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| Mirrors > Home > MPE Home > Th. List > Mathboxes > lclkrlem2u | Structured version Visualization version GIF version | ||
| Description: Lemma for lclkr 41535. lclkrlem2t 41528 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 41112 | . . . . . 6 ⊢ (𝜑 → 𝑈 ∈ LMod) |
| 25 | 8, 9, 10, 24, 14, 13 | ldualvaddcom 39141 | . . . . 5 ⊢ (𝜑 → (𝐸 + 𝐺) = (𝐺 + 𝐸)) |
| 26 | 25 | fveq1d 6908 | . . . 4 ⊢ (𝜑 → ((𝐸 + 𝐺)‘𝑋) = ((𝐺 + 𝐸)‘𝑋)) |
| 27 | lclkrlem2u.n | . . . 4 ⊢ (𝜑 → ((𝐸 + 𝐺)‘𝑋) ≠ 0 ) | |
| 28 | 26, 27 | eqnetrrd 3009 | . . 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 41528 | . 2 ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘(𝐿‘(𝐺 + 𝐸)))) = (𝐿‘(𝐺 + 𝐸))) |
| 30 | 25 | fveq2d 6910 | . . . 4 ⊢ (𝜑 → (𝐿‘(𝐸 + 𝐺)) = (𝐿‘(𝐺 + 𝐸))) |
| 31 | 30 | fveq2d 6910 | . . 3 ⊢ (𝜑 → ( ⊥ ‘(𝐿‘(𝐸 + 𝐺))) = ( ⊥ ‘(𝐿‘(𝐺 + 𝐸)))) |
| 32 | 31 | fveq2d 6910 | . 2 ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘(𝐿‘(𝐸 + 𝐺)))) = ( ⊥ ‘( ⊥ ‘(𝐿‘(𝐺 + 𝐸))))) |
| 33 | 29, 32, 30 | 3eqtr4d 2787 | 1 ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘(𝐿‘(𝐸 + 𝐺)))) = (𝐿‘(𝐸 + 𝐺))) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ≠ wne 2940 {csn 4626 ‘cfv 6561 (class class class)co 7431 Basecbs 17247 +gcplusg 17297 .rcmulr 17298 Scalarcsca 17300 ·𝑠 cvsca 17301 0gc0g 17484 -gcsg 18953 LSSumclsm 19652 invrcinvr 20387 LSpanclspn 20969 LFnlclfn 39058 LKerclk 39086 LDualcld 39124 HLchlt 39351 LHypclh 39986 DVecHcdvh 41080 ocHcoch 41349 |
| 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 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-riotaBAD 38954 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-tp 4631 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-iin 4994 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-om 7888 df-1st 8014 df-2nd 8015 df-tpos 8251 df-undef 8298 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-2o 8507 df-er 8745 df-map 8868 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-5 12332 df-6 12333 df-n0 12527 df-z 12614 df-uz 12879 df-fz 13548 df-struct 17184 df-sets 17201 df-slot 17219 df-ndx 17231 df-base 17248 df-ress 17275 df-plusg 17310 df-mulr 17311 df-sca 17313 df-vsca 17314 df-0g 17486 df-mre 17629 df-mrc 17630 df-acs 17632 df-proset 18340 df-poset 18359 df-plt 18375 df-lub 18391 df-glb 18392 df-join 18393 df-meet 18394 df-p0 18470 df-p1 18471 df-lat 18477 df-clat 18544 df-mgm 18653 df-sgrp 18732 df-mnd 18748 df-submnd 18797 df-grp 18954 df-minusg 18955 df-sbg 18956 df-subg 19141 df-cntz 19335 df-oppg 19364 df-lsm 19654 df-cmn 19800 df-abl 19801 df-mgp 20138 df-rng 20150 df-ur 20179 df-ring 20232 df-oppr 20334 df-dvdsr 20357 df-unit 20358 df-invr 20388 df-dvr 20401 df-drng 20731 df-lmod 20860 df-lss 20930 df-lsp 20970 df-lvec 21102 df-lsatoms 38977 df-lshyp 38978 df-lcv 39020 df-lfl 39059 df-lkr 39087 df-ldual 39125 df-oposet 39177 df-ol 39179 df-oml 39180 df-covers 39267 df-ats 39268 df-atl 39299 df-cvlat 39323 df-hlat 39352 df-llines 39500 df-lplanes 39501 df-lvols 39502 df-lines 39503 df-psubsp 39505 df-pmap 39506 df-padd 39798 df-lhyp 39990 df-laut 39991 df-ldil 40106 df-ltrn 40107 df-trl 40161 df-tgrp 40745 df-tendo 40757 df-edring 40759 df-dveca 41005 df-disoa 41031 df-dvech 41081 df-dib 41141 df-dic 41175 df-dih 41231 df-doch 41350 df-djh 41397 |
| This theorem is referenced by: lclkrlem2x 41532 |
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