eqlkr2 - Mathbox for Norm Megill

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Theorem eqlkr2 34387

Description: Two functionals with the same kernel are the same up to a constant. (Contributed by NM, 10-Oct-2014.)

**Hypotheses**
Ref	Expression
eqlkr.d	⊢ 𝐷 = (Scalar‘𝑊)
eqlkr.k	⊢ 𝐾 = (Base‘𝐷)
eqlkr.t	⊢ · = (._r‘𝐷)
eqlkr.v	⊢ 𝑉 = (Base‘𝑊)
eqlkr.f	⊢ 𝐹 = (LFnl‘𝑊)
eqlkr.l	⊢ 𝐿 = (LKer‘𝑊)

**Assertion**
Ref	Expression
eqlkr2	⊢ ((𝑊 ∈ LVec ∧ (𝐺 ∈ 𝐹 ∧ 𝐻 ∈ 𝐹) ∧ (𝐿‘𝐺) = (𝐿‘𝐻)) → ∃𝑟 ∈ 𝐾 𝐻 = (𝐺 ∘_𝑓 · (𝑉 × {𝑟})))

Distinct variable groups: 𝐷,𝑟 𝐺,𝑟 𝐻,𝑟 𝑉,𝑟 𝐾,𝑟 · ,𝑟 𝐹,𝑟 𝐿,𝑟 𝑊,𝑟

Proof of Theorem eqlkr2 Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqlkr.d	. . 3 ⊢ 𝐷 = (Scalar‘𝑊)
2		eqlkr.k	. . 3 ⊢ 𝐾 = (Base‘𝐷)
3		eqlkr.t	. . 3 ⊢ · = (._r‘𝐷)
4		eqlkr.v	. . 3 ⊢ 𝑉 = (Base‘𝑊)
5		eqlkr.f	. . 3 ⊢ 𝐹 = (LFnl‘𝑊)
6		eqlkr.l	. . 3 ⊢ 𝐿 = (LKer‘𝑊)
7	1, 2, 3, 4, 5, 6	eqlkr 34386	. 2 ⊢ ((𝑊 ∈ LVec ∧ (𝐺 ∈ 𝐹 ∧ 𝐻 ∈ 𝐹) ∧ (𝐿‘𝐺) = (𝐿‘𝐻)) → ∃𝑟 ∈ 𝐾 ∀𝑥 ∈ 𝑉 (𝐻‘𝑥) = ((𝐺‘𝑥) · 𝑟))
8		fvex 6201	. . . . . 6 ⊢ (Base‘𝑊) ∈ V
9	4, 8	eqeltri 2697	. . . . 5 ⊢ 𝑉 ∈ V
10	9	a1i 11	. . . 4 ⊢ (((𝑊 ∈ LVec ∧ (𝐺 ∈ 𝐹 ∧ 𝐻 ∈ 𝐹) ∧ (𝐿‘𝐺) = (𝐿‘𝐻)) ∧ 𝑟 ∈ 𝐾) → 𝑉 ∈ V)
11		simpl1 1064	. . . . . 6 ⊢ (((𝑊 ∈ LVec ∧ (𝐺 ∈ 𝐹 ∧ 𝐻 ∈ 𝐹) ∧ (𝐿‘𝐺) = (𝐿‘𝐻)) ∧ 𝑟 ∈ 𝐾) → 𝑊 ∈ LVec)
12		simpl2l 1114	. . . . . 6 ⊢ (((𝑊 ∈ LVec ∧ (𝐺 ∈ 𝐹 ∧ 𝐻 ∈ 𝐹) ∧ (𝐿‘𝐺) = (𝐿‘𝐻)) ∧ 𝑟 ∈ 𝐾) → 𝐺 ∈ 𝐹)
13	1, 2, 4, 5	lflf 34350	. . . . . 6 ⊢ ((𝑊 ∈ LVec ∧ 𝐺 ∈ 𝐹) → 𝐺:𝑉⟶𝐾)
14	11, 12, 13	syl2anc 693	. . . . 5 ⊢ (((𝑊 ∈ LVec ∧ (𝐺 ∈ 𝐹 ∧ 𝐻 ∈ 𝐹) ∧ (𝐿‘𝐺) = (𝐿‘𝐻)) ∧ 𝑟 ∈ 𝐾) → 𝐺:𝑉⟶𝐾)
15		ffn 6045	. . . . 5 ⊢ (𝐺:𝑉⟶𝐾 → 𝐺 Fn 𝑉)
16	14, 15	syl 17	. . . 4 ⊢ (((𝑊 ∈ LVec ∧ (𝐺 ∈ 𝐹 ∧ 𝐻 ∈ 𝐹) ∧ (𝐿‘𝐺) = (𝐿‘𝐻)) ∧ 𝑟 ∈ 𝐾) → 𝐺 Fn 𝑉)
17		vex 3203	. . . . 5 ⊢ 𝑟 ∈ V
18		fnconstg 6093	. . . . 5 ⊢ (𝑟 ∈ V → (𝑉 × {𝑟}) Fn 𝑉)
19	17, 18	mp1i 13	. . . 4 ⊢ (((𝑊 ∈ LVec ∧ (𝐺 ∈ 𝐹 ∧ 𝐻 ∈ 𝐹) ∧ (𝐿‘𝐺) = (𝐿‘𝐻)) ∧ 𝑟 ∈ 𝐾) → (𝑉 × {𝑟}) Fn 𝑉)
20		simpl2r 1115	. . . . . 6 ⊢ (((𝑊 ∈ LVec ∧ (𝐺 ∈ 𝐹 ∧ 𝐻 ∈ 𝐹) ∧ (𝐿‘𝐺) = (𝐿‘𝐻)) ∧ 𝑟 ∈ 𝐾) → 𝐻 ∈ 𝐹)
21	1, 2, 4, 5	lflf 34350	. . . . . 6 ⊢ ((𝑊 ∈ LVec ∧ 𝐻 ∈ 𝐹) → 𝐻:𝑉⟶𝐾)
