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| Mirrors > Home > MPE Home > Th. List > Mathboxes > lclkrlem2u | Structured version Visualization version GIF version | ||
| Description: Lemma for lclkr 42407. lclkrlem2t 42400 with 𝑋 and 𝑌 swapped. (Contributed by NM, 18-Jan-2015.) |
| Ref | Expression |
|---|---|
| lclkrlem2m.v | ⊢ 𝑉 = (Base‘𝑈) |
| lclkrlem2m.t | ⊢ · = ( ·𝑠 ‘𝑈) |
| lclkrlem2m.s | ⊢ 𝑆 = (Scalar‘𝑈) |
| lclkrlem2m.q | ⊢ × = (.r‘𝑆) |
| lclkrlem2m.z | ⊢ 0 = (0g‘𝑆) |
| lclkrlem2m.i | ⊢ 𝐼 = (invr‘𝑆) |
| lclkrlem2m.m | ⊢ − = (-g‘𝑈) |
| lclkrlem2m.f | ⊢ 𝐹 = (LFnl‘𝑈) |
| lclkrlem2m.d | ⊢ 𝐷 = (LDual‘𝑈) |
| lclkrlem2m.p | ⊢ + = (+g‘𝐷) |
| lclkrlem2m.x | ⊢ (𝜑 → 𝑋 ∈ 𝑉) |
| lclkrlem2m.y | ⊢ (𝜑 → 𝑌 ∈ 𝑉) |
| lclkrlem2m.e | ⊢ (𝜑 → 𝐸 ∈ 𝐹) |
| lclkrlem2m.g | ⊢ (𝜑 → 𝐺 ∈ 𝐹) |
| lclkrlem2n.n | ⊢ 𝑁 = (LSpan‘𝑈) |
| lclkrlem2n.l | ⊢ 𝐿 = (LKer‘𝑈) |
| lclkrlem2o.h | ⊢ 𝐻 = (LHyp‘𝐾) |
| lclkrlem2o.o | ⊢ ⊥ = ((ocH‘𝐾)‘𝑊) |
| lclkrlem2o.u | ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) |
| lclkrlem2o.a | ⊢ ⊕ = (LSSum‘𝑈) |
| lclkrlem2o.k | ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
| lclkrlem2q.le | ⊢ (𝜑 → (𝐿‘𝐸) = ( ⊥ ‘{𝑋})) |
| lclkrlem2q.lg | ⊢ (𝜑 → (𝐿‘𝐺) = ( ⊥ ‘{𝑌})) |
| lclkrlem2u.n | ⊢ (𝜑 → ((𝐸 + 𝐺)‘𝑋) ≠ 0 ) |
| Ref | Expression |
|---|---|
| lclkrlem2u | ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘(𝐿‘(𝐸 + 𝐺)))) = (𝐿‘(𝐸 + 𝐺))) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | lclkrlem2m.v | . . 3 ⊢ 𝑉 = (Base‘𝑈) | |
| 2 | lclkrlem2m.t | . . 3 ⊢ · = ( ·𝑠 ‘𝑈) | |
| 3 | lclkrlem2m.s | . . 3 ⊢ 𝑆 = (Scalar‘𝑈) | |
| 4 | lclkrlem2m.q | . . 3 ⊢ × = (.r‘𝑆) | |
| 5 | lclkrlem2m.z | . . 3 ⊢ 0 = (0g‘𝑆) | |
| 6 | lclkrlem2m.i | . . 3 ⊢ 𝐼 = (invr‘𝑆) | |
| 7 | lclkrlem2m.m | . . 3 ⊢ − = (-g‘𝑈) | |
| 8 | lclkrlem2m.f | . . 3 ⊢ 𝐹 = (LFnl‘𝑈) | |
| 9 | lclkrlem2m.d | . . 3 ⊢ 𝐷 = (LDual‘𝑈) | |
| 10 | lclkrlem2m.p | . . 3 ⊢ + = (+g‘𝐷) | |
| 11 | lclkrlem2m.y | . . 3 ⊢ (𝜑 → 𝑌 ∈ 𝑉) | |
| 12 | lclkrlem2m.x | . . 3 ⊢ (𝜑 → 𝑋 ∈ 𝑉) | |
| 13 | lclkrlem2m.g | . . 3 ⊢ (𝜑 → 𝐺 ∈ 𝐹) | |
| 14 | lclkrlem2m.e | . . 3 ⊢ (𝜑 → 𝐸 ∈ 𝐹) | |
| 15 | lclkrlem2n.n | . . 3 ⊢ 𝑁 = (LSpan‘𝑈) | |
| 16 | lclkrlem2n.l | . . 3 ⊢ 𝐿 = (LKer‘𝑈) | |
| 17 | lclkrlem2o.h | . . 3 ⊢ 𝐻 = (LHyp‘𝐾) | |
| 18 | lclkrlem2o.o | . . 3 ⊢ ⊥ = ((ocH‘𝐾)‘𝑊) | |
