lcfrlem42 - Mathbox for Norm Megill

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Theorem lcfrlem42 42083

Description: Lemma for lcfr 42084. 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 41609	. . . . 5 ⊢ (𝜑 → 𝑈 ∈ LMod)
5		lcfrlem38.o	. . . . . 6 ⊢ ⊥ = ((ocH‘𝐾)‘𝑊)
6		eqid 2740	. . . . . 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 42044	. . . . 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 42044	. . . . 5 ⊢ (𝜑 → 𝑌 ∈ (Base‘𝑈))
16		lcfrlem38.p	. . . . . 6 ⊢ + = (+_g‘𝑈)
17	6, 16	lmodcom 20905	. . . . 5 ⊢ ((𝑈 ∈ LMod ∧ 𝑋 ∈ (Base‘𝑈) ∧ 𝑌 ∈ (Base‘𝑈)) → (𝑋 + 𝑌) = (𝑌 + 𝑋))
18	4, 13, 15, 17	syl3anc 1379	. . . 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 2740	. . . 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 42047	. . 3 ⊢ ((𝜑 ∧ 𝑋 = (0_g‘𝑈)) → (𝑌 + 𝑋) ∈ 𝐸)
26	19, 25	eqeltrd 2840	. 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 42047	. 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 776	. . 3 ⊢ ((𝜑 ∧ (𝑋 ≠ (0_g‘𝑈) ∧ 𝑌 ≠ (0_g‘𝑈))) → 𝑋 ≠ (0_g‘𝑈))
41		simprr 778	. . 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 42082	. 2 ⊢ ((𝜑 ∧ (𝑋 ≠ (0_g‘𝑈) ∧ 𝑌 ≠ (0_g‘𝑈))) → (𝑋 + 𝑌) ∈ 𝐸)
43	26, 31, 42	pm2.61da2ne 3023	1 ⊢ (𝜑 → (𝑋 + 𝑌) ∈ 𝐸)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 396 = wceq 1547 ∈ wcel 2119 ≠ wne 2935 {crab 3392 ⊆ wss 3890 ∪ ciun 4928 ‘cfv 6492 (class class class)co 7363 Basecbs 17177 +_gcplusg 17218 0_gc0g 17400 LModclmod 20857 LSubSpclss 20928 LFnlclfn 39556 LKerclk 39584 LDualcld 39622 HLchlt 39849 LHypclh 40483 DVecHcdvh 41577 ocHcoch 41846

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1974 ax-7 2015 ax-8 2121 ax-9 2129 ax-10 2152 ax-11 2168 ax-12 2189 ax-ext 2712 ax-rep 5206 ax-sep 5225 ax-nul 5235 ax-pow 5301 ax-pr 5369 ax-un 7685 ax-cnex 11092 ax-resscn 11093 ax-1cn 11094 ax-icn 11095 ax-addcl 11096 ax-addrcl 11097 ax-mulcl 11098 ax-mulrcl 11099 ax-mulcom 11100 ax-addass 11101 ax-mulass 11102 ax-distr 11103 ax-i2m1 11104 ax-1ne0 11105 ax-1rid 11106 ax-rnegex 11107 ax-rrecex 11108 ax-cnre 11109 ax-pre-lttri 11110 ax-pre-lttrn 11111 ax-pre-ltadd 11112 ax-pre-mulgt0 11113 ax-riotaBAD 39452

This theorem depends on definitions: df-bi 208 df-an 397 df-or 854 df-3or 1093 df-3an 1094 df-tru 1550 df-fal 1560 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2543 df-eu 2573 df-clab 2719 df-cleq 2732 df-clel 2815 df-nfc 2889 df-ne 2936 df-nel 3040 df-ral 3055 df-rex 3065 df-rmo 3345 df-reu 3346 df-rab 3393 df-v 3434 df-sbc 3731 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-pss 3910 df-nul 4269 df-if 4462 df-pw 4538 df-sn 4563 df-pr 4565 df-tp 4567 df-op 4569 df-uni 4846 df-int 4885 df-iun 4930 df-iin 4931 df-br 5080 df-opab 5142 df-mpt 5161 df-tr 5187 df-id 5520 df-eprel 5525 df-po 5533 df-so 5534 df-fr 5578 df-we 5580 df-xp 5631 df-rel 5632 df-cnv 5633 df-co 5634 df-dm 5635 df-rn 5636 df-res 5637 df-ima 5638 df-pred 6259 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-riota 7320 df-ov 7366 df-oprab 7367 df-mpo 7368 df-of 7627 df-om 7814 df-1st 7938 df-2nd 7939 df-tpos 8173 df-undef 8220 df-frecs 8228 df-wrecs 8259 df-recs 8308 df-rdg 8346 df-1o 8402 df-2o 8403 df-er 8640 df-map 8772 df-en 8891 df-dom 8892 df-sdom 8893 df-fin 8894 df-pnf 11179 df-mnf 11180 df-xr 11181 df-ltxr 11182 df-le 11183 df-sub 11377 df-neg 11378 df-nn 12173 df-2 12242 df-3 12243 df-4 12244 df-5 12245 df-6 12246 df-n0 12436 df-z 12523 df-uz 12787 df-fz 13460 df-struct 17115 df-sets 17132 df-slot 17150 df-ndx 17162 df-base 17178 df-ress 17199 df-plusg 17231 df-mulr 17232 df-sca 17234 df-vsca 17235 df-0g 17402 df-mre 17546 df-mrc 17547 df-acs 17549 df-proset 18258 df-poset 18277 df-plt 18292 df-lub 18308 df-glb 18309 df-join 18310 df-meet 18311 df-p0 18387 df-p1 18388 df-lat 18396 df-clat 18463 df-mgm 18606 df-sgrp 18685 df-mnd 18701 df-submnd 18750 df-grp 18910 df-minusg 18911 df-sbg 18912 df-subg 19097 df-cntz 19290 df-oppg 19319 df-lsm 19609 df-cmn 19755 df-abl 19756 df-mgp 20120 df-rng 20132 df-ur 20161 df-ring 20214 df-oppr 20315 df-dvdsr 20335 df-unit 20336 df-invr 20366 df-dvr 20379 df-nzr 20492 df-rlreg 20673 df-domn 20674 df-drng 20710 df-lmod 20859 df-lss 20929 df-lsp 20969 df-lvec 21100 df-lsatoms 39475 df-lshyp 39476 df-lcv 39518 df-lfl 39557 df-lkr 39585 df-ldual 39623 df-oposet 39675 df-ol 39677 df-oml 39678 df-covers 39765 df-ats 39766 df-atl 39797 df-cvlat 39821 df-hlat 39850 df-llines 39997 df-lplanes 39998 df-lvols 39999 df-lines 40000 df-psubsp 40002 df-pmap 40003 df-padd 40295 df-lhyp 40487 df-laut 40488 df-ldil 40603 df-ltrn 40604 df-trl 40658 df-tgrp 41242 df-tendo 41254 df-edring 41256 df-dveca 41502 df-disoa 41528 df-dvech 41578 df-dib 41638 df-dic 41672 df-dih 41728 df-doch 41847 df-djh 41894

This theorem is referenced by: lcfr 42084

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