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| Mirrors > Home > MPE Home > Th. List > Mathboxes > lclkrlem2x | Structured version Visualization version GIF version | ||
| Description: Lemma for lclkr 41909. Eliminate by cases the hypotheses of lclkrlem2u 41903, lclkrlem2u 41903 and lclkrlem2w 41905. (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 2934 | . . 3 ⊢ (((𝐸 + 𝐺)‘𝑋) ≠ (0g‘(Scalar‘𝑈)) ↔ ¬ ((𝐸 + 𝐺)‘𝑋) = (0g‘(Scalar‘𝑈))) | |
| 2 | lclkrlem2x.v | . . . 4 ⊢ 𝑉 = (Base‘𝑈) | |
| 3 | eqid 2737 | . . . 4 ⊢ ( ·𝑠 ‘𝑈) = ( ·𝑠 ‘𝑈) | |
| 4 | eqid 2737 | . . . 4 ⊢ (Scalar‘𝑈) = (Scalar‘𝑈) | |
| 5 | eqid 2737 | . . . 4 ⊢ (.r‘(Scalar‘𝑈)) = (.r‘(Scalar‘𝑈)) | |
| 6 | eqid 2737 | . . . 4 ⊢ (0g‘(Scalar‘𝑈)) = (0g‘(Scalar‘𝑈)) | |
| 7 | eqid 2737 | . . . 4 ⊢ (invr‘(Scalar‘𝑈)) = (invr‘(Scalar‘𝑈)) | |
| 8 | eqid 2737 | . . . 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 480 | . . . 4 ⊢ ((𝜑 ∧ ((𝐸 + 𝐺)‘𝑋) ≠ (0g‘(Scalar‘𝑈))) → 𝑋 ∈ 𝑉) |
| 14 | lclkrlem2x.y | . . . . 5 ⊢ (𝜑 → 𝑌 ∈ 𝑉) | |
| 15 | 14 | adantr 480 | . . . 4 ⊢ ((𝜑 ∧ ((𝐸 + 𝐺)‘𝑋) ≠ (0g‘(Scalar‘𝑈))) → 𝑌 ∈ 𝑉) |
| 16 | lclkrlem2x.e | . . . . 5 ⊢ (𝜑 → 𝐸 ∈ 𝐹) | |
| 17 | 16 | adantr 480 | . . . 4 ⊢ ((𝜑 ∧ ((𝐸 + 𝐺)‘𝑋) ≠ (0g‘(Scalar‘𝑈))) → 𝐸 ∈ 𝐹) |
| 18 | lclkrlem2x.g | . . . . 5 ⊢ (𝜑 → 𝐺 ∈ 𝐹) | |
| 19 | 18 | adantr 480 | . . . 4 ⊢ ((𝜑 ∧ ((𝐸 + 𝐺)‘𝑋) ≠ (0g‘(Scalar‘𝑈))) → 𝐺 ∈ 𝐹) |
| 20 | eqid 2737 | . . . 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 2737 | . . . 4 ⊢ (LSSum‘𝑈) = (LSSum‘𝑈) | |
| 26 | lclkrlem2x.k | . . . . 5 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
| 27 | 26 | adantr 480 | . . . 4 ⊢ ((𝜑 ∧ ((𝐸 + 𝐺)‘𝑋) ≠ (0g‘(Scalar‘𝑈))) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
| 28 | lclkrlem2x.le | . . . . 5 ⊢ (𝜑 → (𝐿‘𝐸) = ( ⊥ ‘{𝑋})) | |
| 29 | 28 | adantr 480 | . . . 4 ⊢ ((𝜑 ∧ ((𝐸 + 𝐺)‘𝑋) ≠ (0g‘(Scalar‘𝑈))) → (𝐿‘𝐸) = ( ⊥ ‘{𝑋})) |
| 30 | lclkrlem2x.lg | . . . . 5 ⊢ (𝜑 → (𝐿‘𝐺) = ( ⊥ ‘{𝑌})) | |
| 31 | 30 | adantr 480 | . . . 4 ⊢ ((𝜑 ∧ ((𝐸 + 𝐺)‘𝑋) ≠ (0g‘(Scalar‘𝑈))) → (𝐿‘𝐺) = ( ⊥ ‘{𝑌})) |
| 32 | simpr 484 | . . . 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 41903 | . . 3 ⊢ ((𝜑 ∧ ((𝐸 + 𝐺)‘𝑋) ≠ (0g‘(Scalar‘𝑈))) → ( ⊥ ‘( ⊥ ‘(𝐿‘(𝐸 + 𝐺)))) = (𝐿‘(𝐸 + 𝐺))) |
| 34 | 1, 33 | sylan2br 596 | . 2 ⊢ ((𝜑 ∧ ¬ ((𝐸 + 𝐺)‘𝑋) = (0g‘(Scalar‘𝑈))) → ( ⊥ ‘( ⊥ ‘(𝐿‘(𝐸 + 𝐺)))) = (𝐿‘(𝐸 + 𝐺))) |
| 35 | df-ne 2934 | . . 3 ⊢ (((𝐸 + 𝐺)‘𝑌) ≠ (0g‘(Scalar‘𝑈)) ↔ ¬ ((𝐸 + 𝐺)‘𝑌) = (0g‘(Scalar‘𝑈))) | |
| 36 | 12 | adantr 480 | . . . 4 ⊢ ((𝜑 ∧ ((𝐸 + 𝐺)‘𝑌) ≠ (0g‘(Scalar‘𝑈))) → 𝑋 ∈ 𝑉) |
| 37 | 14 | adantr 480 | . . . 4 ⊢ ((𝜑 ∧ ((𝐸 + 𝐺)‘𝑌) ≠ (0g‘(Scalar‘𝑈))) → 𝑌 ∈ 𝑉) |
| 38 | 16 | adantr 480 | . . . 4 ⊢ ((𝜑 ∧ ((𝐸 + 𝐺)‘𝑌) ≠ (0g‘(Scalar‘𝑈))) → 𝐸 ∈ 𝐹) |
| 39 | 18 | adantr 480 | . . . 4 ⊢ ((𝜑 ∧ ((𝐸 + 𝐺)‘𝑌) ≠ (0g‘(Scalar‘𝑈))) → 𝐺 ∈ 𝐹) |
