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Mirrors > Home > MPE Home > Th. List > Mathboxes > mapdpglem22 | Structured version Visualization version GIF version |
Description: Lemma for mapdpg 39967. Baer p. 45, line 9: "(F(x-y))* = ... = G(x'-y')." (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 |
---|---|
mapdpglem22 | ⊢ (𝜑 → (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐺𝑅𝐸)})) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mapdpglem4.jt | . 2 ⊢ (𝜑 → (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{𝑡})) | |
2 | mapdpglem.h | . . . 4 ⊢ 𝐻 = (LHyp‘𝐾) | |
3 | mapdpglem.c | . . . 4 ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) | |
4 | mapdpglem.k | . . . 4 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
5 | 2, 3, 4 | lcdlvec 39852 | . . 3 ⊢ (𝜑 → 𝐶 ∈ LVec) |
6 | mapdpglem.u | . . . . . . 7 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) | |
7 | 2, 6, 4 | dvhlvec 39370 | . . . . . 6 ⊢ (𝜑 → 𝑈 ∈ LVec) |
8 | mapdpglem3.a | . . . . . . 7 ⊢ 𝐴 = (Scalar‘𝑈) | |
9 | 8 | lvecdrng 20465 | . . . . . 6 ⊢ (𝑈 ∈ LVec → 𝐴 ∈ DivRing) |
10 | 7, 9 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ DivRing) |
11 | mapdpglem4.g4 | . . . . 5 ⊢ (𝜑 → 𝑔 ∈ 𝐵) | |
12 | mapdpglem.m | . . . . . 6 ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) | |
13 | mapdpglem.v | . . . . . 6 ⊢ 𝑉 = (Base‘𝑈) | |
14 | mapdpglem.s | . . . . . 6 ⊢ − = (-g‘𝑈) | |
15 | mapdpglem.n | . . . . . 6 ⊢ 𝑁 = (LSpan‘𝑈) | |
16 | mapdpglem.x | . . . . . 6 ⊢ (𝜑 → 𝑋 ∈ 𝑉) | |
17 | mapdpglem.y | . . . . . 6 ⊢ (𝜑 → 𝑌 ∈ 𝑉) | |
18 | mapdpglem1.p | . . . . . 6 ⊢ ⊕ = (LSSum‘𝐶) | |
19 | mapdpglem2.j | . . . . . 6 ⊢ 𝐽 = (LSpan‘𝐶) | |
20 | mapdpglem3.f | . . . . . 6 ⊢ 𝐹 = (Base‘𝐶) | |
21 | mapdpglem3.te | . . . . . 6 ⊢ (𝜑 → 𝑡 ∈ ((𝑀‘(𝑁‘{𝑋})) ⊕ (𝑀‘(𝑁‘{𝑌})))) | |
22 | mapdpglem3.b | . . . . . 6 ⊢ 𝐵 = (Base‘𝐴) | |
23 | mapdpglem3.t | . . . . . 6 ⊢ · = ( ·𝑠 ‘𝐶) | |
24 | mapdpglem3.r | . . . . . 6 ⊢ 𝑅 = (-g‘𝐶) | |
25 | mapdpglem3.g | . . . . . 6 ⊢ (𝜑 → 𝐺 ∈ 𝐹) | |
26 | mapdpglem3.e | . . . . . 6 ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐺})) | |
27 | mapdpglem4.q | . . . . . 6 ⊢ 𝑄 = (0g‘𝑈) | |
28 | mapdpglem.ne | . . . . . 6 ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌})) | |
29 | mapdpglem4.z | . . . . . 6 ⊢ 0 = (0g‘𝐴) | |
30 | mapdpglem4.z4 | . . . . . 6 ⊢ (𝜑 → 𝑧 ∈ (𝑀‘(𝑁‘{𝑌}))) | |
31 | mapdpglem4.t4 | . . . . . 6 ⊢ (𝜑 → 𝑡 = ((𝑔 · 𝐺)𝑅𝑧)) | |
32 | mapdpglem4.xn | . . . . . 6 ⊢ (𝜑 → 𝑋 ≠ 𝑄) | |
33 | 2, 12, 6, 13, 14, 15, 3, 4, 16, 17, 18, 19, 20, 21, 8, 22, 23, 24, 25, 26, 27, 28, 1, 29, 11, 30, 31, 32 | mapdpglem11 39943 | . . . . 5 ⊢ (𝜑 → 𝑔 ≠ 0 ) |
34 | eqid 2736 | . . . . . 6 ⊢ (invr‘𝐴) = (invr‘𝐴) | |
