dihlspsnssN - Mathbox for Norm Megill

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Theorem dihlspsnssN 38462

Description: A subspace included in a 1-dim subspace belongs to the range of isomorphism H. (Contributed by NM, 26-Apr-2014.) (New usage is discouraged.)

**Hypotheses**
Ref	Expression
dih1dor0.h	⊢ 𝐻 = (LHyp‘𝐾)
dih1dor0.u	⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
dihldor0.v	⊢ 𝑉 = (Base‘𝑈)
dih1dor0.s	⊢ 𝑆 = (LSubSp‘𝑈)
dih1dor0.n	⊢ 𝑁 = (LSpan‘𝑈)
dih1dor0.i	⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊)

**Assertion**
Ref	Expression
dihlspsnssN	⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝑉 ∧ 𝑇 ⊆ (𝑁‘{𝑋})) → (𝑇 ∈ 𝑆 ↔ 𝑇 ∈ ran 𝐼))

**Proof of Theorem dihlspsnssN**
Step	Hyp	Ref	Expression
1		simpr 487	. . . . 5 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝑉 ∧ 𝑇 ⊆ (𝑁‘{𝑋})) ∧ 𝑇 ∈ 𝑆) ∧ 𝑇 = (𝑁‘{𝑋})) → 𝑇 = (𝑁‘{𝑋}))
2		dih1dor0.h	. . . . . . . 8 ⊢ 𝐻 = (LHyp‘𝐾)
3		dih1dor0.u	. . . . . . . 8 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
4		dihldor0.v	. . . . . . . 8 ⊢ 𝑉 = (Base‘𝑈)
5		dih1dor0.n	. . . . . . . 8 ⊢ 𝑁 = (LSpan‘𝑈)
6		dih1dor0.i	. . . . . . . 8 ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊)
7	2, 3, 4, 5, 6	dihlsprn 38461	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝑉) → (𝑁‘{𝑋}) ∈ ran 𝐼)
8	7	3adant3 1128	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝑉 ∧ 𝑇 ⊆ (𝑁‘{𝑋})) → (𝑁‘{𝑋}) ∈ ran 𝐼)
9	8	ad2antrr 724	. . . . 5 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝑉 ∧ 𝑇 ⊆ (𝑁‘{𝑋})) ∧ 𝑇 ∈ 𝑆) ∧ 𝑇 = (𝑁‘{𝑋})) → (𝑁‘{𝑋}) ∈ ran 𝐼)
10	1, 9	eqeltrd 2913	. . . 4 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝑉 ∧ 𝑇 ⊆ (𝑁‘{𝑋})) ∧ 𝑇 ∈ 𝑆) ∧ 𝑇 = (𝑁‘{𝑋})) → 𝑇 ∈ ran 𝐼)
11		simpr 487	. . . . . 6 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝑉 ∧ 𝑇 ⊆ (𝑁‘{𝑋})) ∧ 𝑇 ∈ 𝑆) ∧ 𝑇 = {(0_g‘𝑈)}) → 𝑇 = {(0_g‘𝑈)})
12		simpll1 1208	. . . . . . 7 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝑉 ∧ 𝑇 ⊆ (𝑁‘{𝑋})) ∧ 𝑇 ∈ 𝑆) ∧ 𝑇 = {(0_g‘𝑈)}) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
13		eqid 2821	. . . . . . . 8 ⊢ (0.‘𝐾) = (0.‘𝐾)
14		eqid 2821	. . . . . . . 8 ⊢ (0_g‘𝑈) = (0_g‘𝑈)
15	13, 2, 6, 3, 14	dih0 38410	. . . . . . 7 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (𝐼‘(0.‘𝐾)) = {(0_g‘𝑈)})
16	12, 15	syl 17	. . . . . 6 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝑉 ∧ 𝑇 ⊆ (𝑁‘{𝑋})) ∧ 𝑇 ∈ 𝑆) ∧ 𝑇 = {(0_g‘𝑈)}) → (𝐼‘(0.‘𝐾)) = {(0_g‘𝑈)})
17	11, 16	eqtr4d 2859	. . . . 5 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝑉 ∧ 𝑇 ⊆ (𝑁‘{𝑋})) ∧ 𝑇 ∈ 𝑆) ∧ 𝑇 = {(0_g‘𝑈)}) → 𝑇 = (𝐼‘(0.‘𝐾)))
18		eqid 2821	. . . . . . . 8 ⊢ (Base‘𝐾) = (Base‘𝐾)
19	18, 2, 6	dihfn 38398	. . . . . . 7 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 𝐼 Fn (Base‘𝐾))
20	12, 19	syl 17	. . . . . 6 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝑉 ∧ 𝑇 ⊆ (𝑁‘{𝑋})) ∧ 𝑇 ∈ 𝑆) ∧ 𝑇 = {(0_g‘𝑈)}) → 𝐼 Fn (Base‘𝐾))
