hdmaprnlem9N - Mathbox for Norm Megill

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Theorem hdmaprnlem9N 42056

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 41838 and mapdcnv11N 41858. 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 42054	. . . . 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 42055	. . . . . 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 42052	. . . . . 6 ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑡})) = (𝐿‘{𝑠}))
25	23, 24	eleqtrd 2836	. . . . 5 ⊢ (𝜑 → (𝑠(-_g‘𝐶)(𝑆‘𝑡)) ∈ (𝐿‘{𝑠}))
26	22, 25	elind 4150	. . . 4 ⊢ (𝜑 → (𝑠(-_g‘𝐶)(𝑆‘𝑡)) ∈ ((𝐿‘{((𝑆‘𝑢) ✚ 𝑠)}) ∩ (𝐿‘{𝑠})))
27	1, 5, 9	lcdlvec 41790	. . . . 5 ⊢ (𝜑 → 𝐶 ∈ LVec)
28	1, 5, 9	lcdlmod 41791	. . . . . 6 ⊢ (𝜑 → 𝐶 ∈ LMod)
29	1, 2, 3, 5, 15, 8, 9, 13	hdmapcl 42029	. . . . . 6 ⊢ (𝜑 → (𝑆‘𝑢) ∈ 𝐷)
30	10	eldifad 3911	. . . . . 6 ⊢ (𝜑 → 𝑠 ∈ 𝐷)
31	15, 18	lmodvacl 20824	. . . . . 6 ⊢ ((𝐶 ∈ LMod ∧ (𝑆‘𝑢) ∈ 𝐷 ∧ 𝑠 ∈ 𝐷) → ((𝑆‘𝑢) ✚ 𝑠) ∈ 𝐷)
32	28, 29, 30, 31	syl3anc 1373	. . . . 5 ⊢ (𝜑 → ((𝑆‘𝑢) ✚ 𝑠) ∈ 𝐷)
33		eqid 2734	. . . . . . . . . . . . . 14 ⊢ (LSubSp‘𝐶) = (LSubSp‘𝐶)
34	15, 33, 6	lspsncl 20926	. . . . . . . . . . . . 13 ⊢ ((𝐶 ∈ LMod ∧ 𝑠 ∈ 𝐷) → (𝐿‘{𝑠}) ∈ (LSubSp‘𝐶))
35	28, 30, 34	syl2anc 584	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐿‘{𝑠}) ∈ (LSubSp‘𝐶))
36	1, 7, 5, 33, 9	mapdrn2 41850	. . . . . . . . . . . 12 ⊢ (𝜑 → ran 𝑀 = (LSubSp‘𝐶))
37	35, 36	eleqtrrd 2837	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐿‘{𝑠}) ∈ ran 𝑀)
38	1, 7, 9, 37	mapdcnvid2 41856	. . . . . . . . . 10 ⊢ (𝜑 → (𝑀‘(^◡𝑀‘(𝐿‘{𝑠}))) = (𝐿‘{𝑠}))
39	12, 38	eqtr4d 2772	. . . . . . . . 9 ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑣})) = (𝑀‘(^◡𝑀‘(𝐿‘{𝑠}))))
40		eqid 2734	. . . . . . . . . 10 ⊢ (LSubSp‘𝑈) = (LSubSp‘𝑈)
41	1, 2, 9	dvhlmod 41309	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑈 ∈ LMod)
42	3, 40, 4	lspsncl 20926	. . . . . . . . . . 11 ⊢ ((𝑈 ∈ LMod ∧ 𝑣 ∈ 𝑉) → (𝑁‘{𝑣}) ∈ (LSubSp‘𝑈))
43	41, 11, 42	syl2anc 584	. . . . . . . . . 10 ⊢ (𝜑 → (𝑁‘{𝑣}) ∈ (LSubSp‘𝑈))
