hdmaprnlem4N - Mathbox for Norm Megill

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Theorem hdmaprnlem4N 35946

Description: Part of proof of part 12 in [Baer] p. 49 line 19. (T* =) (Ft)* = Gs. (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 }))

**Assertion**
Ref	Expression
hdmaprnlem4N	⊢ (𝜑 → (𝑀‘(𝑁‘{𝑡})) = (𝐿‘{𝑠}))

**Proof of Theorem hdmaprnlem4N**
Step	Hyp	Ref	Expression
1		eqid 2609	. . . . 5 ⊢ (LSubSp‘𝑈) = (LSubSp‘𝑈)
2		hdmaprnlem1.n	. . . . 5 ⊢ 𝑁 = (LSpan‘𝑈)
3		hdmaprnlem1.h	. . . . . 6 ⊢ 𝐻 = (LHyp‘𝐾)
4		hdmaprnlem1.u	. . . . . 6 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
5		hdmaprnlem1.k	. . . . . 6 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
6	3, 4, 5	dvhlmod 35200	. . . . 5 ⊢ (𝜑 → 𝑈 ∈ LMod)
7		hdmaprnlem1.ve	. . . . . 6 ⊢ (𝜑 → 𝑣 ∈ 𝑉)
8		hdmaprnlem1.v	. . . . . . 7 ⊢ 𝑉 = (Base‘𝑈)
9	8, 1, 2	lspsncl 18746	. . . . . 6 ⊢ ((𝑈 ∈ LMod ∧ 𝑣 ∈ 𝑉) → (𝑁‘{𝑣}) ∈ (LSubSp‘𝑈))
10	6, 7, 9	syl2anc 690	. . . . 5 ⊢ (𝜑 → (𝑁‘{𝑣}) ∈ (LSubSp‘𝑈))
11		hdmaprnlem1.t2	. . . . . 6 ⊢ (𝜑 → 𝑡 ∈ ((𝑁‘{𝑣}) ∖ { 0 }))
12	11	eldifad 3551	. . . . 5 ⊢ (𝜑 → 𝑡 ∈ (𝑁‘{𝑣}))
13	1, 2, 6, 10, 12	lspsnel5a 18765	. . . 4 ⊢ (𝜑 → (𝑁‘{𝑡}) ⊆ (𝑁‘{𝑣}))
14		hdmaprnlem1.o	. . . . 5 ⊢ 0 = (0_g‘𝑈)
15	3, 4, 5	dvhlvec 35199	. . . . 5 ⊢ (𝜑 → 𝑈 ∈ LVec)
16	8, 1	lss1 18708	. . . . . . . . 9 ⊢ (𝑈 ∈ LMod → 𝑉 ∈ (LSubSp‘𝑈))
17	6, 16	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝑉 ∈ (LSubSp‘𝑈))
18	1, 2, 6, 17, 7	lspsnel5a 18765	. . . . . . 7 ⊢ (𝜑 → (𝑁‘{𝑣}) ⊆ 𝑉)
19	18	ssdifd 3707	. . . . . 6 ⊢ (𝜑 → ((𝑁‘{𝑣}) ∖ { 0 }) ⊆ (𝑉 ∖ { 0 }))
20	19, 11	sseldd 3568	. . . . 5 ⊢ (𝜑 → 𝑡 ∈ (𝑉 ∖ { 0 }))
21	8, 14, 2, 15, 20, 7	lspsncmp 18885	. . . 4 ⊢ (𝜑 → ((𝑁‘{𝑡}) ⊆ (𝑁‘{𝑣}) ↔ (𝑁‘{𝑡}) = (𝑁‘{𝑣})))
22	13, 21	mpbid 220	. . 3 ⊢ (𝜑 → (𝑁‘{𝑡}) = (𝑁‘{𝑣}))
23	22	fveq2d 6091	. 2 ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑡})) = (𝑀‘(𝑁‘{𝑣})))
24		hdmaprnlem1.e	. 2 ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑣})) = (𝐿‘{𝑠}))
25	23, 24	eqtrd 2643	1 ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑡})) = (𝐿‘{𝑠}))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 382 = wceq 1474 ∈ wcel 1976 ∖ cdif 3536 ⊆ wss 3539 {csn 4124 ‘cfv 5789 Basecbs 15643 +_gcplusg 15716 0_gc0g 15871 LModclmod 18634 LSubSpclss 18701 LSpanclspn 18740 HLchlt 33438 LHypclh 34071 DVecHcdvh 35168 LCDualclcd 35676 mapdcmpd 35714 HDMapchdma 35883

