lcfl9a - Mathbox for Norm Megill

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Theorem lcfl9a 39142

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 38998	. . . 4 ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘𝑉)) = 𝑉)
7	6	adantr 484	. . 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 38747	. . . . . . . 8 ⊢ (𝜑 → 𝑈 ∈ LMod)
11		lcfl9a.g	. . . . . . . 8 ⊢ (𝜑 → 𝐺 ∈ 𝐹)
12	4, 8, 9, 10, 11	lkrssv 36733	. . . . . . 7 ⊢ (𝜑 → (𝐿‘𝐺) ⊆ 𝑉)
13	12	adantr 484	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋 = (0_g‘𝑈)) → (𝐿‘𝐺) ⊆ 𝑉)
14		sneq 4526	. . . . . . . . 9 ⊢ (𝑋 = (0_g‘𝑈) → {𝑋} = {(0_g‘𝑈)})
15	14	fveq2d 6678	. . . . . . . 8 ⊢ (𝑋 = (0_g‘𝑈) → ( ⊥ ‘{𝑋}) = ( ⊥ ‘{(0_g‘𝑈)}))
16		eqid 2738	. . . . . . . . . 10 ⊢ (0_g‘𝑈) = (0_g‘𝑈)
17	1, 2, 3, 4, 16	doch0 38995	. . . . . . . . 9 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → ( ⊥ ‘{(0_g‘𝑈)}) = 𝑉)
18	5, 17	syl 17	. . . . . . . 8 ⊢ (𝜑 → ( ⊥ ‘{(0_g‘𝑈)}) = 𝑉)
19	15, 18	sylan9eqr 2795	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑋 = (0_g‘𝑈)) → ( ⊥ ‘{𝑋}) = 𝑉)
20		lcfl9a.s	. . . . . . . 8 ⊢ (𝜑 → ( ⊥ ‘{𝑋}) ⊆ (𝐿‘𝐺))
21	20	adantr 484	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑋 = (0_g‘𝑈)) → ( ⊥ ‘{𝑋}) ⊆ (𝐿‘𝐺))
22	19, 21	eqsstrrd 3916	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋 = (0_g‘𝑈)) → 𝑉 ⊆ (𝐿‘𝐺))
23	13, 22	eqssd 3894	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 = (0_g‘𝑈)) → (𝐿‘𝐺) = 𝑉)
24	23	fveq2d 6678	. . . 4 ⊢ ((𝜑 ∧ 𝑋 = (0_g‘𝑈)) → ( ⊥ ‘(𝐿‘𝐺)) = ( ⊥ ‘𝑉))
25	24	fveq2d 6678	. . 3 ⊢ ((𝜑 ∧ 𝑋 = (0_g‘𝑈)) → ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) = ( ⊥ ‘( ⊥ ‘𝑉)))
26	7, 25, 23	3eqtr4d 2783	. 2 ⊢ ((𝜑 ∧ 𝑋 = (0_g‘𝑈)) → ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) = (𝐿‘𝐺))
27	6	adantr 484	. . 3 ⊢ ((𝜑 ∧ (𝐿‘𝐺) = 𝑉) → ( ⊥ ‘( ⊥ ‘𝑉)) = 𝑉)
28		simpr 488	. . . . 5 ⊢ ((𝜑 ∧ (𝐿‘𝐺) = 𝑉) → (𝐿‘𝐺) = 𝑉)
29	28	fveq2d 6678	. . . 4 ⊢ ((𝜑 ∧ (𝐿‘𝐺) = 𝑉) → ( ⊥ ‘(𝐿‘𝐺)) = ( ⊥ ‘𝑉))
30	29	fveq2d 6678	. . 3 ⊢ ((𝜑 ∧ (𝐿‘𝐺) = 𝑉) → ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) = ( ⊥ ‘( ⊥ ‘𝑉)))
31	27, 30, 28	3eqtr4d 2783	. 2 ⊢ ((𝜑 ∧ (𝐿‘𝐺) = 𝑉) → ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) = (𝐿‘𝐺))
32		lcfl9a.x	. . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ 𝑉)
33	32	snssd 4697	. . . . . 6 ⊢ (𝜑 → {𝑋} ⊆ 𝑉)
34		eqid 2738	. . . . . . 7 ⊢ ((DIsoH‘𝐾)‘𝑊) = ((DIsoH‘𝐾)‘𝑊)
35	1, 34, 2, 4, 3	dochcl 38990	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ {𝑋} ⊆ 𝑉) → ( ⊥ ‘{𝑋}) ∈ ran ((DIsoH‘𝐾)‘𝑊))
36	5, 33, 35	syl2anc 587	. . . . 5 ⊢ (𝜑 → ( ⊥ ‘{𝑋}) ∈ ran ((DIsoH‘𝐾)‘𝑊))
