hdmap1neglem1N - Mathbox for Norm Megill

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Theorem hdmap1neglem1N 37117

Description: Lemma for hdmapneg 37138. TODO: Not used; delete. (Contributed by NM, 23-May-2015.) (New usage is discouraged.)

**Hypotheses**
Ref	Expression
hdmap1neglem1.h	⊢ 𝐻 = (LHyp‘𝐾)
hdmap1neglem1.u	⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
hdmap1neglem1.v	⊢ 𝑉 = (Base‘𝑈)
hdmap1neglem1.r	⊢ 𝑅 = (inv_g‘𝑈)
hdmap1neglem1.o	⊢ 0 = (0_g‘𝑈)
hdmap1neglem1.n	⊢ 𝑁 = (LSpan‘𝑈)
hdmap1neglem1.c	⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊)
hdmap1neglem1.d	⊢ 𝐷 = (Base‘𝐶)
hdmap1neglem1.s	⊢ 𝑆 = (inv_g‘𝐶)
hdmap1neglem1.l	⊢ 𝐿 = (LSpan‘𝐶)
hdmap1neglem1.m	⊢ 𝑀 = ((mapd‘𝐾)‘𝑊)
hdmap1neglem1.i	⊢ 𝐼 = ((HDMap1‘𝐾)‘𝑊)
hdmap1neglem1.k	⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
hdmap1neglem1.f	⊢ (𝜑 → 𝐹 ∈ 𝐷)
hdmap1neglem1.mn	⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐿‘{𝐹}))
hdmap1neglem1.ne	⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}))
hdmap1neglem1.x	⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))
hdmap1neglem1.y	⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 }))
hdmap1neglem1.e	⊢ (𝜑 → (𝐼‘⟨𝑋, 𝐹, 𝑌⟩) = 𝐺)

