hdmaprnlem9N - Mathbox for Norm Megill

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Theorem hdmaprnlem9N 39153

Description: Part of proof of part 12 in [Baer] p. 49 line 21, s=S(t). TODO: we seem to be going back and forth with mapd11 38935 and mapdcnv11N 38955. Use better hypotheses and/or theorems? (Contributed by NM, 27-May-2015.) (New usage is discouraged.)

**Hypotheses**
Ref	Expression
hdmaprnlem1.h	⊢ 𝐻 = (LHyp‘𝐾)
hdmaprnlem1.u	⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
hdmaprnlem1.v	⊢ 𝑉 = (Base‘𝑈)
hdmaprnlem1.n	⊢ 𝑁 = (LSpan‘𝑈)
hdmaprnlem1.c	⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊)
hdmaprnlem1.l	⊢ 𝐿 = (LSpan‘𝐶)
hdmaprnlem1.m	⊢ 𝑀 = ((mapd‘𝐾)‘𝑊)
hdmaprnlem1.s	⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊)
hdmaprnlem1.k	⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
hdmaprnlem1.se	⊢ (𝜑 → 𝑠 ∈ (𝐷 ∖ {𝑄}))
hdmaprnlem1.ve	⊢ (𝜑 → 𝑣 ∈ 𝑉)
hdmaprnlem1.e	⊢ (𝜑 → (𝑀‘(𝑁‘{𝑣})) = (𝐿‘{𝑠}))
hdmaprnlem1.ue	⊢ (𝜑 → 𝑢 ∈ 𝑉)
hdmaprnlem1.un	⊢ (𝜑 → ¬ 𝑢 ∈ (𝑁‘{𝑣}))
hdmaprnlem1.d	⊢ 𝐷 = (Base‘𝐶)
hdmaprnlem1.q	⊢ 𝑄 = (0_g‘𝐶)
hdmaprnlem1.o	⊢ 0 = (0_g‘𝑈)
hdmaprnlem1.a	⊢ ✚ = (+_g‘𝐶)
hdmaprnlem1.t2	⊢ (𝜑 → 𝑡 ∈ ((𝑁‘{𝑣}) ∖ { 0 }))
hdmaprnlem1.p	⊢ + = (+_g‘𝑈)
hdmaprnlem1.pt	⊢ (𝜑 → (𝐿‘{((𝑆‘𝑢) ✚ 𝑠)}) = (𝑀‘(𝑁‘{(𝑢 + 𝑡)})))

