lcfrlem12N - Mathbox for Norm Megill

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Theorem lcfrlem12N 42720

Description: Lemma for lcfr 42751. (Contributed by NM, 23-Feb-2015.) (New usage is discouraged.)

**Hypotheses**
Ref	Expression
lcf1o.h	⊢ 𝐻 = (LHyp‘𝐾)
lcf1o.o	⊢ ⊥ = ((ocH‘𝐾)‘𝑊)
lcf1o.u	⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
lcf1o.v	⊢ 𝑉 = (Base‘𝑈)
lcf1o.a	⊢ + = (+_g‘𝑈)
lcf1o.t	⊢ · = ( ·_𝑠 ‘𝑈)
lcf1o.s	⊢ 𝑆 = (Scalar‘𝑈)
lcf1o.r	⊢ 𝑅 = (Base‘𝑆)
lcf1o.z	⊢ 0 = (0_g‘𝑈)
lcf1o.f	⊢ 𝐹 = (LFnl‘𝑈)
lcf1o.l	⊢ 𝐿 = (LKer‘𝑈)
lcf1o.d	⊢ 𝐷 = (LDual‘𝑈)
lcf1o.q	⊢ 𝑄 = (0_g‘𝐷)
lcf1o.c	⊢ 𝐶 = {𝑓 ∈ 𝐹 ∣ ( ⊥ ‘( ⊥ ‘(𝐿‘𝑓))) = (𝐿‘𝑓)}
lcf1o.j	⊢ 𝐽 = (𝑥 ∈ (𝑉 ∖ { 0 }) ↦ (𝑣 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑥})𝑣 = (𝑤 + (𝑘 · 𝑥)))))
lcflo.k	⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
lcfrlem10.x	⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))
lcfrlem12.b	⊢ 𝐵 = (0_g‘𝑆)
lcfrlem12.y	⊢ (𝜑 → 𝑌 ∈ ( ⊥ ‘{𝑋}))

**Assertion**
Ref	Expression
lcfrlem12N	⊢ (𝜑 → ((𝐽‘𝑋)‘𝑌) = 𝐵)

Distinct variable groups: 𝑥,𝑤, ⊥ 𝑥, 0 𝑥,𝑣,𝑉 𝑥, · 𝑣,𝑘,𝑤,𝑥,𝑋 𝑥, + 𝑥,𝑅 + ,𝑘,𝑣,𝑤 ⊥ ,𝑘,𝑣 𝑅,𝑘,𝑣 𝑆,𝑘 · ,𝑘,𝑣,𝑤 Allowed substitution hints: 𝜑(𝑥, 𝑤, 𝑣, 𝑓, 𝑘) 𝐵(𝑥, 𝑤, 𝑣, 𝑓, 𝑘) 𝐶(𝑥, 𝑤, 𝑣, 𝑓, 𝑘) 𝐷(𝑥, 𝑤, 𝑣, 𝑓, 𝑘) + (𝑓) 𝑄(𝑥, 𝑤, 𝑣, 𝑓, 𝑘) 𝑅(𝑤, 𝑓) 𝑆(𝑥, 𝑤, 𝑣, 𝑓) · (𝑓) 𝑈(𝑥, 𝑤, 𝑣, 𝑓, 𝑘) 𝐹(𝑥, 𝑤, 𝑣, 𝑓, 𝑘) 𝐻(𝑥, 𝑤, 𝑣, 𝑓, 𝑘) 𝐽(𝑥, 𝑤, 𝑣, 𝑓, 𝑘) 𝐾(𝑥, 𝑤, 𝑣, 𝑓, 𝑘) 𝐿(𝑥, 𝑤, 𝑣, 𝑓, 𝑘) ⊥ (𝑓) 𝑉(𝑤, 𝑓, 𝑘) 𝑊(𝑥, 𝑤, 𝑣, 𝑓, 𝑘) 𝑋(𝑓) 𝑌(𝑥, 𝑤, 𝑣, 𝑓, 𝑘) 0 (𝑤, 𝑣, 𝑓, 𝑘)

