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| Mirrors > Home > MPE Home > Th. List > Mathboxes > lclkrlem2x | Structured version Visualization version GIF version | ||
| Description: Lemma for lclkr 41527. Eliminate by cases the hypotheses of lclkrlem2u 41521, lclkrlem2u 41521 and lclkrlem2w 41523. (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 2926 | . . 3 ⊢ (((𝐸 + 𝐺)‘𝑋) ≠ (0g‘(Scalar‘𝑈)) ↔ ¬ ((𝐸 + 𝐺)‘𝑋) = (0g‘(Scalar‘𝑈))) | |
| 2 | lclkrlem2x.v | . . . 4 ⊢ 𝑉 = (Base‘𝑈) | |
| 3 | eqid 2729 | . . . 4 ⊢ ( ·𝑠 ‘𝑈) = ( ·𝑠 ‘𝑈) | |
| 4 | eqid 2729 | . . . 4 ⊢ (Scalar‘𝑈) = (Scalar‘𝑈) | |
| 5 | eqid 2729 | . . . 4 ⊢ (.r‘(Scalar‘𝑈)) = (.r‘(Scalar‘𝑈)) | |
| 6 | eqid 2729 | . . . 4 ⊢ (0g‘(Scalar‘𝑈)) = (0g‘(Scalar‘𝑈)) | |
| 7 | eqid 2729 | . . . 4 ⊢ (invr‘(Scalar‘𝑈)) = (invr‘(Scalar‘𝑈)) | |
| 8 | eqid 2729 | . . . 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 2729 | . . . 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 2729 | . . . 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 41521 | . . 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 2926 | . . 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 41520 | . . 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 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 770 | . . 3 ⊢ ((𝜑 ∧ (((𝐸 + 𝐺)‘𝑋) = (0g‘(Scalar‘𝑈)) ∧ ((𝐸 + 𝐺)‘𝑌) = (0g‘(Scalar‘𝑈)))) → ((𝐸 + 𝐺)‘𝑋) = (0g‘(Scalar‘𝑈))) | |
| 54 | simprr 772 | . . 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 41523 | . 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 814 | 1 ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘(𝐿‘(𝐸 + 𝐺)))) = (𝐿‘(𝐸 + 𝐺))) |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2109 ≠ wne 2925 {csn 4589 ‘cfv 6511 (class class class)co 7387 Basecbs 17179 +gcplusg 17220 .rcmulr 17221 Scalarcsca 17223 ·𝑠 cvsca 17224 0gc0g 17402 -gcsg 18867 LSSumclsm 19564 invrcinvr 20296 LSpanclspn 20877 LFnlclfn 39050 LKerclk 39078 LDualcld 39116 HLchlt 39343 LHypclh 39978 DVecHcdvh 41072 ocHcoch 41341 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5234 ax-sep 5251 ax-nul 5261 ax-pow 5320 ax-pr 5387 ax-un 7711 ax-cnex 11124 ax-resscn 11125 ax-1cn 11126 ax-icn 11127 ax-addcl 11128 ax-addrcl 11129 ax-mulcl 11130 ax-mulrcl 11131 ax-mulcom 11132 ax-addass 11133 ax-mulass 11134 ax-distr 11135 ax-i2m1 11136 ax-1ne0 11137 ax-1rid 11138 ax-rnegex 11139 ax-rrecex 11140 ax-cnre 11141 ax-pre-lttri 11142 ax-pre-lttrn 11143 ax-pre-ltadd 11144 ax-pre-mulgt0 11145 ax-riotaBAD 38946 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3354 df-reu 3355 df-rab 3406 df-v 3449 df-sbc 3754 df-csb 3863 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3934 df-nul 4297 df-if 4489 df-pw 4565 df-sn 4590 df-pr 4592 df-tp 4594 df-op 4596 df-uni 4872 df-int 4911 df-iun 4957 df-iin 4958 df-br 5108 df-opab 5170 df-mpt 5189 df-tr 5215 df-id 5533 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5591 df-we 5593 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6274 df-ord 6335 df-on 6336 df-lim 6337 df-suc 6338 df-iota 6464 df-fun 6513 df-fn 6514 df-f 6515 df-f1 6516 df-fo 6517 df-f1o 6518 df-fv 6519 df-riota 7344 df-ov 7390 df-oprab 7391 df-mpo 7392 df-of 7653 df-om 7843 df-1st 7968 df-2nd 7969 df-tpos 8205 df-undef 8252 df-frecs 8260 df-wrecs 8291 df-recs 8340 df-rdg 8378 df-1o 8434 df-2o 8435 df-er 8671 df-map 8801 df-en 8919 df-dom 8920 df-sdom 8921 df-fin 8922 df-pnf 11210 df-mnf 11211 df-xr 11212 df-ltxr 11213 df-le 11214 df-sub 11407 df-neg 11408 df-nn 12187 df-2 12249 df-3 12250 df-4 12251 df-5 12252 df-6 12253 df-n0 12443 df-z 12530 df-uz 12794 df-fz 13469 df-struct 17117 df-sets 17134 df-slot 17152 df-ndx 17164 df-base 17180 df-ress 17201 df-plusg 17233 df-mulr 17234 df-sca 17236 df-vsca 17237 df-0g 17404 df-mre 17547 df-mrc 17548 df-acs 17550 df-proset 18255 df-poset 18274 df-plt 18289 df-lub 18305 df-glb 18306 df-join 18307 df-meet 18308 df-p0 18384 df-p1 18385 df-lat 18391 df-clat 18458 df-mgm 18567 df-sgrp 18646 df-mnd 18662 df-submnd 18711 df-grp 18868 df-minusg 18869 df-sbg 18870 df-subg 19055 df-cntz 19249 df-oppg 19278 df-lsm 19566 df-cmn 19712 df-abl 19713 df-mgp 20050 df-rng 20062 df-ur 20091 df-ring 20144 df-oppr 20246 df-dvdsr 20266 df-unit 20267 df-invr 20297 df-dvr 20310 df-drng 20640 df-lmod 20768 df-lss 20838 df-lsp 20878 df-lvec 21010 df-lsatoms 38969 df-lshyp 38970 df-lcv 39012 df-lfl 39051 df-lkr 39079 df-ldual 39117 df-oposet 39169 df-ol 39171 df-oml 39172 df-covers 39259 df-ats 39260 df-atl 39291 df-cvlat 39315 df-hlat 39344 df-llines 39492 df-lplanes 39493 df-lvols 39494 df-lines 39495 df-psubsp 39497 df-pmap 39498 df-padd 39790 df-lhyp 39982 df-laut 39983 df-ldil 40098 df-ltrn 40099 df-trl 40153 df-tgrp 40737 df-tendo 40749 df-edring 40751 df-dveca 40997 df-disoa 41023 df-dvech 41073 df-dib 41133 df-dic 41167 df-dih 41223 df-doch 41342 df-djh 41389 |
| This theorem is referenced by: lclkrlem2y 41525 |
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