lcfrlem42 - Mathbox for Norm Megill

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Theorem lcfrlem42 39598

Description: Lemma for lcfr 39599. Eliminate nonzero condition. (Contributed by NM, 11-Mar-2015.)

**Hypotheses**
Ref	Expression
lcfrlem38.h	⊢ 𝐻 = (LHyp‘𝐾)
lcfrlem38.o	⊢ ⊥ = ((ocH‘𝐾)‘𝑊)
lcfrlem38.u	⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
lcfrlem38.p	⊢ + = (+_g‘𝑈)
lcfrlem38.f	⊢ 𝐹 = (LFnl‘𝑈)
lcfrlem38.l	⊢ 𝐿 = (LKer‘𝑈)
lcfrlem38.d	⊢ 𝐷 = (LDual‘𝑈)
lcfrlem38.q	⊢ 𝑄 = (LSubSp‘𝐷)
lcfrlem38.c	⊢ 𝐶 = {𝑓 ∈ (LFnl‘𝑈) ∣ ( ⊥ ‘( ⊥ ‘(𝐿‘𝑓))) = (𝐿‘𝑓)}
lcfrlem38.e	⊢ 𝐸 = ∪ 𝑔 ∈ 𝐺 ( ⊥ ‘(𝐿‘𝑔))
lcfrlem38.k	⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
lcfrlem38.g	⊢ (𝜑 → 𝐺 ∈ 𝑄)
lcfrlem38.gs	⊢ (𝜑 → 𝐺 ⊆ 𝐶)
lcfrlem38.xe	⊢ (𝜑 → 𝑋 ∈ 𝐸)
lcfrlem38.ye	⊢ (𝜑 → 𝑌 ∈ 𝐸)

**Assertion**
Ref	Expression
lcfrlem42	⊢ (𝜑 → (𝑋 + 𝑌) ∈ 𝐸)

Distinct variable groups: 𝐷,𝑔 𝑔,𝐺 𝑓,𝑔,𝐿 ⊥ ,𝑓,𝑔 + ,𝑓,𝑔 𝑈,𝑓,𝑔 𝑓,𝑋,𝑔 𝑓,𝑌,𝑔 𝜑,𝑔 Allowed substitution hints: 𝜑(𝑓) 𝐶(𝑓,𝑔) 𝐷(𝑓) 𝑄(𝑓,𝑔) 𝐸(𝑓,𝑔) 𝐹(𝑓,𝑔) 𝐺(𝑓) 𝐻(𝑓,𝑔) 𝐾(𝑓,𝑔) 𝑊(𝑓,𝑔)

