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Mirrors > Home > MPE Home > Th. List > Mathboxes > lcfrlem40 | Structured version Visualization version GIF version |
Description: Lemma for lcfr 41190. Eliminate 𝐵 and 𝐼. (Contributed by NM, 11-Mar-2015.) |
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 = (0g‘𝑈) |
lcfrlem38.x | ⊢ (𝜑 → 𝑋 ≠ 0 ) |
lcfrlem38.y | ⊢ (𝜑 → 𝑌 ≠ 0 ) |
lcfrlem38.sp | ⊢ 𝑁 = (LSpan‘𝑈) |
lcfrlem38.ne | ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌})) |
Ref | Expression |
---|---|
lcfrlem40 | ⊢ (𝜑 → (𝑋 + 𝑌) ∈ 𝐸) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lcfrlem38.z | . . 3 ⊢ 0 = (0g‘𝑈) | |
2 | eqid 2725 | . . 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 40715 | . . 3 ⊢ (𝜑 → 𝑈 ∈ LMod) |
7 | lcfrlem38.o | . . . 4 ⊢ ⊥ = ((ocH‘𝐾)‘𝑊) | |
8 | eqid 2725 | . . . 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 41150 | . . . . 5 ⊢ (𝜑 → 𝑋 ∈ (Base‘𝑈)) |
18 | lcfrlem38.x | . . . . 5 ⊢ (𝜑 → 𝑋 ≠ 0 ) | |
19 | eldifsn 4792 | . . . . 5 ⊢ (𝑋 ∈ ((Base‘𝑈) ∖ { 0 }) ↔ (𝑋 ∈ (Base‘𝑈) ∧ 𝑋 ≠ 0 )) | |
20 | 17, 18, 19 | sylanbrc 581 | . . . 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 41150 | . . . . 5 ⊢ (𝜑 → 𝑌 ∈ (Base‘𝑈)) |
23 | lcfrlem38.y | . . . . 5 ⊢ (𝜑 → 𝑌 ≠ 0 ) | |
24 | eldifsn 4792 | . . . . 5 ⊢ (𝑌 ∈ ((Base‘𝑈) ∖ { 0 }) ↔ (𝑌 ∈ (Base‘𝑈) ∧ 𝑌 ≠ 0 )) | |
25 | 22, 23, 24 | sylanbrc 581 | . . . 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 41168 | . . 3 ⊢ (𝜑 → ((𝑁‘{𝑋, 𝑌}) ∩ ( ⊥ ‘{(𝑋 + 𝑌)})) ∈ (LSAtoms‘𝑈)) |
28 | 1, 2, 6, 27 | lsateln0 38599 | . 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 1130 | . . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ ((𝑁‘{𝑋, 𝑌}) ∩ ( ⊥ ‘{(𝑋 + 𝑌)})) ∧ 𝑖 ≠ 0 ) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
32 | 15 | 3ad2ant1 1130 | . . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ ((𝑁‘{𝑋, 𝑌}) ∩ ( ⊥ ‘{(𝑋 + 𝑌)})) ∧ 𝑖 ≠ 0 ) → 𝐺 ∈ 𝑄) |
33 | lcfrlem38.gs | . . . . 5 ⊢ (𝜑 → 𝐺 ⊆ 𝐶) | |
34 | 33 | 3ad2ant1 1130 | . . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ ((𝑁‘{𝑋, 𝑌}) ∩ ( ⊥ ‘{(𝑋 + 𝑌)})) ∧ 𝑖 ≠ 0 ) → 𝐺 ⊆ 𝐶) |
35 | 16 | 3ad2ant1 1130 | . . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ ((𝑁‘{𝑋, 𝑌}) ∩ ( ⊥ ‘{(𝑋 + 𝑌)})) ∧ 𝑖 ≠ 0 ) → 𝑋 ∈ 𝐸) |
36 | 21 | 3ad2ant1 1130 | . . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ ((𝑁‘{𝑋, 𝑌}) ∩ ( ⊥ ‘{(𝑋 + 𝑌)})) ∧ 𝑖 ≠ 0 ) → 𝑌 ∈ 𝐸) |
37 | 18 | 3ad2ant1 1130 | . . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ ((𝑁‘{𝑋, 𝑌}) ∩ ( ⊥ ‘{(𝑋 + 𝑌)})) ∧ 𝑖 ≠ 0 ) → 𝑋 ≠ 0 ) |
38 | 23 | 3ad2ant1 1130 | . . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ ((𝑁‘{𝑋, 𝑌}) ∩ ( ⊥ ‘{(𝑋 + 𝑌)})) ∧ 𝑖 ≠ 0 ) → 𝑌 ≠ 0 ) |
39 | 26 | 3ad2ant1 1130 | . . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ ((𝑁‘{𝑋, 𝑌}) ∩ ( ⊥ ‘{(𝑋 + 𝑌)})) ∧ 𝑖 ≠ 0 ) → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌})) |
40 | eqid 2725 | . . . 4 ⊢ ((𝑁‘{𝑋, 𝑌}) ∩ ( ⊥ ‘{(𝑋 + 𝑌)})) = ((𝑁‘{𝑋, 𝑌}) ∩ ( ⊥ ‘{(𝑋 + 𝑌)})) | |
41 | simp2 1134 | . . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ ((𝑁‘{𝑋, 𝑌}) ∩ ( ⊥ ‘{(𝑋 + 𝑌)})) ∧ 𝑖 ≠ 0 ) → 𝑖 ∈ ((𝑁‘{𝑋, 𝑌}) ∩ ( ⊥ ‘{(𝑋 + 𝑌)}))) | |
42 | simp3 1135 | . . . 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 41186 | . . 3 ⊢ ((𝜑 ∧ 𝑖 ∈ ((𝑁‘{𝑋, 𝑌}) ∩ ( ⊥ ‘{(𝑋 + 𝑌)})) ∧ 𝑖 ≠ 0 ) → (𝑋 + 𝑌) ∈ 𝐸) |
