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Mirrors > Home > MPE Home > Th. List > Mathboxes > mapdpglem17N | Structured version Visualization version GIF version |
Description: Lemma for mapdpg 39716. Baer p. 45, line 7: "Hence we may form y' = g^-1 z." (Contributed by NM, 20-Mar-2015.) (New usage is discouraged.) |
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 |
---|---|
mapdpglem17N | ⊢ (𝜑 → 𝐸 ∈ 𝐹) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mapdpglem17.ep | . 2 ⊢ 𝐸 = (((invr‘𝐴)‘𝑔) · 𝑧) | |
2 | mapdpglem.h | . . 3 ⊢ 𝐻 = (LHyp‘𝐾) | |
3 | mapdpglem.u | . . 3 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) | |
4 | mapdpglem3.a | . . 3 ⊢ 𝐴 = (Scalar‘𝑈) | |
5 | mapdpglem3.b | . . 3 ⊢ 𝐵 = (Base‘𝐴) | |
6 | mapdpglem.c | . . 3 ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) | |
7 | mapdpglem3.f | . . 3 ⊢ 𝐹 = (Base‘𝐶) | |
8 | mapdpglem3.t | . . 3 ⊢ · = ( ·𝑠 ‘𝐶) | |
9 | mapdpglem.k | . . 3 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
10 | 2, 3, 9 | dvhlvec 39119 | . . . . 5 ⊢ (𝜑 → 𝑈 ∈ LVec) |
11 | 4 | lvecdrng 20365 | . . . . 5 ⊢ (𝑈 ∈ LVec → 𝐴 ∈ DivRing) |
12 | 10, 11 | syl 17 | . . . 4 ⊢ (𝜑 → 𝐴 ∈ DivRing) |
13 | mapdpglem4.g4 | . . . 4 ⊢ (𝜑 → 𝑔 ∈ 𝐵) | |
14 | mapdpglem.m | . . . . 5 ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) | |
15 | mapdpglem.v | . . . . 5 ⊢ 𝑉 = (Base‘𝑈) | |
16 | mapdpglem.s | . . . . 5 ⊢ − = (-g‘𝑈) | |
17 | mapdpglem.n | . . . . 5 ⊢ 𝑁 = (LSpan‘𝑈) | |
18 | mapdpglem.x | . . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝑉) | |
19 | mapdpglem.y | . . . . 5 ⊢ (𝜑 → 𝑌 ∈ 𝑉) | |
20 | mapdpglem1.p | . . . . 5 ⊢ ⊕ = (LSSum‘𝐶) | |
21 | mapdpglem2.j | . . . . 5 ⊢ 𝐽 = (LSpan‘𝐶) | |
22 | mapdpglem3.te | . . . . 5 ⊢ (𝜑 → 𝑡 ∈ ((𝑀‘(𝑁‘{𝑋})) ⊕ (𝑀‘(𝑁‘{𝑌})))) | |
23 | mapdpglem3.r | . . . . 5 ⊢ 𝑅 = (-g‘𝐶) | |
24 | mapdpglem3.g | . . . . 5 ⊢ (𝜑 → 𝐺 ∈ 𝐹) | |
25 | mapdpglem3.e | . . . . 5 ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐺})) | |
26 | mapdpglem4.q | . . . . 5 ⊢ 𝑄 = (0g‘𝑈) | |
27 | mapdpglem.ne | . . . . 5 ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌})) | |
28 | mapdpglem4.jt | . . . . 5 ⊢ (𝜑 → (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{𝑡})) | |
29 | mapdpglem4.z | . . . . 5 ⊢ 0 = (0g‘𝐴) | |
30 | mapdpglem4.z4 | . . . . 5 ⊢ (𝜑 → 𝑧 ∈ (𝑀‘(𝑁‘{𝑌}))) | |
31 | mapdpglem4.t4 | . . . . 5 ⊢ (𝜑 → 𝑡 = ((𝑔 · 𝐺)𝑅𝑧)) | |
