lcfrlem42 - Mathbox for Norm Megill

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Theorem lcfrlem42 41563

Description: Lemma for lcfr 41564. 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 41089	. . . . 5 ⊢ (𝜑 → 𝑈 ∈ LMod)
5		lcfrlem38.o	. . . . . 6 ⊢ ⊥ = ((ocH‘𝐾)‘𝑊)
6		eqid 2729	. . . . . 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 41524	. . . . 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 41524	. . . . 5 ⊢ (𝜑 → 𝑌 ∈ (Base‘𝑈))
16		lcfrlem38.p	. . . . . 6 ⊢ + = (+_g‘𝑈)
17	6, 16	lmodcom 20829	. . . . 5 ⊢ ((𝑈 ∈ LMod ∧ 𝑋 ∈ (Base‘𝑈) ∧ 𝑌 ∈ (Base‘𝑈)) → (𝑋 + 𝑌) = (𝑌 + 𝑋))
18	4, 13, 15, 17	syl3anc 1373	. . . 4 ⊢ (𝜑 → (𝑋 + 𝑌) = (𝑌 + 𝑋))
19	18	adantr 480	. . 3 ⊢ ((𝜑 ∧ 𝑋 = (0_g‘𝑈)) → (𝑋 + 𝑌) = (𝑌 + 𝑋))
20	3	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑋 = (0_g‘𝑈)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
21	11	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑋 = (0_g‘𝑈)) → 𝐺 ∈ 𝑄)
22	14	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑋 = (0_g‘𝑈)) → 𝑌 ∈ 𝐸)
23		eqid 2729	. . . 4 ⊢ (0_g‘𝑈) = (0_g‘𝑈)
24		simpr 484	. . . 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 41527	. . 3 ⊢ ((𝜑 ∧ 𝑋 = (0_g‘𝑈)) → (𝑌 + 𝑋) ∈ 𝐸)
26	19, 25	eqeltrd 2828	. 2 ⊢ ((𝜑 ∧ 𝑋 = (0_g‘𝑈)) → (𝑋 + 𝑌) ∈ 𝐸)
27	3	adantr 480	. . 3 ⊢ ((𝜑 ∧ 𝑌 = (0_g‘𝑈)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
28	11	adantr 480	. . 3 ⊢ ((𝜑 ∧ 𝑌 = (0_g‘𝑈)) → 𝐺 ∈ 𝑄)
29	12	adantr 480	. . 3 ⊢ ((𝜑 ∧ 𝑌 = (0_g‘𝑈)) → 𝑋 ∈ 𝐸)
30		simpr 484	. . 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 41527	. 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 480	. . 3 ⊢ ((𝜑 ∧ (𝑋 ≠ (0_g‘𝑈) ∧ 𝑌 ≠ (0_g‘𝑈))) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
35	11	adantr 480	. . 3 ⊢ ((𝜑 ∧ (𝑋 ≠ (0_g‘𝑈) ∧ 𝑌 ≠ (0_g‘𝑈))) → 𝐺 ∈ 𝑄)
36		lcfrlem38.gs	. . . 4 ⊢ (𝜑 → 𝐺 ⊆ 𝐶)
37	36	adantr 480	. . 3 ⊢ ((𝜑 ∧ (𝑋 ≠ (0_g‘𝑈) ∧ 𝑌 ≠ (0_g‘𝑈))) → 𝐺 ⊆ 𝐶)
38	12	adantr 480	. . 3 ⊢ ((𝜑 ∧ (𝑋 ≠ (0_g‘𝑈) ∧ 𝑌 ≠ (0_g‘𝑈))) → 𝑋 ∈ 𝐸)
39	14	adantr 480	. . 3 ⊢ ((𝜑 ∧ (𝑋 ≠ (0_g‘𝑈) ∧ 𝑌 ≠ (0_g‘𝑈))) → 𝑌 ∈ 𝐸)
40		simprl 770	. . 3 ⊢ ((𝜑 ∧ (𝑋 ≠ (0_g‘𝑈) ∧ 𝑌 ≠ (0_g‘𝑈))) → 𝑋 ≠ (0_g‘𝑈))
41		simprr 772	. . 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 41562	. 2 ⊢ ((𝜑 ∧ (𝑋 ≠ (0_g‘𝑈) ∧ 𝑌 ≠ (0_g‘𝑈))) → (𝑋 + 𝑌) ∈ 𝐸)
43	26, 31, 42	pm2.61da2ne 3013	1 ⊢ (𝜑 → (𝑋 + 𝑌) ∈ 𝐸)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2109 ≠ wne 2925 {crab 3396 ⊆ wss 3905 ∪ ciun 4944 ‘cfv 6486 (class class class)co 7353 Basecbs 17138 +_gcplusg 17179 0_gc0g 17361 LModclmod 20781 LSubSpclss 20852 LFnlclfn 39035 LKerclk 39063 LDualcld 39101 HLchlt 39328 LHypclh 39963 DVecHcdvh 41057 ocHcoch 41326

