hdmap1neglem1N - Mathbox for Norm Megill

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Theorem hdmap1neglem1N 36135

Description: Lemma for hdmapneg 36156. 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 2610	. . . . . 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 2610	. . . . . 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 3552	. . . . . . . 8 ⊢ (𝜑 → 𝑌 ∈ 𝑉)
21	2, 3, 4, 6, 7, 8, 9, 11, 12, 13, 14, 16, 18, 19, 15, 20	hdmap1cl 36112	. . . . . . 7 ⊢ (𝜑 → (𝐼‘⟨𝑋, 𝐹, 𝑌⟩) ∈ 𝐷)
22	1, 21	eqeltrrd 2689	. . . . . 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 36109	. . . . 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 221	. . . 4 ⊢ (𝜑 → ((𝑀‘(𝑁‘{𝑌})) = (𝐿‘{𝐺}) ∧ (𝑀‘(𝑁‘{(𝑋(-_g‘𝑈)𝑌)})) = (𝐿‘{(𝐹(-_g‘𝐶)𝐺)})))
25	24	simpld 474	. . 3 ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑌})) = (𝐿‘{𝐺}))
26	2, 3, 14	dvhlmod 35417	. . . . 5 ⊢ (𝜑 → 𝑈 ∈ LMod)
27		hdmap1neglem1.r	. . . . . 6 ⊢ 𝑅 = (inv_g‘𝑈)
28	4, 27, 7	lspsnneg 18827	. . . . 5 ⊢ ((𝑈 ∈ LMod ∧ 𝑌 ∈ 𝑉) → (𝑁‘{(𝑅‘𝑌)}) = (𝑁‘{𝑌}))
29	26, 20, 28	syl2anc 691	. . . 4 ⊢ (𝜑 → (𝑁‘{(𝑅‘𝑌)}) = (𝑁‘{𝑌}))
30	29	fveq2d 6107	. . 3 ⊢ (𝜑 → (𝑀‘(𝑁‘{(𝑅‘𝑌)})) = (𝑀‘(𝑁‘{𝑌})))
31	2, 8, 14	lcdlmod 35899	. . . 4 ⊢ (𝜑 → 𝐶 ∈ LMod)
32		hdmap1neglem1.s	. . . . 5 ⊢ 𝑆 = (inv_g‘𝐶)
33	9, 32, 11	lspsnneg 18827	. . . 4 ⊢ ((𝐶 ∈ LMod ∧ 𝐺 ∈ 𝐷) → (𝐿‘{(𝑆‘𝐺)}) = (𝐿‘{𝐺}))
34	31, 22, 33	syl2anc 691	. . 3 ⊢ (𝜑 → (𝐿‘{(𝑆‘𝐺)}) = (𝐿‘{𝐺}))
35	25, 30, 34	3eqtr4d 2654	. 2 ⊢ (𝜑 → (𝑀‘(𝑁‘{(𝑅‘𝑌)})) = (𝐿‘{(𝑆‘𝐺)}))
36	24	simprd 478	. . 3 ⊢ (𝜑 → (𝑀‘(𝑁‘{(𝑋(-_g‘𝑈)𝑌)})) = (𝐿‘{(𝐹(-_g‘𝐶)𝐺)}))
37		lmodabl 18733	. . . . . . . . 9 ⊢ (𝑈 ∈ LMod → 𝑈 ∈ Abel)
38	26, 37	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝑈 ∈ Abel)
39	15	eldifad 3552	. . . . . . . 8 ⊢ (𝜑 → 𝑋 ∈ 𝑉)
40	4, 5, 27, 38, 39, 20	ablsub2inv 18039	. . . . . . 7 ⊢ (𝜑 → ((𝑅‘𝑋)(-_g‘𝑈)(𝑅‘𝑌)) = (𝑌(-_g‘𝑈)𝑋))
41	40	sneqd 4137	. . . . . 6 ⊢ (𝜑 → {((𝑅‘𝑋)(-_g‘𝑈)(𝑅‘𝑌))} = {(𝑌(-_g‘𝑈)𝑋)})
42	41	fveq2d 6107	. . . . 5 ⊢ (𝜑 → (𝑁‘{((𝑅‘𝑋)(-_g‘𝑈)(𝑅‘𝑌))}) = (𝑁‘{(𝑌(-_g‘𝑈)𝑋)}))
43	4, 5, 7, 26, 20, 39	lspsnsub 18828	. . . . 5 ⊢ (𝜑 → (𝑁‘{(𝑌(-_g‘𝑈)𝑋)}) = (𝑁‘{(𝑋(-_g‘𝑈)𝑌)}))
44	42, 43	eqtrd 2644	. . . 4 ⊢ (𝜑 → (𝑁‘{((𝑅‘𝑋)(-_g‘𝑈)(𝑅‘𝑌))}) = (𝑁‘{(𝑋(-_g‘𝑈)𝑌)}))
45	44	fveq2d 6107	. . 3 ⊢ (𝜑 → (𝑀‘(𝑁‘{((𝑅‘𝑋)(-_g‘𝑈)(𝑅‘𝑌))})) = (𝑀‘(𝑁‘{(𝑋(-_g‘𝑈)𝑌)})))
