lkreqN - Mathbox for Norm Megill

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Theorem lkreqN 38029

Description: Proportional functionals have equal kernels. (Contributed by NM, 28-Mar-2015.) (New usage is discouraged.)

**Hypotheses**
Ref	Expression
lkreq.s	⊢ 𝑆 = (Scalar‘𝑊)
lkreq.r	⊢ 𝑅 = (Base‘𝑆)
lkreq.o	⊢ 0 = (0_g‘𝑆)
lkreq.f	⊢ 𝐹 = (LFnl‘𝑊)
lkreq.k	⊢ 𝐾 = (LKer‘𝑊)
lkreq.d	⊢ 𝐷 = (LDual‘𝑊)
lkreq.t	⊢ · = ( ·_𝑠 ‘𝐷)
lkreq.w	⊢ (𝜑 → 𝑊 ∈ LVec)
lkreq.a	⊢ (𝜑 → 𝐴 ∈ (𝑅 ∖ { 0 }))
lkreq.h	⊢ (𝜑 → 𝐻 ∈ 𝐹)
lkreq.g	⊢ (𝜑 → 𝐺 = (𝐴 · 𝐻))

**Assertion**
Ref	Expression
lkreqN	⊢ (𝜑 → (𝐾‘𝐺) = (𝐾‘𝐻))

**Proof of Theorem lkreqN**
Step	Hyp	Ref	Expression
1		lkreq.g	. . . . . . . . 9 ⊢ (𝜑 → 𝐺 = (𝐴 · 𝐻))
2	1	eqeq1d 2735	. . . . . . . 8 ⊢ (𝜑 → (𝐺 = (0_g‘𝐷) ↔ (𝐴 · 𝐻) = (0_g‘𝐷)))
3		eqid 2733	. . . . . . . . . 10 ⊢ (Base‘𝐷) = (Base‘𝐷)
4		lkreq.t	. . . . . . . . . 10 ⊢ · = ( ·_𝑠 ‘𝐷)
5		eqid 2733	. . . . . . . . . 10 ⊢ (Scalar‘𝐷) = (Scalar‘𝐷)
6		eqid 2733	. . . . . . . . . 10 ⊢ (Base‘(Scalar‘𝐷)) = (Base‘(Scalar‘𝐷))
7		eqid 2733	. . . . . . . . . 10 ⊢ (0_g‘(Scalar‘𝐷)) = (0_g‘(Scalar‘𝐷))
8		eqid 2733	. . . . . . . . . 10 ⊢ (0_g‘𝐷) = (0_g‘𝐷)
9		lkreq.d	. . . . . . . . . . 11 ⊢ 𝐷 = (LDual‘𝑊)
10		lkreq.w	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑊 ∈ LVec)
11	9, 10	lduallvec 38013	. . . . . . . . . 10 ⊢ (𝜑 → 𝐷 ∈ LVec)
12		lkreq.a	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐴 ∈ (𝑅 ∖ { 0 }))
13	12	eldifad 3960	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐴 ∈ 𝑅)
14		lkreq.s	. . . . . . . . . . . 12 ⊢ 𝑆 = (Scalar‘𝑊)
15		lkreq.r	. . . . . . . . . . . 12 ⊢ 𝑅 = (Base‘𝑆)
16	14, 15, 9, 5, 6, 10	ldualsbase 37992	. . . . . . . . . . 11 ⊢ (𝜑 → (Base‘(Scalar‘𝐷)) = 𝑅)
17	13, 16	eleqtrrd 2837	. . . . . . . . . 10 ⊢ (𝜑 → 𝐴 ∈ (Base‘(Scalar‘𝐷)))
18		lkreq.f	. . . . . . . . . . 11 ⊢ 𝐹 = (LFnl‘𝑊)
19		lkreq.h	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐻 ∈ 𝐹)
20	18, 9, 3, 10, 19	ldualelvbase 37986	. . . . . . . . . 10 ⊢ (𝜑 → 𝐻 ∈ (Base‘𝐷))
21	3, 4, 5, 6, 7, 8, 11, 17, 20	lvecvs0or 20714	. . . . . . . . 9 ⊢ (𝜑 → ((𝐴 · 𝐻) = (0_g‘𝐷) ↔ (𝐴 = (0_g‘(Scalar‘𝐷)) ∨ 𝐻 = (0_g‘𝐷))))
22		lkreq.o	. . . . . . . . . . . . 13 ⊢ 0 = (0_g‘𝑆)
23		lveclmod 20710	. . . . . . . . . . . . . 14 ⊢ (𝑊 ∈ LVec → 𝑊 ∈ LMod)
24	10, 23	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑊 ∈ LMod)
25	14, 22, 9, 5, 7, 24	ldual0 38006	. . . . . . . . . . . 12 ⊢ (𝜑 → (0_g‘(Scalar‘𝐷)) = 0 )
26	25	eqeq2d 2744	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐴 = (0_g‘(Scalar‘𝐷)) ↔ 𝐴 = 0 ))
