lcfrlem41 - Mathbox for Norm Megill

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Theorem lcfrlem41 41050

Description: Lemma for lcfr 41052. Eliminate span 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)
lcfrlem38.z	⊢ 0 = (0_g‘𝑈)
lcfrlem38.x	⊢ (𝜑 → 𝑋 ≠ 0 )
lcfrlem38.y	⊢ (𝜑 → 𝑌 ≠ 0 )

**Assertion**
Ref	Expression
lcfrlem41	⊢ (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 0 ,𝑓,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 lcfrlem41**
Step	Hyp	Ref	Expression
1		lcfrlem38.h	. . 3 ⊢ 𝐻 = (LHyp‘𝐾)
2		lcfrlem38.o	. . 3 ⊢ ⊥ = ((ocH‘𝐾)‘𝑊)
3		lcfrlem38.u	. . 3 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
4		lcfrlem38.p	. . 3 ⊢ + = (+_g‘𝑈)
5		eqid 2728	. . 3 ⊢ (LSpan‘𝑈) = (LSpan‘𝑈)
6		lcfrlem38.l	. . 3 ⊢ 𝐿 = (LKer‘𝑈)
7		lcfrlem38.d	. . 3 ⊢ 𝐷 = (LDual‘𝑈)
8		lcfrlem38.q	. . 3 ⊢ 𝑄 = (LSubSp‘𝐷)
9		lcfrlem38.k	. . . 4 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
10	9	adantr 480	. . 3 ⊢ ((𝜑 ∧ ((LSpan‘𝑈)‘{𝑋}) = ((LSpan‘𝑈)‘{𝑌})) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
11		lcfrlem38.g	. . . 4 ⊢ (𝜑 → 𝐺 ∈ 𝑄)
12	11	adantr 480	. . 3 ⊢ ((𝜑 ∧ ((LSpan‘𝑈)‘{𝑋}) = ((LSpan‘𝑈)‘{𝑌})) → 𝐺 ∈ 𝑄)
13		lcfrlem38.e	. . 3 ⊢ 𝐸 = ∪ 𝑔 ∈ 𝐺 ( ⊥ ‘(𝐿‘𝑔))
14		lcfrlem38.xe	. . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝐸)
15	14	adantr 480	. . 3 ⊢ ((𝜑 ∧ ((LSpan‘𝑈)‘{𝑋}) = ((LSpan‘𝑈)‘{𝑌})) → 𝑋 ∈ 𝐸)
16		lcfrlem38.ye	. . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝐸)
17	16	adantr 480	. . 3 ⊢ ((𝜑 ∧ ((LSpan‘𝑈)‘{𝑋}) = ((LSpan‘𝑈)‘{𝑌})) → 𝑌 ∈ 𝐸)
18		simpr 484	. . 3 ⊢ ((𝜑 ∧ ((LSpan‘𝑈)‘{𝑋}) = ((LSpan‘𝑈)‘{𝑌})) → ((LSpan‘𝑈)‘{𝑋}) = ((LSpan‘𝑈)‘{𝑌}))
19	1, 2, 3, 4, 5, 6, 7, 8, 10, 12, 13, 15, 17, 18	lcfrlem6 41014	. 2 ⊢ ((𝜑 ∧ ((LSpan‘𝑈)‘{𝑋}) = ((LSpan‘𝑈)‘{𝑌})) → (𝑋 + 𝑌) ∈ 𝐸)
20		lcfrlem38.f	. . 3 ⊢ 𝐹 = (LFnl‘𝑈)
21		lcfrlem38.c	. . 3 ⊢ 𝐶 = {𝑓 ∈ (LFnl‘𝑈) ∣ ( ⊥ ‘( ⊥ ‘(𝐿‘𝑓))) = (𝐿‘𝑓)}
22	9	adantr 480	. . 3 ⊢ ((𝜑 ∧ ((LSpan‘𝑈)‘{𝑋}) ≠ ((LSpan‘𝑈)‘{𝑌})) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
23	11	adantr 480	. . 3 ⊢ ((𝜑 ∧ ((LSpan‘𝑈)‘{𝑋}) ≠ ((LSpan‘𝑈)‘{𝑌})) → 𝐺 ∈ 𝑄)
24		lcfrlem38.gs	. . . 4 ⊢ (𝜑 → 𝐺 ⊆ 𝐶)
25	24	adantr 480	. . 3 ⊢ ((𝜑 ∧ ((LSpan‘𝑈)‘{𝑋}) ≠ ((LSpan‘𝑈)‘{𝑌})) → 𝐺 ⊆ 𝐶)
26	14	adantr 480	. . 3 ⊢ ((𝜑 ∧ ((LSpan‘𝑈)‘{𝑋}) ≠ ((LSpan‘𝑈)‘{𝑌})) → 𝑋 ∈ 𝐸)
27	16	adantr 480	. . 3 ⊢ ((𝜑 ∧ ((LSpan‘𝑈)‘{𝑋}) ≠ ((LSpan‘𝑈)‘{𝑌})) → 𝑌 ∈ 𝐸)
28		lcfrlem38.z	. . 3 ⊢ 0 = (0_g‘𝑈)
29		lcfrlem38.x	. . . 4 ⊢ (𝜑 → 𝑋 ≠ 0 )
30	29	adantr 480	. . 3 ⊢ ((𝜑 ∧ ((LSpan‘𝑈)‘{𝑋}) ≠ ((LSpan‘𝑈)‘{𝑌})) → 𝑋 ≠ 0 )
31		lcfrlem38.y	. . . 4 ⊢ (𝜑 → 𝑌 ≠ 0 )
32	31	adantr 480	. . 3 ⊢ ((𝜑 ∧ ((LSpan‘𝑈)‘{𝑋}) ≠ ((LSpan‘𝑈)‘{𝑌})) → 𝑌 ≠ 0 )
33		simpr 484	. . 3 ⊢ ((𝜑 ∧ ((LSpan‘𝑈)‘{𝑋}) ≠ ((LSpan‘𝑈)‘{𝑌})) → ((LSpan‘𝑈)‘{𝑋}) ≠ ((LSpan‘𝑈)‘{𝑌}))
34	1, 2, 3, 4, 20, 6, 7, 8, 21, 13, 22, 23, 25, 26, 27, 28, 30, 32, 5, 33	lcfrlem40 41049	. 2 ⊢ ((𝜑 ∧ ((LSpan‘𝑈)‘{𝑋}) ≠ ((LSpan‘𝑈)‘{𝑌})) → (𝑋 + 𝑌) ∈ 𝐸)
35	19, 34	pm2.61dane 3025	1 ⊢ (𝜑 → (𝑋 + 𝑌) ∈ 𝐸)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1534 ∈ wcel 2099 ≠ wne 2936 {crab 3428 ⊆ wss 3945 {csn 4624 ∪ ciun 4991 ‘cfv 6542 (class class class)co 7414 +_gcplusg 17226 0_gc0g 17414 LSubSpclss 20808 LSpanclspn 20848 LFnlclfn 38523 LKerclk 38551 LDualcld 38589 HLchlt 38816 LHypclh 39451 DVecHcdvh 40545 ocHcoch 40814

