lcfl9a - Mathbox for Norm Megill

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Theorem lcfl9a 41488

Description: Property implying that a functional has a closed kernel. (Contributed by NM, 16-Feb-2015.)

**Hypotheses**
Ref	Expression
lcfl9a.h	⊢ 𝐻 = (LHyp‘𝐾)
lcfl9a.o	⊢ ⊥ = ((ocH‘𝐾)‘𝑊)
lcfl9a.u	⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
lcfl9a.v	⊢ 𝑉 = (Base‘𝑈)
lcfl9a.f	⊢ 𝐹 = (LFnl‘𝑈)
lcfl9a.l	⊢ 𝐿 = (LKer‘𝑈)
lcfl9a.k	⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
lcfl9a.g	⊢ (𝜑 → 𝐺 ∈ 𝐹)
lcfl9a.x	⊢ (𝜑 → 𝑋 ∈ 𝑉)
lcfl9a.s	⊢ (𝜑 → ( ⊥ ‘{𝑋}) ⊆ (𝐿‘𝐺))

**Assertion**
Ref	Expression
lcfl9a	⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) = (𝐿‘𝐺))

**Proof of Theorem lcfl9a**
Step	Hyp	Ref	Expression
1		lcfl9a.h	. . . . 5 ⊢ 𝐻 = (LHyp‘𝐾)
2		lcfl9a.u	. . . . 5 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
3		lcfl9a.o	. . . . 5 ⊢ ⊥ = ((ocH‘𝐾)‘𝑊)
4		lcfl9a.v	. . . . 5 ⊢ 𝑉 = (Base‘𝑈)
5		lcfl9a.k	. . . . 5 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
6	1, 2, 3, 4, 5	dochoc1 41344	. . . 4 ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘𝑉)) = 𝑉)
7	6	adantr 480	. . 3 ⊢ ((𝜑 ∧ 𝑋 = (0_g‘𝑈)) → ( ⊥ ‘( ⊥ ‘𝑉)) = 𝑉)
8		lcfl9a.f	. . . . . . . 8 ⊢ 𝐹 = (LFnl‘𝑈)
9		lcfl9a.l	. . . . . . . 8 ⊢ 𝐿 = (LKer‘𝑈)
10	1, 2, 5	dvhlmod 41093	. . . . . . . 8 ⊢ (𝜑 → 𝑈 ∈ LMod)
11		lcfl9a.g	. . . . . . . 8 ⊢ (𝜑 → 𝐺 ∈ 𝐹)
12	4, 8, 9, 10, 11	lkrssv 39078	. . . . . . 7 ⊢ (𝜑 → (𝐿‘𝐺) ⊆ 𝑉)
13	12	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋 = (0_g‘𝑈)) → (𝐿‘𝐺) ⊆ 𝑉)
14		sneq 4641	. . . . . . . . 9 ⊢ (𝑋 = (0_g‘𝑈) → {𝑋} = {(0_g‘𝑈)})
15	14	fveq2d 6911	. . . . . . . 8 ⊢ (𝑋 = (0_g‘𝑈) → ( ⊥ ‘{𝑋}) = ( ⊥ ‘{(0_g‘𝑈)}))
16		eqid 2735	. . . . . . . . . 10 ⊢ (0_g‘𝑈) = (0_g‘𝑈)
17	1, 2, 3, 4, 16	doch0 41341	. . . . . . . . 9 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → ( ⊥ ‘{(0_g‘𝑈)}) = 𝑉)
18	5, 17	syl 17	. . . . . . . 8 ⊢ (𝜑 → ( ⊥ ‘{(0_g‘𝑈)}) = 𝑉)
19	15, 18	sylan9eqr 2797	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑋 = (0_g‘𝑈)) → ( ⊥ ‘{𝑋}) = 𝑉)
20		lcfl9a.s	. . . . . . . 8 ⊢ (𝜑 → ( ⊥ ‘{𝑋}) ⊆ (𝐿‘𝐺))
21	20	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑋 = (0_g‘𝑈)) → ( ⊥ ‘{𝑋}) ⊆ (𝐿‘𝐺))
22	19, 21	eqsstrrd 4035	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋 = (0_g‘𝑈)) → 𝑉 ⊆ (𝐿‘𝐺))
23	13, 22	eqssd 4013	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 = (0_g‘𝑈)) → (𝐿‘𝐺) = 𝑉)
24	23	fveq2d 6911	. . . 4 ⊢ ((𝜑 ∧ 𝑋 = (0_g‘𝑈)) → ( ⊥ ‘(𝐿‘𝐺)) = ( ⊥ ‘𝑉))
25	24	fveq2d 6911	. . 3 ⊢ ((𝜑 ∧ 𝑋 = (0_g‘𝑈)) → ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) = ( ⊥ ‘( ⊥ ‘𝑉)))
26	7, 25, 23	3eqtr4d 2785	. 2 ⊢ ((𝜑 ∧ 𝑋 = (0_g‘𝑈)) → ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) = (𝐿‘𝐺))
27	6	adantr 480	. . 3 ⊢ ((𝜑 ∧ (𝐿‘𝐺) = 𝑉) → ( ⊥ ‘( ⊥ ‘𝑉)) = 𝑉)
28		simpr 484	. . . . 5 ⊢ ((𝜑 ∧ (𝐿‘𝐺) = 𝑉) → (𝐿‘𝐺) = 𝑉)
29	28	fveq2d 6911	. . . 4 ⊢ ((𝜑 ∧ (𝐿‘𝐺) = 𝑉) → ( ⊥ ‘(𝐿‘𝐺)) = ( ⊥ ‘𝑉))
30	29	fveq2d 6911	. . 3 ⊢ ((𝜑 ∧ (𝐿‘𝐺) = 𝑉) → ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) = ( ⊥ ‘( ⊥ ‘𝑉)))
31	27, 30, 28	3eqtr4d 2785	. 2 ⊢ ((𝜑 ∧ (𝐿‘𝐺) = 𝑉) → ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) = (𝐿‘𝐺))
32		lcfl9a.x	. . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ 𝑉)