| 19 | lclkrlem2o.u | . . 3 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) | |
| 20 | lclkrlem2o.a | . . 3 ⊢ ⊕ = (LSSum‘𝑈) | |
| 21 | lclkrlem2o.k | . . 3 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
| 22 | lclkrlem2q.lg | . . 3 ⊢ (𝜑 → (𝐿‘𝐺) = ( ⊥ ‘{𝑌})) | |
| 23 | lclkrlem2q.le | . . 3 ⊢ (𝜑 → (𝐿‘𝐸) = ( ⊥ ‘{𝑋})) | |
| 24 | 17, 19, 21 | dvhlmod 41984 | . . . . . 6 ⊢ (𝜑 → 𝑈 ∈ LMod) |
| 25 | 8, 9, 10, 24, 14, 13 | ldualvaddcom 40014 | . . . . 5 ⊢ (𝜑 → (𝐸 + 𝐺) = (𝐺 + 𝐸)) |
| 26 | 25 | fveq1d 6881 | . . . 4 ⊢ (𝜑 → ((𝐸 + 𝐺)‘𝑋) = ((𝐺 + 𝐸)‘𝑋)) |
| 27 | lclkrlem2u.n | . . . 4 ⊢ (𝜑 → ((𝐸 + 𝐺)‘𝑋) ≠ 0 ) | |
| 28 | 26, 27 | eqnetrrd 3023 | . . 3 ⊢ (𝜑 → ((𝐺 + 𝐸)‘𝑋) ≠ 0 ) |
| 29 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 28 | lclkrlem2t 42400 | . 2 ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘(𝐿‘(𝐺 + 𝐸)))) = (𝐿‘(𝐺 + 𝐸))) |
| 30 | 25 | fveq2d 6883 | . . . 4 ⊢ (𝜑 → (𝐿‘(𝐸 + 𝐺)) = (𝐿‘(𝐺 + 𝐸))) |
| 31 | 30 | fveq2d 6883 | . . 3 ⊢ (𝜑 → ( ⊥ ‘(𝐿‘(𝐸 + 𝐺))) = ( ⊥ ‘(𝐿‘(𝐺 + 𝐸)))) |
| 32 | 31 | fveq2d 6883 | . 2 ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘(𝐿‘(𝐸 + 𝐺)))) = ( ⊥ ‘( ⊥ ‘(𝐿‘(𝐺 + 𝐸))))) |
| 33 | 29, 32, 30 | 3eqtr4d 2805 | 1 ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘(𝐿‘(𝐸 + 𝐺)))) = (𝐿‘(𝐸 + 𝐺))) |
| Colors of variables: wff setvar class |
| This proof depends on syntax axioms: → wi 4 ∧ wa 401 = wceq 1570 ∈ wcel 2145 ≠ wne 2955 {csn 4584 ‘cfv 6533 (class class class)co 7414 Basecbs 17302 +gcplusg 17343 .rcmulr 17344 Scalarcsca 17346 ·𝑠 cvsca 17347 0gc0g 17525 -gcsg 19060 LSSumclsm 19762 invrcinvr 20529 LSpanclspn 21156 LFnlclfn 39931 LKerclk 39959 LDualcld 39997 HLchlt 40224 LHypclh 40858 DVecHcdvh 41952 ocHcoch 42221 |
| This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2732 ax-rep 5232 ax-sep 5251 ax-nul 5263 ax-pow 5330 ax-pr 5398 ax-un 7737 ax-cnex 11181 ax-resscn 11182 ax-1cn 11183 ax-icn 11184 ax-addcl 11185 ax-addrcl 11186 ax-mulcl 11187 ax-mulrcl 11188 ax-mulcom 11189 ax-addass 11190 ax-mulass 11191 ax-distr 11192 ax-i2m1 11193 ax-1ne0 11194 ax-1rid 11195 ax-rnegex 11196 ax-rrecex 11197 ax-cnre 11198 ax-pre-lttri 11199 ax-pre-lttrn 11200 ax-pre-ltadd 11201 ax-pre-mulgt0 11202 ax-riotaBAD 39827 |