21		simp1l 1193	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝑉 ∧ 𝑇 ⊆ (𝑁‘{𝑋})) → 𝐾 ∈ HL)
22	21	ad2antrr 724	. . . . . . 7 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝑉 ∧ 𝑇 ⊆ (𝑁‘{𝑋})) ∧ 𝑇 ∈ 𝑆) ∧ 𝑇 = {(0_g‘𝑈)}) → 𝐾 ∈ HL)
23		hlop 36492	. . . . . . 7 ⊢ (𝐾 ∈ HL → 𝐾 ∈ OP)
24	18, 13	op0cl 36314	. . . . . . 7 ⊢ (𝐾 ∈ OP → (0.‘𝐾) ∈ (Base‘𝐾))
25	22, 23, 24	3syl 18	. . . . . 6 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝑉 ∧ 𝑇 ⊆ (𝑁‘{𝑋})) ∧ 𝑇 ∈ 𝑆) ∧ 𝑇 = {(0_g‘𝑈)}) → (0.‘𝐾) ∈ (Base‘𝐾))
26		fnfvelrn 6842	. . . . . 6 ⊢ ((𝐼 Fn (Base‘𝐾) ∧ (0.‘𝐾) ∈ (Base‘𝐾)) → (𝐼‘(0.‘𝐾)) ∈ ran 𝐼)
27	20, 25, 26	syl2anc 586	. . . . 5 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝑉 ∧ 𝑇 ⊆ (𝑁‘{𝑋})) ∧ 𝑇 ∈ 𝑆) ∧ 𝑇 = {(0_g‘𝑈)}) → (𝐼‘(0.‘𝐾)) ∈ ran 𝐼)
28	17, 27	eqeltrd 2913	. . . 4 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝑉 ∧ 𝑇 ⊆ (𝑁‘{𝑋})) ∧ 𝑇 ∈ 𝑆) ∧ 𝑇 = {(0_g‘𝑈)}) → 𝑇 ∈ ran 𝐼)
29		simpl1 1187	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝑉 ∧ 𝑇 ⊆ (𝑁‘{𝑋})) ∧ 𝑇 ∈ 𝑆) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
30	2, 3, 29	dvhlvec 38239	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝑉 ∧ 𝑇 ⊆ (𝑁‘{𝑋})) ∧ 𝑇 ∈ 𝑆) → 𝑈 ∈ LVec)
31		simpr 487	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝑉 ∧ 𝑇 ⊆ (𝑁‘{𝑋})) ∧ 𝑇 ∈ 𝑆) → 𝑇 ∈ 𝑆)
32		simpl2 1188	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝑉 ∧ 𝑇 ⊆ (𝑁‘{𝑋})) ∧ 𝑇 ∈ 𝑆) → 𝑋 ∈ 𝑉)
33		simpl3 1189	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝑉 ∧ 𝑇 ⊆ (𝑁‘{𝑋})) ∧ 𝑇 ∈ 𝑆) → 𝑇 ⊆ (𝑁‘{𝑋}))
34		dih1dor0.s	. . . . . 6 ⊢ 𝑆 = (LSubSp‘𝑈)
35	4, 14, 34, 5	lspsnat 19911	. . . . 5 ⊢ (((𝑈 ∈ LVec ∧ 𝑇 ∈ 𝑆 ∧ 𝑋 ∈ 𝑉) ∧ 𝑇 ⊆ (𝑁‘{𝑋})) → (𝑇 = (𝑁‘{𝑋}) ∨ 𝑇 = {(0_g‘𝑈)}))
36	30, 31, 32, 33, 35	syl31anc 1369	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝑉 ∧ 𝑇 ⊆ (𝑁‘{𝑋})) ∧ 𝑇 ∈ 𝑆) → (𝑇 = (𝑁‘{𝑋}) ∨ 𝑇 = {(0_g‘𝑈)}))
37	10, 28, 36	mpjaodan 955	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝑉 ∧ 𝑇 ⊆ (𝑁‘{𝑋})) ∧ 𝑇 ∈ 𝑆) → 𝑇 ∈ ran 𝐼)
38	37	ex 415	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝑉 ∧ 𝑇 ⊆ (𝑁‘{𝑋})) → (𝑇 ∈ 𝑆 → 𝑇 ∈ ran 𝐼))
39	2, 3, 6, 34	dihsslss 38406	. . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → ran 𝐼 ⊆ 𝑆)
40	39	3ad2ant1 1129	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝑉 ∧ 𝑇 ⊆ (𝑁‘{𝑋})) → ran 𝐼 ⊆ 𝑆)
41	40	sseld 3965	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝑉 ∧ 𝑇 ⊆ (𝑁‘{𝑋})) → (𝑇 ∈ ran 𝐼 → 𝑇 ∈ 𝑆))
42	38, 41	impbid 214	1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝑉 ∧ 𝑇 ⊆ (𝑁‘{𝑋})) → (𝑇 ∈ 𝑆 ↔ 𝑇 ∈ ran 𝐼))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∨ wo 843 ∧ w3a 1083 = wceq 1533 ∈ wcel 2110 ⊆ wss 3935 {csn 4560 ran crn 5550 Fn wfn 6344 ‘cfv 6349 Basecbs 16477 0_gc0g 16707 0.cp0 17641 LSubSpclss 19697 LSpanclspn 19737 LVecclvec 19868 OPcops 36302 HLchlt 36480 LHypclh 37114 DVecHcdvh 38208 DIsoHcdih 38358