33	32	snssd 4814	. . . . . 6 ⊢ (𝜑 → {𝑋} ⊆ 𝑉)
34		eqid 2735	. . . . . . 7 ⊢ ((DIsoH‘𝐾)‘𝑊) = ((DIsoH‘𝐾)‘𝑊)
35	1, 34, 2, 4, 3	dochcl 41336	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ {𝑋} ⊆ 𝑉) → ( ⊥ ‘{𝑋}) ∈ ran ((DIsoH‘𝐾)‘𝑊))
36	5, 33, 35	syl2anc 584	. . . . 5 ⊢ (𝜑 → ( ⊥ ‘{𝑋}) ∈ ran ((DIsoH‘𝐾)‘𝑊))
37	1, 34, 3	dochoc 41350	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ( ⊥ ‘{𝑋}) ∈ ran ((DIsoH‘𝐾)‘𝑊)) → ( ⊥ ‘( ⊥ ‘( ⊥ ‘{𝑋}))) = ( ⊥ ‘{𝑋}))
38	5, 36, 37	syl2anc 584	. . . 4 ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘( ⊥ ‘{𝑋}))) = ( ⊥ ‘{𝑋}))
39	38	adantr 480	. . 3 ⊢ ((𝜑 ∧ (𝑋 ≠ (0_g‘𝑈) ∧ (𝐿‘𝐺) ≠ 𝑉)) → ( ⊥ ‘( ⊥ ‘( ⊥ ‘{𝑋}))) = ( ⊥ ‘{𝑋}))
40	20	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ (𝑋 ≠ (0_g‘𝑈) ∧ (𝐿‘𝐺) ≠ 𝑉)) → ( ⊥ ‘{𝑋}) ⊆ (𝐿‘𝐺))
41		eqid 2735	. . . . . . 7 ⊢ (LSHyp‘𝑈) = (LSHyp‘𝑈)
42	1, 2, 5	dvhlvec 41092	. . . . . . . 8 ⊢ (𝜑 → 𝑈 ∈ LVec)
43	42	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑋 ≠ (0_g‘𝑈) ∧ (𝐿‘𝐺) ≠ 𝑉)) → 𝑈 ∈ LVec)
44	5	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑋 ≠ (0_g‘𝑈) ∧ (𝐿‘𝐺) ≠ 𝑉)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
45	32	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑋 ≠ (0_g‘𝑈) ∧ (𝐿‘𝐺) ≠ 𝑉)) → 𝑋 ∈ 𝑉)
46		simprl 771	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑋 ≠ (0_g‘𝑈) ∧ (𝐿‘𝐺) ≠ 𝑉)) → 𝑋 ≠ (0_g‘𝑈))
47		eldifsn 4791	. . . . . . . . 9 ⊢ (𝑋 ∈ (𝑉 ∖ {(0_g‘𝑈)}) ↔ (𝑋 ∈ 𝑉 ∧ 𝑋 ≠ (0_g‘𝑈)))
48	45, 46, 47	sylanbrc 583	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑋 ≠ (0_g‘𝑈) ∧ (𝐿‘𝐺) ≠ 𝑉)) → 𝑋 ∈ (𝑉 ∖ {(0_g‘𝑈)}))
49	1, 3, 2, 4, 16, 41, 44, 48	dochsnshp 41436	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑋 ≠ (0_g‘𝑈) ∧ (𝐿‘𝐺) ≠ 𝑉)) → ( ⊥ ‘{𝑋}) ∈ (LSHyp‘𝑈))
50		simprr 773	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑋 ≠ (0_g‘𝑈) ∧ (𝐿‘𝐺) ≠ 𝑉)) → (𝐿‘𝐺) ≠ 𝑉)
51	4, 41, 8, 9, 42, 11	lkrshp4 39090	. . . . . . . . 9 ⊢ (𝜑 → ((𝐿‘𝐺) ≠ 𝑉 ↔ (𝐿‘𝐺) ∈ (LSHyp‘𝑈)))
52	51	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑋 ≠ (0_g‘𝑈) ∧ (𝐿‘𝐺) ≠ 𝑉)) → ((𝐿‘𝐺) ≠ 𝑉 ↔ (𝐿‘𝐺) ∈ (LSHyp‘𝑈)))
53	50, 52	mpbid 232	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑋 ≠ (0_g‘𝑈) ∧ (𝐿‘𝐺) ≠ 𝑉)) → (𝐿‘𝐺) ∈ (LSHyp‘𝑈))
54	41, 43, 49, 53	lshpcmp 38970	. . . . . 6 ⊢ ((𝜑 ∧ (𝑋 ≠ (0_g‘𝑈) ∧ (𝐿‘𝐺) ≠ 𝑉)) → (( ⊥ ‘{𝑋}) ⊆ (𝐿‘𝐺) ↔ ( ⊥ ‘{𝑋}) = (𝐿‘𝐺)))
55	40, 54	mpbid 232	. . . . 5 ⊢ ((𝜑 ∧ (𝑋 ≠ (0_g‘𝑈) ∧ (𝐿‘𝐺) ≠ 𝑉)) → ( ⊥ ‘{𝑋}) = (𝐿‘𝐺))
56	55	fveq2d 6911	. . . 4 ⊢ ((𝜑 ∧ (𝑋 ≠ (0_g‘𝑈) ∧ (𝐿‘𝐺) ≠ 𝑉)) → ( ⊥ ‘( ⊥ ‘{𝑋})) = ( ⊥ ‘(𝐿‘𝐺)))
57	56	fveq2d 6911	. . 3 ⊢ ((𝜑 ∧ (𝑋 ≠ (0_g‘𝑈) ∧ (𝐿‘𝐺) ≠ 𝑉)) → ( ⊥ ‘( ⊥ ‘( ⊥ ‘{𝑋}))) = ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))))
58	39, 57, 55	3eqtr3d 2783	. 2 ⊢ ((𝜑 ∧ (𝑋 ≠ (0_g‘𝑈) ∧ (𝐿‘𝐺) ≠ 𝑉)) → ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) = (𝐿‘𝐺))
59	26, 31, 58	pm2.61da2ne 3028	1 ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) = (𝐿‘𝐺))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1537 ∈ wcel 2106 ≠ wne 2938 ∖ cdif 3960 ⊆ wss 3963 {csn 4631 ran crn 5690 ‘cfv 6563 Basecbs 17245 0_gc0g 17486 LVecclvec 21119 LSHypclsh 38957 LFnlclfn 39039 LKerclk 39067 HLchlt 39332 LHypclh 39967 DVecHcdvh 41061 DIsoHcdih 41211 ocHcoch 41330

