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| Mirrors > Home > MPE Home > Th. List > Mathboxes > lclkrlem2x | Structured version Visualization version GIF version | ||
| Description: Lemma for lclkr 39555. Eliminate by cases the hypotheses of lclkrlem2u 39549, lclkrlem2u 39549 and lclkrlem2w 39551. (Contributed by NM, 18-Jan-2015.) |
| Ref | Expression |
|---|---|
| lclkrlem2x.l | ⊢ 𝐿 = (LKer‘𝑈) |
| lclkrlem2x.h | ⊢ 𝐻 = (LHyp‘𝐾) |
| lclkrlem2x.o | ⊢ ⊥ = ((ocH‘𝐾)‘𝑊) |
| lclkrlem2x.u | ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) |
| lclkrlem2x.v | ⊢ 𝑉 = (Base‘𝑈) |
| lclkrlem2x.f | ⊢ 𝐹 = (LFnl‘𝑈) |
| lclkrlem2x.d | ⊢ 𝐷 = (LDual‘𝑈) |
| lclkrlem2x.p | ⊢ + = (+g‘𝐷) |
| lclkrlem2x.k | ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
| lclkrlem2x.x | ⊢ (𝜑 → 𝑋 ∈ 𝑉) |
| lclkrlem2x.y | ⊢ (𝜑 → 𝑌 ∈ 𝑉) |
| lclkrlem2x.e | ⊢ (𝜑 → 𝐸 ∈ 𝐹) |
| lclkrlem2x.g | ⊢ (𝜑 → 𝐺 ∈ 𝐹) |
| lclkrlem2x.le | ⊢ (𝜑 → (𝐿‘𝐸) = ( ⊥ ‘{𝑋})) |
| lclkrlem2x.lg | ⊢ (𝜑 → (𝐿‘𝐺) = ( ⊥ ‘{𝑌})) |
| Ref | Expression |
|---|---|
| lclkrlem2x | ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘(𝐿‘(𝐸 + 𝐺)))) = (𝐿‘(𝐸 + 𝐺))) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | df-ne 2944 | . . 3 ⊢ (((𝐸 + 𝐺)‘𝑋) ≠ (0g‘(Scalar‘𝑈)) ↔ ¬ ((𝐸 + 𝐺)‘𝑋) = (0g‘(Scalar‘𝑈))) | |
| 2 | lclkrlem2x.v | . . . 4 ⊢ 𝑉 = (Base‘𝑈) | |
| 3 | eqid 2738 | . . . 4 ⊢ ( ·𝑠 ‘𝑈) = ( ·𝑠 ‘𝑈) | |
| 4 | eqid 2738 | . . . 4 ⊢ (Scalar‘𝑈) = (Scalar‘𝑈) | |
| 5 | eqid 2738 | . . . 4 ⊢ (.r‘(Scalar‘𝑈)) = (.r‘(Scalar‘𝑈)) | |
| 6 | eqid 2738 | . . . 4 ⊢ (0g‘(Scalar‘𝑈)) = (0g‘(Scalar‘𝑈)) | |
| 7 | eqid 2738 | . . . 4 ⊢ (invr‘(Scalar‘𝑈)) = (invr‘(Scalar‘𝑈)) | |
| 8 | eqid 2738 | . . . 4 ⊢ (-g‘𝑈) = (-g‘𝑈) | |
| 9 | lclkrlem2x.f | . . . 4 ⊢ 𝐹 = (LFnl‘𝑈) | |
| 10 | lclkrlem2x.d | . . . 4 ⊢ 𝐷 = (LDual‘𝑈) | |
| 11 | lclkrlem2x.p | . . . 4 ⊢ + = (+g‘𝐷) | |
| 12 | lclkrlem2x.x | . . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝑉) | |
| 13 | 12 | adantr 481 | . . . 4 ⊢ ((𝜑 ∧ ((𝐸 + 𝐺)‘𝑋) ≠ (0g‘(Scalar‘𝑈))) → 𝑋 ∈ 𝑉) |
