lcfrlem40 - Mathbox for Norm Megill

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Theorem lcfrlem40 42397

Description: Lemma for lcfr 42400. Eliminate 𝐵 and 𝐼. (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 )
lcfrlem38.sp	⊢ 𝑁 = (LSpan‘𝑈)
lcfrlem38.ne	⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}))

**Assertion**
Ref	Expression
lcfrlem40	⊢ (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 𝑔,𝑁 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) 𝑁(𝑓) 𝑊(𝑓, 𝑔)

Proof of Theorem lcfrlem40 Dummy variable 𝑖 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		lcfrlem38.z	. . 3 ⊢ 0 = (0_g‘𝑈)
2		eqid 2766	. . 3 ⊢ (LSAtoms‘𝑈) = (LSAtoms‘𝑈)
3		lcfrlem38.h	. . . 4 ⊢ 𝐻 = (LHyp‘𝐾)
4		lcfrlem38.u	. . . 4 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
5		lcfrlem38.k	. . . 4 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
6	3, 4, 5	dvhlmod 41925	. . 3 ⊢ (𝜑 → 𝑈 ∈ LMod)
7		lcfrlem38.o	. . . 4 ⊢ ⊥ = ((ocH‘𝐾)‘𝑊)
8		eqid 2766	. . . 4 ⊢ (Base‘𝑈) = (Base‘𝑈)
9		lcfrlem38.p	. . . 4 ⊢ + = (+_g‘𝑈)
10		lcfrlem38.sp	. . . 4 ⊢ 𝑁 = (LSpan‘𝑈)
11		lcfrlem38.l	. . . . . 6 ⊢ 𝐿 = (LKer‘𝑈)
12		lcfrlem38.d	. . . . . 6 ⊢ 𝐷 = (LDual‘𝑈)
13		lcfrlem38.q	. . . . . 6 ⊢ 𝑄 = (LSubSp‘𝐷)
14		lcfrlem38.e	. . . . . 6 ⊢ 𝐸 = ∪ 𝑔 ∈ 𝐺 ( ⊥ ‘(𝐿‘𝑔))
15		lcfrlem38.g	. . . . . 6 ⊢ (𝜑 → 𝐺 ∈ 𝑄)
16		lcfrlem38.xe	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ 𝐸)
17	3, 7, 4, 8, 11, 12, 13, 14, 5, 15, 16	lcfrlem4 42360	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ (Base‘𝑈))
18		lcfrlem38.x	. . . . 5 ⊢ (𝜑 → 𝑋 ≠ 0 )
19		eldifsn 4758	. . . . 5 ⊢ (𝑋 ∈ ((Base‘𝑈) ∖ { 0 }) ↔ (𝑋 ∈ (Base‘𝑈) ∧ 𝑋 ≠ 0 ))
20	17, 18, 19	sylanbrc 595	. . . 4 ⊢ (𝜑 → 𝑋 ∈ ((Base‘𝑈) ∖ { 0 }))
21		lcfrlem38.ye	. . . . . 6 ⊢ (𝜑 → 𝑌 ∈ 𝐸)
22	3, 7, 4, 8, 11, 12, 13, 14, 5, 15, 21	lcfrlem4 42360	. . . . 5 ⊢ (𝜑 → 𝑌 ∈ (Base‘𝑈))
23		lcfrlem38.y	. . . . 5 ⊢ (𝜑 → 𝑌 ≠ 0 )
24		eldifsn 4758	. . . . 5 ⊢ (𝑌 ∈ ((Base‘𝑈) ∖ { 0 }) ↔ (𝑌 ∈ (Base‘𝑈) ∧ 𝑌 ≠ 0 ))
25	22, 23, 24	sylanbrc 595	. . . 4 ⊢ (𝜑 → 𝑌 ∈ ((Base‘𝑈) ∖ { 0 }))
26		lcfrlem38.ne	. . . 4 ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}))
