lkrss2N - Mathbox for Norm Megill

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Theorem lkrss2N 39187

Description: Two functionals with kernels in a subset relationship. (Contributed by NM, 17-Feb-2015.) (New usage is discouraged.)

**Hypotheses**
Ref	Expression
lkrss2.s	⊢ 𝑆 = (Scalar‘𝑊)
lkrss2.r	⊢ 𝑅 = (Base‘𝑆)
lkrss2.f	⊢ 𝐹 = (LFnl‘𝑊)
lkrss2.k	⊢ 𝐾 = (LKer‘𝑊)
lkrss2.d	⊢ 𝐷 = (LDual‘𝑊)
lkrss2.t	⊢ · = ( ·_𝑠 ‘𝐷)
lkrss2.w	⊢ (𝜑 → 𝑊 ∈ LVec)
lkrss2.g	⊢ (𝜑 → 𝐺 ∈ 𝐹)
lkrss2.h	⊢ (𝜑 → 𝐻 ∈ 𝐹)

**Assertion**
Ref	Expression
lkrss2N	⊢ (𝜑 → ((𝐾‘𝐺) ⊆ (𝐾‘𝐻) ↔ ∃𝑟 ∈ 𝑅 𝐻 = (𝑟 · 𝐺)))

Distinct variable groups: 𝐹,𝑟 𝐺,𝑟 𝐻,𝑟 𝐾,𝑟 𝑅,𝑟 𝑆,𝑟 𝑊,𝑟 𝜑,𝑟 · ,𝑟 Allowed substitution hint: 𝐷(𝑟)

