 ∧ (𝑆‘𝑡) ∈ 𝐷) → (𝑧 ∙ (𝑆‘𝑡)) ∈ 𝐷)
17	4, 5, 12, 16	syl3anc 1398	. . . 4 ⊢ (𝜑 → (𝑧 ∙ (𝑆‘𝑡)) ∈ 𝐷)
18	1, 2, 8, 9, 3	hdmaprnN 43030	. . . 4 ⊢ (𝜑 → ran 𝑆 = 𝐷)
19	17, 18	eleqtrrd 2863	. . 3 ⊢ (𝜑 → (𝑧 ∙ (𝑆‘𝑡)) ∈ ran 𝑆)
20	1, 6, 7, 9, 3	hdmapfnN 42995	. . . 4 ⊢ (𝜑 → 𝑆 Fn 𝑉)
21		fvelrnb 6950	. . . 4 ⊢ (𝑆 Fn 𝑉 → ((𝑧 ∙ (𝑆‘𝑡)) ∈ ran 𝑆 ↔ ∃𝑠 ∈ 𝑉 (𝑆‘𝑠) = (𝑧 ∙ (𝑆‘𝑡))))
22	20, 21	syl 18	. . 3 ⊢ (𝜑 → ((𝑧 ∙ (𝑆‘𝑡)) ∈ ran 𝑆 ↔ ∃𝑠 ∈ 𝑉 (𝑆‘𝑠) = (𝑧 ∙ (𝑆‘𝑡))))
23	19, 22	mpbid 235	. 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 1151	. . . 4 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑉 ∧ (𝑆‘𝑠) = (𝑧 ∙ (𝑆‘𝑡))) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
31	5	3ad2ant1 1151	. . . 4 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑉 ∧ (𝑆‘𝑠) = (𝑧 ∙ (𝑆‘𝑡))) → 𝑧 ∈ 𝐴)
32	10	3ad2ant1 1151	. . . 4 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑉 ∧ (𝑆‘𝑠) = (𝑧 ∙ (𝑆‘𝑡))) → 𝑡 ∈ (𝑉 ∖ { 0 }))
33		simp2 1155	. . . 4 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑉 ∧ (𝑆‘𝑠) = (𝑧 ∙ (𝑆‘𝑡))) → 𝑠 ∈ 𝑉)
34		simp3 1156	. . . 4 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑉 ∧ (𝑆‘𝑠) = (𝑧 ∙ (𝑆‘𝑡))) → (𝑆‘𝑠) = (𝑧 ∙ (𝑆‘𝑡)))
35		eqid 2760	. . . 4 ⊢ ((mapd‘𝐾)‘𝑊) = ((mapd‘𝐾)‘𝑊)
36		eqid 2760	. . . 4 ⊢ (LSpan‘𝑈) = (LSpan‘𝑈)
37		eqid 2760	. . . 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 43064	. . 3 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑉 ∧ (𝑆‘𝑠) = (𝑧 ∙ (𝑆‘𝑡))) → 𝑧 ∈ ran 𝐺)
39	38	rexlimdv3a 3167	. 2 ⊢ (𝜑 → (∃𝑠 ∈ 𝑉 (𝑆‘𝑠) = (𝑧 ∙ (𝑆‘𝑡)) → 𝑧 ∈ ran 𝐺))
40	23, 39	mpd 16	1 ⊢ (𝜑 → 𝑧 ∈ ran 𝐺)

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 ∧ w3a 1103 = wceq 1569 ∈ wcel 2144 ∃wrex 3086 ∖ cdif 4044 {csn 4584 ran crn 5652 Fn wfn 6538 ‘cfv 6543 (class class class)co 7424 Basecbs 17396 Scalarcsca 17440 ·_𝑠 cvsca 17441 0_gc0g 17619 LModclmod 21144 LSpanclspn 21255 HLchlt 40516 LHypclh 41150 DVecHcdvh 42244 LCDualclcd 42752 mapdcmpd 42790 HDMapchdma 42958 HGMapchg 43049