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 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5221 ax-sep 5238 ax-nul 5248 ax-pow 5307 ax-pr 5374 ax-un 7675 ax-cnex 11084 ax-resscn 11085 ax-1cn 11086 ax-icn 11087 ax-addcl 11088 ax-addrcl 11089 ax-mulcl 11090 ax-mulrcl 11091 ax-mulcom 11092 ax-addass 11093 ax-mulass 11094 ax-distr 11095 ax-i2m1 11096 ax-1ne0 11097 ax-1rid 11098 ax-rnegex 11099 ax-rrecex 11100 ax-cnre 11101 ax-pre-lttri 11102 ax-pre-lttrn 11103 ax-pre-ltadd 11104 ax-pre-mulgt0 11105 ax-riotaBAD 38931

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 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3345 df-reu 3346 df-rab 3397 df-v 3440 df-sbc 3745 df-csb 3854 df-dif 3908 df-un 3910 df-in 3912 df-ss 3922 df-pss 3925 df-nul 4287 df-if 4479 df-pw 4555 df-sn 4580 df-pr 4582 df-tp 4584 df-op 4586 df-uni 4862 df-int 4900 df-iun 4946 df-iin 4947 df-br 5096 df-opab 5158 df-mpt 5177 df-tr 5203 df-id 5518 df-eprel 5523 df-po 5531 df-so 5532 df-fr 5576 df-we 5578 df-xp 5629 df-rel 5630 df-cnv 5631 df-co 5632 df-dm 5633 df-rn 5634 df-res 5635 df-ima 5636 df-pred 6253 df-ord 6314 df-on 6315 df-lim 6316 df-suc 6317 df-iota 6442 df-fun 6488 df-fn 6489 df-f 6490 df-f1 6491 df-fo 6492 df-f1o 6493 df-fv 6494 df-riota 7310 df-ov 7356 df-oprab 7357 df-mpo 7358 df-of 7617 df-om 7807 df-1st 7931 df-2nd 7932 df-tpos 8166 df-undef 8213 df-frecs 8221 df-wrecs 8252 df-recs 8301 df-rdg 8339 df-1o 8395 df-2o 8396 df-er 8632 df-map 8762 df-en 8880 df-dom 8881 df-sdom 8882 df-fin 8883 df-pnf 11170 df-mnf 11171 df-xr 11172 df-ltxr 11173 df-le 11174 df-sub 11367 df-neg 11368 df-nn 12147 df-2 12209 df-3 12210 df-4 12211 df-5 12212 df-6 12213 df-n0 12403 df-z 12490 df-uz 12754 df-fz 13429 df-struct 17076 df-sets 17093 df-slot 17111 df-ndx 17123 df-base 17139 df-ress 17160 df-plusg 17192 df-mulr 17193 df-sca 17195 df-vsca 17196 df-0g 17363 df-mre 17506 df-mrc 17507 df-acs 17509 df-proset 18218 df-poset 18237 df-plt 18252 df-lub 18268 df-glb 18269 df-join 18270 df-meet 18271 df-p0 18347 df-p1 18348 df-lat 18356 df-clat 18423 df-mgm 18532 df-sgrp 18611 df-mnd 18627 df-submnd 18676 df-grp 18833 df-minusg 18834 df-sbg 18835 df-subg 19020 df-cntz 19214 df-oppg 19243 df-lsm 19533 df-cmn 19679 df-abl 19680 df-mgp 20044 df-rng 20056 df-ur 20085 df-ring 20138 df-oppr 20240 df-dvdsr 20260 df-unit 20261 df-invr 20291 df-dvr 20304 df-nzr 20416 df-rlreg 20597 df-domn 20598 df-drng 20634 df-lmod 20783 df-lss 20853 df-lsp 20893 df-lvec 21025 df-lsatoms 38954 df-lshyp 38955 df-lcv 38997 df-lfl 39036 df-lkr 39064 df-ldual 39102 df-oposet 39154 df-ol 39156 df-oml 39157 df-covers 39244 df-ats 39245 df-atl 39276 df-cvlat 39300 df-hlat 39329 df-llines 39477 df-lplanes 39478 df-lvols 39479 df-lines 39480 df-psubsp 39482 df-pmap 39483 df-padd 39775 df-lhyp 39967 df-laut 39968 df-ldil 40083 df-ltrn 40084 df-trl 40138 df-tgrp 40722 df-tendo 40734 df-edring 40736 df-dveca 40982 df-disoa 41008 df-dvech 41058 df-dib 41118 df-dic 41152 df-dih 41208 df-doch 41327 df-djh 41374

This theorem is referenced by: lcfr 41564

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