| 40 | 26 | adantr 480 | . . . 4 ⊢ ((𝜑 ∧ ((𝐸 + 𝐺)‘𝑌) ≠ (0g‘(Scalar‘𝑈))) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
| 41 | 28 | adantr 480 | . . . 4 ⊢ ((𝜑 ∧ ((𝐸 + 𝐺)‘𝑌) ≠ (0g‘(Scalar‘𝑈))) → (𝐿‘𝐸) = ( ⊥ ‘{𝑋})) |
| 42 | 30 | adantr 480 | . . . 4 ⊢ ((𝜑 ∧ ((𝐸 + 𝐺)‘𝑌) ≠ (0g‘(Scalar‘𝑈))) → (𝐿‘𝐺) = ( ⊥ ‘{𝑌})) |
| 43 | simpr 484 | . . . 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 41902 | . . 3 ⊢ ((𝜑 ∧ ((𝐸 + 𝐺)‘𝑌) ≠ (0g‘(Scalar‘𝑈))) → ( ⊥ ‘( ⊥ ‘(𝐿‘(𝐸 + 𝐺)))) = (𝐿‘(𝐸 + 𝐺))) |
| 45 | 35, 44 | sylan2br 596 | . 2 ⊢ ((𝜑 ∧ ¬ ((𝐸 + 𝐺)‘𝑌) = (0g‘(Scalar‘𝑈))) → ( ⊥ ‘( ⊥ ‘(𝐿‘(𝐸 + 𝐺)))) = (𝐿‘(𝐸 + 𝐺))) |
| 46 | 12 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ (((𝐸 + 𝐺)‘𝑋) = (0g‘(Scalar‘𝑈)) ∧ ((𝐸 + 𝐺)‘𝑌) = (0g‘(Scalar‘𝑈)))) → 𝑋 ∈ 𝑉) |
| 47 | 14 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ (((𝐸 + 𝐺)‘𝑋) = (0g‘(Scalar‘𝑈)) ∧ ((𝐸 + 𝐺)‘𝑌) = (0g‘(Scalar‘𝑈)))) → 𝑌 ∈ 𝑉) |
| 48 | 16 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ (((𝐸 + 𝐺)‘𝑋) = (0g‘(Scalar‘𝑈)) ∧ ((𝐸 + 𝐺)‘𝑌) = (0g‘(Scalar‘𝑈)))) → 𝐸 ∈ 𝐹) |
| 49 | 18 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ (((𝐸 + 𝐺)‘𝑋) = (0g‘(Scalar‘𝑈)) ∧ ((𝐸 + 𝐺)‘𝑌) = (0g‘(Scalar‘𝑈)))) → 𝐺 ∈ 𝐹) |
| 50 | 26 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ (((𝐸 + 𝐺)‘𝑋) = (0g‘(Scalar‘𝑈)) ∧ ((𝐸 + 𝐺)‘𝑌) = (0g‘(Scalar‘𝑈)))) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
| 51 | 28 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ (((𝐸 + 𝐺)‘𝑋) = (0g‘(Scalar‘𝑈)) ∧ ((𝐸 + 𝐺)‘𝑌) = (0g‘(Scalar‘𝑈)))) → (𝐿‘𝐸) = ( ⊥ ‘{𝑋})) |
| 52 | 30 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ (((𝐸 + 𝐺)‘𝑋) = (0g‘(Scalar‘𝑈)) ∧ ((𝐸 + 𝐺)‘𝑌) = (0g‘(Scalar‘𝑈)))) → (𝐿‘𝐺) = ( ⊥ ‘{𝑌})) |
| 53 | simprl 771 | . . 3 ⊢ ((𝜑 ∧ (((𝐸 + 𝐺)‘𝑋) = (0g‘(Scalar‘𝑈)) ∧ ((𝐸 + 𝐺)‘𝑌) = (0g‘(Scalar‘𝑈)))) → ((𝐸 + 𝐺)‘𝑋) = (0g‘(Scalar‘𝑈))) | |
| 54 | simprr 773 | . . 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 41905 | . 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 815 | 1 ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘(𝐿‘(𝐸 + 𝐺)))) = (𝐿‘(𝐸 + 𝐺))) |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 = wceq 1542 ∈ wcel 2114 ≠ wne 2933 {csn 4582 ‘cfv 6500 (class class class)co 7368 Basecbs 17148 +gcplusg 17189 .rcmulr 17190 Scalarcsca 17192 ·𝑠 cvsca 17193 0gc0g 17371 -gcsg 18877 LSSumclsm 19575 invrcinvr 20335 LSpanclspn 20934 LFnlclfn 39433 LKerclk 39461 LDualcld 39499 HLchlt 39726 LHypclh 40360 DVecHcdvh 41454 ocHcoch 41723 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5226 ax-sep 5243 ax-nul 5253 ax-pow 5312 ax-pr 5379 ax-un 7690 ax-cnex 11094 ax-resscn 11095 ax-1cn 11096 ax-icn 11097 ax-addcl 11098 ax-addrcl 11099 ax-mulcl 11100 ax-mulrcl 11101 