| 14 | lclkrlem2x.y | . . . . 5 ⊢ (𝜑 → 𝑌 ∈ 𝑉) | |
| 15 | 14 | adantr 481 | . . . 4 ⊢ ((𝜑 ∧ ((𝐸 + 𝐺)‘𝑋) ≠ (0g‘(Scalar‘𝑈))) → 𝑌 ∈ 𝑉) |
| 16 | lclkrlem2x.e | . . . . 5 ⊢ (𝜑 → 𝐸 ∈ 𝐹) | |
| 17 | 16 | adantr 481 | . . . 4 ⊢ ((𝜑 ∧ ((𝐸 + 𝐺)‘𝑋) ≠ (0g‘(Scalar‘𝑈))) → 𝐸 ∈ 𝐹) |
| 18 | lclkrlem2x.g | . . . . 5 ⊢ (𝜑 → 𝐺 ∈ 𝐹) | |
| 19 | 18 | adantr 481 | . . . 4 ⊢ ((𝜑 ∧ ((𝐸 + 𝐺)‘𝑋) ≠ (0g‘(Scalar‘𝑈))) → 𝐺 ∈ 𝐹) |
| 20 | eqid 2738 | . . . 4 ⊢ (LSpan‘𝑈) = (LSpan‘𝑈) | |
| 21 | lclkrlem2x.l | . . . 4 ⊢ 𝐿 = (LKer‘𝑈) | |
| 22 | lclkrlem2x.h | . . . 4 ⊢ 𝐻 = (LHyp‘𝐾) | |
| 23 | lclkrlem2x.o | . . . 4 ⊢ ⊥ = ((ocH‘𝐾)‘𝑊) | |
| 24 | lclkrlem2x.u | . . . 4 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) | |
| 25 | eqid 2738 | . . . 4 ⊢ (LSSum‘𝑈) = (LSSum‘𝑈) | |
| 26 | lclkrlem2x.k | . . . . 5 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
| 27 | 26 | adantr 481 | . . . 4 ⊢ ((𝜑 ∧ ((𝐸 + 𝐺)‘𝑋) ≠ (0g‘(Scalar‘𝑈))) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
| 28 | lclkrlem2x.le | . . . . 5 ⊢ (𝜑 → (𝐿‘𝐸) = ( ⊥ ‘{𝑋})) | |
| 29 | 28 | adantr 481 | . . . 4 ⊢ ((𝜑 ∧ ((𝐸 + 𝐺)‘𝑋) ≠ (0g‘(Scalar‘𝑈))) → (𝐿‘𝐸) = ( ⊥ ‘{𝑋})) |
| 30 | lclkrlem2x.lg | . . . . 5 ⊢ (𝜑 → (𝐿‘𝐺) = ( ⊥ ‘{𝑌})) | |
| 31 | 30 | adantr 481 | . . . 4 ⊢ ((𝜑 ∧ ((𝐸 + 𝐺)‘𝑋) ≠ (0g‘(Scalar‘𝑈))) → (𝐿‘𝐺) = ( ⊥ ‘{𝑌})) |
| 32 | simpr 485 | . . . 4 ⊢ ((𝜑 ∧ ((𝐸 + 𝐺)‘𝑋) ≠ (0g‘(Scalar‘𝑈))) → ((𝐸 + 𝐺)‘𝑋) ≠ (0g‘(Scalar‘𝑈))) | |
| 33 | 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 15, 17, 19, 20, 21, 22, 23, 24, 25, 27, 29, 31, 32 | lclkrlem2u 39549 | . . 3 ⊢ ((𝜑 ∧ ((𝐸 + 𝐺)‘𝑋) ≠ (0g‘(Scalar‘𝑈))) → ( ⊥ ‘( ⊥ ‘(𝐿‘(𝐸 + 𝐺)))) = (𝐿‘(𝐸 + 𝐺))) |
| 34 | 1, 33 | sylan2br 595 | . 2 ⊢ ((𝜑 ∧ ¬ ((𝐸 + 𝐺)‘𝑋) = (0g‘(Scalar‘𝑈))) → ( ⊥ ‘( ⊥ ‘(𝐿‘(𝐸 + 𝐺)))) = (𝐿‘(𝐸 + 𝐺))) |
| 35 | df-ne 2944 | . . 3 ⊢ (((𝐸 + 𝐺)‘𝑌) ≠ (0g‘(Scalar‘𝑈)) ↔ ¬ ((𝐸 + 𝐺)‘𝑌) = (0g‘(Scalar‘𝑈))) | |
| 36 | 12 | adantr 481 | . . . 4 ⊢ ((𝜑 ∧ ((𝐸 + 𝐺)‘𝑌) ≠ (0g‘(Scalar‘𝑈))) → 𝑋 ∈ 𝑉) |