**Proof of Theorem lkrss2N**
Step	Hyp	Ref	Expression
1		sspss 4077	. . 3 ⊢ ((𝐾‘𝐺) ⊆ (𝐾‘𝐻) ↔ ((𝐾‘𝐺) ⊊ (𝐾‘𝐻) ∨ (𝐾‘𝐺) = (𝐾‘𝐻)))
2		lkrss2.f	. . . . . . 7 ⊢ 𝐹 = (LFnl‘𝑊)
3		lkrss2.k	. . . . . . 7 ⊢ 𝐾 = (LKer‘𝑊)
4		lkrss2.d	. . . . . . 7 ⊢ 𝐷 = (LDual‘𝑊)
5		eqid 2735	. . . . . . 7 ⊢ (0_g‘𝐷) = (0_g‘𝐷)
6		lkrss2.w	. . . . . . 7 ⊢ (𝜑 → 𝑊 ∈ LVec)
7		lkrss2.g	. . . . . . 7 ⊢ (𝜑 → 𝐺 ∈ 𝐹)
8		lkrss2.h	. . . . . . 7 ⊢ (𝜑 → 𝐻 ∈ 𝐹)
9	2, 3, 4, 5, 6, 7, 8	lkrpssN 39181	. . . . . 6 ⊢ (𝜑 → ((𝐾‘𝐺) ⊊ (𝐾‘𝐻) ↔ (𝐺 ≠ (0_g‘𝐷) ∧ 𝐻 = (0_g‘𝐷))))
10		lveclmod 21064	. . . . . . . . . . . 12 ⊢ (𝑊 ∈ LVec → 𝑊 ∈ LMod)
11	6, 10	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑊 ∈ LMod)
12		lkrss2.s	. . . . . . . . . . . 12 ⊢ 𝑆 = (Scalar‘𝑊)
13		lkrss2.r	. . . . . . . . . . . 12 ⊢ 𝑅 = (Base‘𝑆)
14		eqid 2735	. . . . . . . . . . . 12 ⊢ (0_g‘𝑆) = (0_g‘𝑆)
15	12, 13, 14	lmod0cl 20845	. . . . . . . . . . 11 ⊢ (𝑊 ∈ LMod → (0_g‘𝑆) ∈ 𝑅)
16	11, 15	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → (0_g‘𝑆) ∈ 𝑅)
17	16	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐻 = (0_g‘𝐷)) → (0_g‘𝑆) ∈ 𝑅)
18		simpr 484	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐻 = (0_g‘𝐷)) → 𝐻 = (0_g‘𝐷))
19		lkrss2.t	. . . . . . . . . . . 12 ⊢ · = ( ·_𝑠 ‘𝐷)
20	2, 12, 14, 4, 19, 5, 11, 7	ldual0vs 39178	. . . . . . . . . . 11 ⊢ (𝜑 → ((0_g‘𝑆) · 𝐺) = (0_g‘𝐷))
21	20	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐻 = (0_g‘𝐷)) → ((0_g‘𝑆) · 𝐺) = (0_g‘𝐷))
22	18, 21	eqtr4d 2773	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐻 = (0_g‘𝐷)) → 𝐻 = ((0_g‘𝑆) · 𝐺))
23		oveq1 7412	. . . . . . . . . 10 ⊢ (𝑟 = (0_g‘𝑆) → (𝑟 · 𝐺) = ((0_g‘𝑆) · 𝐺))
24	23	rspceeqv 3624	. . . . . . . . 9 ⊢ (((0_g‘𝑆) ∈ 𝑅 ∧ 𝐻 = ((0_g‘𝑆) · 𝐺)) → ∃𝑟 ∈ 𝑅 𝐻 = (𝑟 · 𝐺))
25	17, 22, 24	syl2anc 584	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐻 = (0_g‘𝐷)) → ∃𝑟 ∈ 𝑅 𝐻 = (𝑟 · 𝐺))
26	25	ex 412	. . . . . . 7 ⊢ (𝜑 → (𝐻 = (0_g‘𝐷) → ∃𝑟 ∈ 𝑅 𝐻 = (𝑟 · 𝐺)))
27	26	adantld 490	. . . . . 6 ⊢ (𝜑 → ((𝐺 ≠ (0_g‘𝐷) ∧ 𝐻 = (0_g‘𝐷)) → ∃𝑟 ∈ 𝑅 𝐻 = (𝑟 · 𝐺)))
28	9, 27	sylbid 240	. . . . 5 ⊢ (𝜑 → ((𝐾‘𝐺) ⊊ (𝐾‘𝐻) → ∃𝑟 ∈ 𝑅 𝐻 = (𝑟 · 𝐺)))
29	28	imp 406	. . . 4 ⊢ ((𝜑 ∧ (𝐾‘𝐺) ⊊ (𝐾‘𝐻)) → ∃𝑟 ∈ 𝑅 𝐻 = (𝑟 · 𝐺))
30	6	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ (𝐾‘𝐺) = (𝐾‘𝐻)) → 𝑊 ∈ LVec)
31	7	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ (𝐾‘𝐺) = (𝐾‘𝐻)) → 𝐺 ∈ 𝐹)
32	8	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ (𝐾‘𝐺) = (𝐾‘𝐻)) → 𝐻 ∈ 𝐹)
33		simpr 484	. . . . 5 ⊢ ((𝜑 ∧ (𝐾‘𝐺) = (𝐾‘𝐻)) → (𝐾‘𝐺) = (𝐾‘𝐻))
34	12, 13, 2, 3, 4, 19, 30, 31, 32, 33	eqlkr4 39183	. . . 4 ⊢ ((𝜑 ∧ (𝐾‘𝐺) = (𝐾‘𝐻)) → ∃𝑟 ∈ 𝑅 𝐻 = (𝑟 · 𝐺))
35	29, 34	jaodan 959	. . 3 ⊢ ((𝜑 ∧ ((𝐾‘𝐺) ⊊ (𝐾‘𝐻) ∨ (𝐾‘𝐺) = (𝐾‘𝐻))) → ∃𝑟 ∈ 𝑅 𝐻 = (𝑟 · 𝐺))
36	1, 35	sylan2b 594	. 2 ⊢ ((𝜑 ∧ (𝐾‘𝐺) ⊆ (𝐾‘𝐻)) → ∃𝑟 ∈ 𝑅 𝐻 = (𝑟 · 𝐺))
37	6	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑟 ∈ 𝑅) → 𝑊 ∈ LVec)
38	7	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑟 ∈ 𝑅) → 𝐺 ∈ 𝐹)
39		simpr 484	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑟 ∈ 𝑅) → 𝑟 ∈ 𝑅)
40	12, 13, 2, 3, 4, 19, 37, 38, 39	lkrss 39186	. . . . . 6 ⊢ ((𝜑 ∧ 𝑟 ∈ 𝑅) → (𝐾‘𝐺) ⊆ (𝐾‘(𝑟 · 𝐺)))
41	40	ex 412	. . . . 5 ⊢ (𝜑 → (𝑟 ∈ 𝑅 → (𝐾‘𝐺) ⊆ (𝐾‘(𝑟 · 𝐺))))
42		fveq2 6876	. . . . . . 7 ⊢ (𝐻 = (𝑟 · 𝐺) → (𝐾‘𝐻) = (𝐾‘(𝑟 · 𝐺)))
43	42	sseq2d 3991	. . . . . 6 ⊢ (𝐻 = (𝑟 · 𝐺) → ((𝐾‘𝐺) ⊆ (𝐾‘𝐻) ↔ (𝐾‘𝐺) ⊆ (𝐾‘(𝑟 · 𝐺))))
44	43	biimprcd 250	. . . . 5 ⊢ ((𝐾‘𝐺) ⊆ (𝐾‘(𝑟 · 𝐺)) → (𝐻 = (𝑟 · 𝐺) → (𝐾‘𝐺) ⊆ (𝐾‘𝐻)))
45	41, 44	syl6 35	. . . 4 ⊢ (𝜑 → (𝑟 ∈ 𝑅 → (𝐻 = (𝑟 · 𝐺) → (𝐾‘𝐺) ⊆ (𝐾‘𝐻))))
46	45	rexlimdv 3139	. . 3 ⊢ (𝜑 → (∃𝑟 ∈ 𝑅 𝐻 = (𝑟 · 𝐺) → (𝐾‘𝐺) ⊆ (𝐾‘𝐻)))
47	46	imp 406	. 2 ⊢ ((𝜑 ∧ ∃𝑟 ∈ 𝑅 𝐻 = (𝑟 · 𝐺)) → (𝐾‘𝐺) ⊆ (𝐾‘𝐻))
48	36, 47	impbida 800	1 ⊢ (𝜑 → ((𝐾‘𝐺) ⊆ (𝐾‘𝐻) ↔ ∃𝑟 ∈ 𝑅 𝐻 = (𝑟 · 𝐺)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∨ wo 847 = wceq 1540 ∈ wcel 2108 ≠ wne 2932 ∃wrex 3060 ⊆ wss 3926 ⊊ wpss 3927 ‘cfv 6531 (class class class)co 7405 Basecbs 17228 Scalarcsca 17274 ·_𝑠 cvsca 17275 0_gc0g 17453 LModclmod 20817 LVecclvec 21060 LFnlclfn 39075 LKerclk 39103 LDualcld 39141

