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Mirrors > Home > MPE Home > Th. List > Mathboxes > mapdpglem22 | Structured version Visualization version GIF version |
Description: Lemma for mapdpg 40169. 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 40054 | . . 3 ⊢ (𝜑 → 𝐶 ∈ LVec) |
6 | mapdpglem.u | . . . . . . 7 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) | |
7 | 2, 6, 4 | dvhlvec 39572 | . . . . . 6 ⊢ (𝜑 → 𝑈 ∈ LVec) |
8 | mapdpglem3.a | . . . . . . 7 ⊢ 𝐴 = (Scalar‘𝑈) | |
9 | 8 | lvecdrng 20566 | . . . . . 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 40145 | . . . . 5 ⊢ (𝜑 → 𝑔 ≠ 0 ) |
34 | eqid 2736 | . . . . . 6 ⊢ (invr‘𝐴) = (invr‘𝐴) | |
35 | 22, 29, 34 | drnginvrcl 20205 | . . . . 5 ⊢ ((𝐴 ∈ DivRing ∧ 𝑔 ∈ 𝐵 ∧ 𝑔 ≠ 0 ) → ((invr‘𝐴)‘𝑔) ∈ 𝐵) |
36 | 10, 11, 33, 35 | syl3anc 1371 | . . . 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 40063 | . . . 4 ⊢ (𝜑 → (Base‘(Scalar‘𝐶)) = 𝐵) |
40 | 36, 39 | eleqtrrd 2841 | . . 3 ⊢ (𝜑 → ((invr‘𝐴)‘𝑔) ∈ (Base‘(Scalar‘𝐶))) |
41 | 22, 29, 34 | drnginvrn0 20206 | . . . . 5 ⊢ ((𝐴 ∈ DivRing ∧ 𝑔 ∈ 𝐵 ∧ 𝑔 ≠ 0 ) → ((invr‘𝐴)‘𝑔) ≠ 0 ) |
42 | 10, 11, 33, 41 | syl3anc 1371 | . . . 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 40071 | . . . 4 ⊢ (𝜑 → (0g‘(Scalar‘𝐶)) = 0 ) |
45 | 42, 44 | neeqtrrd 3018 | . . 3 ⊢ (𝜑 → ((invr‘𝐴)‘𝑔) ≠ (0g‘(Scalar‘𝐶))) |
46 | 2, 12, 6, 13, 14, 15, 3, 4, 16, 17, 18, 19, 20, 21 | mapdpglem2a 40137 | . . 3 ⊢ (𝜑 → 𝑡 ∈ 𝐹) |
47 | 20, 37, 23, 38, 43, 19 | lspsnvs 20575 | . . 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 1375 | . 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 40155 | . . . 4 ⊢ (𝜑 → (((invr‘𝐴)‘𝑔) · 𝑡) = (𝐺𝑅𝐸)) |
52 | 51 | sneqd 4598 | . . 3 ⊢ (𝜑 → {(((invr‘𝐴)‘𝑔) · 𝑡)} = {(𝐺𝑅𝐸)}) |
53 | 52 | fveq2d 6846 | . 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 1541 ∈ wcel 2106 ≠ wne 2943 {csn 4586 ‘cfv 6496 (class class class)co 7357 Basecbs 17083 Scalarcsca 17136 ·𝑠 cvsca 17137 0gc0g 17321 -gcsg 18750 LSSumclsm 19416 invrcinvr 20100 DivRingcdr 20185 LSpanclspn 20432 LVecclvec 20563 HLchlt 37812 LHypclh 38447 DVecHcdvh 39541 LCDualclcd 40049 mapdcmpd 40087 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-rep 5242 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7672 ax-cnex 11107 ax-resscn 11108 ax-1cn 11109 ax-icn 11110 ax-addcl 11111 ax-addrcl 11112 ax-mulcl 11113 ax-mulrcl 11114 ax-mulcom 11115 ax-addass 11116 ax-mulass 11117 ax-distr 11118 ax-i2m1 11119 ax-1ne0 11120 ax-1rid 11121 ax-rnegex 11122 ax-rrecex 11123 ax-cnre 11124 ax-pre-lttri 11125 ax-pre-lttrn 11126 ax-pre-ltadd 11127 ax-pre-mulgt0 11128 ax-riotaBAD 37415 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3065 df-rex 3074 df-rmo 3353 df-reu 3354 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-pss 3929 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-tp 4591 df-op 4593 df-uni 4866 df-int 4908 df-iun 4956 df-iin 4957 df-br 5106 df-opab 5168 df-mpt 5189 df-tr 5223 df-id 5531 df-eprel 5537 df-po 5545 df-so 5546 df-fr 5588 df-we 5590 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-pred 6253 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-riota 7313 df-ov 7360 df-oprab 7361 df-mpo 7362 df-of 7617 df-om 7803 df-1st 7921 df-2nd 7922 df-tpos 8157 df-undef 8204 df-frecs 8212 df-wrecs 8243 df-recs 8317 df-rdg 8356 df-1o 8412 df-er 8648 df-map 8767 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-pnf 11191 df-mnf 11192 df-xr 11193 df-ltxr 11194 df-le 11195 df-sub 11387 df-neg 11388 df-nn 12154 df-2 12216 df-3 12217 df-4 12218 df-5 12219 df-6 12220 df-n0 12414 df-z 12500 df-uz 12764 df-fz 13425 df-struct 17019 df-sets 17036 df-slot 17054 df-ndx 17066 df-base 17084 df-ress 17113 df-plusg 17146 df-mulr 17147 df-sca 17149 df-vsca 17150 df-0g 17323 df-mre 17466 df-mrc 17467 df-acs 17469 df-proset 18184 df-poset 18202 df-plt 18219 df-lub 18235 df-glb 18236 df-join 18237 df-meet 18238 df-p0 18314 df-p1 18315 df-lat 18321 df-clat 18388 df-mgm 18497 df-sgrp 18546 df-mnd 18557 df-submnd 18602 df-grp 18751 df-minusg 18752 df-sbg 18753 df-subg 18925 df-cntz 19097 df-oppg 19124 df-lsm 19418 df-cmn 19564 df-abl 19565 df-mgp 19897 df-ur 19914 df-ring 19966 df-oppr 20049 df-dvdsr 20070 df-unit 20071 df-invr 20101 df-dvr 20112 df-drng 20187 df-lmod 20324 df-lss 20393 df-lsp 20433 df-lvec 20564 df-lsatoms 37438 df-lshyp 37439 df-lcv 37481 df-lfl 37520 df-lkr 37548 df-ldual 37586 df-oposet 37638 df-ol 37640 df-oml 37641 df-covers 37728 df-ats 37729 df-atl 37760 df-cvlat 37784 df-hlat 37813 df-llines 37961 df-lplanes 37962 df-lvols 37963 df-lines 37964 df-psubsp 37966 df-pmap 37967 df-padd 38259 df-lhyp 38451 df-laut 38452 df-ldil 38567 df-ltrn 38568 df-trl 38622 df-tgrp 39206 df-tendo 39218 df-edring 39220 df-dveca 39466 df-disoa 39492 df-dvech 39542 df-dib 39602 df-dic 39636 df-dih 39692 df-doch 39811 df-djh 39858 df-lcdual 40050 df-mapd 40088 |
This theorem is referenced by: mapdpglem23 40157 |
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