| 37 | 14 | adantr 481 | . . . 4 ⊢ ((𝜑 ∧ ((𝐸 + 𝐺)‘𝑌) ≠ (0g‘(Scalar‘𝑈))) → 𝑌 ∈ 𝑉) |
| 38 | 16 | adantr 481 | . . . 4 ⊢ ((𝜑 ∧ ((𝐸 + 𝐺)‘𝑌) ≠ (0g‘(Scalar‘𝑈))) → 𝐸 ∈ 𝐹) |
| 39 | 18 | adantr 481 | . . . 4 ⊢ ((𝜑 ∧ ((𝐸 + 𝐺)‘𝑌) ≠ (0g‘(Scalar‘𝑈))) → 𝐺 ∈ 𝐹) |
| 40 | 26 | adantr 481 | . . . 4 ⊢ ((𝜑 ∧ ((𝐸 + 𝐺)‘𝑌) ≠ (0g‘(Scalar‘𝑈))) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
| 41 | 28 | adantr 481 | . . . 4 ⊢ ((𝜑 ∧ ((𝐸 + 𝐺)‘𝑌) ≠ (0g‘(Scalar‘𝑈))) → (𝐿‘𝐸) = ( ⊥ ‘{𝑋})) |
| 42 | 30 | adantr 481 | . . . 4 ⊢ ((𝜑 ∧ ((𝐸 + 𝐺)‘𝑌) ≠ (0g‘(Scalar‘𝑈))) → (𝐿‘𝐺) = ( ⊥ ‘{𝑌})) |
| 43 | simpr 485 | . . . 4 ⊢ ((𝜑 ∧ ((𝐸 + 𝐺)‘𝑌) ≠ (0g‘(Scalar‘𝑈))) → ((𝐸 + 𝐺)‘𝑌) ≠ (0g‘(Scalar‘𝑈))) | |
| 44 | 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 36, 37, 38, 39, 20, 21, 22, 23, 24, 25, 40, 41, 42, 43 | lclkrlem2t 39548 | . . 3 ⊢ ((𝜑 ∧ ((𝐸 + 𝐺)‘𝑌) ≠ (0g‘(Scalar‘𝑈))) → ( ⊥ ‘( ⊥ ‘(𝐿‘(𝐸 + 𝐺)))) = (𝐿‘(𝐸 + 𝐺))) |
| 45 | 35, 44 | sylan2br 595 | . 2 ⊢ ((𝜑 ∧ ¬ ((𝐸 + 𝐺)‘𝑌) = (0g‘(Scalar‘𝑈))) → ( ⊥ ‘( ⊥ ‘(𝐿‘(𝐸 + 𝐺)))) = (𝐿‘(𝐸 + 𝐺))) |
| 46 | 12 | adantr 481 | . . 3 ⊢ ((𝜑 ∧ (((𝐸 + 𝐺)‘𝑋) = (0g‘(Scalar‘𝑈)) ∧ ((𝐸 + 𝐺)‘𝑌) = (0g‘(Scalar‘𝑈)))) → 𝑋 ∈ 𝑉) |
| 47 | 14 | adantr 481 | . . 3 ⊢ ((𝜑 ∧ (((𝐸 + 𝐺)‘𝑋) = (0g‘(Scalar‘𝑈)) ∧ ((𝐸 + 𝐺)‘𝑌) = (0g‘(Scalar‘𝑈)))) → 𝑌 ∈ 𝑉) |
| 48 | 16 | adantr 481 | . . 3 ⊢ ((𝜑 ∧ (((𝐸 + 𝐺)‘𝑋) = (0g‘(Scalar‘𝑈)) ∧ ((𝐸 + 𝐺)‘𝑌) = (0g‘(Scalar‘𝑈)))) → 𝐸 ∈ 𝐹) |
| 49 | 18 | adantr 481 | . . 3 ⊢ ((𝜑 ∧ (((𝐸 + 𝐺)‘𝑋) = (0g‘(Scalar‘𝑈)) ∧ ((𝐸 + 𝐺)‘𝑌) = (0g‘(Scalar‘𝑈)))) → 𝐺 ∈ 𝐹) |
| 50 | 26 | adantr 481 | . . 3 ⊢ ((𝜑 ∧ (((𝐸 + 𝐺)‘𝑋) = (0g‘(Scalar‘𝑈)) ∧ ((𝐸 + 𝐺)‘𝑌) = (0g‘(Scalar‘𝑈)))) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
| 51 | 28 | adantr 481 | . . 3 ⊢ ((𝜑 ∧ (((𝐸 + 𝐺)‘𝑋) = (0g‘(Scalar‘𝑈)) ∧ ((𝐸 + 𝐺)‘𝑌) = (0g‘(Scalar‘𝑈)))) → (𝐿‘𝐸) = ( ⊥ ‘{𝑋})) |
| 52 | 30 | adantr 481 | . . 3 ⊢ ((𝜑 ∧ (((𝐸 + 𝐺)‘𝑋) = (0g‘(Scalar‘𝑈)) ∧ ((𝐸 + 𝐺)‘𝑌) = (0g‘(Scalar‘𝑈)))) → (𝐿‘𝐺) = ( ⊥ ‘{𝑌})) |
| 53 | simprl 768 | . . 3 ⊢ ((𝜑 ∧ (((𝐸 + 𝐺)‘𝑋) = (0g‘(Scalar‘𝑈)) ∧ ((𝐸 + 𝐺)‘𝑌) = (0g‘(Scalar‘𝑈)))) → ((𝐸 + 𝐺)‘𝑋) = (0g‘(Scalar‘𝑈))) | |