22	11, 20, 21	syl2anc 693	. . . . 5 ⊢ (((𝑊 ∈ LVec ∧ (𝐺 ∈ 𝐹 ∧ 𝐻 ∈ 𝐹) ∧ (𝐿‘𝐺) = (𝐿‘𝐻)) ∧ 𝑟 ∈ 𝐾) → 𝐻:𝑉⟶𝐾)
23		ffn 6045	. . . . 5 ⊢ (𝐻:𝑉⟶𝐾 → 𝐻 Fn 𝑉)
24	22, 23	syl 17	. . . 4 ⊢ (((𝑊 ∈ LVec ∧ (𝐺 ∈ 𝐹 ∧ 𝐻 ∈ 𝐹) ∧ (𝐿‘𝐺) = (𝐿‘𝐻)) ∧ 𝑟 ∈ 𝐾) → 𝐻 Fn 𝑉)
25		eqidd 2623	. . . 4 ⊢ ((((𝑊 ∈ LVec ∧ (𝐺 ∈ 𝐹 ∧ 𝐻 ∈ 𝐹) ∧ (𝐿‘𝐺) = (𝐿‘𝐻)) ∧ 𝑟 ∈ 𝐾) ∧ 𝑥 ∈ 𝑉) → (𝐺‘𝑥) = (𝐺‘𝑥))
26	17	fvconst2 6469	. . . . 5 ⊢ (𝑥 ∈ 𝑉 → ((𝑉 × {𝑟})‘𝑥) = 𝑟)
27	26	adantl 482	. . . 4 ⊢ ((((𝑊 ∈ LVec ∧ (𝐺 ∈ 𝐹 ∧ 𝐻 ∈ 𝐹) ∧ (𝐿‘𝐺) = (𝐿‘𝐻)) ∧ 𝑟 ∈ 𝐾) ∧ 𝑥 ∈ 𝑉) → ((𝑉 × {𝑟})‘𝑥) = 𝑟)
28	10, 16, 19, 24, 25, 27	offveqb 6919	. . 3 ⊢ (((𝑊 ∈ LVec ∧ (𝐺 ∈ 𝐹 ∧ 𝐻 ∈ 𝐹) ∧ (𝐿‘𝐺) = (𝐿‘𝐻)) ∧ 𝑟 ∈ 𝐾) → (𝐻 = (𝐺 ∘_𝑓 · (𝑉 × {𝑟})) ↔ ∀𝑥 ∈ 𝑉 (𝐻‘𝑥) = ((𝐺‘𝑥) · 𝑟)))
29	28	rexbidva 3049	. 2 ⊢ ((𝑊 ∈ LVec ∧ (𝐺 ∈ 𝐹 ∧ 𝐻 ∈ 𝐹) ∧ (𝐿‘𝐺) = (𝐿‘𝐻)) → (∃𝑟 ∈ 𝐾 𝐻 = (𝐺 ∘_𝑓 · (𝑉 × {𝑟})) ↔ ∃𝑟 ∈ 𝐾 ∀𝑥 ∈ 𝑉 (𝐻‘𝑥) = ((𝐺‘𝑥) · 𝑟)))
30	7, 29	mpbird 247	1 ⊢ ((𝑊 ∈ LVec ∧ (𝐺 ∈ 𝐹 ∧ 𝐻 ∈ 𝐹) ∧ (𝐿‘𝐺) = (𝐿‘𝐻)) → ∃𝑟 ∈ 𝐾 𝐻 = (𝐺 ∘_𝑓 · (𝑉 × {𝑟})))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 384 ∧ w3a 1037 = wceq 1483 ∈ wcel 1990 ∀wral 2912 ∃wrex 2913 Vcvv 3200 {csn 4177 × cxp 5112 Fn wfn 5883 ⟶wf 5884 ‘cfv 5888 (class class class)co 6650 ∘_𝑓 cof 6895 Basecbs 15857 ._rcmulr 15942 Scalarcsca 15944 LVecclvec 19102 LFnlclfn 34344 LKerclk 34372

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-of 6897 df-om 7066 df-1st 7168 df-2nd 7169 df-tpos 7352 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-er 7742 df-map 7859 df-en 7956 df-dom 7957 df-sdom 7958 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-2 11079 df-3 11080 df-ndx 15860 df-slot 15861 df-base 15863 df-sets 15864 df-ress 15865 df-plusg 15954 df-mulr 15955 df-0g 16102 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-grp 17425 df-minusg 17426 df-sbg 17427 df-cmn 18195 df-abl 18196 df-mgp 18490 df-ur 18502 df-ring 18549 df-oppr 18623 df-dvdsr 18641 df-unit 18642 df-invr 18672 df-drng 18749 df-lmod 18865 df-lvec 19103 df-lfl 34345 df-lkr 34373

This theorem is referenced by: lfl1dim 34408 lfl1dim2N 34409 eqlkr4 34452

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