lcfrlem12N - Mathbox for Norm Megill

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Theorem lcfrlem12N 42002

Description: Lemma for lcfr 42033. (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 41558	. 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 42000	. 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 42001	. . 3 ⊢ (𝜑 → (𝐿‘(𝐽‘𝑋)) = ( ⊥ ‘{𝑋}))
22	20, 21	eleqtrrd 2840	. 2 ⊢ (𝜑 → 𝑌 ∈ (𝐿‘(𝐽‘𝑋)))
23		lcfrlem12.b	. . 3 ⊢ 𝐵 = (0_g‘𝑆)
24	9, 23, 12, 13	lkrf0 39541	. 2 ⊢ ((𝑈 ∈ LMod ∧ (𝐽‘𝑋) ∈ 𝐹 ∧ 𝑌 ∈ (𝐿‘(𝐽‘𝑋))) → ((𝐽‘𝑋)‘𝑌) = 𝐵)
25	4, 19, 22, 24	syl3anc 1374	1 ⊢ (𝜑 → ((𝐽‘𝑋)‘𝑌) = 𝐵)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1542 ∈ wcel 2114 ∃wrex 3062 {crab 3390 ∖ cdif 3887 {csn 4568 ↦ cmpt 5167 ‘cfv 6500 ℩crio 7325 (class class class)co 7369 Basecbs 17181 +_gcplusg 17222 Scalarcsca 17225 ·_𝑠 cvsca 17226 0_gc0g 17404 LModclmod 20857 LFnlclfn 39505 LKerclk 39533 LDualcld 39571 HLchlt 39798 LHypclh 40432 DVecHcdvh 41526 ocHcoch 41795

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5213 ax-sep 5232 ax-nul 5242 ax-pow 5308 ax-pr 5376 ax-un 7691 ax-cnex 11096 ax-resscn 11097 ax-1cn 11098 ax-icn 11099 ax-addcl 11100 ax-addrcl 11101 ax-mulcl 11102 ax-mulrcl 11103 ax-mulcom 11104 ax-addass 11105 ax-mulass 11106 ax-distr 11107 ax-i2m1 11108 ax-1ne0 11109 ax-1rid 11110 ax-rnegex 11111 ax-rrecex 11112 ax-cnre 11113 ax-pre-lttri 11114 ax-pre-lttrn 11115 ax-pre-ltadd 11116 ax-pre-mulgt0 11117 ax-riotaBAD 39401

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rmo 3343 df-reu 3344 df-rab 3391 df-v 3432 df-sbc 3730 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-pss 3910 df-nul 4275 df-if 4468 df-pw 4544 df-sn 4569 df-pr 4571 df-tp 4573 df-op 4575 df-uni 4852 df-int 4891 df-iun 4936 df-iin 4937 df-br 5087 df-opab 5149 df-mpt 5168 df-tr 5194 df-id 5527 df-eprel 5532 df-po 5540 df-so 5541 df-fr 5585 df-we 5587 df-xp 5638 df-rel 5639 df-cnv 5640 df-co 5641 df-dm 5642 df-rn 5643 df-res 5644 df-ima 5645 df-pred 6267 df-ord 6328 df-on 6329 df-lim 6330 df-suc 6331 df-iota 6456 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-riota 7326 df-ov 7372 df-oprab 7373 df-mpo 7374 df-om 7820 df-1st 7944 df-2nd 7945 df-tpos 8178 df-undef 8225 df-frecs 8233 df-wrecs 8264 df-recs 8313 df-rdg 8351 df-1o 8407 df-er 8645 df-map 8777 df-en 8896 df-dom 8897 df-sdom 8898 df-fin 8899 df-pnf 11183 df-mnf 11184 df-xr 11185 df-ltxr 11186 df-le 11187 df-sub 11381 df-neg 11382 df-nn 12177 df-2 12246 df-3 12247 df-4 12248 df-5 12249 df-6 12250 df-n0 12440 df-z 12527 df-uz 12791 df-fz 13464 df-struct 17119 df-sets 17136 df-slot 17154 df-ndx 17166 df-base 17182 df-ress 17203 df-plusg 17235 df-mulr 17236 df-sca 17238 df-vsca 17239 df-0g 17406 df-proset 18262 df-poset 18281 df-plt 18296 df-lub 18312 df-glb 18313 df-join 18314 df-meet 18315 df-p0 18391 df-p1 18392 df-lat 18400 df-clat 18467 df-mgm 18610 df-sgrp 18689 df-mnd 18705 df-submnd 18754 df-grp 18914 df-minusg 18915 df-sbg 18916 df-subg 19101 df-cntz 19294 df-lsm 19613 df-cmn 19759 df-abl 19760 df-mgp 20124 df-rng 20136 df-ur 20165 df-ring 20218 df-oppr 20319 df-dvdsr 20339 df-unit 20340 df-invr 20370 df-dvr 20383 df-drng 20710 df-lmod 20859 df-lss 20929 df-lsp 20969 df-lvec 21100 df-lsatoms 39424 df-lshyp 39425 df-lfl 39506 df-lkr 39534 df-oposet 39624 df-ol 39626 df-oml 39627 df-covers 39714 df-ats 39715 df-atl 39746 df-cvlat 39770 df-hlat 39799 df-llines 39946 df-lplanes 39947 df-lvols 39948 df-lines 39949 df-psubsp 39951 df-pmap 39952 df-padd 40244 df-lhyp 40436 df-laut 40437 df-ldil 40552 df-ltrn 40553 df-trl 40607 df-tgrp 41191 df-tendo 41203 df-edring 41205 df-dveca 41451 df-disoa 41477 df-dvech 41527 df-dib 41587 df-dic 41621 df-dih 41677 df-doch 41796 df-djh 41843

This theorem is referenced by: (None)

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