**Assertion**
Ref	Expression
hdmaprnlem9N	⊢ (𝜑 → 𝑠 = (𝑆‘𝑡))

**Proof of Theorem hdmaprnlem9N**
Step	Hyp	Ref	Expression
1		hdmaprnlem1.h	. . . . . 6 ⊢ 𝐻 = (LHyp‘𝐾)
2		hdmaprnlem1.u	. . . . . 6 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
3		hdmaprnlem1.v	. . . . . 6 ⊢ 𝑉 = (Base‘𝑈)
4		hdmaprnlem1.n	. . . . . 6 ⊢ 𝑁 = (LSpan‘𝑈)
5		hdmaprnlem1.c	. . . . . 6 ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊)
6		hdmaprnlem1.l	. . . . . 6 ⊢ 𝐿 = (LSpan‘𝐶)
7		hdmaprnlem1.m	. . . . . 6 ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊)
8		hdmaprnlem1.s	. . . . . 6 ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊)
9		hdmaprnlem1.k	. . . . . 6 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
10		hdmaprnlem1.se	. . . . . 6 ⊢ (𝜑 → 𝑠 ∈ (𝐷 ∖ {𝑄}))
11		hdmaprnlem1.ve	. . . . . 6 ⊢ (𝜑 → 𝑣 ∈ 𝑉)
12		hdmaprnlem1.e	. . . . . 6 ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑣})) = (𝐿‘{𝑠}))
13		hdmaprnlem1.ue	. . . . . 6 ⊢ (𝜑 → 𝑢 ∈ 𝑉)
14		hdmaprnlem1.un	. . . . . 6 ⊢ (𝜑 → ¬ 𝑢 ∈ (𝑁‘{𝑣}))
15		hdmaprnlem1.d	. . . . . 6 ⊢ 𝐷 = (Base‘𝐶)
16		hdmaprnlem1.q	. . . . . 6 ⊢ 𝑄 = (0_g‘𝐶)
17		hdmaprnlem1.o	. . . . . 6 ⊢ 0 = (0_g‘𝑈)
18		hdmaprnlem1.a	. . . . . 6 ⊢ ✚ = (+_g‘𝐶)
19		hdmaprnlem1.t2	. . . . . 6 ⊢ (𝜑 → 𝑡 ∈ ((𝑁‘{𝑣}) ∖ { 0 }))
20		hdmaprnlem1.p	. . . . . 6 ⊢ + = (+_g‘𝑈)
21		hdmaprnlem1.pt	. . . . . 6 ⊢ (𝜑 → (𝐿‘{((𝑆‘𝑢) ✚ 𝑠)}) = (𝑀‘(𝑁‘{(𝑢 + 𝑡)})))
22	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21	hdmaprnlem7N 39151	. . . . 5 ⊢ (𝜑 → (𝑠(-_g‘𝐶)(𝑆‘𝑡)) ∈ (𝐿‘{((𝑆‘𝑢) ✚ 𝑠)}))
23	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21	hdmaprnlem8N 39152	. . . . . 6 ⊢ (𝜑 → (𝑠(-_g‘𝐶)(𝑆‘𝑡)) ∈ (𝑀‘(𝑁‘{𝑡})))
24	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19	hdmaprnlem4N 39149	. . . . . 6 ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑡})) = (𝐿‘{𝑠}))
25	23, 24	eleqtrd 2892	. . . . 5 ⊢ (𝜑 → (𝑠(-_g‘𝐶)(𝑆‘𝑡)) ∈ (𝐿‘{𝑠}))
26	22, 25	elind 4121	. . . 4 ⊢ (𝜑 → (𝑠(-_g‘𝐶)(𝑆‘𝑡)) ∈ ((𝐿‘{((𝑆‘𝑢) ✚ 𝑠)}) ∩ (𝐿‘{𝑠})))
27	1, 5, 9	lcdlvec 38887	. . . . 5 ⊢ (𝜑 → 𝐶 ∈ LVec)
28	1, 5, 9	lcdlmod 38888	. . . . . 6 ⊢ (𝜑 → 𝐶 ∈ LMod)
29	1, 2, 3, 5, 15, 8, 9, 13	hdmapcl 39126	. . . . . 6 ⊢ (𝜑 → (𝑆‘𝑢) ∈ 𝐷)
30	10	eldifad 3893	. . . . . 6 ⊢ (𝜑 → 𝑠 ∈ 𝐷)
31	15, 18	lmodvacl 19641	. . . . . 6 ⊢ ((𝐶 ∈ LMod ∧ (𝑆‘𝑢) ∈ 𝐷 ∧ 𝑠 ∈ 𝐷) → ((𝑆‘𝑢) ✚ 𝑠) ∈ 𝐷)
32	28, 29, 30, 31	syl3anc 1368	. . . . 5 ⊢ (𝜑 → ((𝑆‘𝑢) ✚ 𝑠) ∈ 𝐷)
33		eqid 2798	. . . . . . . . . . . . . 14 ⊢ (LSubSp‘𝐶) = (LSubSp‘𝐶)
34	15, 33, 6	lspsncl 19742	. . . . . . . . . . . . 13 ⊢ ((𝐶 ∈ LMod ∧ 𝑠 ∈ 𝐷) → (𝐿‘{𝑠}) ∈ (LSubSp‘𝐶))
35	28, 30, 34	syl2anc 587	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐿‘{𝑠}) ∈ (LSubSp‘𝐶))
36	1, 7, 5, 33, 9	mapdrn2 38947	. . . . . . . . . . . 12 ⊢ (𝜑 → ran 𝑀 = (LSubSp‘𝐶))
37	35, 36	eleqtrrd 2893	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐿‘{𝑠}) ∈ ran 𝑀)