46		lmodabl 18733	. . . . . . . 8 ⊢ (𝐶 ∈ LMod → 𝐶 ∈ Abel)
47	31, 46	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝐶 ∈ Abel)
48	9, 10, 32, 47, 16, 22	ablsub2inv 18039	. . . . . 6 ⊢ (𝜑 → ((𝑆‘𝐹)(-_g‘𝐶)(𝑆‘𝐺)) = (𝐺(-_g‘𝐶)𝐹))
49	48	sneqd 4137	. . . . 5 ⊢ (𝜑 → {((𝑆‘𝐹)(-_g‘𝐶)(𝑆‘𝐺))} = {(𝐺(-_g‘𝐶)𝐹)})
50	49	fveq2d 6107	. . . 4 ⊢ (𝜑 → (𝐿‘{((𝑆‘𝐹)(-_g‘𝐶)(𝑆‘𝐺))}) = (𝐿‘{(𝐺(-_g‘𝐶)𝐹)}))
51	9, 10, 11, 31, 22, 16	lspsnsub 18828	. . . 4 ⊢ (𝜑 → (𝐿‘{(𝐺(-_g‘𝐶)𝐹)}) = (𝐿‘{(𝐹(-_g‘𝐶)𝐺)}))
52	50, 51	eqtrd 2644	. . 3 ⊢ (𝜑 → (𝐿‘{((𝑆‘𝐹)(-_g‘𝐶)(𝑆‘𝐺))}) = (𝐿‘{(𝐹(-_g‘𝐶)𝐺)}))
53	36, 45, 52	3eqtr4d 2654	. 2 ⊢ (𝜑 → (𝑀‘(𝑁‘{((𝑅‘𝑋)(-_g‘𝑈)(𝑅‘𝑌))})) = (𝐿‘{((𝑆‘𝐹)(-_g‘𝐶)(𝑆‘𝐺))}))
54		lmodgrp 18693	. . . . 5 ⊢ (𝑈 ∈ LMod → 𝑈 ∈ Grp)
55	26, 54	syl 17	. . . 4 ⊢ (𝜑 → 𝑈 ∈ Grp)
56	4, 6, 27	grpinvnzcl 17310	. . . 4 ⊢ ((𝑈 ∈ Grp ∧ 𝑋 ∈ (𝑉 ∖ { 0 })) → (𝑅‘𝑋) ∈ (𝑉 ∖ { 0 }))
57	55, 15, 56	syl2anc 691	. . 3 ⊢ (𝜑 → (𝑅‘𝑋) ∈ (𝑉 ∖ { 0 }))
58	9, 32	lmodvnegcl 18727	. . . 4 ⊢ ((𝐶 ∈ LMod ∧ 𝐹 ∈ 𝐷) → (𝑆‘𝐹) ∈ 𝐷)
59	31, 16, 58	syl2anc 691	. . 3 ⊢ (𝜑 → (𝑆‘𝐹) ∈ 𝐷)
60	4, 6, 27	grpinvnzcl 17310	. . . 4 ⊢ ((𝑈 ∈ Grp ∧ 𝑌 ∈ (𝑉 ∖ { 0 })) → (𝑅‘𝑌) ∈ (𝑉 ∖ { 0 }))
61	55, 17, 60	syl2anc 691	. . 3 ⊢ (𝜑 → (𝑅‘𝑌) ∈ (𝑉 ∖ { 0 }))
62	9, 32	lmodvnegcl 18727	. . . 4 ⊢ ((𝐶 ∈ LMod ∧ 𝐺 ∈ 𝐷) → (𝑆‘𝐺) ∈ 𝐷)
63	31, 22, 62	syl2anc 691	. . 3 ⊢ (𝜑 → (𝑆‘𝐺) ∈ 𝐷)
64	4, 27, 7	lspsnneg 18827	. . . . 5 ⊢ ((𝑈 ∈ LMod ∧ 𝑋 ∈ 𝑉) → (𝑁‘{(𝑅‘𝑋)}) = (𝑁‘{𝑋}))
65	26, 39, 64	syl2anc 691	. . . 4 ⊢ (𝜑 → (𝑁‘{(𝑅‘𝑋)}) = (𝑁‘{𝑋}))
66	19, 65, 29	3netr4d 2859	. . 3 ⊢ (𝜑 → (𝑁‘{(𝑅‘𝑋)}) ≠ (𝑁‘{(𝑅‘𝑌)}))
67	65	fveq2d 6107	. . . 4 ⊢ (𝜑 → (𝑀‘(𝑁‘{(𝑅‘𝑋)})) = (𝑀‘(𝑁‘{𝑋})))
68	9, 32, 11	lspsnneg 18827	. . . . 5 ⊢ ((𝐶 ∈ LMod ∧ 𝐹 ∈ 𝐷) → (𝐿‘{(𝑆‘𝐹)}) = (𝐿‘{𝐹}))
69	31, 16, 68	syl2anc 691	. . . 4 ⊢ (𝜑 → (𝐿‘{(𝑆‘𝐹)}) = (𝐿‘{𝐹}))
70	18, 67, 69	3eqtr4d 2654	. . 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 36109	. 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 959	1 ⊢ (𝜑 → (𝐼‘⟨(𝑅‘𝑋), (𝑆‘𝐹), (𝑅‘𝑌)⟩) = (𝑆‘𝐺))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 ∖ cdif 3537 {csn 4125 ⟨cotp 4133 ‘cfv 5804 (class class class)co 6549 Basecbs 15695 0_gc0g 15923 Grpcgrp 17245 inv_gcminusg 17246 -_gcsg 17247 Abelcabl 18017 LModclmod 18686 LSpanclspn 18792 HLchlt 33655 LHypclh 34288 DVecHcdvh 35385 LCDualclcd 35893 mapdcmpd 35931 HDMap1chdma1 36099