27		eldifsni 4793	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ (𝑅 ∖ { 0 }) → 𝐴 ≠ 0 )
28	12, 27	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐴 ≠ 0 )
29	28	a1d 25	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐻 ≠ (0_g‘𝐷) → 𝐴 ≠ 0 ))
30	29	necon4d 2965	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐴 = 0 → 𝐻 = (0_g‘𝐷)))
31	26, 30	sylbid 239	. . . . . . . . . 10 ⊢ (𝜑 → (𝐴 = (0_g‘(Scalar‘𝐷)) → 𝐻 = (0_g‘𝐷)))
32		idd 24	. . . . . . . . . 10 ⊢ (𝜑 → (𝐻 = (0_g‘𝐷) → 𝐻 = (0_g‘𝐷)))
33	31, 32	jaod 858	. . . . . . . . 9 ⊢ (𝜑 → ((𝐴 = (0_g‘(Scalar‘𝐷)) ∨ 𝐻 = (0_g‘𝐷)) → 𝐻 = (0_g‘𝐷)))
34	21, 33	sylbid 239	. . . . . . . 8 ⊢ (𝜑 → ((𝐴 · 𝐻) = (0_g‘𝐷) → 𝐻 = (0_g‘𝐷)))
35	2, 34	sylbid 239	. . . . . . 7 ⊢ (𝜑 → (𝐺 = (0_g‘𝐷) → 𝐻 = (0_g‘𝐷)))
36		nne 2945	. . . . . . 7 ⊢ (¬ 𝐻 ≠ (0_g‘𝐷) ↔ 𝐻 = (0_g‘𝐷))
37	35, 36	syl6ibr 252	. . . . . 6 ⊢ (𝜑 → (𝐺 = (0_g‘𝐷) → ¬ 𝐻 ≠ (0_g‘𝐷)))
38	37	con3d 152	. . . . 5 ⊢ (𝜑 → (¬ ¬ 𝐻 ≠ (0_g‘𝐷) → ¬ 𝐺 = (0_g‘𝐷)))
39	38	orrd 862	. . . 4 ⊢ (𝜑 → (¬ 𝐻 ≠ (0_g‘𝐷) ∨ ¬ 𝐺 = (0_g‘𝐷)))
40		ianor 981	. . . 4 ⊢ (¬ (𝐻 ≠ (0_g‘𝐷) ∧ 𝐺 = (0_g‘𝐷)) ↔ (¬ 𝐻 ≠ (0_g‘𝐷) ∨ ¬ 𝐺 = (0_g‘𝐷)))
41	39, 40	sylibr 233	. . 3 ⊢ (𝜑 → ¬ (𝐻 ≠ (0_g‘𝐷) ∧ 𝐺 = (0_g‘𝐷)))
42		lkreq.k	. . . . . . 7 ⊢ 𝐾 = (LKer‘𝑊)
43	18, 14, 15, 9, 4, 24, 13, 19	ldualvscl 37998	. . . . . . . 8 ⊢ (𝜑 → (𝐴 · 𝐻) ∈ 𝐹)
44	1, 43	eqeltrd 2834	. . . . . . 7 ⊢ (𝜑 → 𝐺 ∈ 𝐹)
45	18, 42, 9, 8, 10, 19, 44	lkrpssN 38022	. . . . . 6 ⊢ (𝜑 → ((𝐾‘𝐻) ⊊ (𝐾‘𝐺) ↔ (𝐻 ≠ (0_g‘𝐷) ∧ 𝐺 = (0_g‘𝐷))))
46		df-pss 3967	. . . . . 6 ⊢ ((𝐾‘𝐻) ⊊ (𝐾‘𝐺) ↔ ((𝐾‘𝐻) ⊆ (𝐾‘𝐺) ∧ (𝐾‘𝐻) ≠ (𝐾‘𝐺)))
47	45, 46	bitr3di 286	. . . . 5 ⊢ (𝜑 → ((𝐻 ≠ (0_g‘𝐷) ∧ 𝐺 = (0_g‘𝐷)) ↔ ((𝐾‘𝐻) ⊆ (𝐾‘𝐺) ∧ (𝐾‘𝐻) ≠ (𝐾‘𝐺))))
48	14, 15, 18, 42, 9, 4, 10, 19, 13	lkrss 38027	. . . . . . 7 ⊢ (𝜑 → (𝐾‘𝐻) ⊆ (𝐾‘(𝐴 · 𝐻)))
49	1	fveq2d 6893	. . . . . . 7 ⊢ (𝜑 → (𝐾‘𝐺) = (𝐾‘(𝐴 · 𝐻)))
50	48, 49	sseqtrrd 4023	. . . . . 6 ⊢ (𝜑 → (𝐾‘𝐻) ⊆ (𝐾‘𝐺))
51	50	biantrurd 534	. . . . 5 ⊢ (𝜑 → ((𝐾‘𝐻) ≠ (𝐾‘𝐺) ↔ ((𝐾‘𝐻) ⊆ (𝐾‘𝐺) ∧ (𝐾‘𝐻) ≠ (𝐾‘𝐺))))
52	47, 51	bitr4d 282	. . . 4 ⊢ (𝜑 → ((𝐻 ≠ (0_g‘𝐷) ∧ 𝐺 = (0_g‘𝐷)) ↔ (𝐾‘𝐻) ≠ (𝐾‘𝐺)))
53	52	necon2bbid 2985	. . 3 ⊢ (𝜑 → ((𝐾‘𝐻) = (𝐾‘𝐺) ↔ ¬ (𝐻 ≠ (0_g‘𝐷) ∧ 𝐺 = (0_g‘𝐷))))
54	41, 53	mpbird 257	. 2 ⊢ (𝜑 → (𝐾‘𝐻) = (𝐾‘𝐺))
55	54	eqcomd 2739	1 ⊢ (𝜑 → (𝐾‘𝐺) = (𝐾‘𝐻))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 397 ∨ wo 846 = wceq 1542 ∈ wcel 2107 ≠ wne 2941 ∖ cdif 3945 ⊆ wss 3948 ⊊ wpss 3949 {csn 4628 ‘cfv 6541 (class class class)co 7406 Basecbs 17141 Scalarcsca 17197 ·_𝑠 cvsca 17198 0_gc0g 17382 LModclmod 20464 LVecclvec 20706 LFnlclfn 37916 LKerclk 37944 LDualcld 37982