38	1, 7, 9, 37	mapdcnvid2 38953	. . . . . . . . . 10 ⊢ (𝜑 → (𝑀‘(^◡𝑀‘(𝐿‘{𝑠}))) = (𝐿‘{𝑠}))
39	12, 38	eqtr4d 2836	. . . . . . . . 9 ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑣})) = (𝑀‘(^◡𝑀‘(𝐿‘{𝑠}))))
40		eqid 2798	. . . . . . . . . 10 ⊢ (LSubSp‘𝑈) = (LSubSp‘𝑈)
41	1, 2, 9	dvhlmod 38406	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑈 ∈ LMod)
42	3, 40, 4	lspsncl 19742	. . . . . . . . . . 11 ⊢ ((𝑈 ∈ LMod ∧ 𝑣 ∈ 𝑉) → (𝑁‘{𝑣}) ∈ (LSubSp‘𝑈))
43	41, 11, 42	syl2anc 587	. . . . . . . . . 10 ⊢ (𝜑 → (𝑁‘{𝑣}) ∈ (LSubSp‘𝑈))
44	1, 7, 2, 40, 9, 37	mapdcnvcl 38948	. . . . . . . . . 10 ⊢ (𝜑 → (^◡𝑀‘(𝐿‘{𝑠})) ∈ (LSubSp‘𝑈))
45	1, 2, 40, 7, 9, 43, 44	mapd11 38935	. . . . . . . . 9 ⊢ (𝜑 → ((𝑀‘(𝑁‘{𝑣})) = (𝑀‘(^◡𝑀‘(𝐿‘{𝑠}))) ↔ (𝑁‘{𝑣}) = (^◡𝑀‘(𝐿‘{𝑠}))))
46	39, 45	mpbid 235	. . . . . . . 8 ⊢ (𝜑 → (𝑁‘{𝑣}) = (^◡𝑀‘(𝐿‘{𝑠})))
47	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18	hdmaprnlem3N 39146	. . . . . . . 8 ⊢ (𝜑 → (𝑁‘{𝑣}) ≠ (^◡𝑀‘(𝐿‘{((𝑆‘𝑢) ✚ 𝑠)})))
48	46, 47	eqnetrrd 3055	. . . . . . 7 ⊢ (𝜑 → (^◡𝑀‘(𝐿‘{𝑠})) ≠ (^◡𝑀‘(𝐿‘{((𝑆‘𝑢) ✚ 𝑠)})))
49	15, 33, 6	lspsncl 19742	. . . . . . . . . . 11 ⊢ ((𝐶 ∈ LMod ∧ ((𝑆‘𝑢) ✚ 𝑠) ∈ 𝐷) → (𝐿‘{((𝑆‘𝑢) ✚ 𝑠)}) ∈ (LSubSp‘𝐶))
50	28, 32, 49	syl2anc 587	. . . . . . . . . 10 ⊢ (𝜑 → (𝐿‘{((𝑆‘𝑢) ✚ 𝑠)}) ∈ (LSubSp‘𝐶))
51	50, 36	eleqtrrd 2893	. . . . . . . . 9 ⊢ (𝜑 → (𝐿‘{((𝑆‘𝑢) ✚ 𝑠)}) ∈ ran 𝑀)
52	1, 7, 9, 37, 51	mapdcnv11N 38955	. . . . . . . 8 ⊢ (𝜑 → ((^◡𝑀‘(𝐿‘{𝑠})) = (^◡𝑀‘(𝐿‘{((𝑆‘𝑢) ✚ 𝑠)})) ↔ (𝐿‘{𝑠}) = (𝐿‘{((𝑆‘𝑢) ✚ 𝑠)})))
53	52	necon3bid 3031	. . . . . . 7 ⊢ (𝜑 → ((^◡𝑀‘(𝐿‘{𝑠})) ≠ (^◡𝑀‘(𝐿‘{((𝑆‘𝑢) ✚ 𝑠)})) ↔ (𝐿‘{𝑠}) ≠ (𝐿‘{((𝑆‘𝑢) ✚ 𝑠)})))
54	48, 53	mpbid 235	. . . . . 6 ⊢ (𝜑 → (𝐿‘{𝑠}) ≠ (𝐿‘{((𝑆‘𝑢) ✚ 𝑠)}))
55	54	necomd 3042	. . . . 5 ⊢ (𝜑 → (𝐿‘{((𝑆‘𝑢) ✚ 𝑠)}) ≠ (𝐿‘{𝑠}))
56	15, 16, 6, 27, 32, 30, 55	lspdisj2 19892	. . . 4 ⊢ (𝜑 → ((𝐿‘{((𝑆‘𝑢) ✚ 𝑠)}) ∩ (𝐿‘{𝑠})) = {𝑄})
57	26, 56	eleqtrd 2892	. . 3 ⊢ (𝜑 → (𝑠(-_g‘𝐶)(𝑆‘𝑡)) ∈ {𝑄})
58		elsni 4542	. . 3 ⊢ ((𝑠(-_g‘𝐶)(𝑆‘𝑡)) ∈ {𝑄} → (𝑠(-_g‘𝐶)(𝑆‘𝑡)) = 𝑄)
59	57, 58	syl 17	. 2 ⊢ (𝜑 → (𝑠(-_g‘𝐶)(𝑆‘𝑡)) = 𝑄)
60	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19	hdmaprnlem4tN 39148	. . . 4 ⊢ (𝜑 → 𝑡 ∈ 𝑉)
61	1, 2, 3, 5, 15, 8, 9, 60	hdmapcl 39126	. . 3 ⊢ (𝜑 → (𝑆‘𝑡) ∈ 𝐷)
62		eqid 2798	. . . 4 ⊢ (-_g‘𝐶) = (-_g‘𝐶)
63	15, 16, 62	lmodsubeq0 19686	. . 3 ⊢ ((𝐶 ∈ LMod ∧ 𝑠 ∈ 𝐷 ∧ (𝑆‘𝑡) ∈ 𝐷) → ((𝑠(-_g‘𝐶)(𝑆‘𝑡)) = 𝑄 ↔ 𝑠 = (𝑆‘𝑡)))
64	28, 30, 61, 63	syl3anc 1368	. 2 ⊢ (𝜑 → ((𝑠(-_g‘𝐶)(𝑆‘𝑡)) = 𝑄 ↔ 𝑠 = (𝑆‘𝑡)))
65	59, 64	mpbid 235	1 ⊢ (𝜑 → 𝑠 = (𝑆‘𝑡))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ≠ wne 2987 ∖ cdif 3878 ∩ cin 3880 {csn 4525 ^◡ccnv 5518 ran crn 5520 ‘cfv 6324 (class class class)co 7135 Basecbs 16475 +_gcplusg 16557 0_gc0g 16705 -_gcsg 18097 LModclmod 19627 LSubSpclss 19696 LSpanclspn 19736 HLchlt 36646 LHypclh 37280 DVecHcdvh 38374 LCDualclcd 38882 mapdcmpd 38920 HDMapchdma 39088