ax-mulcom 11102 ax-addass 11103 ax-mulass 11104 ax-distr 11105 ax-i2m1 11106 ax-1ne0 11107 ax-1rid 11108 ax-rnegex 11109 ax-rrecex 11110 ax-cnre 11111 ax-pre-lttri 11112 ax-pre-lttrn 11113 ax-pre-ltadd 11114 ax-pre-mulgt0 11115 ax-riotaBAD 39329 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rmo 3352 df-reu 3353 df-rab 3402 df-v 3444 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-pss 3923 df-nul 4288 df-if 4482 df-pw 4558 df-sn 4583 df-pr 4585 df-tp 4587 df-op 4589 df-uni 4866 df-int 4905 df-iun 4950 df-iin 4951 df-br 5101 df-opab 5163 df-mpt 5182 df-tr 5208 df-id 5527 df-eprel 5532 df-po 5540 df-so 5541 df-fr 5585 df-we 5587 df-xp 5638 df-rel 5639 df-cnv 5640 df-co 5641 df-dm 5642 df-rn 5643 df-res 5644 df-ima 5645 df-pred 6267 df-ord 6328 df-on 6329 df-lim 6330 df-suc 6331 df-iota 6456 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-riota 7325 df-ov 7371 df-oprab 7372 df-mpo 7373 df-of 7632 df-om 7819 df-1st 7943 df-2nd 7944 df-tpos 8178 df-undef 8225 df-frecs 8233 df-wrecs 8264 df-recs 8313 df-rdg 8351 df-1o 8407 df-2o 8408 df-er 8645 df-map 8777 df-en 8896 df-dom 8897 df-sdom 8898 df-fin 8899 df-pnf 11180 df-mnf 11181 df-xr 11182 df-ltxr 11183 df-le 11184 df-sub 11378 df-neg 11379 df-nn 12158 df-2 12220 df-3 12221 df-4 12222 df-5 12223 df-6 12224 df-n0 12414 df-z 12501 df-uz 12764 df-fz 13436 df-struct 17086 df-sets 17103 df-slot 17121 df-ndx 17133 df-base 17149 df-ress 17170 df-plusg 17202 df-mulr 17203 df-sca 17205 df-vsca 17206 df-0g 17373 df-mre 17517 df-mrc 17518 df-acs 17520 df-proset 18229 df-poset 18248 df-plt 18263 df-lub 18279 df-glb 18280 df-join 18281 df-meet 18282 df-p0 18358 df-p1 18359 df-lat 18367 df-clat 18434 df-mgm 18577 df-sgrp 18656 df-mnd 18672 df-submnd 18721 df-grp 18878 df-minusg 18879 df-sbg 18880 df-subg 19065 df-cntz 19258 df-oppg 19287 df-lsm 19577 df-cmn 19723 df-abl 19724 df-mgp 20088 df-rng 20100 df-ur 20129 df-ring 20182 df-oppr 20285 df-dvdsr 20305 df-unit 20306 df-invr 20336 df-dvr 20349 df-drng 20676 df-lmod 20825 df-lss 20895 df-lsp 20935 df-lvec 21067 df-lsatoms 39352 df-lshyp 39353 df-lcv 39395 df-lfl 39434 df-lkr 39462 df-ldual 39500 df-oposet 39552 df-ol 39554 df-oml 39555 df-covers 39642 df-ats 39643 df-atl 39674 df-cvlat 39698 df-hlat 39727 df-llines 39874 df-lplanes 39875 df-lvols 39876 df-lines 39877 df-psubsp 39879 df-pmap 39880 df-padd 40172 df-lhyp 40364 df-laut 40365 df-ldil 40480 df-ltrn 40481 df-trl 40535 df-tgrp 41119 df-tendo 41131 df-edring 41133 df-dveca 41379 df-disoa 41405 df-dvech 41455 df-dib 41515 df-dic 41549 df-dih 41605 df-doch 41724 df-djh 41771 |
| This theorem is referenced by: lclkrlem2y 41907 |
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