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Mirrors > Home > MPE Home > Th. List > Mathboxes > mapdpglem20 | Structured version Visualization version GIF version |
Description: Lemma for mapdpg 41041. Baer p. 45, line 8: "...so that (Fy)*=Gy'." (Contributed by NM, 20-Mar-2015.) |
Ref | Expression |
---|---|
mapdpglem.h | ⊢ 𝐻 = (LHyp‘𝐾) |
mapdpglem.m | ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) |
mapdpglem.u | ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) |
mapdpglem.v | ⊢ 𝑉 = (Base‘𝑈) |
mapdpglem.s | ⊢ − = (-g‘𝑈) |
mapdpglem.n | ⊢ 𝑁 = (LSpan‘𝑈) |
mapdpglem.c | ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) |
mapdpglem.k | ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
mapdpglem.x | ⊢ (𝜑 → 𝑋 ∈ 𝑉) |
mapdpglem.y | ⊢ (𝜑 → 𝑌 ∈ 𝑉) |
mapdpglem1.p | ⊢ ⊕ = (LSSum‘𝐶) |
mapdpglem2.j | ⊢ 𝐽 = (LSpan‘𝐶) |
mapdpglem3.f | ⊢ 𝐹 = (Base‘𝐶) |
mapdpglem3.te | ⊢ (𝜑 → 𝑡 ∈ ((𝑀‘(𝑁‘{𝑋})) ⊕ (𝑀‘(𝑁‘{𝑌})))) |
mapdpglem3.a | ⊢ 𝐴 = (Scalar‘𝑈) |
mapdpglem3.b | ⊢ 𝐵 = (Base‘𝐴) |
mapdpglem3.t | ⊢ · = ( ·𝑠 ‘𝐶) |
mapdpglem3.r | ⊢ 𝑅 = (-g‘𝐶) |
mapdpglem3.g | ⊢ (𝜑 → 𝐺 ∈ 𝐹) |
mapdpglem3.e | ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐺})) |
mapdpglem4.q | ⊢ 𝑄 = (0g‘𝑈) |
mapdpglem.ne | ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌})) |
mapdpglem4.jt | ⊢ (𝜑 → (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{𝑡})) |
mapdpglem4.z | ⊢ 0 = (0g‘𝐴) |
mapdpglem4.g4 | ⊢ (𝜑 → 𝑔 ∈ 𝐵) |
mapdpglem4.z4 | ⊢ (𝜑 → 𝑧 ∈ (𝑀‘(𝑁‘{𝑌}))) |
mapdpglem4.t4 | ⊢ (𝜑 → 𝑡 = ((𝑔 · 𝐺)𝑅𝑧)) |
mapdpglem4.xn | ⊢ (𝜑 → 𝑋 ≠ 𝑄) |
mapdpglem12.yn | ⊢ (𝜑 → 𝑌 ≠ 𝑄) |
mapdpglem17.ep | ⊢ 𝐸 = (((invr‘𝐴)‘𝑔) · 𝑧) |
Ref | Expression |
---|---|
mapdpglem20 | ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑌})) = (𝐽‘{𝐸})) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2731 | . 2 ⊢ (0g‘𝐶) = (0g‘𝐶) | |
2 | mapdpglem2.j | . 2 ⊢ 𝐽 = (LSpan‘𝐶) | |
3 | eqid 2731 | . 2 ⊢ (LSAtoms‘𝐶) = (LSAtoms‘𝐶) | |
4 | mapdpglem.h | . . 3 ⊢ 𝐻 = (LHyp‘𝐾) | |
5 | mapdpglem.c | . . 3 ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) | |
6 | mapdpglem.k | . . 3 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
7 | 4, 5, 6 | lcdlvec 40926 | . 2 ⊢ (𝜑 → 𝐶 ∈ LVec) |
8 | mapdpglem.m | . . 3 ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) | |
9 | mapdpglem.u | . . 3 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) | |
10 | eqid 2731 | . . 3 ⊢ (LSAtoms‘𝑈) = (LSAtoms‘𝑈) | |
11 | mapdpglem.v | . . . 4 ⊢ 𝑉 = (Base‘𝑈) | |