**Proof of Theorem lcfrlem12N**
Step	Hyp	Ref	Expression
1		lcf1o.h	. . 3 ⊢ 𝐻 = (LHyp‘𝐾)
2		lcf1o.u	. . 3 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
3		lcflo.k	. . 3 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
4	1, 2, 3	dvhlmod 42276	. 2 ⊢ (𝜑 → 𝑈 ∈ LMod)
5		lcf1o.o	. . 3 ⊢ ⊥ = ((ocH‘𝐾)‘𝑊)
6		lcf1o.v	. . 3 ⊢ 𝑉 = (Base‘𝑈)
7		lcf1o.a	. . 3 ⊢ + = (+_g‘𝑈)
8		lcf1o.t	. . 3 ⊢ · = ( ·_𝑠 ‘𝑈)
9		lcf1o.s	. . 3 ⊢ 𝑆 = (Scalar‘𝑈)
10		lcf1o.r	. . 3 ⊢ 𝑅 = (Base‘𝑆)
11		lcf1o.z	. . 3 ⊢ 0 = (0_g‘𝑈)
12		lcf1o.f	. . 3 ⊢ 𝐹 = (LFnl‘𝑈)
13		lcf1o.l	. . 3 ⊢ 𝐿 = (LKer‘𝑈)
14		lcf1o.d	. . 3 ⊢ 𝐷 = (LDual‘𝑈)
15		lcf1o.q	. . 3 ⊢ 𝑄 = (0_g‘𝐷)
16		lcf1o.c	. . 3 ⊢ 𝐶 = {𝑓 ∈ 𝐹 ∣ ( ⊥ ‘( ⊥ ‘(𝐿‘𝑓))) = (𝐿‘𝑓)}
17		lcf1o.j	. . 3 ⊢ 𝐽 = (𝑥 ∈ (𝑉 ∖ { 0 }) ↦ (𝑣 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑥})𝑣 = (𝑤 + (𝑘 · 𝑥)))))
18		lcfrlem10.x	. . 3 ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))
19	1, 5, 2, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 3, 18	lcfrlem10 42718	. 2 ⊢ (𝜑 → (𝐽‘𝑋) ∈ 𝐹)
20		lcfrlem12.y	. . 3 ⊢ (𝜑 → 𝑌 ∈ ( ⊥ ‘{𝑋}))
21	1, 5, 2, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 3, 18	lcfrlem11 42719	. . 3 ⊢ (𝜑 → (𝐿‘(𝐽‘𝑋)) = ( ⊥ ‘{𝑋}))
22	20, 21	eleqtrrd 2863	. 2 ⊢ (𝜑 → 𝑌 ∈ (𝐿‘(𝐽‘𝑋)))
23		lcfrlem12.b	. . 3 ⊢ 𝐵 = (0_g‘𝑆)
24	9, 23, 12, 13	lkrf0 40259	. 2 ⊢ ((𝑈 ∈ LMod ∧ (𝐽‘𝑋) ∈ 𝐹 ∧ 𝑌 ∈ (𝐿‘(𝐽‘𝑋))) → ((𝐽‘𝑋)‘𝑌) = 𝐵)
25	4, 19, 22, 24	syl3anc 1398	1 ⊢ (𝜑 → ((𝐽‘𝑋)‘𝑌) = 𝐵)

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∧ wa 401 = wceq 1569 ∈ wcel 2144 ∃wrex 3086 {crab 3412 ∖ cdif 4044 {csn 4584 ↦ cmpt 5185 ‘cfv 6543 ℩crio 7380 (class class class)co 7424 Basecbs 17396 +_gcplusg 17437 Scalarcsca 17440 ·_𝑠 cvsca 17441 0_gc0g 17619 LModclmod 21144 LFnlclfn 40223 LKerclk 40251 LDualcld 40289 HLchlt 40516 LHypclh 41150 DVecHcdvh 42244 ocHcoch 42513