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1712 ax-4 1727 ax-5 1826 ax-6 1874 ax-7 1921 ax-8 1978 ax-9 1985 ax-10 2005 ax-11 2020 ax-12 2033 ax-13 2233 ax-ext 2589 ax-rep 4693 ax-sep 4703 ax-nul 4711 ax-pow 4763 ax-pr 4827 ax-un 6824 ax-cnex 9848 ax-resscn 9849 ax-1cn 9850 ax-icn 9851 ax-addcl 9852 ax-addrcl 9853 ax-mulcl 9854 ax-mulrcl 9855 ax-mulcom 9856 ax-addass 9857 ax-mulass 9858 ax-distr 9859 ax-i2m1 9860 ax-1ne0 9861 ax-1rid 9862 ax-rnegex 9863 ax-rrecex 9864 ax-cnre 9865 ax-pre-lttri 9866 ax-pre-lttrn 9867 ax-pre-ltadd 9868 ax-pre-mulgt0 9869 ax-riotaBAD 33040

This theorem depends on definitions: df-bi 195 df-or 383 df-an 384 df-3or 1031 df-3an 1032 df-tru 1477 df-fal 1480 df-ex 1695 df-nf 1700 df-sb 1867 df-eu 2461 df-mo 2462 df-clab 2596 df-cleq 2602 df-clel 2605 df-nfc 2739 df-ne 2781 df-nel 2782 df-ral 2900 df-rex 2901 df-reu 2902 df-rmo 2903 df-rab 2904 df-v 3174 df-sbc 3402 df-csb 3499 df-dif 3542 df-un 3544 df-in 3546 df-ss 3553 df-pss 3555 df-nul 3874 df-if 4036 df-pw 4109 df-sn 4125 df-pr 4127 df-tp 4129 df-op 4131 df-uni 4367 df-int 4405 df-iun 4451 df-iin 4452 df-br 4578 df-opab 4638 df-mpt 4639 df-tr 4675 df-eprel 4938 df-id 4942 df-po 4948 df-so 4949 df-fr 4986 df-we 4988 df-xp 5033 df-rel 5034 df-cnv 5035 df-co 5036 df-dm 5037 df-rn 5038 df-res 5039 df-ima 5040 df-pred 5582 df-ord 5628 df-on 5629 df-lim 5630 df-suc 5631 df-iota 5753 df-fun 5791 df-fn 5792 df-f 5793 df-f1 5794 df-fo 5795 df-f1o 5796 df-fv 5797 df-riota 6488 df-ov 6529 df-oprab 6530 df-mpt2 6531 df-om 6935 df-1st 7036 df-2nd 7037 df-tpos 7216 df-undef 7263 df-wrecs 7271 df-recs 7332 df-rdg 7370 df-1o 7424 df-oadd 7428 df-er 7606 df-map 7723 df-en 7819 df-dom 7820 df-sdom 7821 df-fin 7822 df-pnf 9932 df-mnf 9933 df-xr 9934 df-ltxr 9935 df-le 9936 df-sub 10119 df-neg 10120 df-nn 10870 df-2 10928 df-3 10929 df-4 10930 df-5 10931 df-6 10932 df-n0 11142 df-z 11213 df-uz 11522 df-fz 12155 df-struct 15645 df-ndx 15646 df-slot 15647 df-base 15648 df-sets 15649 df-ress 15650 df-plusg 15729 df-mulr 15730 df-sca 15732 df-vsca 15733 df-0g 15873 df-preset 16699 df-poset 16717 df-plt 16729 df-lub 16745 df-glb 16746 df-join 16747 df-meet 16748 df-p0 16810 df-p1 16811 df-lat 16817 df-clat 16879 df-mgm 17013 df-sgrp 17055 df-mnd 17066 df-grp 17196 df-minusg 17197 df-sbg 17198 df-mgp 18261 df-ur 18273 df-ring 18320 df-oppr 18394 df-dvdsr 18412 df-unit 18413 df-invr 18443 df-dvr 18454 df-drng 18520 df-lmod 18636 df-lss 18702 df-lsp 18741 df-lvec 18872 df-oposet 33264 df-ol 33266 df-oml 33267 df-covers 33354 df-ats 33355 df-atl 33386 df-cvlat 33410 df-hlat 33439 df-llines 33585 df-lplanes 33586 df-lvols 33587 df-lines 33588 df-psubsp 33590 df-pmap 33591 df-padd 33883 df-lhyp 34075 df-laut 34076 df-ldil 34191 df-ltrn 34192 df-trl 34247 df-tendo 34844 df-edring 34846 df-dvech 35169

This theorem is referenced by: hdmaprnlem8N 35949 hdmaprnlem9N 35950

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