**Proof of Theorem lcfrlem42**
Step	Hyp	Ref	Expression
1		lcfrlem38.h	. . . . . 6 ⊢ 𝐻 = (LHyp‘𝐾)
2		lcfrlem38.u	. . . . . 6 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
3		lcfrlem38.k	. . . . . 6 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
4	1, 2, 3	dvhlmod 39124	. . . . 5 ⊢ (𝜑 → 𝑈 ∈ LMod)
5		lcfrlem38.o	. . . . . 6 ⊢ ⊥ = ((ocH‘𝐾)‘𝑊)
6		eqid 2738	. . . . . 6 ⊢ (Base‘𝑈) = (Base‘𝑈)
7		lcfrlem38.l	. . . . . 6 ⊢ 𝐿 = (LKer‘𝑈)
8		lcfrlem38.d	. . . . . 6 ⊢ 𝐷 = (LDual‘𝑈)
9		lcfrlem38.q	. . . . . 6 ⊢ 𝑄 = (LSubSp‘𝐷)
10		lcfrlem38.e	. . . . . 6 ⊢ 𝐸 = ∪ 𝑔 ∈ 𝐺 ( ⊥ ‘(𝐿‘𝑔))
11		lcfrlem38.g	. . . . . 6 ⊢ (𝜑 → 𝐺 ∈ 𝑄)
12		lcfrlem38.xe	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ 𝐸)
13	1, 5, 2, 6, 7, 8, 9, 10, 3, 11, 12	lcfrlem4 39559	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ (Base‘𝑈))
14		lcfrlem38.ye	. . . . . 6 ⊢ (𝜑 → 𝑌 ∈ 𝐸)
15	1, 5, 2, 6, 7, 8, 9, 10, 3, 11, 14	lcfrlem4 39559	. . . . 5 ⊢ (𝜑 → 𝑌 ∈ (Base‘𝑈))
16		lcfrlem38.p	. . . . . 6 ⊢ + = (+_g‘𝑈)
17	6, 16	lmodcom 20169	. . . . 5 ⊢ ((𝑈 ∈ LMod ∧ 𝑋 ∈ (Base‘𝑈) ∧ 𝑌 ∈ (Base‘𝑈)) → (𝑋 + 𝑌) = (𝑌 + 𝑋))
18	4, 13, 15, 17	syl3anc 1370	. . . 4 ⊢ (𝜑 → (𝑋 + 𝑌) = (𝑌 + 𝑋))
19	18	adantr 481	. . 3 ⊢ ((𝜑 ∧ 𝑋 = (0_g‘𝑈)) → (𝑋 + 𝑌) = (𝑌 + 𝑋))
20	3	adantr 481	. . . 4 ⊢ ((𝜑 ∧ 𝑋 = (0_g‘𝑈)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
21	11	adantr 481	. . . 4 ⊢ ((𝜑 ∧ 𝑋 = (0_g‘𝑈)) → 𝐺 ∈ 𝑄)
22	14	adantr 481	. . . 4 ⊢ ((𝜑 ∧ 𝑋 = (0_g‘𝑈)) → 𝑌 ∈ 𝐸)
23		eqid 2738	. . . 4 ⊢ (0_g‘𝑈) = (0_g‘𝑈)
24		simpr 485	. . . 4 ⊢ ((𝜑 ∧ 𝑋 = (0_g‘𝑈)) → 𝑋 = (0_g‘𝑈))
25	1, 5, 2, 16, 7, 8, 9, 20, 21, 10, 22, 23, 24	lcfrlem7 39562	. . 3 ⊢ ((𝜑 ∧ 𝑋 = (0_g‘𝑈)) → (𝑌 + 𝑋) ∈ 𝐸)
26	19, 25	eqeltrd 2839	. 2 ⊢ ((𝜑 ∧ 𝑋 = (0_g‘𝑈)) → (𝑋 + 𝑌) ∈ 𝐸)
27	3	adantr 481	. . 3 ⊢ ((𝜑 ∧ 𝑌 = (0_g‘𝑈)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
28	11	adantr 481	. . 3 ⊢ ((𝜑 ∧ 𝑌 = (0_g‘𝑈)) → 𝐺 ∈ 𝑄)
29	12	adantr 481	. . 3 ⊢ ((𝜑 ∧ 𝑌 = (0_g‘𝑈)) → 𝑋 ∈ 𝐸)
30		simpr 485	. . 3 ⊢ ((𝜑 ∧ 𝑌 = (0_g‘𝑈)) → 𝑌 = (0_g‘𝑈))
31	1, 5, 2, 16, 7, 8, 9, 27, 28, 10, 29, 23, 30	lcfrlem7 39562	. 2 ⊢ ((𝜑 ∧ 𝑌 = (0_g‘𝑈)) → (𝑋 + 𝑌) ∈ 𝐸)
32		lcfrlem38.f	. . 3 ⊢ 𝐹 = (LFnl‘𝑈)
33		lcfrlem38.c	. . 3 ⊢ 𝐶 = {𝑓 ∈ (LFnl‘𝑈) ∣ ( ⊥ ‘( ⊥ ‘(𝐿‘𝑓))) = (𝐿‘𝑓)}
34	3	adantr 481	. . 3 ⊢ ((𝜑 ∧ (𝑋 ≠ (0_g‘𝑈) ∧ 𝑌 ≠ (0_g‘𝑈))) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
35	11	adantr 481	. . 3 ⊢ ((𝜑 ∧ (𝑋 ≠ (0_g‘𝑈) ∧ 𝑌 ≠ (0_g‘𝑈))) → 𝐺 ∈ 𝑄)
36		lcfrlem38.gs	. . . 4 ⊢ (𝜑 → 𝐺 ⊆ 𝐶)
37	36	adantr 481	. . 3 ⊢ ((𝜑 ∧ (𝑋 ≠ (0_g‘𝑈) ∧ 𝑌 ≠ (0_g‘𝑈))) → 𝐺 ⊆ 𝐶)
38	12	adantr 481	. . 3 ⊢ ((𝜑 ∧ (𝑋 ≠ (0_g‘𝑈) ∧ 𝑌 ≠ (0_g‘𝑈))) → 𝑋 ∈ 𝐸)
39	14	adantr 481	. . 3 ⊢ ((𝜑 ∧ (𝑋 ≠ (0_g‘𝑈) ∧ 𝑌 ≠ (0_g‘𝑈))) → 𝑌 ∈ 𝐸)
40		simprl 768	. . 3 ⊢ ((𝜑 ∧ (𝑋 ≠ (0_g‘𝑈) ∧ 𝑌 ≠ (0_g‘𝑈))) → 𝑋 ≠ (0_g‘𝑈))
41		simprr 770	. . 3 ⊢ ((𝜑 ∧ (𝑋 ≠ (0_g‘𝑈) ∧ 𝑌 ≠ (0_g‘𝑈))) → 𝑌 ≠ (0_g‘𝑈))
42	1, 5, 2, 16, 32, 7, 8, 9, 33, 10, 34, 35, 37, 38, 39, 23, 40, 41	lcfrlem41 39597	. 2 ⊢ ((𝜑 ∧ (𝑋 ≠ (0_g‘𝑈) ∧ 𝑌 ≠ (0_g‘𝑈))) → (𝑋 + 𝑌) ∈ 𝐸)
43	26, 31, 42	pm2.61da2ne 3033	1 ⊢ (𝜑 → (𝑋 + 𝑌) ∈ 𝐸)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 396 = wceq 1539 ∈ wcel 2106 ≠ wne 2943 {crab 3068 ⊆ wss 3887 ∪ ciun 4924 ‘cfv 6433 (class class class)co 7275 Basecbs 16912 +_gcplusg 16962 0_gc0g 17150 LModclmod 20123 LSubSpclss 20193 LFnlclfn 37071 LKerclk 37099 LDualcld 37137 HLchlt 37364 LHypclh 37998 DVecHcdvh 39092 ocHcoch 39361