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-riotaBAD 33257

This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-fal 1481 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-ot 4134 df-uni 4373 df-int 4411 df-iun 4457 df-iin 4458 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-of 6795 df-om 6958 df-1st 7059 df-2nd 7060 df-tpos 7239 df-undef 7286 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-oadd 7451 df-er 7629 df-map 7746 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-2 10956 df-3 10957 df-4 10958 df-5 10959 df-6 10960 df-n0 11170 df-z 11255 df-uz 11564 df-fz 12198 df-struct 15697 df-ndx 15698 df-slot 15699 df-base 15700 df-sets 15701 df-ress 15702 df-plusg 15781 df-mulr 15782 df-sca 15784 df-vsca 15785 df-0g 15925 df-mre 16069 df-mrc 16070 df-acs 16072 df-preset 16751 df-poset 16769 df-plt 16781 df-lub 16797 df-glb 16798 df-join 16799 df-meet 16800 df-p0 16862 df-p1 16863 df-lat 16869 df-clat 16931 df-mgm 17065 df-sgrp 17107 df-mnd 17118 df-submnd 17159 df-grp 17248 df-minusg 17249 df-sbg 17250 df-subg 17414 df-cntz 17573 df-oppg 17599 df-lsm 17874 df-cmn 18018 df-abl 18019 df-mgp 18313 df-ur 18325 df-ring 18372 df-oppr 18446 df-dvdsr 18464 df-unit 18465 df-invr 18495 df-dvr 18506 df-drng 18572 df-lmod 18688 df-lss 18754 df-lsp 18793 df-lvec 18924 df-lsatoms 33281 df-lshyp 33282 df-lcv 33324 df-lfl 33363 df-lkr 33391 df-ldual 33429 df-oposet 33481 df-ol 33483 df-oml 33484 df-covers 33571 df-ats 33572 df-atl 33603 df-cvlat 33627 df-hlat 33656 df-llines 33802 df-lplanes 33803 df-lvols 33804 df-lines 33805 df-psubsp 33807 df-pmap 33808 df-padd 34100 df-lhyp 34292 df-laut 34293 df-ldil 34408 df-ltrn 34409 df-trl 34464 df-tgrp 35049 df-tendo 35061 df-edring 35063 df-dveca 35309 df-disoa 35336 df-dvech 35386 df-dib 35446 df-dic 35480 df-dih 35536 df-doch 35655 df-djh 35702 df-lcdual 35894 df-mapd 35932 df-hdmap1 36101

This theorem is referenced by: (None)

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