**Assertion**
Ref	Expression
hdmap1neglem1N	⊢ (𝜑 → (𝐼‘⟨(𝑅‘𝑋), (𝑆‘𝐹), (𝑅‘𝑌)⟩) = (𝑆‘𝐺))

**Proof of Theorem hdmap1neglem1N**
Step	Hyp	Ref	Expression
1		hdmap1neglem1.e	. . . . 5 ⊢ (𝜑 → (𝐼‘⟨𝑋, 𝐹, 𝑌⟩) = 𝐺)
2		hdmap1neglem1.h	. . . . . 6 ⊢ 𝐻 = (LHyp‘𝐾)
3		hdmap1neglem1.u	. . . . . 6 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
4		hdmap1neglem1.v	. . . . . 6 ⊢ 𝑉 = (Base‘𝑈)
5		eqid 2622	. . . . . 6 ⊢ (-_g‘𝑈) = (-_g‘𝑈)
6		hdmap1neglem1.o	. . . . . 6 ⊢ 0 = (0_g‘𝑈)
7		hdmap1neglem1.n	. . . . . 6 ⊢ 𝑁 = (LSpan‘𝑈)
8		hdmap1neglem1.c	. . . . . 6 ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊)
9		hdmap1neglem1.d	. . . . . 6 ⊢ 𝐷 = (Base‘𝐶)
10		eqid 2622	. . . . . 6 ⊢ (-_g‘𝐶) = (-_g‘𝐶)
11		hdmap1neglem1.l	. . . . . 6 ⊢ 𝐿 = (LSpan‘𝐶)
12		hdmap1neglem1.m	. . . . . 6 ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊)
13		hdmap1neglem1.i	. . . . . 6 ⊢ 𝐼 = ((HDMap1‘𝐾)‘𝑊)
14		hdmap1neglem1.k	. . . . . 6 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
15		hdmap1neglem1.x	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))
16		hdmap1neglem1.f	. . . . . 6 ⊢ (𝜑 → 𝐹 ∈ 𝐷)
17		hdmap1neglem1.y	. . . . . 6 ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 }))
18		hdmap1neglem1.mn	. . . . . . . 8 ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐿‘{𝐹}))
19		hdmap1neglem1.ne	. . . . . . . 8 ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}))
20	17	eldifad 3586	. . . . . . . 8 ⊢ (𝜑 → 𝑌 ∈ 𝑉)
21	2, 3, 4, 6, 7, 8, 9, 11, 12, 13, 14, 16, 18, 19, 15, 20	hdmap1cl 37094	. . . . . . 7 ⊢ (𝜑 → (𝐼‘⟨𝑋, 𝐹, 𝑌⟩) ∈ 𝐷)
22	1, 21	eqeltrrd 2702	. . . . . 6 ⊢ (𝜑 → 𝐺 ∈ 𝐷)
23	2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 22, 19, 18	hdmap1eq 37091	. . . . 5 ⊢ (𝜑 → ((𝐼‘⟨𝑋, 𝐹, 𝑌⟩) = 𝐺 ↔ ((𝑀‘(𝑁‘{𝑌})) = (𝐿‘{𝐺}) ∧ (𝑀‘(𝑁‘{(𝑋(-_g‘𝑈)𝑌)})) = (𝐿‘{(𝐹(-_g‘𝐶)𝐺)}))))
24	1, 23	mpbid 222	. . . 4 ⊢ (𝜑 → ((𝑀‘(𝑁‘{𝑌})) = (𝐿‘{𝐺}) ∧ (𝑀‘(𝑁‘{(𝑋(-_g‘𝑈)𝑌)})) = (𝐿‘{(𝐹(-_g‘𝐶)𝐺)})))
25	24	simpld 475	. . 3 ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑌})) = (𝐿‘{𝐺}))
26	2, 3, 14	dvhlmod 36399	. . . . 5 ⊢ (𝜑 → 𝑈 ∈ LMod)
27		hdmap1neglem1.r	. . . . . 6 ⊢ 𝑅 = (inv_g‘𝑈)
28	4, 27, 7	lspsnneg 19006	. . . . 5 ⊢ ((𝑈 ∈ LMod ∧ 𝑌 ∈ 𝑉) → (𝑁‘{(𝑅‘𝑌)}) = (𝑁‘{𝑌}))
29	26, 20, 28	syl2anc 693	. . . 4 ⊢ (𝜑 → (𝑁‘{(𝑅‘𝑌)}) = (𝑁‘{𝑌}))
30	29	fveq2d 6195	. . 3 ⊢ (𝜑 → (𝑀‘(𝑁‘{(𝑅‘𝑌)})) = (𝑀‘(𝑁‘{𝑌})))
31	2, 8, 14	lcdlmod 36881	. . . 4 ⊢ (𝜑 → 𝐶 ∈ LMod)
32		hdmap1neglem1.s	. . . . 5 ⊢ 𝑆 = (inv_g‘𝐶)
33	9, 32, 11	lspsnneg 19006	. . . 4 ⊢ ((𝐶 ∈ LMod ∧ 𝐺 ∈ 𝐷) → (𝐿‘{(𝑆‘𝐺)}) = (𝐿‘{𝐺}))
34	31, 22, 33	syl2anc 693	. . 3 ⊢ (𝜑 → (𝐿‘{(𝑆‘𝐺)}) = (𝐿‘{𝐺}))
35	25, 30, 34	3eqtr4d 2666	. 2 ⊢ (𝜑 → (𝑀‘(𝑁‘{(𝑅‘𝑌)})) = (𝐿‘{(𝑆‘𝐺)}))
36	24	simprd 479	. . 3 ⊢ (𝜑 → (𝑀‘(𝑁‘{(𝑋(-_g‘𝑈)𝑌)})) = (𝐿‘{(𝐹(-_g‘𝐶)𝐺)}))
37		lmodabl 18910	. . . . . . . . 9 ⊢ (𝑈 ∈ LMod → 𝑈 ∈ Abel)
38	26, 37	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝑈 ∈ Abel)
39	15	eldifad 3586	. . . . . . . 8 ⊢ (𝜑 → 𝑋 ∈ 𝑉)
40	4, 5, 27, 38, 39, 20	ablsub2inv 18216	. . . . . . 7 ⊢ (𝜑 → ((𝑅‘𝑋)(-_g‘𝑈)(𝑅‘𝑌)) = (𝑌(-_g‘𝑈)𝑋))