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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 ax-riotaBAD 38935

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-tp 4636 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-iin 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8013 df-2nd 8014 df-tpos 8250 df-undef 8297 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-er 8744 df-map 8867 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-nn 12265 df-2 12327 df-3 12328 df-4 12329 df-5 12330 df-6 12331 df-n0 12525 df-z 12612 df-uz 12877 df-fz 13545 df-struct 17181 df-sets 17198 df-slot 17216 df-ndx 17228 df-base 17246 df-ress 17275 df-plusg 17311 df-mulr 17312 df-sca 17314 df-vsca 17315 df-0g 17488 df-proset 18352 df-poset 18371 df-plt 18388 df-lub 18404 df-glb 18405 df-join 18406 df-meet 18407 df-p0 18483 df-p1 18484 df-lat 18490 df-clat 18557 df-mgm 18666 df-sgrp 18745 df-mnd 18761 df-submnd 18810 df-grp 18967 df-minusg 18968 df-sbg 18969 df-subg 19154 df-cntz 19348 df-lsm 19669 df-cmn 19815 df-abl 19816 df-mgp 20153 df-rng 20171 df-ur 20200 df-ring 20253 df-oppr 20351 df-dvdsr 20374 df-unit 20375 df-invr 20405 df-dvr 20418 df-drng 20748 df-lmod 20877 df-lss 20948 df-lsp 20988 df-lvec 21120 df-lsatoms 38958 df-lshyp 38959 df-lfl 39040 df-lkr 39068 df-oposet 39158 df-ol 39160 df-oml 39161 df-covers 39248 df-ats 39249 df-atl 39280 df-cvlat 39304 df-hlat 39333 df-llines 39481 df-lplanes 39482 df-lvols 39483 df-lines 39484 df-psubsp 39486 df-pmap 39487 df-padd 39779 df-lhyp 39971 df-laut 39972 df-ldil 40087 df-ltrn 40088 df-trl 40142 df-tgrp 40726 df-tendo 40738 df-edring 40740 df-dveca 40986 df-disoa 41012 df-dvech 41062 df-dib 41122 df-dic 41156 df-dih 41212 df-doch 41331 df-djh 41378

This theorem is referenced by: mapdsn 41624

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