35 | 22, 29, 34 | drnginvrcl 20105 | . . . . 5 ⊢ ((𝐴 ∈ DivRing ∧ 𝑔 ∈ 𝐵 ∧ 𝑔 ≠ 0 ) → ((invr‘𝐴)‘𝑔) ∈ 𝐵) |
36 | 10, 11, 33, 35 | syl3anc 1370 | . . . 4 ⊢ (𝜑 → ((invr‘𝐴)‘𝑔) ∈ 𝐵) |
37 | eqid 2736 | . . . . 5 ⊢ (Scalar‘𝐶) = (Scalar‘𝐶) | |
38 | eqid 2736 | . . . . 5 ⊢ (Base‘(Scalar‘𝐶)) = (Base‘(Scalar‘𝐶)) | |
39 | 2, 6, 8, 22, 3, 37, 38, 4 | lcdsbase 39861 | . . . 4 ⊢ (𝜑 → (Base‘(Scalar‘𝐶)) = 𝐵) |
40 | 36, 39 | eleqtrrd 2840 | . . 3 ⊢ (𝜑 → ((invr‘𝐴)‘𝑔) ∈ (Base‘(Scalar‘𝐶))) |
41 | 22, 29, 34 | drnginvrn0 20106 | . . . . 5 ⊢ ((𝐴 ∈ DivRing ∧ 𝑔 ∈ 𝐵 ∧ 𝑔 ≠ 0 ) → ((invr‘𝐴)‘𝑔) ≠ 0 ) |
42 | 10, 11, 33, 41 | syl3anc 1370 | . . . 4 ⊢ (𝜑 → ((invr‘𝐴)‘𝑔) ≠ 0 ) |
43 | eqid 2736 | . . . . 5 ⊢ (0g‘(Scalar‘𝐶)) = (0g‘(Scalar‘𝐶)) | |
44 | 2, 6, 8, 29, 3, 37, 43, 4 | lcd0 39869 | . . . 4 ⊢ (𝜑 → (0g‘(Scalar‘𝐶)) = 0 ) |
45 | 42, 44 | neeqtrrd 3015 | . . 3 ⊢ (𝜑 → ((invr‘𝐴)‘𝑔) ≠ (0g‘(Scalar‘𝐶))) |
46 | 2, 12, 6, 13, 14, 15, 3, 4, 16, 17, 18, 19, 20, 21 | mapdpglem2a 39935 | . . 3 ⊢ (𝜑 → 𝑡 ∈ 𝐹) |
47 | 20, 37, 23, 38, 43, 19 | lspsnvs 20474 | . . 3 ⊢ ((𝐶 ∈ LVec ∧ (((invr‘𝐴)‘𝑔) ∈ (Base‘(Scalar‘𝐶)) ∧ ((invr‘𝐴)‘𝑔) ≠ (0g‘(Scalar‘𝐶))) ∧ 𝑡 ∈ 𝐹) → (𝐽‘{(((invr‘𝐴)‘𝑔) · 𝑡)}) = (𝐽‘{𝑡})) |
48 | 5, 40, 45, 46, 47 | syl121anc 1374 | . 2 ⊢ (𝜑 → (𝐽‘{(((invr‘𝐴)‘𝑔) · 𝑡)}) = (𝐽‘{𝑡})) |
49 | mapdpglem12.yn | . . . . 5 ⊢ (𝜑 → 𝑌 ≠ 𝑄) | |
50 | mapdpglem17.ep | . . . . 5 ⊢ 𝐸 = (((invr‘𝐴)‘𝑔) · 𝑧) | |
51 | 2, 12, 6, 13, 14, 15, 3, 4, 16, 17, 18, 19, 20, 21, 8, 22, 23, 24, 25, 26, 27, 28, 1, 29, 11, 30, 31, 32, 49, 50 | mapdpglem21 39953 | . . . 4 ⊢ (𝜑 → (((invr‘𝐴)‘𝑔) · 𝑡) = (𝐺𝑅𝐸)) |
52 | 51 | sneqd 4584 | . . 3 ⊢ (𝜑 → {(((invr‘𝐴)‘𝑔) · 𝑡)} = {(𝐺𝑅𝐸)}) |
53 | 52 | fveq2d 6823 | . 2 ⊢ (𝜑 → (𝐽‘{(((invr‘𝐴)‘𝑔) · 𝑡)}) = (𝐽‘{(𝐺𝑅𝐸)})) |
54 | 1, 48, 53 | 3eqtr2d 2782 | 1 ⊢ (𝜑 → (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐺𝑅𝐸)})) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1540 ∈ wcel 2105 ≠ wne 2940 {csn 4572 ‘cfv 6473 (class class class)co 7329 Basecbs 17001 Scalarcsca 17054 ·𝑠 cvsca 17055 0gc0g 17239 -gcsg 18667 LSSumclsm 19327 invrcinvr 20000 DivRingcdr 20085 LSpanclspn 20331 LVecclvec 20462 HLchlt 37610 LHypclh 38245 DVecHcdvh 39339 LCDualclcd 39847 mapdcmpd 39885 |
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 2707 ax-rep 5226 ax-sep 5240 ax-nul 5247 ax-pow 5305 ax-pr 5369 ax-un 7642 ax-cnex 11020 ax-resscn 11021 ax-1cn 11022 ax-icn 11023 ax-addcl 11024 ax-addrcl 11025 ax-mulcl 11026 ax-mulrcl 11027 ax-mulcom 11028 ax-addass 11029 ax-mulass 11030 ax-distr 11031 ax-i2m1 11032 ax-1ne0 11033 ax-1rid 11034 ax-rnegex 11035 ax-rrecex 11036 ax-cnre 11037 ax-pre-lttri 11038 ax-pre-lttrn 11039 ax-pre-ltadd 11040 ax-pre-mulgt0 11041 ax-riotaBAD 37213 |