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 1942 ax-6 1999 ax-7 2040 ax-8 2146 ax-9 2154 ax-10 2177 ax-11 2193 ax-12 2212 ax-ext 2732 ax-rep 5231 ax-sep 5248 ax-nul 5259 ax-pow 5327 ax-pr 5391 ax-un 7755 ax-cnex 11263 ax-resscn 11264 ax-1cn 11265 ax-icn 11266 ax-addcl 11267 ax-addrcl 11268 ax-mulcl 11269 ax-mulrcl 11270 ax-mulcom 11271 ax-addass 11272 ax-mulass 11273 ax-distr 11274 ax-i2m1 11275 ax-1ne0 11276 ax-1rid 11277 ax-rnegex 11278 ax-rrecex 11279 ax-cnre 11280 ax-pre-lttri 11281 ax-pre-lttrn 11282 ax-pre-ltadd 11283 ax-pre-mulgt0 11284 ax-riotaBAD 40119

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 2099 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3365 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3739 df-csb 3848 df-ss 3897 df-pss 4013 df-dif 4046 df-in 4090 df-un 4150 df-nul 4278 df-if 4482 df-pw 4558 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4835 df-ot 4858 df-uni 4867 df-int 4907 df-iun 4952 df-iin 4953 df-br 5103 df-opab 5167 df-mpt 5186 df-tr 5212 df-id 5546 df-eprel 5551 df-po 5559 df-so 5560 df-fr 5604 df-we 5606 df-xp 5657 df-rel 5658 df-cnv 5659 df-co 5660 df-dm 5661 df-rn 5662 df-res 5663 df-ima 5664 df-pred 6307 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6497 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7381 df-ov 7427 df-oprab 7428 df-mpo 7429 df-of 7697 df-om 7882 df-1st 8005 df-2nd 8006 df-tpos 8250 df-undef 8297 df-frecs 8306 df-wrecs 8337 df-recs 8386 df-rdg 8425 df-1o 8483 df-2o 8484 df-er 8724 df-map 8856 df-en 8981 df-dom 8982 df-sdom 8983 df-fin 8984 df-pnf 11352 df-mnf 11353 df-xr 11354 df-ltxr 11355 df-le 11356 df-sub 11550 df-neg 11551 df-nn 12343 df-2 12412 df-3 12413 df-4 12414 df-5 12415 df-6 12416 df-n0 12614 df-z 12701 df-uz 12973 df-fz 13647 df-struct 17334 df-sets 17351 df-slot 17369 df-ndx 17381 df-base 17397 df-ress 17418 df-plusg 17450 df-mulr 17451 df-sca 17453 df-vsca 17454 df-0g 17621 df-mre 17765 df-mrc 17766 df-acs 17768 df-proset 18477 df-poset 18496 df-plt 18511 df-lub 18527 df-glb 18528 df-join 18529 df-meet 18530 df-p0 18606 df-p1 18607 df-lat 18615 df-clat 18682 df-mgm 18825 df-sgrp 18917 df-mnd 18933 df-submnd 18988 df-grp 19156 df-minusg 19157 df-sbg 19158 df-subg 19342 df-cntz 19540 df-oppg 19569 df-lsm 19859 df-cmn 20005 df-abl 20006 df-mgp 20370 df-rng 20384 df-ur 20417 df-ring 20470 df-oppr 20576 df-dvdsr 20596 df-unit 20597 df-invr 20627 df-dvr 20640 df-nzr 20772 df-rlreg 20955 df-domn 20956 df-drng 20991 df-lmod 21146 df-lss 21216 df-lsp 21256 df-lvec 21387 df-lsatoms 40142 df-lshyp 40143 df-lcv 40185 df-lfl 40224 df-lkr 40252 df-ldual 40290 df-oposet 40342 df-ol 40344 df-oml 40345 df-covers 40432 df-ats 40433 df-atl 40464 df-cvlat 40488 df-hlat 40517 df-llines 40664 df-lplanes 40665 df-lvols 40666 df-lines 40667 df-psubsp 40669 df-pmap 40670 df-padd 40962 df-lhyp 41154 df-laut 41155 df-ldil 41270 df-ltrn 41271 df-trl 41325 df-tgrp 41909 df-tendo 41921 df-edring 41923 df-dveca 42169 df-disoa 42195 df-dvech 42245 df-dib 42305 df-dic 42339 df-dih 42395 df-doch 42514 df-djh 42561 df-lcdual 42753 df-mapd 42791 df-hvmap 42923 df-hdmap1 42959 df-hdmap 42960 df-hgmap 43050

This theorem is used by: hgmaprnlem5N 43066

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