41	40	sneqd 4189	. . . . . 6 ⊢ (𝜑 → {((𝑅‘𝑋)(-_g‘𝑈)(𝑅‘𝑌))} = {(𝑌(-_g‘𝑈)𝑋)})
42	41	fveq2d 6195	. . . . 5 ⊢ (𝜑 → (𝑁‘{((𝑅‘𝑋)(-_g‘𝑈)(𝑅‘𝑌))}) = (𝑁‘{(𝑌(-_g‘𝑈)𝑋)}))
43	4, 5, 7, 26, 20, 39	lspsnsub 19007	. . . . 5 ⊢ (𝜑 → (𝑁‘{(𝑌(-_g‘𝑈)𝑋)}) = (𝑁‘{(𝑋(-_g‘𝑈)𝑌)}))
44	42, 43	eqtrd 2656	. . . 4 ⊢ (𝜑 → (𝑁‘{((𝑅‘𝑋)(-_g‘𝑈)(𝑅‘𝑌))}) = (𝑁‘{(𝑋(-_g‘𝑈)𝑌)}))
45	44	fveq2d 6195	. . 3 ⊢ (𝜑 → (𝑀‘(𝑁‘{((𝑅‘𝑋)(-_g‘𝑈)(𝑅‘𝑌))})) = (𝑀‘(𝑁‘{(𝑋(-_g‘𝑈)𝑌)})))
46		lmodabl 18910	. . . . . . . 8 ⊢ (𝐶 ∈ LMod → 𝐶 ∈ Abel)
47	31, 46	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝐶 ∈ Abel)
48	9, 10, 32, 47, 16, 22	ablsub2inv 18216	. . . . . 6 ⊢ (𝜑 → ((𝑆‘𝐹)(-_g‘𝐶)(𝑆‘𝐺)) = (𝐺(-_g‘𝐶)𝐹))
49	48	sneqd 4189	. . . . 5 ⊢ (𝜑 → {((𝑆‘𝐹)(-_g‘𝐶)(𝑆‘𝐺))} = {(𝐺(-_g‘𝐶)𝐹)})
50	49	fveq2d 6195	. . . 4 ⊢ (𝜑 → (𝐿‘{((𝑆‘𝐹)(-_g‘𝐶)(𝑆‘𝐺))}) = (𝐿‘{(𝐺(-_g‘𝐶)𝐹)}))
51	9, 10, 11, 31, 22, 16	lspsnsub 19007	. . . 4 ⊢ (𝜑 → (𝐿‘{(𝐺(-_g‘𝐶)𝐹)}) = (𝐿‘{(𝐹(-_g‘𝐶)𝐺)}))
52	50, 51	eqtrd 2656	. . 3 ⊢ (𝜑 → (𝐿‘{((𝑆‘𝐹)(-_g‘𝐶)(𝑆‘𝐺))}) = (𝐿‘{(𝐹(-_g‘𝐶)𝐺)}))
53	36, 45, 52	3eqtr4d 2666	. 2 ⊢ (𝜑 → (𝑀‘(𝑁‘{((𝑅‘𝑋)(-_g‘𝑈)(𝑅‘𝑌))})) = (𝐿‘{((𝑆‘𝐹)(-_g‘𝐶)(𝑆‘𝐺))}))
54		lmodgrp 18870	. . . . 5 ⊢ (𝑈 ∈ LMod → 𝑈 ∈ Grp)
55	26, 54	syl 17	. . . 4 ⊢ (𝜑 → 𝑈 ∈ Grp)
56	4, 6, 27	grpinvnzcl 17487	. . . 4 ⊢ ((𝑈 ∈ Grp ∧ 𝑋 ∈ (𝑉 ∖ { 0 })) → (𝑅‘𝑋) ∈ (𝑉 ∖ { 0 }))
57	55, 15, 56	syl2anc 693	. . 3 ⊢ (𝜑 → (𝑅‘𝑋) ∈ (𝑉 ∖ { 0 }))
58	9, 32	lmodvnegcl 18904	. . . 4 ⊢ ((𝐶 ∈ LMod ∧ 𝐹 ∈ 𝐷) → (𝑆‘𝐹) ∈ 𝐷)
59	31, 16, 58	syl2anc 693	. . 3 ⊢ (𝜑 → (𝑆‘𝐹) ∈ 𝐷)
60	4, 6, 27	grpinvnzcl 17487	. . . 4 ⊢ ((𝑈 ∈ Grp ∧ 𝑌 ∈ (𝑉 ∖ { 0 })) → (𝑅‘𝑌) ∈ (𝑉 ∖ { 0 }))
61	55, 17, 60	syl2anc 693	. . 3 ⊢ (𝜑 → (𝑅‘𝑌) ∈ (𝑉 ∖ { 0 }))
62	9, 32	lmodvnegcl 18904	. . . 4 ⊢ ((𝐶 ∈ LMod ∧ 𝐺 ∈ 𝐷) → (𝑆‘𝐺) ∈ 𝐷)
63	31, 22, 62	syl2anc 693	. . 3 ⊢ (𝜑 → (𝑆‘𝐺) ∈ 𝐷)
64	4, 27, 7	lspsnneg 19006	. . . . 5 ⊢ ((𝑈 ∈ LMod ∧ 𝑋 ∈ 𝑉) → (𝑁‘{(𝑅‘𝑋)}) = (𝑁‘{𝑋}))
65	26, 39, 64	syl2anc 693	. . . 4 ⊢ (𝜑 → (𝑁‘{(𝑅‘𝑋)}) = (𝑁‘{𝑋}))
66	19, 65, 29	3netr4d 2871	. . 3 ⊢ (𝜑 → (𝑁‘{(𝑅‘𝑋)}) ≠ (𝑁‘{(𝑅‘𝑌)}))
67	65	fveq2d 6195	. . . 4 ⊢ (𝜑 → (𝑀‘(𝑁‘{(𝑅‘𝑋)})) = (𝑀‘(𝑁‘{𝑋})))
68	9, 32, 11	lspsnneg 19006	. . . . 5 ⊢ ((𝐶 ∈ LMod ∧ 𝐹 ∈ 𝐷) → (𝐿‘{(𝑆‘𝐹)}) = (𝐿‘{𝐹}))
69	31, 16, 68	syl2anc 693	. . . 4 ⊢ (𝜑 → (𝐿‘{(𝑆‘𝐹)}) = (𝐿‘{𝐹}))
70	18, 67, 69	3eqtr4d 2666	. . 3 ⊢ (𝜑 → (𝑀‘(𝑁‘{(𝑅‘𝑋)})) = (𝐿‘{(𝑆‘𝐹)}))
71	2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 57, 59, 61, 63, 66, 70	hdmap1eq 37091	. 2 ⊢ (𝜑 → ((𝐼‘⟨(𝑅‘𝑋), (𝑆‘𝐹), (𝑅‘𝑌)⟩) = (𝑆‘𝐺) ↔ ((𝑀‘(𝑁‘{(𝑅‘𝑌)})) = (𝐿‘{(𝑆‘𝐺)}) ∧ (𝑀‘(𝑁‘{((𝑅‘𝑋)(-_g‘𝑈)(𝑅‘𝑌))})) = (𝐿‘{((𝑆‘𝐹)(-_g‘𝐶)(𝑆‘𝐺))}))))
72	35, 53, 71	mpbir2and 957	1 ⊢ (𝜑 → (𝐼‘⟨(𝑅‘𝑋), (𝑆‘𝐹), (𝑅‘𝑌)⟩) = (𝑆‘𝐺))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ≠ wne 2794 ∖ cdif 3571 {csn 4177 ⟨cotp 4185 ‘cfv 5888 (class class class)co 6650 Basecbs 15857 0_gc0g 16100 Grpcgrp 17422 inv_gcminusg 17423 -_gcsg 17424 Abelcabl 18194 LModclmod 18863 LSpanclspn 18971 HLchlt 34637 LHypclh 35270 DVecHcdvh 36367 LCDualclcd 36875 mapdcmpd 36913 HDMap1chdma1 37081