| This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3365 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4591 df-uni 4868 df-int 4908 df-iun 4953 df-iin 4954 df-br 5104 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5550 df-eprel 5555 df-po 5563 df-so 5564 df-fr 5608 df-we 5610 df-xp 5661 df-rel 5662 df-cnv 5663 df-co 5664 df-dm 5665 df-rn 5666 df-res 5667 df-ima 5668 df-pred 6299 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6489 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-riota 7371 df-ov 7417 df-oprab 7418 df-mpo 7419 df-of 7679 df-om 7864 df-1st 7987 df-2nd 7988 df-tpos 8225 df-undef 8272 df-frecs 8281 df-wrecs 8312 df-recs 8361 df-rdg 8400 df-1o 8456 df-2o 8457 df-er 8697 df-map 8829 df-en 8954 df-dom 8955 df-sdom 8956 df-fin 8957 df-pnf 11270 df-mnf 11271 df-xr 11272 df-ltxr 11273 df-le 11274 df-sub 11468 df-neg 11469 df-nn 12259 df-2 12328 df-3 12329 df-4 12330 df-5 12331 df-6 12332 df-n0 12530 df-z 12617 df-uz 12889 df-fz 13563 df-struct 17240 df-sets 17257 df-slot 17275 df-ndx 17287 df-base 17303 df-ress 17324 df-plusg 17356 df-mulr 17357 df-sca 17359 df-vsca 17360 df-0g 17527 df-mre 17671 df-mrc 17672 df-acs 17674 df-proset 18383 df-poset 18402 df-plt 18417 df-lub 18433 df-glb 18434 df-join 18435 df-meet 18436 df-p0 18512 df-p1 18513 df-lat 18521 df-clat 18588 df-mgm 18731 df-sgrp 18822 df-mnd 18838 df-submnd 18893 df-grp 19061 df-minusg 19062 df-sbg 19063 df-subg 19247 df-cntz 19445 df-oppg 19474 df-lsm 19764 df-cmn 19910 df-abl 19911 df-mgp 20275 df-rng 20289 df-ur 20322 df-ring 20375 df-oppr 20479 df-dvdsr 20499 df-unit 20500 df-invr 20530 df-dvr 20543 df-drng 20893 df-lmod 21047 df-lss 21117 df-lsp 21157 df-lvec 21288 df-lsatoms 39850 df-lshyp 39851 df-lcv 39893 df-lfl 39932 df-lkr 39960 df-ldual 39998 df-oposet 40050 df-ol 40052 df-oml 40053 df-covers 40140 df-ats 40141 df-atl 40172 df-cvlat 40196 df-hlat 40225 df-llines 40372 df-lplanes 40373 df-lvols 40374 df-lines 40375 df-psubsp 40377 df-pmap 40378 df-padd 40670 df-lhyp 40862 df-laut 40863 df-ldil 40978 df-ltrn 40979 df-trl 41033 df-tgrp 41617 df-tendo 41629 df-edring 41631 df-dveca 41877 df-disoa 41903 df-dvech 41953 df-dib 42013 df-dic 42047 df-dih 42103 df-doch 42222 df-djh 42269 |
| This theorem is used by: lclkrlem2x 42404 |
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