| 54 | simprr 770 | . . 3 ⊢ ((𝜑 ∧ (((𝐸 + 𝐺)‘𝑋) = (0g‘(Scalar‘𝑈)) ∧ ((𝐸 + 𝐺)‘𝑌) = (0g‘(Scalar‘𝑈)))) → ((𝐸 + 𝐺)‘𝑌) = (0g‘(Scalar‘𝑈))) | |
| 55 | 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 46, 47, 48, 49, 20, 21, 22, 23, 24, 25, 50, 51, 52, 53, 54 | lclkrlem2w 39551 | . 2 ⊢ ((𝜑 ∧ (((𝐸 + 𝐺)‘𝑋) = (0g‘(Scalar‘𝑈)) ∧ ((𝐸 + 𝐺)‘𝑌) = (0g‘(Scalar‘𝑈)))) → ( ⊥ ‘( ⊥ ‘(𝐿‘(𝐸 + 𝐺)))) = (𝐿‘(𝐸 + 𝐺))) |
| 56 | 34, 45, 55 | pm2.61dda 812 | 1 ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘(𝐿‘(𝐸 + 𝐺)))) = (𝐿‘(𝐸 + 𝐺))) |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 → wi 4 ∧ wa 396 = wceq 1539 ∈ wcel 2106 ≠ wne 2943 {csn 4561 ‘cfv 6426 (class class class)co 7267 Basecbs 16922 +gcplusg 16972 .rcmulr 16973 Scalarcsca 16975 ·𝑠 cvsca 16976 0gc0g 17160 -gcsg 18589 LSSumclsm 19249 invrcinvr 19923 LSpanclspn 20243 LFnlclfn 37079 LKerclk 37107 LDualcld 37145 HLchlt 37372 LHypclh 38006 DVecHcdvh 39100 ocHcoch 39369 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5208 ax-sep 5221 ax-nul 5228 ax-pow 5286 ax-pr 5350 ax-un 7578 ax-cnex 10937 ax-resscn 10938 ax-1cn 10939 ax-icn 10940 ax-addcl 10941 ax-addrcl 10942 ax-mulcl 10943 ax-mulrcl 10944 ax-mulcom 10945 ax-addass 10946 ax-mulass 10947 ax-distr 10948 ax-i2m1 10949 ax-1ne0 10950 ax-1rid 10951 ax-rnegex 10952 ax-rrecex 10953 ax-cnre 10954 ax-pre-lttri 10955 ax-pre-lttrn 10956 ax-pre-ltadd 10957 ax-pre-mulgt0 10958 ax-riotaBAD 36975 |
| This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-reu 3071 df-rmo 3072 df-rab 3073 df-v 3431 df-sbc 3716 df-csb 3832 df-dif 3889 df-un 3891 df-in 3893 df-ss 3903 df-pss 3905 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4840 df-int 4880 df-iun 4926 df-iin 4927 df-br 5074 df-opab 5136 df-mpt 5157 df-tr 5191 df-id 5484 df-eprel 5490 df-po 5498 df-so 5499 df-fr 5539 df-we 5541 df-xp 5590 df-rel 5591 df-cnv 5592 df-co 5593 df-dm 5594 df-rn 5595 df-res 5596 df-ima 5597 