44 | 43 | rexlimdv3a 3148 | . 2 ⊢ (𝜑 → (∃𝑖 ∈ ((𝑁‘{𝑋, 𝑌}) ∩ ( ⊥ ‘{(𝑋 + 𝑌)}))𝑖 ≠ 0 → (𝑋 + 𝑌) ∈ 𝐸)) |
45 | 28, 44 | mpd 15 | 1 ⊢ (𝜑 → (𝑋 + 𝑌) ∈ 𝐸) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 ≠ wne 2929 ∃wrex 3059 {crab 3418 ∖ cdif 3941 ∩ cin 3943 ⊆ wss 3944 {csn 4630 {cpr 4632 ∪ ciun 4997 ‘cfv 6549 (class class class)co 7419 Basecbs 17188 +gcplusg 17241 0gc0g 17429 LSubSpclss 20832 LSpanclspn 20872 LSAtomsclsa 38578 LFnlclfn 38661 LKerclk 38689 LDualcld 38727 HLchlt 38954 LHypclh 39589 DVecHcdvh 40683 ocHcoch 40952 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 ax-cnex 11201 ax-resscn 11202 ax-1cn 11203 ax-icn 11204 ax-addcl 11205 ax-addrcl 11206 ax-mulcl 11207 ax-mulrcl 11208 ax-mulcom 11209 ax-addass 11210 ax-mulass 11211 ax-distr 11212 ax-i2m1 11213 ax-1ne0 11214 ax-1rid 11215 ax-rnegex 11216 ax-rrecex 11217 ax-cnre 11218 ax-pre-lttri 11219 ax-pre-lttrn 11220 ax-pre-ltadd 11221 ax-pre-mulgt0 11222 ax-riotaBAD 38557 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-nel 3036 df-ral 3051 df-rex 3060 df-rmo 3363 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3964 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-tp 4635 df-op 4637 df-uni 4910 df-int 4951 df-iun 4999 df-iin 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6307 df-ord 6374 df-on 6375 df-lim 6376 df-suc 6377 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-riota 7375 df-ov 7422 df-oprab 7423 df-mpo 7424 df-of 7685 df-om 7872 df-1st 7994 df-2nd 7995 df-tpos 8232 df-undef 8279 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-1o 8487 df-2o 8488 df-er 8725 df-map 8847 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-pnf 11287 df-mnf 11288 df-xr 11289 df-ltxr 11290 df-le 11291 df-sub 11483 df-neg 11484 df-nn 12251 df-2 12313 df-3 12314 df-4 12315 df-5 12316 df-6 12317 df-n0 12511 df-z 12597 df-uz 12861 df-fz 13525 df-struct 17124 df-sets 17141 df-slot 17159 df-ndx 17171 df-base 17189 df-ress 17218 df-plusg 17254 df-mulr 17255 df-sca 17257 df-vsca 17258 df-0g 17431 df-mre 17574 df-mrc 17575 df-acs 17577 df-proset 18295 df-poset 18313 df-plt 18330 df-lub 18346 df-glb 18347 df-join 18348 df-meet 18349 df-p0 18425 df-p1 18426 df-lat 18432 df-clat 18499 df-mgm 18608 df-sgrp 18687 df-mnd 18703 df-submnd 18749 df-grp 18906 df-minusg 18907 df-sbg 18908 df-subg 19091 df-cntz 19285 df-oppg 19314 df-lsm 19608 df-cmn 19754 df-abl 19755 df-mgp 20092 df-rng 20110 df-ur 20139 df-ring 20192 df-oppr 20290 df-dvdsr 20313 df-unit 20314 df-invr 20344 df-dvr 20357 df-drng 20643 df-lmod 20762 df-lss 20833 df-lsp 20873 df-lvec 21005 df-lsatoms 38580 df-lshyp 38581 df-lcv 38623 df-lfl 38662 df-lkr 38690 df-ldual 38728 df-oposet 38780 df-ol 38782 df-oml 38783 df-covers 38870 df-ats 38871 df-atl 38902 df-cvlat 38926 df-hlat 38955 df-llines 39103 df-lplanes 39104 df-lvols 39105 df-lines 39106 df-psubsp 39108 df-pmap 39109 df-padd 39401 df-lhyp 39593 df-laut 39594 df-ldil 39709 df-ltrn 39710 df-trl 39764 df-tgrp 40348 df-tendo 40360 df-edring 40362 df-dveca 40608 df-disoa 40634 df-dvech 40684 df-dib 40744 df-dic 40778 df-dih 40834 df-doch 40953 df-djh 41000 |
This theorem is referenced by: lcfrlem41 41188 |
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