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7722 ax-cnex 11163 ax-resscn 11164 ax-1cn 11165 ax-icn 11166 ax-addcl 11167 ax-addrcl 11168 ax-mulcl 11169 ax-mulrcl 11170 ax-mulcom 11171 ax-addass 11172 ax-mulass 11173 ax-distr 11174 ax-i2m1 11175 ax-1ne0 11176 ax-1rid 11177 ax-rnegex 11178 ax-rrecex 11179 ax-cnre 11180 ax-pre-lttri 11181 ax-pre-lttrn 11182 ax-pre-ltadd 11183 ax-pre-mulgt0 11184

This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-tp 4633 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6298 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6493 df-fun 6543 df-fn 6544 df-f 6545 df-f1 6546 df-fo 6547 df-f1o 6548 df-fv 6549 df-riota 7362 df-ov 7409 df-oprab 7410 df-mpo 7411 df-of 7667 df-om 7853 df-1st 7972 df-2nd 7973 df-tpos 8208 df-frecs 8263 df-wrecs 8294 df-recs 8368 df-rdg 8407 df-1o 8463 df-er 8700 df-map 8819 df-en 8937 df-dom 8938 df-sdom 8939 df-fin 8940 df-pnf 11247 df-mnf 11248 df-xr 11249 df-ltxr 11250 df-le 11251 df-sub 11443 df-neg 11444 df-nn 12210 df-2 12272 df-3 12273 df-4 12274 df-5 12275 df-6 12276 df-n0 12470 df-z 12556 df-uz 12820 df-fz 13482 df-struct 17077 df-sets 17094 df-slot 17112 df-ndx 17124 df-base 17142 df-ress 17171 df-plusg 17207 df-mulr 17208 df-sca 17210 df-vsca 17211 df-0g 17384 df-mgm 18558 df-sgrp 18607 df-mnd 18623 df-submnd 18669 df-grp 18819 df-minusg 18820 df-sbg 18821 df-subg 18998 df-cntz 19176 df-lsm 19499 df-cmn 19645 df-abl 19646 df-mgp 19983 df-ur 20000 df-ring 20052 df-oppr 20143 df-dvdsr 20164 df-unit 20165 df-invr 20195 df-drng 20310 df-lmod 20466 df-lss 20536 df-lsp 20576 df-lvec 20707 df-lshyp 37836 df-lfl 37917 df-lkr 37945 df-ldual 37983

This theorem is referenced by: lkrlspeqN 38030 lcdlkreq2N 40483

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