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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5209 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-cnex 10927 ax-resscn 10928 ax-1cn 10929 ax-icn 10930 ax-addcl 10931 ax-addrcl 10932 ax-mulcl 10933 ax-mulrcl 10934 ax-mulcom 10935 ax-addass 10936 ax-mulass 10937 ax-distr 10938 ax-i2m1 10939 ax-1ne0 10940 ax-1rid 10941 ax-rnegex 10942 ax-rrecex 10943 ax-cnre 10944 ax-pre-lttri 10945 ax-pre-lttrn 10946 ax-pre-ltadd 10947 ax-pre-mulgt0 10948 ax-riotaBAD 36967

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-rmo 3071 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4840 df-int 4880 df-iun 4926 df-iin 4927 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6202 df-ord 6269 df-on 6270 df-lim 6271 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-riota 7232 df-ov 7278 df-oprab 7279 df-mpo 7280 df-of 7533 df-om 7713 df-1st 7831 df-2nd 7832 df-tpos 8042 df-undef 8089 df-frecs 8097 df-wrecs 8128 df-recs 8202 df-rdg 8241 df-1o 8297 df-er 8498 df-map 8617 df-en 8734 df-dom 8735 df-sdom 8736 df-fin 8737 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014 df-le 11015 df-sub 11207 df-neg 11208 df-nn 11974 df-2 12036 df-3 12037 df-4 12038 df-5 12039 df-6 12040 df-n0 12234 df-z 12320 df-uz 12583 df-fz 13240 df-struct 16848 df-sets 16865 df-slot 16883 df-ndx 16895 df-base 16913 df-ress 16942 df-plusg 16975 df-mulr 16976 df-sca 16978 df-vsca 16979 df-0g 17152 df-mre 17295 df-mrc 17296 df-acs 17298 df-proset 18013 df-poset 18031 df-plt 18048 df-lub 18064 df-glb 18065 df-join 18066 df-meet 18067 df-p0 18143 df-p1 18144 df-lat 18150 df-clat 18217 df-mgm 18326 df-sgrp 18375 df-mnd 18386 df-submnd 18431 df-grp 18580 df-minusg 18581 df-sbg 18582 df-subg 18752 df-cntz 18923 df-oppg 18950 df-lsm 19241 df-cmn 19388 df-abl 19389 df-mgp 19721 df-ur 19738 df-ring 19785 df-oppr 19862 df-dvdsr 19883 df-unit 19884 df-invr 19914 df-dvr 19925 df-drng 19993 df-lmod 20125 df-lss 20194 df-lsp 20234 df-lvec 20365 df-lsatoms 36990 df-lshyp 36991 df-lcv 37033 df-lfl 37072 df-lkr 37100 df-ldual 37138 df-oposet 37190 df-ol 37192 df-oml 37193 df-covers 37280 df-ats 37281 df-atl 37312 df-cvlat 37336 df-hlat 37365 df-llines 37512 df-lplanes 37513 df-lvols 37514 df-lines 37515 df-psubsp 37517 df-pmap 37518 df-padd 37810 df-lhyp 38002 df-laut 38003 df-ldil 38118 df-ltrn 38119 df-trl 38173 df-tgrp 38757 df-tendo 38769 df-edring 38771 df-dveca 39017 df-disoa 39043 df-dvech 39093 df-dib 39153 df-dic 39187 df-dih 39243 df-doch 39362 df-djh 39409

This theorem is referenced by: lcfr 39599

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