27	3, 7, 4, 8, 9, 1, 10, 2, 5, 20, 25, 26	lcfrlem21 42378	. . 3 ⊢ (𝜑 → ((𝑁‘{𝑋, 𝑌}) ∩ ( ⊥ ‘{(𝑋 + 𝑌)})) ∈ (LSAtoms‘𝑈))
28	1, 2, 6, 27	lsateln0 39810	. 2 ⊢ (𝜑 → ∃𝑖 ∈ ((𝑁‘{𝑋, 𝑌}) ∩ ( ⊥ ‘{(𝑋 + 𝑌)}))𝑖 ≠ 0 )
29		lcfrlem38.f	. . . 4 ⊢ 𝐹 = (LFnl‘𝑈)
30		lcfrlem38.c	. . . 4 ⊢ 𝐶 = {𝑓 ∈ (LFnl‘𝑈) ∣ ( ⊥ ‘( ⊥ ‘(𝐿‘𝑓))) = (𝐿‘𝑓)}
31	5	3ad2ant1 1151	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ ((𝑁‘{𝑋, 𝑌}) ∩ ( ⊥ ‘{(𝑋 + 𝑌)})) ∧ 𝑖 ≠ 0 ) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
32	15	3ad2ant1 1151	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ ((𝑁‘{𝑋, 𝑌}) ∩ ( ⊥ ‘{(𝑋 + 𝑌)})) ∧ 𝑖 ≠ 0 ) → 𝐺 ∈ 𝑄)
33		lcfrlem38.gs	. . . . 5 ⊢ (𝜑 → 𝐺 ⊆ 𝐶)
34	33	3ad2ant1 1151	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ ((𝑁‘{𝑋, 𝑌}) ∩ ( ⊥ ‘{(𝑋 + 𝑌)})) ∧ 𝑖 ≠ 0 ) → 𝐺 ⊆ 𝐶)
35	16	3ad2ant1 1151	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ ((𝑁‘{𝑋, 𝑌}) ∩ ( ⊥ ‘{(𝑋 + 𝑌)})) ∧ 𝑖 ≠ 0 ) → 𝑋 ∈ 𝐸)
36	21	3ad2ant1 1151	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ ((𝑁‘{𝑋, 𝑌}) ∩ ( ⊥ ‘{(𝑋 + 𝑌)})) ∧ 𝑖 ≠ 0 ) → 𝑌 ∈ 𝐸)
37	18	3ad2ant1 1151	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ ((𝑁‘{𝑋, 𝑌}) ∩ ( ⊥ ‘{(𝑋 + 𝑌)})) ∧ 𝑖 ≠ 0 ) → 𝑋 ≠ 0 )
38	23	3ad2ant1 1151	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ ((𝑁‘{𝑋, 𝑌}) ∩ ( ⊥ ‘{(𝑋 + 𝑌)})) ∧ 𝑖 ≠ 0 ) → 𝑌 ≠ 0 )
39	26	3ad2ant1 1151	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ ((𝑁‘{𝑋, 𝑌}) ∩ ( ⊥ ‘{(𝑋 + 𝑌)})) ∧ 𝑖 ≠ 0 ) → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}))
40		eqid 2766	. . . 4 ⊢ ((𝑁‘{𝑋, 𝑌}) ∩ ( ⊥ ‘{(𝑋 + 𝑌)})) = ((𝑁‘{𝑋, 𝑌}) ∩ ( ⊥ ‘{(𝑋 + 𝑌)}))
41		simp2 1155	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ ((𝑁‘{𝑋, 𝑌}) ∩ ( ⊥ ‘{(𝑋 + 𝑌)})) ∧ 𝑖 ≠ 0 ) → 𝑖 ∈ ((𝑁‘{𝑋, 𝑌}) ∩ ( ⊥ ‘{(𝑋 + 𝑌)})))
42		simp3 1156	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ ((𝑁‘{𝑋, 𝑌}) ∩ ( ⊥ ‘{(𝑋 + 𝑌)})) ∧ 𝑖 ≠ 0 ) → 𝑖 ≠ 0 )
43	3, 7, 4, 9, 29, 11, 12, 13, 30, 14, 31, 32, 34, 35, 36, 1, 37, 38, 10, 39, 40, 41, 42	lcfrlem39 42396	. . 3 ⊢ ((𝜑 ∧ 𝑖 ∈ ((𝑁‘{𝑋, 𝑌}) ∩ ( ⊥ ‘{(𝑋 + 𝑌)})) ∧ 𝑖 ≠ 0 ) → (𝑋 + 𝑌) ∈ 𝐸)
44	43	rexlimdv3a 3173	. 2 ⊢ (𝜑 → (∃𝑖 ∈ ((𝑁‘{𝑋, 𝑌}) ∩ ( ⊥ ‘{(𝑋 + 𝑌)}))𝑖 ≠ 0 → (𝑋 + 𝑌) ∈ 𝐸))
45	28, 44	mpd 16	1 ⊢ (𝜑 → (𝑋 + 𝑌) ∈ 𝐸)