df-pred 6195 df-ord 6262 df-on 6263 df-lim 6264 df-suc 6265 df-iota 6384 df-fun 6428 df-fn 6429 df-f 6430 df-f1 6431 df-fo 6432 df-f1o 6433 df-fv 6434 df-riota 7224 df-ov 7270 df-oprab 7271 df-mpo 7272 df-of 7523 df-om 7703 df-1st 7820 df-2nd 7821 df-tpos 8029 df-undef 8076 df-frecs 8084 df-wrecs 8115 df-recs 8189 df-rdg 8228 df-1o 8284 df-er 8485 df-map 8604 df-en 8721 df-dom 8722 df-sdom 8723 df-fin 8724 df-pnf 11021 df-mnf 11022 df-xr 11023 df-ltxr 11024 df-le 11025 df-sub 11217 df-neg 11218 df-nn 11984 df-2 12046 df-3 12047 df-4 12048 df-5 12049 df-6 12050 df-n0 12244 df-z 12330 df-uz 12593 df-fz 13250 df-struct 16858 df-sets 16875 df-slot 16893 df-ndx 16905 df-base 16923 df-ress 16952 df-plusg 16985 df-mulr 16986 df-sca 16988 df-vsca 16989 df-0g 17162 df-mre 17305 df-mrc 17306 df-acs 17308 df-proset 18023 df-poset 18041 df-plt 18058 df-lub 18074 df-glb 18075 df-join 18076 df-meet 18077 df-p0 18153 df-p1 18154 df-lat 18160 df-clat 18227 df-mgm 18336 df-sgrp 18385 df-mnd 18396 df-submnd 18441 df-grp 18590 df-minusg 18591 df-sbg 18592 df-subg 18762 df-cntz 18933 df-oppg 18960 df-lsm 19251 df-cmn 19398 df-abl 19399 df-mgp 19731 df-ur 19748 df-ring 19795 df-oppr 19872 df-dvdsr 19893 df-unit 19894 df-invr 19924 df-dvr 19935 df-drng 20003 df-lmod 20135 df-lss 20204 df-lsp 20244 df-lvec 20375 df-lsatoms 36998 df-lshyp 36999 df-lcv 37041 df-lfl 37080 df-lkr 37108 df-ldual 37146 df-oposet 37198 df-ol 37200 df-oml 37201 df-covers 37288 df-ats 37289 df-atl 37320 df-cvlat 37344 df-hlat 37373 df-llines 37520 df-lplanes 37521 df-lvols 37522 df-lines 37523 df-psubsp 37525 df-pmap 37526 df-padd 37818 df-lhyp 38010 df-laut 38011 df-ldil 38126 df-ltrn 38127 df-trl 38181 df-tgrp 38765 df-tendo 38777 df-edring 38779 df-dveca 39025 df-disoa 39051 df-dvech 39101 df-dib 39161 df-dic 39195 df-dih 39251 df-doch 39370 df-djh 39417 |
| This theorem is referenced by: lclkrlem2y 39553 |
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