12 | mapdpglem.n | . . . 4 ⊢ 𝑁 = (LSpan‘𝑈) | |
13 | mapdpglem4.q | . . . 4 ⊢ 𝑄 = (0g‘𝑈) | |
14 | 4, 9, 6 | dvhlmod 40445 | . . . 4 ⊢ (𝜑 → 𝑈 ∈ LMod) |
15 | mapdpglem.y | . . . . 5 ⊢ (𝜑 → 𝑌 ∈ 𝑉) | |
16 | mapdpglem12.yn | . . . . 5 ⊢ (𝜑 → 𝑌 ≠ 𝑄) | |
17 | eldifsn 4790 | . . . . 5 ⊢ (𝑌 ∈ (𝑉 ∖ {𝑄}) ↔ (𝑌 ∈ 𝑉 ∧ 𝑌 ≠ 𝑄)) | |
18 | 15, 16, 17 | sylanbrc 582 | . . . 4 ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ {𝑄})) |
19 | 11, 12, 13, 10, 14, 18 | lsatlspsn 38327 | . . 3 ⊢ (𝜑 → (𝑁‘{𝑌}) ∈ (LSAtoms‘𝑈)) |
20 | 4, 8, 9, 10, 5, 3, 6, 19 | mapdat 41002 | . 2 ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑌})) ∈ (LSAtoms‘𝐶)) |
21 | mapdpglem.s | . . 3 ⊢ − = (-g‘𝑈) | |
22 | mapdpglem.x | . . 3 ⊢ (𝜑 → 𝑋 ∈ 𝑉) | |
23 | mapdpglem1.p | . . 3 ⊢ ⊕ = (LSSum‘𝐶) | |
24 | mapdpglem3.f | . . 3 ⊢ 𝐹 = (Base‘𝐶) | |
25 | mapdpglem3.te | . . 3 ⊢ (𝜑 → 𝑡 ∈ ((𝑀‘(𝑁‘{𝑋})) ⊕ (𝑀‘(𝑁‘{𝑌})))) | |
26 | mapdpglem3.a | . . 3 ⊢ 𝐴 = (Scalar‘𝑈) | |
27 | mapdpglem3.b | . . 3 ⊢ 𝐵 = (Base‘𝐴) | |
28 | mapdpglem3.t | . . 3 ⊢ · = ( ·𝑠 ‘𝐶) | |
29 | mapdpglem3.r | . . 3 ⊢ 𝑅 = (-g‘𝐶) | |
30 | mapdpglem3.g | . . 3 ⊢ (𝜑 → 𝐺 ∈ 𝐹) | |
31 | mapdpglem3.e | . . 3 ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐺})) | |
32 | mapdpglem.ne | . . 3 ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌})) | |
33 | mapdpglem4.jt | . . 3 ⊢ (𝜑 → (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{𝑡})) | |
34 | mapdpglem4.z | . . 3 ⊢ 0 = (0g‘𝐴) | |
35 | mapdpglem4.g4 | . . 3 ⊢ (𝜑 → 𝑔 ∈ 𝐵) | |
36 | mapdpglem4.z4 | . . 3 ⊢ (𝜑 → 𝑧 ∈ (𝑀‘(𝑁‘{𝑌}))) | |
37 | mapdpglem4.t4 | . . 3 ⊢ (𝜑 → 𝑡 = ((𝑔 · 𝐺)𝑅𝑧)) | |
38 | mapdpglem4.xn | . . 3 ⊢ (𝜑 → 𝑋 ≠ 𝑄) | |
39 | mapdpglem17.ep | . . 3 ⊢ 𝐸 = (((invr‘𝐴)‘𝑔) · 𝑧) | |
40 | 4, 8, 9, 11, 21, 12, 5, 6, 22, 15, 23, 2, 24, 25, 26, 27, 28, 29, 30, 31, 13, 32, 33, 34, 35, 36, 37, 38, 16, 39 | mapdpglem19 41025 | . 2 ⊢ (𝜑 → 𝐸 ∈ (𝑀‘(𝑁‘{𝑌}))) |
41 | 4, 8, 9, 11, 21, 12, 5, 6, 22, 15, 23, 2, 24, 25, 26, 27, 28, 29, 30, 31, 13, 32, 33, 34, 35, 36, 37, 38, 16, 39 | mapdpglem18 41024 | . 2 ⊢ (𝜑 → 𝐸 ≠ (0g‘𝐶)) |
42 | 1, 2, 3, 7, 20, 40, 41 | lsatel 38339 | 1 ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑌})) = (𝐽‘{𝐸})) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2105 ≠ wne 2939 ∖ cdif 3945 {csn 4628 ‘cfv 6543 (class class class)co 7412 Basecbs 17151 Scalarcsca 17207 ·𝑠 cvsca 17208 0gc0g 17392 -gcsg 18863 LSSumclsm 19550 invrcinvr 20285 LSpanclspn 20814 LSAtomsclsa 38308 HLchlt 38684 LHypclh 39319 