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-riotaBAD 34239

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-fal 1489 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-ot 4186 df-uni 4437 df-int 4476 df-iun 4522 df-iin 4523 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-of 6897 df-om 7066 df-1st 7168 df-2nd 7169 df-tpos 7352 df-undef 7399 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-oadd 7564 df-er 7742 df-map 7859 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-2 11079 df-3 11080 df-4 11081 df-5 11082 df-6 11083 df-n0 11293 df-z 11378 df-uz 11688 df-fz 12327 df-struct 15859 df-ndx 15860 df-slot 15861 df-base 15863 df-sets 15864 df-ress 15865 df-plusg 15954 df-mulr 15955 df-sca 15957 df-vsca 15958 df-0g 16102 df-mre 16246 df-mrc 16247 df-acs 16249 df-preset 16928 df-poset 16946 df-plt 16958 df-lub 16974 df-glb 16975 df-join 16976 df-meet 16977 df-p0 17039 df-p1 17040 df-lat 17046 df-clat 17108 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-submnd 17336 df-grp 17425 df-minusg 17426 df-sbg 17427 df-subg 17591 df-cntz 17750 df-oppg 17776 df-lsm 18051 df-cmn 18195 df-abl 18196 df-mgp 18490 df-ur 18502 df-ring 18549 df-oppr 18623 df-dvdsr 18641 df-unit 18642 df-invr 18672 df-dvr 18683 df-drng 18749 df-lmod 18865 df-lss 18933 df-lsp 18972 df-lvec 19103 df-lsatoms 34263 df-lshyp 34264 df-lcv 34306 df-lfl 34345 df-lkr 34373 df-ldual 34411 df-oposet 34463 df-ol 34465 df-oml 34466 df-covers 34553 df-ats 34554 df-atl 34585 df-cvlat 34609 df-hlat 34638 df-llines 34784 df-lplanes 34785 df-lvols 34786 df-lines 34787 df-psubsp 34789 df-pmap 34790 df-padd 35082 df-lhyp 35274 df-laut 35275 df-ldil 35390 df-ltrn 35391 df-trl 35446 df-tgrp 36031 df-tendo 36043 df-edring 36045 df-dveca 36291 df-disoa 36318 df-dvech 36368 df-dib 36428 df-dic 36462 df-dih 36518 df-doch 36637 df-djh 36684 df-lcdual 36876 df-mapd 36914 df-hdmap1 37083

This theorem is referenced by: (None)

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