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2707 ax-rep 5249 ax-sep 5266 ax-nul 5276 ax-pow 5335 ax-pr 5402 ax-un 7729 ax-cnex 11185 ax-resscn 11186 ax-1cn 11187 ax-icn 11188 ax-addcl 11189 ax-addrcl 11190 ax-mulcl 11191 ax-mulrcl 11192 ax-mulcom 11193 ax-addass 11194 ax-mulass 11195 ax-distr 11196 ax-i2m1 11197 ax-1ne0 11198 ax-1rid 11199 ax-rnegex 11200 ax-rrecex 11201 ax-cnre 11202 ax-pre-lttri 11203 ax-pre-lttrn 11204 ax-pre-ltadd 11205 ax-pre-mulgt0 11206

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2727 df-clel 2809 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3359 df-reu 3360 df-rab 3416 df-v 3461 df-sbc 3766 df-csb 3875 df-dif 3929 df-un 3931 df-in 3933 df-ss 3943 df-pss 3946 df-nul 4309 df-if 4501 df-pw 4577 df-sn 4602 df-pr 4604 df-tp 4606 df-op 4608 df-uni 4884 df-int 4923 df-iun 4969 df-br 5120 df-opab 5182 df-mpt 5202 df-tr 5230 df-id 5548 df-eprel 5553 df-po 5561 df-so 5562 df-fr 5606 df-we 5608 df-xp 5660 df-rel 5661 df-cnv 5662 df-co 5663 df-dm 5664 df-rn 5665 df-res 5666 df-ima 5667 df-pred 6290 df-ord 6355 df-on 6356 df-lim 6357 df-suc 6358 df-iota 6484 df-fun 6533 df-fn 6534 df-f 6535 df-f1 6536 df-fo 6537 df-f1o 6538 df-fv 6539 df-riota 7362 df-ov 7408 df-oprab 7409 df-mpo 7410 df-of 7671 df-om 7862 df-1st 7988 df-2nd 7989 df-tpos 8225 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-1o 8480 df-er 8719 df-map 8842 df-en 8960 df-dom 8961 df-sdom 8962 df-fin 8963 df-pnf 11271 df-mnf 11272 df-xr 11273 df-ltxr 11274 df-le 11275 df-sub 11468 df-neg 11469 df-nn 12241 df-2 12303 df-3 12304 df-4 12305 df-5 12306 df-6 12307 df-n0 12502 df-z 12589 df-uz 12853 df-fz 13525 df-struct 17166 df-sets 17183 df-slot 17201 df-ndx 17213 df-base 17229 df-ress 17252 df-plusg 17284 df-mulr 17285 df-sca 17287 df-vsca 17288 df-0g 17455 df-mgm 18618 df-sgrp 18697 df-mnd 18713 df-submnd 18762 df-grp 18919 df-minusg 18920 df-sbg 18921 df-subg 19106 df-cntz 19300 df-lsm 19617 df-cmn 19763 df-abl 19764 df-mgp 20101 df-rng 20113 df-ur 20142 df-ring 20195 df-oppr 20297 df-dvdsr 20317 df-unit 20318 df-invr 20348 df-nzr 20473 df-rlreg 20654 df-domn 20655 df-drng 20691 df-lmod 20819 df-lss 20889 df-lsp 20929 df-lvec 21061 df-lshyp 38995 df-lfl 39076 df-lkr 39104 df-ldual 39142

This theorem is referenced by: lcfrvalsnN 41560

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