32 | mapdpglem4.xn | . . . . 5 ⊢ (𝜑 → 𝑋 ≠ 𝑄) | |
33 | 2, 14, 3, 15, 16, 17, 6, 9, 18, 19, 20, 21, 7, 22, 4, 5, 8, 23, 24, 25, 26, 27, 28, 29, 13, 30, 31, 32 | mapdpglem11 39692 | . . . 4 ⊢ (𝜑 → 𝑔 ≠ 0 ) |
34 | eqid 2740 | . . . . 5 ⊢ (invr‘𝐴) = (invr‘𝐴) | |
35 | 5, 29, 34 | drnginvrcl 20006 | . . . 4 ⊢ ((𝐴 ∈ DivRing ∧ 𝑔 ∈ 𝐵 ∧ 𝑔 ≠ 0 ) → ((invr‘𝐴)‘𝑔) ∈ 𝐵) |
36 | 12, 13, 33, 35 | syl3anc 1370 | . . 3 ⊢ (𝜑 → ((invr‘𝐴)‘𝑔) ∈ 𝐵) |
37 | eqid 2740 | . . . . . 6 ⊢ (LSubSp‘𝑈) = (LSubSp‘𝑈) | |
38 | eqid 2740 | . . . . . 6 ⊢ (LSubSp‘𝐶) = (LSubSp‘𝐶) | |
39 | 2, 3, 9 | dvhlmod 39120 | . . . . . . 7 ⊢ (𝜑 → 𝑈 ∈ LMod) |
40 | 15, 37, 17 | lspsncl 20237 | . . . . . . 7 ⊢ ((𝑈 ∈ LMod ∧ 𝑌 ∈ 𝑉) → (𝑁‘{𝑌}) ∈ (LSubSp‘𝑈)) |
41 | 39, 19, 40 | syl2anc 584 | . . . . . 6 ⊢ (𝜑 → (𝑁‘{𝑌}) ∈ (LSubSp‘𝑈)) |
42 | 2, 14, 3, 37, 6, 38, 9, 41 | mapdcl2 39666 | . . . . 5 ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑌})) ∈ (LSubSp‘𝐶)) |
43 | 7, 38 | lssss 20196 | . . . . 5 ⊢ ((𝑀‘(𝑁‘{𝑌})) ∈ (LSubSp‘𝐶) → (𝑀‘(𝑁‘{𝑌})) ⊆ 𝐹) |
44 | 42, 43 | syl 17 | . . . 4 ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑌})) ⊆ 𝐹) |
45 | 44, 30 | sseldd 3927 | . . 3 ⊢ (𝜑 → 𝑧 ∈ 𝐹) |
46 | 2, 3, 4, 5, 6, 7, 8, 9, 36, 45 | lcdvscl 39615 | . 2 ⊢ (𝜑 → (((invr‘𝐴)‘𝑔) · 𝑧) ∈ 𝐹) |
47 | 1, 46 | eqeltrid 2845 | 1 ⊢ (𝜑 → 𝐸 ∈ 𝐹) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1542 ∈ wcel 2110 ≠ wne 2945 ⊆ wss 3892 {csn 4567 ‘cfv 6432 (class class class)co 7271 Basecbs 16910 Scalarcsca 16963 ·𝑠 cvsca 16964 0gc0g 17148 -gcsg 18577 LSSumclsm 19237 invrcinvr 19911 DivRingcdr 19989 LModclmod 20121 LSubSpclss 20191 LSpanclspn 20231 LVecclvec 20362 HLchlt 37360 LHypclh 37994 DVecHcdvh 39088 LCDualclcd 39596 mapdcmpd 39634 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2015 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2711 ax-rep 5214 ax-sep 5227 ax-nul 5234 ax-pow 5292 ax-pr 5356 ax-un 7582 ax-cnex 10928 ax-resscn 10929 ax-1cn 10930 ax-icn 10931 ax-addcl 10932 ax-addrcl 10933 ax-mulcl 10934 ax-mulrcl 10935 ax-mulcom 10936 ax-addass 10937 ax-mulass 10938 ax-distr 10939 ax-i2m1 10940 ax-1ne0 10941 ax-1rid 10942 ax-rnegex 10943 ax-rrecex 10944 ax-cnre 10945 ax-pre-lttri 10946 ax-pre-lttrn 10947 ax-pre-ltadd 10948 ax-pre-mulgt0 10949 ax-riotaBAD 36963 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2072 df-mo 2542 df-eu 2571 df-clab 2718 df-cleq 2732 df-clel 2818 df-nfc 2891 df-ne 2946 df-nel 3052 