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| Mirrors > Home > MPE Home > Th. List > Mathboxes > lclkrlem2x | Structured version Visualization version GIF version | ||
| Description: Lemma for lclkr 41534. Eliminate by cases the hypotheses of lclkrlem2u 41528, lclkrlem2u 41528 and lclkrlem2w 41530. (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 2927 | . . 3 ⊢ (((𝐸 + 𝐺)‘𝑋) ≠ (0g‘(Scalar‘𝑈)) ↔ ¬ ((𝐸 + 𝐺)‘𝑋) = (0g‘(Scalar‘𝑈))) | |
| 2 | lclkrlem2x.v | . . . 4 ⊢ 𝑉 = (Base‘𝑈) | |
| 3 | eqid 2730 | . . . 4 ⊢ ( ·𝑠 ‘𝑈) = ( ·𝑠 ‘𝑈) | |
| 4 | eqid 2730 | . . . 4 ⊢ (Scalar‘𝑈) = (Scalar‘𝑈) | |
| 5 | eqid 2730 | . . . 4 ⊢ (.r‘(Scalar‘𝑈)) = (.r‘(Scalar‘𝑈)) | |
| 6 | eqid 2730 | . . . 4 ⊢ (0g‘(Scalar‘𝑈)) = (0g‘(Scalar‘𝑈)) | |
| 7 | eqid 2730 | . . . 4 ⊢ (invr‘(Scalar‘𝑈)) = (invr‘(Scalar‘𝑈)) | |
| 8 | eqid 2730 | . . . 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 2730 | . . . 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 2730 | . . . 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 41528 | . . 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 2927 | . . 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 41527 | . . 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 41530 | . 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 2926 {csn 4592 ‘cfv 6514 (class class class)co 7390 Basecbs 17186 +gcplusg 17227 .rcmulr 17228 Scalarcsca 17230 ·𝑠 cvsca 17231 0gc0g 17409 -gcsg 18874 LSSumclsm 19571 invrcinvr 20303 LSpanclspn 20884 LFnlclfn 39057 LKerclk 39085 LDualcld 39123 HLchlt 39350 LHypclh 39985 DVecHcdvh 41079 ocHcoch 41348 |
| 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 2702 ax-rep 5237 ax-sep 5254 ax-nul 5264 ax-pow 5323 ax-pr 5390 ax-un 7714 ax-cnex 11131 ax-resscn 11132 ax-1cn 11133 ax-icn 11134 ax-addcl 11135 ax-addrcl 11136 ax-mulcl 11137 ax-mulrcl 11138 ax-mulcom 11139 ax-addass 11140 ax-mulass 11141 ax-distr 11142 ax-i2m1 11143 ax-1ne0 11144 ax-1rid 11145 ax-rnegex 11146 ax-rrecex 11147 ax-cnre 11148 ax-pre-lttri 11149 ax-pre-lttrn 11150 ax-pre-ltadd 11151 ax-pre-mulgt0 11152 ax-riotaBAD 38953 |
| 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 2534 df-eu 2563 df-clab 2709 df-cleq 2722 df-clel 2804 df-nfc 2879 df-ne 2927 df-nel 3031 df-ral 3046 df-rex 3055 df-rmo 3356 df-reu 3357 df-rab 3409 df-v 3452 df-sbc 3757 df-csb 3866 df-dif 3920 df-un 3922 df-in 3924 df-ss 3934 df-pss 3937 df-nul 4300 df-if 4492 df-pw 4568 df-sn 4593 df-pr 4595 df-tp 4597 df-op 4599 df-uni 4875 df-int 4914 df-iun 4960 df-iin 4961 df-br 5111 df-opab 5173 df-mpt 5192 df-tr 5218 df-id 5536 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5594 df-we 5596 df-xp 5647 df-rel 5648 df-cnv 5649 df-co 5650 df-dm 5651 df-rn 5652 df-res 5653 df-ima 5654 df-pred 6277 df-ord 6338 df-on 6339 df-lim 6340 df-suc 6341 df-iota 6467 df-fun 6516 df-fn 6517 df-f 6518 df-f1 6519 df-fo 6520 df-f1o 6521 df-fv 6522 df-riota 7347 df-ov 7393 df-oprab 7394 df-mpo 7395 df-of 7656 df-om 7846 df-1st 7971 df-2nd 7972 df-tpos 8208 df-undef 8255 df-frecs 8263 df-wrecs 8294 df-recs 8343 df-rdg 8381 df-1o 8437 df-2o 8438 df-er 8674 df-map 8804 df-en 8922 df-dom 8923 df-sdom 8924 df-fin 8925 df-pnf 11217 df-mnf 11218 df-xr 11219 df-ltxr 11220 df-le 11221 df-sub 11414 df-neg 11415 df-nn 12194 df-2 12256 df-3 12257 df-4 12258 df-5 12259 df-6 12260 df-n0 12450 df-z 12537 df-uz 12801 df-fz 13476 df-struct 17124 df-sets 17141 df-slot 17159 df-ndx 17171 df-base 17187 df-ress 17208 df-plusg 17240 df-mulr 17241 df-sca 17243 df-vsca 17244 df-0g 17411 df-mre 17554 df-mrc 17555 df-acs 17557 df-proset 18262 df-poset 18281 df-plt 18296 df-lub 18312 df-glb 18313 df-join 18314 df-meet 18315 df-p0 18391 df-p1 18392 df-lat 18398 df-clat 18465 df-mgm 18574 df-sgrp 18653 df-mnd 18669 df-submnd 18718 df-grp 18875 df-minusg 18876 df-sbg 18877 df-subg 19062 df-cntz 19256 df-oppg 19285 df-lsm 19573 df-cmn 19719 df-abl 19720 df-mgp 20057 df-rng 20069 df-ur 20098 df-ring 20151 df-oppr 20253 df-dvdsr 20273 df-unit 20274 df-invr 20304 df-dvr 20317 df-drng 20647 df-lmod 20775 df-lss 20845 df-lsp 20885 df-lvec 21017 df-lsatoms 38976 df-lshyp 38977 df-lcv 39019 df-lfl 39058 df-lkr 39086 df-ldual 39124 df-oposet 39176 df-ol 39178 df-oml 39179 df-covers 39266 df-ats 39267 df-atl 39298 df-cvlat 39322 df-hlat 39351 df-llines 39499 df-lplanes 39500 df-lvols 39501 df-lines 39502 df-psubsp 39504 df-pmap 39505 df-padd 39797 df-lhyp 39989 df-laut 39990 df-ldil 40105 df-ltrn 40106 df-trl 40160 df-tgrp 40744 df-tendo 40756 df-edring 40758 df-dveca 41004 df-disoa 41030 df-dvech 41080 df-dib 41140 df-dic 41174 df-dih 41230 df-doch 41349 df-djh 41396 |
| This theorem is referenced by: lclkrlem2y 41532 |
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