37	1, 34, 3	dochoc 39004	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ( ⊥ ‘{𝑋}) ∈ ran ((DIsoH‘𝐾)‘𝑊)) → ( ⊥ ‘( ⊥ ‘( ⊥ ‘{𝑋}))) = ( ⊥ ‘{𝑋}))
38	5, 36, 37	syl2anc 587	. . . 4 ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘( ⊥ ‘{𝑋}))) = ( ⊥ ‘{𝑋}))
39	38	adantr 484	. . 3 ⊢ ((𝜑 ∧ (𝑋 ≠ (0_g‘𝑈) ∧ (𝐿‘𝐺) ≠ 𝑉)) → ( ⊥ ‘( ⊥ ‘( ⊥ ‘{𝑋}))) = ( ⊥ ‘{𝑋}))
40	20	adantr 484	. . . . . 6 ⊢ ((𝜑 ∧ (𝑋 ≠ (0_g‘𝑈) ∧ (𝐿‘𝐺) ≠ 𝑉)) → ( ⊥ ‘{𝑋}) ⊆ (𝐿‘𝐺))
41		eqid 2738	. . . . . . 7 ⊢ (LSHyp‘𝑈) = (LSHyp‘𝑈)
42	1, 2, 5	dvhlvec 38746	. . . . . . . 8 ⊢ (𝜑 → 𝑈 ∈ LVec)
43	42	adantr 484	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑋 ≠ (0_g‘𝑈) ∧ (𝐿‘𝐺) ≠ 𝑉)) → 𝑈 ∈ LVec)
44	5	adantr 484	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑋 ≠ (0_g‘𝑈) ∧ (𝐿‘𝐺) ≠ 𝑉)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
45	32	adantr 484	. . . . . . . . 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 4675	. . . . . . . . 9 ⊢ (𝑋 ∈ (𝑉 ∖ {(0_g‘𝑈)}) ↔ (𝑋 ∈ 𝑉 ∧ 𝑋 ≠ (0_g‘𝑈)))
48	45, 46, 47	sylanbrc 586	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑋 ≠ (0_g‘𝑈) ∧ (𝐿‘𝐺) ≠ 𝑉)) → 𝑋 ∈ (𝑉 ∖ {(0_g‘𝑈)}))
49	1, 3, 2, 4, 16, 41, 44, 48	dochsnshp 39090	. . . . . . 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 36745	. . . . . . . . 9 ⊢ (𝜑 → ((𝐿‘𝐺) ≠ 𝑉 ↔ (𝐿‘𝐺) ∈ (LSHyp‘𝑈)))
52	51	adantr 484	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑋 ≠ (0_g‘𝑈) ∧ (𝐿‘𝐺) ≠ 𝑉)) → ((𝐿‘𝐺) ≠ 𝑉 ↔ (𝐿‘𝐺) ∈ (LSHyp‘𝑈)))
53	50, 52	mpbid 235	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑋 ≠ (0_g‘𝑈) ∧ (𝐿‘𝐺) ≠ 𝑉)) → (𝐿‘𝐺) ∈ (LSHyp‘𝑈))
54	41, 43, 49, 53	lshpcmp 36625	. . . . . 6 ⊢ ((𝜑 ∧ (𝑋 ≠ (0_g‘𝑈) ∧ (𝐿‘𝐺) ≠ 𝑉)) → (( ⊥ ‘{𝑋}) ⊆ (𝐿‘𝐺) ↔ ( ⊥ ‘{𝑋}) = (𝐿‘𝐺)))
55	40, 54	mpbid 235	. . . . 5 ⊢ ((𝜑 ∧ (𝑋 ≠ (0_g‘𝑈) ∧ (𝐿‘𝐺) ≠ 𝑉)) → ( ⊥ ‘{𝑋}) = (𝐿‘𝐺))
56	55	fveq2d 6678	. . . 4 ⊢ ((𝜑 ∧ (𝑋 ≠ (0_g‘𝑈) ∧ (𝐿‘𝐺) ≠ 𝑉)) → ( ⊥ ‘( ⊥ ‘{𝑋})) = ( ⊥ ‘(𝐿‘𝐺)))
57	56	fveq2d 6678	. . 3 ⊢ ((𝜑 ∧ (𝑋 ≠ (0_g‘𝑈) ∧ (𝐿‘𝐺) ≠ 𝑉)) → ( ⊥ ‘( ⊥ ‘( ⊥ ‘{𝑋}))) = ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))))
58	39, 57, 55	3eqtr3d 2781	. 2 ⊢ ((𝜑 ∧ (𝑋 ≠ (0_g‘𝑈) ∧ (𝐿‘𝐺) ≠ 𝑉)) → ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) = (𝐿‘𝐺))
59	26, 31, 58	pm2.61da2ne 3022	1 ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) = (𝐿‘𝐺))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1542 ∈ wcel 2114 ≠ wne 2934 ∖ cdif 3840 ⊆ wss 3843 {csn 4516 ran crn 5526 ‘cfv 6339 Basecbs 16586 0_gc0g 16816 LVecclvec 19993 LSHypclsh 36612 LFnlclfn 36694 LKerclk 36722 HLchlt 36987 LHypclh 37621 DVecHcdvh 38715 DIsoHcdih 38865 ocHcoch 38984