44	1, 7, 2, 40, 9, 37	mapdcnvcl 41851	. . . . . . . . . 10 ⊢ (𝜑 → (^◡𝑀‘(𝐿‘{𝑠})) ∈ (LSubSp‘𝑈))
45	1, 2, 40, 7, 9, 43, 44	mapd11 41838	. . . . . . . . 9 ⊢ (𝜑 → ((𝑀‘(𝑁‘{𝑣})) = (𝑀‘(^◡𝑀‘(𝐿‘{𝑠}))) ↔ (𝑁‘{𝑣}) = (^◡𝑀‘(𝐿‘{𝑠}))))
46	39, 45	mpbid 232	. . . . . . . 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 42049	. . . . . . . 8 ⊢ (𝜑 → (𝑁‘{𝑣}) ≠ (^◡𝑀‘(𝐿‘{((𝑆‘𝑢) ✚ 𝑠)})))
48	46, 47	eqnetrrd 2998	. . . . . . 7 ⊢ (𝜑 → (^◡𝑀‘(𝐿‘{𝑠})) ≠ (^◡𝑀‘(𝐿‘{((𝑆‘𝑢) ✚ 𝑠)})))
49	15, 33, 6	lspsncl 20926	. . . . . . . . . . 11 ⊢ ((𝐶 ∈ LMod ∧ ((𝑆‘𝑢) ✚ 𝑠) ∈ 𝐷) → (𝐿‘{((𝑆‘𝑢) ✚ 𝑠)}) ∈ (LSubSp‘𝐶))
50	28, 32, 49	syl2anc 584	. . . . . . . . . 10 ⊢ (𝜑 → (𝐿‘{((𝑆‘𝑢) ✚ 𝑠)}) ∈ (LSubSp‘𝐶))
51	50, 36	eleqtrrd 2837	. . . . . . . . 9 ⊢ (𝜑 → (𝐿‘{((𝑆‘𝑢) ✚ 𝑠)}) ∈ ran 𝑀)
52	1, 7, 9, 37, 51	mapdcnv11N 41858	. . . . . . . 8 ⊢ (𝜑 → ((^◡𝑀‘(𝐿‘{𝑠})) = (^◡𝑀‘(𝐿‘{((𝑆‘𝑢) ✚ 𝑠)})) ↔ (𝐿‘{𝑠}) = (𝐿‘{((𝑆‘𝑢) ✚ 𝑠)})))
53	52	necon3bid 2974	. . . . . . 7 ⊢ (𝜑 → ((^◡𝑀‘(𝐿‘{𝑠})) ≠ (^◡𝑀‘(𝐿‘{((𝑆‘𝑢) ✚ 𝑠)})) ↔ (𝐿‘{𝑠}) ≠ (𝐿‘{((𝑆‘𝑢) ✚ 𝑠)})))
54	48, 53	mpbid 232	. . . . . 6 ⊢ (𝜑 → (𝐿‘{𝑠}) ≠ (𝐿‘{((𝑆‘𝑢) ✚ 𝑠)}))
55	54	necomd 2985	. . . . 5 ⊢ (𝜑 → (𝐿‘{((𝑆‘𝑢) ✚ 𝑠)}) ≠ (𝐿‘{𝑠}))
56	15, 16, 6, 27, 32, 30, 55	lspdisj2 21080	. . . 4 ⊢ (𝜑 → ((𝐿‘{((𝑆‘𝑢) ✚ 𝑠)}) ∩ (𝐿‘{𝑠})) = {𝑄})
57	26, 56	eleqtrd 2836	. . 3 ⊢ (𝜑 → (𝑠(-_g‘𝐶)(𝑆‘𝑡)) ∈ {𝑄})
58		elsni 4595	. . 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 42051	. . . 4 ⊢ (𝜑 → 𝑡 ∈ 𝑉)
61	1, 2, 3, 5, 15, 8, 9, 60	hdmapcl 42029	. . 3 ⊢ (𝜑 → (𝑆‘𝑡) ∈ 𝐷)
62		eqid 2734	. . . 4 ⊢ (-_g‘𝐶) = (-_g‘𝐶)
63	15, 16, 62	lmodsubeq0 20870	. . 3 ⊢ ((𝐶 ∈ LMod ∧ 𝑠 ∈ 𝐷 ∧ (𝑆‘𝑡) ∈ 𝐷) → ((𝑠(-_g‘𝐶)(𝑆‘𝑡)) = 𝑄 ↔ 𝑠 = (𝑆‘𝑡)))
64	28, 30, 61, 63	syl3anc 1373	. 2 ⊢ (𝜑 → ((𝑠(-_g‘𝐶)(𝑆‘𝑡)) = 𝑄 ↔ 𝑠 = (𝑆‘𝑡)))
65	59, 64	mpbid 232	1 ⊢ (𝜑 → 𝑠 = (𝑆‘𝑡))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1541 ∈ wcel 2113 ≠ wne 2930 ∖ cdif 3896 ∩ cin 3898 {csn 4578 ^◡ccnv 5621 ran crn 5623 ‘cfv 6490 (class class class)co 7356 Basecbs 17134 +_gcplusg 17175 0_gc0g 17357 -_gcsg 18863 LModclmod 20809 LSubSpclss 20880 LSpanclspn 20920 HLchlt 39549 LHypclh 40183 DVecHcdvh 41277 LCDualclcd 41785 mapdcmpd 41823 HDMapchdma 41991