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 ax-riotaBAD 36083

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-fal 1546 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-int 4869 df-iun 4913 df-iin 4914 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-1st 7683 df-2nd 7684 df-tpos 7886 df-undef 7933 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-1o 8096 df-oadd 8100 df-er 8283 df-map 8402 df-en 8504 df-dom 8505 df-sdom 8506 df-fin 8507 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-nn 11633 df-2 11694 df-3 11695 df-4 11696 df-5 11697 df-6 11698 df-n0 11892 df-z 11976 df-uz 12238 df-fz 12887 df-struct 16479 df-ndx 16480 df-slot 16481 df-base 16483 df-sets 16484 df-ress 16485 df-plusg 16572 df-mulr 16573 df-sca 16575 df-vsca 16576 df-0g 16709 df-proset 17532 df-poset 17550 df-plt 17562 df-lub 17578 df-glb 17579 df-join 17580 df-meet 17581 df-p0 17643 df-p1 17644 df-lat 17650 df-clat 17712 df-mgm 17846 df-sgrp 17895 df-mnd 17906 df-submnd 17951 df-grp 18100 df-minusg 18101 df-sbg 18102 df-subg 18270 df-cntz 18441 df-lsm 18755 df-cmn 18902 df-abl 18903 df-mgp 19234 df-ur 19246 df-ring 19293 df-oppr 19367 df-dvdsr 19385 df-unit 19386 df-invr 19416 df-dvr 19427 df-drng 19498 df-lmod 19630 df-lss 19698 df-lsp 19738 df-lvec 19869 df-lsatoms 36106 df-oposet 36306 df-ol 36308 df-oml 36309 df-covers 36396 df-ats 36397 df-atl 36428 df-cvlat 36452 df-hlat 36481 df-llines 36628 df-lplanes 36629 df-lvols 36630 df-lines 36631 df-psubsp 36633 df-pmap 36634 df-padd 36926 df-lhyp 37118 df-laut 37119 df-ldil 37234 df-ltrn 37235 df-trl 37289 df-tendo 37885 df-edring 37887 df-disoa 38159 df-dvech 38209 df-dib 38269 df-dic 38303 df-dih 38359

This theorem is referenced by: (None)

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