This theorem depends on definitions: df-bi 206 df-an 397 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 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3349 df-reu 3350 df-rab 3404 df-v 3443 df-sbc 3727 df-csb 3843 df-dif 3900 df-un 3902 df-in 3904 df-ss 3914 df-pss 3916 df-nul 4269 df-if 4473 df-pw 4548 df-sn 4573 df-pr 4575 df-tp 4577 df-op 4579 df-uni 4852 df-int 4894 df-iun 4940 df-iin 4941 df-br 5090 df-opab 5152 df-mpt 5173 df-tr 5207 df-id 5512 df-eprel 5518 df-po 5526 df-so 5527 df-fr 5569 df-we 5571 df-xp 5620 df-rel 5621 df-cnv 5622 df-co 5623 df-dm 5624 df-rn 5625 df-res 5626 df-ima 5627 df-pred 6232 df-ord 6299 df-on 6300 df-lim 6301 df-suc 6302 df-iota 6425 df-fun 6475 df-fn 6476 df-f 6477 df-f1 6478 df-fo 6479 df-f1o 6480 df-fv 6481 df-riota 7286 df-ov 7332 df-oprab 7333 df-mpo 7334 df-of 7587 df-om 7773 df-1st 7891 df-2nd 7892 df-tpos 8104 df-undef 8151 df-frecs 8159 df-wrecs 8190 df-recs 8264 df-rdg 8303 df-1o 8359 df-er 8561 df-map 8680 df-en 8797 df-dom 8798 df-sdom 8799 df-fin 8800 df-pnf 11104 df-mnf 11105 df-xr 11106 df-ltxr 11107 df-le 11108 df-sub 11300 df-neg 11301 df-nn 12067 df-2 12129 df-3 12130 df-4 12131 df-5 12132 df-6 12133 df-n0 12327 df-z 12413 df-uz 12676 df-fz 13333 df-struct 16937 df-sets 16954 df-slot 16972 df-ndx 16984 df-base 17002 df-ress 17031 df-plusg 17064 df-mulr 17065 df-sca 17067 df-vsca 17068 df-0g 17241 df-mre 17384 df-mrc 17385 df-acs 17387 df-proset 18102 df-poset 18120 df-plt 18137 df-lub 18153 df-glb 18154 df-join 18155 df-meet 18156 df-p0 18232 df-p1 18233 df-lat 18239 df-clat 18306 df-mgm 18415 df-sgrp 18464 df-mnd 18475 df-submnd 18520 df-grp 18668 df-minusg 18669 df-sbg 18670 df-subg 18840 df-cntz 19011 df-oppg 19038 df-lsm 19329 df-cmn 19475 df-abl 19476 df-mgp 19808 df-ur 19825 df-ring 19872 df-oppr 19949 df-dvdsr 19970 df-unit 19971 df-invr 20001 df-dvr 20012 df-drng 20087 df-lmod 20223 df-lss 20292 df-lsp 20332 df-lvec 20463 df-lsatoms 37236 df-lshyp 37237 df-lcv 37279 df-lfl 37318 df-lkr 37346 df-ldual 37384 df-oposet 37436 df-ol 37438 df-oml 37439 df-covers 37526 df-ats 37527 df-atl 37558 df-cvlat 37582 df-hlat 37611 df-llines 37759 df-lplanes 37760 df-lvols 37761 df-lines 37762 df-psubsp 37764 df-pmap 37765 df-padd 38057 df-lhyp 38249 df-laut 38250 df-ldil 38365 df-ltrn 38366 df-trl 38420 df-tgrp 39004 df-tendo 39016 df-edring 39018 df-dveca 39264 df-disoa 39290 df-dvech 39340 df-dib 39400 df-dic 39434 df-dih 39490 df-doch 39609 df-djh 39656 df-lcdual 39848 df-mapd 39886 |
This theorem is referenced by: mapdpglem23 39955 |
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