DVecHcdvh 40413 LCDualclcd 40921 mapdcmpd 40959 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 ax-cnex 11172 ax-resscn 11173 ax-1cn 11174 ax-icn 11175 ax-addcl 11176 ax-addrcl 11177 ax-mulcl 11178 ax-mulrcl 11179 ax-mulcom 11180 ax-addass 11181 ax-mulass 11182 ax-distr 11183 ax-i2m1 11184 ax-1ne0 11185 ax-1rid 11186 ax-rnegex 11187 ax-rrecex 11188 ax-cnre 11189 ax-pre-lttri 11190 ax-pre-lttrn 11191 ax-pre-ltadd 11192 ax-pre-mulgt0 11193 ax-riotaBAD 38287 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-tp 4633 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-iin 5000 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-of 7674 df-om 7860 df-1st 7979 df-2nd 7980 df-tpos 8217 df-undef 8264 df-frecs 8272 df-wrecs 8303 df-recs 8377 df-rdg 8416 df-1o 8472 df-er 8709 df-map 8828 df-en 8946 df-dom 8947 df-sdom 8948 df-fin 8949 df-pnf 11257 df-mnf 11258 df-xr 11259 df-ltxr 11260 df-le 11261 df-sub 11453 df-neg 11454 df-nn 12220 df-2 12282 df-3 12283 df-4 12284 df-5 12285 df-6 12286 df-n0 12480 df-z 12566 df-uz 12830 df-fz 13492 df-struct 17087 df-sets 17104 df-slot 17122 df-ndx 17134 df-base 17152 df-ress 17181 df-plusg 17217 df-mulr 17218 df-sca 17220 df-vsca 17221 df-0g 17394 df-mre 17537 df-mrc 17538 df-acs 17540 df-proset 18258 df-poset 18276 df-plt 18293 df-lub 18309 df-glb 18310 df-join 18311 df-meet 18312 df-p0 18388 df-p1 18389 df-lat 18395 df-clat 18462 df-mgm 18571 df-sgrp 18650 df-mnd 18666 df-submnd 18712 df-grp 18864 df-minusg 18865 df-sbg 18866 df-subg 19046 df-cntz 19229 df-oppg 19258 df-lsm 19552 df-cmn 19698 df-abl 19699 df-mgp 20036 df-rng 20054 df-ur 20083 df-ring 20136 df-oppr 20232 df-dvdsr 20255 df-unit 20256 df-invr 20286 df-dvr 20299 df-drng 20585 df-lmod 20704 df-lss 20775 df-lsp 20815 df-lvec 20947 df-lsatoms 38310 df-lshyp 38311 df-lcv 38353 df-lfl 38392 df-lkr 38420 df-ldual 38458 df-oposet 38510 df-ol 38512 df-oml 38513 df-covers 38600 df-ats 38601 df-atl 38632 df-cvlat 38656 df-hlat 38685 df-llines 38833 df-lplanes 38834 df-lvols 38835 df-lines 38836 df-psubsp 38838 df-pmap 38839 df-padd 39131 df-lhyp 39323 df-laut 39324 df-ldil 39439 df-ltrn 39440 df-trl 39494 df-tgrp 40078 df-tendo 40090 df-edring 40092 df-dveca 40338 df-disoa 40364 df-dvech 40414 df-dib 40474 df-dic 40508 df-dih 40564 df-doch 40683 df-djh 40730 df-lcdual 40922 df-mapd 40960 |
This theorem is referenced by: mapdpglem23 41029 |
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