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∧ wa 401 ∧ w3a 1103 = wceq 1570 ∈ wcel 2146 ≠ wne 2961 ∃wrex 3092 {crab 3419 ∖ cdif 3905 ∩ cin 3907 ⊆ wss 3908 {csn 4594 {cpr 4596 ∪ ciun 4961 ‘cfv 6543 (class class class)co 7423 Basecbs 17294 +_gcplusg 17335 0_gc0g 17517 LSubSpclss 21089 LSpanclspn 21129 LSAtomsclsa 39789 LFnlclfn 39872 LKerclk 39900 LDualcld 39938 HLchlt 40165 LHypclh 40799 DVecHcdvh 41893 ocHcoch 42162

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2148 ax-9 2156 ax-10 2179 ax-11 2195 ax-12 2216 ax-ext 2738 ax-rep 5243 ax-sep 5262 ax-nul 5274 ax-pow 5341 ax-pr 5409 ax-un 7745 ax-cnex 11174 ax-resscn 11175 ax-1cn 11176 ax-icn 11177 ax-addcl 11178 ax-addrcl 11179 ax-mulcl 11180 ax-mulrcl 11181 ax-mulcom 11182 ax-addass 11183 ax-mulass 11184 ax-distr 11185 ax-i2m1 11186 ax-1ne0 11187 ax-1rid 11188 ax-rnegex 11189 ax-rrecex 11190 ax-cnre 11191 ax-pre-lttri 11192 ax-pre-lttrn 11193 ax-pre-ltadd 11194 ax-pre-mulgt0 11195 ax-riotaBAD 39768

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2570 df-eu 2600 df-clab 2745 df-cleq 2758 df-clel 2841 df-nfc 2915 df-ne 2962 df-nel 3068 df-ral 3083 df-rex 3093 df-rmo 3372 df-reu 3373 df-rab 3420 df-v 3460 df-sbc 3748 df-csb 3857 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-pss 3928 df-nul 4290 df-if 4493 df-pw 4569 df-sn 4595 df-pr 4597 df-tp 4599 df-op 4601 df-uni 4878 df-int 4918 df-iun 4963 df-iin 4964 df-br 5115 df-opab 5179 df-mpt 5198 df-tr 5224 df-id 5561 df-eprel 5566 df-po 5574 df-so 5575 df-fr 5619 df-we 5621 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-pred 6309 df-ord 6370 df-on 6371 df-lim 6372 df-suc 6373 df-iota 6499 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7380 df-ov 7426 df-oprab 7427 df-mpo 7428 df-of 7687 df-om 7872 df-1st 7995 df-2nd 7996 df-tpos 8231 df-undef 8278 df-frecs 8287 df-wrecs 8318 df-recs 8367 df-rdg 8406 df-1o 8462 df-2o 8463 df-er 8703 df-map 8835 df-en 8953 df-dom 8954 df-sdom 8955 df-fin 8956 df-pnf 11263 df-mnf 11264 df-xr 11265 df-ltxr 11266 df-le 11267 df-sub 11461 df-neg 11462 df-nn 12252 df-2 12321 df-3 12322 df-4 12323 df-5 12324 df-6 12325 df-n0 12523 df-z 12610 df-uz 12881 df-fz 13554 df-struct 17232 df-sets 17249 df-slot 17267 df-ndx 17279 df-base 17295 df-ress 17316 df-plusg 17348 df-mulr 17349 df-sca 17351 df-vsca 17352 df-0g 17519 df-mre 17663 df-mrc 17664 df-acs 17666 df-proset 18375 df-poset 18394 df-plt 18409 df-lub 18425 df-glb 18426 df-join 18427 df-meet 18428 df-p0 18504 df-p1 18505 df-lat 18513 df-clat 18580 df-mgm 18723 df-sgrp 18806 df-mnd 18822 df-submnd 18873 df-grp 19034 df-minusg 19035 df-sbg 19036 df-subg 19220 df-cntz 19418 df-oppg 19447 df-lsm 19737 df-cmn 19883 df-abl 19884 df-mgp 20248 df-rng 20262 df-ur 20295 df-ring 20348 df-oppr 20452 df-dvdsr 20472 df-unit 20473 df-invr 20503 df-dvr 20516 df-nzr 20647 df-rlreg 20830 df-domn 20831 df-drng 20866 df-lmod 21020 df-lss 21090 df-lsp 21130 df-lvec 21261 df-lsatoms 39791 df-lshyp 39792 df-lcv 39834 df-lfl 39873 df-lkr 39901 df-ldual 39939 df-oposet 39991 df-ol 39993 df-oml 39994 df-covers 40081 df-ats 40082 df-atl 40113 df-cvlat 40137 df-hlat 40166 df-llines 40313 df-lplanes 40314 df-lvols 40315 df-lines 40316 df-psubsp 40318 df-pmap 40319 df-padd 40611 df-lhyp 40803 df-laut 40804 df-ldil 40919 df-ltrn 40920 df-trl 40974 df-tgrp 41558 df-tendo 41570 df-edring 41572 df-dveca 41818 df-disoa 41844 df-dvech 41894 df-dib 41954 df-dic 41988 df-dih 42044 df-doch 42163 df-djh 42210

This theorem is used by: lcfrlem41 42398

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