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2710 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5232 ax-pr 5296 ax-un 7479 ax-cnex 10671 ax-resscn 10672 ax-1cn 10673 ax-icn 10674 ax-addcl 10675 ax-addrcl 10676 ax-mulcl 10677 ax-mulrcl 10678 ax-mulcom 10679 ax-addass 10680 ax-mulass 10681 ax-distr 10682 ax-i2m1 10683 ax-1ne0 10684 ax-1rid 10685 ax-rnegex 10686 ax-rrecex 10687 ax-cnre 10688 ax-pre-lttri 10689 ax-pre-lttrn 10690 ax-pre-ltadd 10691 ax-pre-mulgt0 10692 ax-riotaBAD 36590

This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-nel 3039 df-ral 3058 df-rex 3059 df-reu 3060 df-rmo 3061 df-rab 3062 df-v 3400 df-sbc 3681 df-csb 3791 df-dif 3846 df-un 3848 df-in 3850 df-ss 3860 df-pss 3862 df-nul 4212 df-if 4415 df-pw 4490 df-sn 4517 df-pr 4519 df-tp 4521 df-op 4523 df-uni 4797 df-int 4837 df-iun 4883 df-iin 4884 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5429 df-eprel 5434 df-po 5442 df-so 5443 df-fr 5483 df-we 5485 df-xp 5531 df-rel 5532 df-cnv 5533 df-co 5534 df-dm 5535 df-rn 5536 df-res 5537 df-ima 5538 df-pred 6129 df-ord 6175 df-on 6176 df-lim 6177 df-suc 6178 df-iota 6297 df-fun 6341 df-fn 6342 df-f 6343 df-f1 6344 df-fo 6345 df-f1o 6346 df-fv 6347 df-riota 7127 df-ov 7173 df-oprab 7174 df-mpo 7175 df-om 7600 df-1st 7714 df-2nd 7715 df-tpos 7921 df-undef 7968 df-wrecs 7976 df-recs 8037 df-rdg 8075 df-1o 8131 df-er 8320 df-map 8439 df-en 8556 df-dom 8557 df-sdom 8558 df-fin 8559 df-pnf 10755 df-mnf 10756 df-xr 10757 df-ltxr 10758 df-le 10759 df-sub 10950 df-neg 10951 df-nn 11717 df-2 11779 df-3 11780 df-4 11781 df-5 11782 df-6 11783 df-n0 11977 df-z 12063 df-uz 12325 df-fz 12982 df-struct 16588 df-ndx 16589 df-slot 16590 df-base 16592 df-sets 16593 df-ress 16594 df-plusg 16681 df-mulr 16682 df-sca 16684 df-vsca 16685 df-0g 16818 df-proset 17654 df-poset 17672 df-plt 17684 df-lub 17700 df-glb 17701 df-join 17702 df-meet 17703 df-p0 17765 df-p1 17766 df-lat 17772 df-clat 17834 df-mgm 17968 df-sgrp 18017 df-mnd 18028 df-submnd 18073 df-grp 18222 df-minusg 18223 df-sbg 18224 df-subg 18394 df-cntz 18565 df-lsm 18879 df-cmn 19026 df-abl 19027 df-mgp 19359 df-ur 19371 df-ring 19418 df-oppr 19495 df-dvdsr 19513 df-unit 19514 df-invr 19544 df-dvr 19555 df-drng 19623 df-lmod 19755 df-lss 19823 df-lsp 19863 df-lvec 19994 df-lsatoms 36613 df-lshyp 36614 df-lfl 36695 df-lkr 36723 df-oposet 36813 df-ol 36815 df-oml 36816 df-covers 36903 df-ats 36904 df-atl 36935 df-cvlat 36959 df-hlat 36988 df-llines 37135 df-lplanes 37136 df-lvols 37137 df-lines 37138 df-psubsp 37140 df-pmap 37141 df-padd 37433 df-lhyp 37625 df-laut 37626 df-ldil 37741 df-ltrn 37742 df-trl 37796 df-tgrp 38380 df-tendo 38392 df-edring 38394 df-dveca 38640 df-disoa 38666 df-dvech 38716 df-dib 38776 df-dic 38810 df-dih 38866 df-doch 38985 df-djh 39032

This theorem is referenced by: mapdsn 39278

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