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1827 ax-4 1841 ax-5 1942 ax-6 1999 ax-7 2040 ax-8 2146 ax-9 2154 ax-10 2177 ax-11 2193 ax-12 2212 ax-ext 2732 ax-rep 5231 ax-sep 5248 ax-nul 5259 ax-pow 5327 ax-pr 5391 ax-un 7755 ax-cnex 11263 ax-resscn 11264 ax-1cn 11265 ax-icn 11266 ax-addcl 11267 ax-addrcl 11268 ax-mulcl 11269 ax-mulrcl 11270 ax-mulcom 11271 ax-addass 11272 ax-mulass 11273 ax-distr 11274 ax-i2m1 11275 ax-1ne0 11276 ax-1rid 11277 ax-rnegex 11278 ax-rrecex 11279 ax-cnre 11280 ax-pre-lttri 11281 ax-pre-lttrn 11282 ax-pre-ltadd 11283 ax-pre-mulgt0 11284 ax-riotaBAD 40119

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1572 df-fal 1582 df-ex 1812 df-nf 1816 df-sb 2099 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3365 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3739 df-csb 3848 df-ss 3897 df-pss 4013 df-dif 4046 df-in 4090 df-un 4150 df-nul 4278 df-if 4482 df-pw 4558 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4835 df-uni 4867 df-int 4907 df-iun 4952 df-iin 4953 df-br 5103 df-opab 5167 df-mpt 5186 df-tr 5212 df-id 5546 df-eprel 5551 df-po 5559 df-so 5560 df-fr 5604 df-we 5606 df-xp 5657 df-rel 5658 df-cnv 5659 df-co 5660 df-dm 5661 df-rn 5662 df-res 5663 df-ima 5664 df-pred 6307 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6497 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7381 df-ov 7427 df-oprab 7428 df-mpo 7429 df-om 7882 df-1st 8005 df-2nd 8006 df-tpos 8250 df-undef 8297 df-frecs 8306 df-wrecs 8337 df-recs 8386 df-rdg 8425 df-1o 8483 df-er 8724 df-map 8856 df-en 8981 df-dom 8982 df-sdom 8983 df-fin 8984 df-pnf 11352 df-mnf 11353 df-xr 11354 df-ltxr 11355 df-le 11356 df-sub 11550 df-neg 11551 df-nn 12343 df-2 12412 df-3 12413 df-4 12414 df-5 12415 df-6 12416 df-n0 12614 df-z 12701 df-uz 12973 df-fz 13647 df-struct 17334 df-sets 17351 df-slot 17369 df-ndx 17381 df-base 17397 df-ress 17418 df-plusg 17450 df-mulr 17451 df-sca 17453 df-vsca 17454 df-0g 17621 df-proset 18477 df-poset 18496 df-plt 18511 df-lub 18527 df-glb 18528 df-join 18529 df-meet 18530 df-p0 18606 df-p1 18607 df-lat 18615 df-clat 18682 df-mgm 18825 df-sgrp 18917 df-mnd 18933 df-submnd 18988 df-grp 19156 df-minusg 19157 df-sbg 19158 df-subg 19342 df-cntz 19540 df-lsm 19859 df-cmn 20005 df-abl 20006 df-mgp 20370 df-rng 20384 df-ur 20417 df-ring 20470 df-oppr 20576 df-dvdsr 20596 df-unit 20597 df-invr 20627 df-dvr 20640 df-drng 20991 df-lmod 21146 df-lss 21216 df-lsp 21256 df-lvec 21387 df-lsatoms 40142 df-lshyp 40143 df-lfl 40224 df-lkr 40252 df-oposet 40342 df-ol 40344 df-oml 40345 df-covers 40432 df-ats 40433 df-atl 40464 df-cvlat 40488 df-hlat 40517 df-llines 40664 df-lplanes 40665 df-lvols 40666 df-lines 40667 df-psubsp 40669 df-pmap 40670 df-padd 40962 df-lhyp 41154 df-laut 41155 df-ldil 41270 df-ltrn 41271 df-trl 41325 df-tgrp 41909 df-tendo 41921 df-edring 41923 df-dveca 42169 df-disoa 42195 df-dvech 42245 df-dib 42305 df-dic 42339 df-dih 42395 df-doch 42514 df-djh 42561

This theorem is used by: (None)

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