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 ax-cnex 11188 ax-resscn 11189 ax-1cn 11190 ax-icn 11191 ax-addcl 11192 ax-addrcl 11193 ax-mulcl 11194 ax-mulrcl 11195 ax-mulcom 11196 ax-addass 11197 ax-mulass 11198 ax-distr 11199 ax-i2m1 11200 ax-1ne0 11201 ax-1rid 11202 ax-rnegex 11203 ax-rrecex 11204 ax-cnre 11205 ax-pre-lttri 11206 ax-pre-lttrn 11207 ax-pre-ltadd 11208 ax-pre-mulgt0 11209 ax-riotaBAD 38419

This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2937 df-nel 3043 df-ral 3058 df-rex 3067 df-rmo 3372 df-reu 3373 df-rab 3429 df-v 3472 df-sbc 3776 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-pss 3964 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-tp 4629 df-op 4631 df-uni 4904 df-int 4945 df-iun 4993 df-iin 4994 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-of 7679 df-om 7865 df-1st 7987 df-2nd 7988 df-tpos 8225 df-undef 8272 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-1o 8480 df-er 8718 df-map 8840 df-en 8958 df-dom 8959 df-sdom 8960 df-fin 8961 df-pnf 11274 df-mnf 11275 df-xr 11276 df-ltxr 11277 df-le 11278 df-sub 11470 df-neg 11471 df-nn 12237 df-2 12299 df-3 12300 df-4 12301 df-5 12302 df-6 12303 df-n0 12497 df-z 12583 df-uz 12847 df-fz 13511 df-struct 17109 df-sets 17126 df-slot 17144 df-ndx 17156 df-base 17174 df-ress 17203 df-plusg 17239 df-mulr 17240 df-sca 17242 df-vsca 17243 df-0g 17416 df-mre 17559 df-mrc 17560 df-acs 17562 df-proset 18280 df-poset 18298 df-plt 18315 df-lub 18331 df-glb 18332 df-join 18333 df-meet 18334 df-p0 18410 df-p1 18411 df-lat 18417 df-clat 18484 df-mgm 18593 df-sgrp 18672 df-mnd 18688 df-submnd 18734 df-grp 18886 df-minusg 18887 df-sbg 18888 df-subg 19071 df-cntz 19261 df-oppg 19290 df-lsm 19584 df-cmn 19730 df-abl 19731 df-mgp 20068 df-rng 20086 df-ur 20115 df-ring 20168 df-oppr 20266 df-dvdsr 20289 df-unit 20290 df-invr 20320 df-dvr 20333 df-drng 20619 df-lmod 20738 df-lss 20809 df-lsp 20849 df-lvec 20981 df-lsatoms 38442 df-lshyp 38443 df-lcv 38485 df-lfl 38524 df-lkr 38552 df-ldual 38590 df-oposet 38642 df-ol 38644 df-oml 38645 df-covers 38732 df-ats 38733 df-atl 38764 df-cvlat 38788 df-hlat 38817 df-llines 38965 df-lplanes 38966 df-lvols 38967 df-lines 38968 df-psubsp 38970 df-pmap 38971 df-padd 39263 df-lhyp 39455 df-laut 39456 df-ldil 39571 df-ltrn 39572 df-trl 39626 df-tgrp 40210 df-tendo 40222 df-edring 40224 df-dveca 40470 df-disoa 40496 df-dvech 40546 df-dib 40606 df-dic 40640 df-dih 40696 df-doch 40815 df-djh 40862

This theorem is referenced by: lcfrlem42 41051

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