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2182 ax-ext 2706 ax-rep 5222 ax-sep 5239 ax-nul 5249 ax-pow 5308 ax-pr 5375 ax-un 7678 ax-cnex 11080 ax-resscn 11081 ax-1cn 11082 ax-icn 11083 ax-addcl 11084 ax-addrcl 11085 ax-mulcl 11086 ax-mulrcl 11087 ax-mulcom 11088 ax-addass 11089 ax-mulass 11090 ax-distr 11091 ax-i2m1 11092 ax-1ne0 11093 ax-1rid 11094 ax-rnegex 11095 ax-rrecex 11096 ax-cnre 11097 ax-pre-lttri 11098 ax-pre-lttrn 11099 ax-pre-ltadd 11100 ax-pre-mulgt0 11101 ax-riotaBAD 39152

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2537 df-eu 2567 df-clab 2713 df-cleq 2726 df-clel 2809 df-nfc 2883 df-ne 2931 df-nel 3035 df-ral 3050 df-rex 3059 df-rmo 3348 df-reu 3349 df-rab 3398 df-v 3440 df-sbc 3739 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4284 df-if 4478 df-pw 4554 df-sn 4579 df-pr 4581 df-tp 4583 df-op 4585 df-ot 4587 df-uni 4862 df-int 4901 df-iun 4946 df-iin 4947 df-br 5097 df-opab 5159 df-mpt 5178 df-tr 5204 df-id 5517 df-eprel 5522 df-po 5530 df-so 5531 df-fr 5575 df-we 5577 df-xp 5628 df-rel 5629 df-cnv 5630 df-co 5631 df-dm 5632 df-rn 5633 df-res 5634 df-ima 5635 df-pred 6257 df-ord 6318 df-on 6319 df-lim 6320 df-suc 6321 df-iota 6446 df-fun 6492 df-fn 6493 df-f 6494 df-f1 6495 df-fo 6496 df-f1o 6497 df-fv 6498 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 8763 df-en 8882 df-dom 8883 df-sdom 8884 df-fin 8885 df-pnf 11166 df-mnf 11167 df-xr 11168 df-ltxr 11169 df-le 11170 df-sub 11364 df-neg 11365 df-nn 12144 df-2 12206 df-3 12207 df-4 12208 df-5 12209 df-6 12210 df-n0 12400 df-z 12487 df-uz 12750 df-fz 13422 df-struct 17072 df-sets 17089 df-slot 17107 df-ndx 17119 df-base 17135 df-ress 17156 df-plusg 17188 df-mulr 17189 df-sca 17191 df-vsca 17192 df-0g 17359 df-mre 17503 df-mrc 17504 df-acs 17506 df-proset 18215 df-poset 18234 df-plt 18249 df-lub 18265 df-glb 18266 df-join 18267 df-meet 18268 df-p0 18344 df-p1 18345 df-lat 18353 df-clat 18420 df-mgm 18563 df-sgrp 18642 df-mnd 18658 df-submnd 18707 df-grp 18864 df-minusg 18865 df-sbg 18866 df-subg 19051 df-cntz 19244 df-oppg 19273 df-lsm 19563 df-cmn 19709 df-abl 19710 df-mgp 20074 df-rng 20086 df-ur 20115 df-ring 20168 df-oppr 20271 df-dvdsr 20291 df-unit 20292 df-invr 20322 df-dvr 20335 df-nzr 20444 df-rlreg 20625 df-domn 20626 df-drng 20662 df-lmod 20811 df-lss 20881 df-lsp 20921 df-lvec 21053 df-lsatoms 39175 df-lshyp 39176 df-lcv 39218 df-lfl 39257 df-lkr 39285 df-ldual 39323 df-oposet 39375 df-ol 39377 df-oml 39378 df-covers 39465 df-ats 39466 df-atl 39497 df-cvlat 39521 df-hlat 39550 df-llines 39697 df-lplanes 39698 df-lvols 39699 df-lines 39700 df-psubsp 39702 df-pmap 39703 df-padd 39995 df-lhyp 40187 df-laut 40188 df-ldil 40303 df-ltrn 40304 df-trl 40358 df-tgrp 40942 df-tendo 40954 df-edring 40956 df-dveca 41202 df-disoa 41228 df-dvech 41278 df-dib 41338 df-dic 41372 df-dih 41428 df-doch 41547 df-djh 41594 df-lcdual 41786 df-mapd 41824 df-hvmap 41956 df-hdmap1 41992 df-hdmap 41993

This theorem is referenced by: hdmaprnlem10N 42058

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