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| Mirrors > Home > MPE Home > Th. List > Mathboxes > lclkrlem2x | Structured version Visualization version GIF version | ||
| Description: Lemma for lclkr 41512. Eliminate by cases the hypotheses of lclkrlem2u 41506, lclkrlem2u 41506 and lclkrlem2w 41508. (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 41506 | . . 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 41505 | . . 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 41508 | . 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 4577 ‘cfv 6482 (class class class)co 7349 Basecbs 17120 +gcplusg 17161 .rcmulr 17162 Scalarcsca 17164 ·𝑠 cvsca 17165 0gc0g 17343 -gcsg 18814 LSSumclsm 19513 invrcinvr 20272 LSpanclspn 20874 LFnlclfn 39036 LKerclk 39064 LDualcld 39102 HLchlt 39329 LHypclh 39963 DVecHcdvh 41057 ocHcoch 41326 |
| 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 5218 ax-sep 5235 ax-nul 5245 ax-pow 5304 ax-pr 5371 ax-un 7671 ax-cnex 11065 ax-resscn 11066 ax-1cn 11067 ax-icn 11068 ax-addcl 11069 ax-addrcl 11070 ax-mulcl 11071 ax-mulrcl 11072 ax-mulcom 11073 ax-addass 11074 ax-mulass 11075 ax-distr 11076 ax-i2m1 11077 ax-1ne0 11078 ax-1rid 11079 ax-rnegex 11080 ax-rrecex 11081 ax-cnre 11082 ax-pre-lttri 11083 ax-pre-lttrn 11084 ax-pre-ltadd 11085 ax-pre-mulgt0 11086 ax-riotaBAD 38932 |
| 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 3343 df-reu 3344 df-rab 3395 df-v 3438 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-pss 3923 df-nul 4285 df-if 4477 df-pw 4553 df-sn 4578 df-pr 4580 df-tp 4582 df-op 4584 df-uni 4859 df-int 4897 df-iun 4943 df-iin 4944 df-br 5093 df-opab 5155 df-mpt 5174 df-tr 5200 df-id 5514 df-eprel 5519 df-po 5527 df-so 5528 df-fr 5572 df-we 5574 df-xp 5625 df-rel 5626 df-cnv 5627 df-co 5628 df-dm 5629 df-rn 5630 df-res 5631 df-ima 5632 df-pred 6249 df-ord 6310 df-on 6311 df-lim 6312 df-suc 6313 df-iota 6438 df-fun 6484 df-fn 6485 df-f 6486 df-f1 6487 df-fo 6488 df-f1o 6489 df-fv 6490 df-riota 7306 df-ov 7352 df-oprab 7353 df-mpo 7354 df-of 7613 df-om 7800 df-1st 7924 df-2nd 7925 df-tpos 8159 df-undef 8206 df-frecs 8214 df-wrecs 8245 df-recs 8294 df-rdg 8332 df-1o 8388 df-2o 8389 df-er 8625 df-map 8755 df-en 8873 df-dom 8874 df-sdom 8875 df-fin 8876 df-pnf 11151 df-mnf 11152 df-xr 11153 df-ltxr 11154 df-le 11155 df-sub 11349 df-neg 11350 df-nn 12129 df-2 12191 df-3 12192 df-4 12193 df-5 12194 df-6 12195 df-n0 12385 df-z 12472 df-uz 12736 df-fz 13411 df-struct 17058 df-sets 17075 df-slot 17093 df-ndx 17105 df-base 17121 df-ress 17142 df-plusg 17174 df-mulr 17175 df-sca 17177 df-vsca 17178 df-0g 17345 df-mre 17488 df-mrc 17489 df-acs 17491 df-proset 18200 df-poset 18219 df-plt 18234 df-lub 18250 df-glb 18251 df-join 18252 df-meet 18253 df-p0 18329 df-p1 18330 df-lat 18338 df-clat 18405 df-mgm 18514 df-sgrp 18593 df-mnd 18609 df-submnd 18658 df-grp 18815 df-minusg 18816 df-sbg 18817 df-subg 19002 df-cntz 19196 df-oppg 19225 df-lsm 19515 df-cmn 19661 df-abl 19662 df-mgp 20026 df-rng 20038 df-ur 20067 df-ring 20120 df-oppr 20222 df-dvdsr 20242 df-unit 20243 df-invr 20273 df-dvr 20286 df-drng 20616 df-lmod 20765 df-lss 20835 df-lsp 20875 df-lvec 21007 df-lsatoms 38955 df-lshyp 38956 df-lcv 38998 df-lfl 39037 df-lkr 39065 df-ldual 39103 df-oposet 39155 df-ol 39157 df-oml 39158 df-covers 39245 df-ats 39246 df-atl 39277 df-cvlat 39301 df-hlat 39330 df-llines 39477 df-lplanes 39478 df-lvols 39479 df-lines 39480 df-psubsp 39482 df-pmap 39483 df-padd 39775 df-lhyp 39967 df-laut 39968 df-ldil 40083 df-ltrn 40084 df-trl 40138 df-tgrp 40722 df-tendo 40734 df-edring 40736 df-dveca 40982 df-disoa 41008 df-dvech 41058 df-dib 41118 df-dic 41152 df-dih 41208 df-doch 41327 df-djh 41374 |
| This theorem is referenced by: lclkrlem2y 41510 |
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