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 ax-riotaBAD 36249

This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-fal 1551 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-ot 4534 df-uni 4801 df-int 4839 df-iun 4883 df-iin 4884 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-of 7389 df-om 7561 df-1st 7671 df-2nd 7672 df-tpos 7875 df-undef 7922 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-oadd 8089 df-er 8272 df-map 8391 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-nn 11626 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-n0 11886 df-z 11970 df-uz 12232 df-fz 12886 df-struct 16477 df-ndx 16478 df-slot 16479 df-base 16481 df-sets 16482 df-ress 16483 df-plusg 16570 df-mulr 16571 df-sca 16573 df-vsca 16574 df-0g 16707 df-mre 16849 df-mrc 16850 df-acs 16852 df-proset 17530 df-poset 17548 df-plt 17560 df-lub 17576 df-glb 17577 df-join 17578 df-meet 17579 df-p0 17641 df-p1 17642 df-lat 17648 df-clat 17710 df-mgm 17844 df-sgrp 17893 df-mnd 17904 df-submnd 17949 df-grp 18098 df-minusg 18099 df-sbg 18100 df-subg 18268 df-cntz 18439 df-oppg 18466 df-lsm 18753 df-cmn 18900 df-abl 18901 df-mgp 19233 df-ur 19245 df-ring 19292 df-oppr 19369 df-dvdsr 19387 df-unit 19388 df-invr 19418 df-dvr 19429 df-drng 19497 df-lmod 19629 df-lss 19697 df-lsp 19737 df-lvec 19868 df-lsatoms 36272 df-lshyp 36273 df-lcv 36315 df-lfl 36354 df-lkr 36382 df-ldual 36420 df-oposet 36472 df-ol 36474 df-oml 36475 df-covers 36562 df-ats 36563 df-atl 36594 df-cvlat 36618 df-hlat 36647 df-llines 36794 df-lplanes 36795 df-lvols 36796 df-lines 36797 df-psubsp 36799 df-pmap 36800 df-padd 37092 df-lhyp 37284 df-laut 37285 df-ldil 37400 df-ltrn 37401 df-trl 37455 df-tgrp 38039 df-tendo 38051 df-edring 38053 df-dveca 38299 df-disoa 38325 df-dvech 38375 df-dib 38435 df-dic 38469 df-dih 38525 df-doch 38644 df-djh 38691 df-lcdual 38883 df-mapd 38921 df-hvmap 39053 df-hdmap1 39089 df-hdmap 39090

This theorem is referenced by: hdmaprnlem10N 39155

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