df-ral 3071 df-rex 3072 df-reu 3073 df-rmo 3074 df-rab 3075 df-v 3433 df-sbc 3721 df-csb 3838 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-pss 3911 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4846 df-int 4886 df-iun 4932 df-iin 4933 df-br 5080 df-opab 5142 df-mpt 5163 df-tr 5197 df-id 5490 df-eprel 5496 df-po 5504 df-so 5505 df-fr 5545 df-we 5547 df-xp 5596 df-rel 5597 df-cnv 5598 df-co 5599 df-dm 5600 df-rn 5601 df-res 5602 df-ima 5603 df-pred 6201 df-ord 6268 df-on 6269 df-lim 6270 df-suc 6271 df-iota 6390 df-fun 6434 df-fn 6435 df-f 6436 df-f1 6437 df-fo 6438 df-f1o 6439 df-fv 6440 df-riota 7228 df-ov 7274 df-oprab 7275 df-mpo 7276 df-of 7527 df-om 7707 df-1st 7824 df-2nd 7825 df-tpos 8033 df-undef 8080 df-frecs 8088 df-wrecs 8119 df-recs 8193 df-rdg 8232 df-1o 8288 df-er 8481 df-map 8600 df-en 8717 df-dom 8718 df-sdom 8719 df-fin 8720 df-pnf 11012 df-mnf 11013 df-xr 11014 df-ltxr 11015 df-le 11016 df-sub 11207 df-neg 11208 df-nn 11974 df-2 12036 df-3 12037 df-4 12038 df-5 12039 df-6 12040 df-n0 12234 df-z 12320 df-uz 12582 df-fz 13239 df-struct 16846 df-sets 16863 df-slot 16881 df-ndx 16893 df-base 16911 df-ress 16940 df-plusg 16973 df-mulr 16974 df-sca 16976 df-vsca 16977 df-0g 17150 df-mre 17293 df-mrc 17294 df-acs 17296 df-proset 18011 df-poset 18029 df-plt 18046 df-lub 18062 df-glb 18063 df-join 18064 df-meet 18065 df-p0 18141 df-p1 18142 df-lat 18148 df-clat 18215 df-mgm 18324 df-sgrp 18373 df-mnd 18384 df-submnd 18429 df-grp 18578 df-minusg 18579 df-sbg 18580 df-subg 18750 df-cntz 18921 df-oppg 18948 df-lsm 19239 df-cmn 19386 df-abl 19387 df-mgp 19719 df-ur 19736 df-ring 19783 df-oppr 19860 df-dvdsr 19881 df-unit 19882 df-invr 19912 df-dvr 19923 df-drng 19991 df-lmod 20123 df-lss 20192 df-lsp 20232 df-lvec 20363 df-lsatoms 36986 df-lshyp 36987 df-lcv 37029 df-lfl 37068 df-lkr 37096 df-ldual 37134 df-oposet 37186 df-ol 37188 df-oml 37189 df-covers 37276 df-ats 37277 df-atl 37308 df-cvlat 37332 df-hlat 37361 df-llines 37508 df-lplanes 37509 df-lvols 37510 df-lines 37511 df-psubsp 37513 df-pmap 37514 df-padd 37806 df-lhyp 37998 df-laut 37999 df-ldil 38114 df-ltrn 38115 df-trl 38169 df-tgrp 38753 df-tendo 38765 df-edring 38767 df-dveca 39013 df-disoa 39039 df-dvech 39089 df-dib 39149 df-dic 39183 df-dih 39239 df-doch 39358 df-djh 39405 df-lcdual 39597 df-mapd 39635 |
This theorem is referenced by: (None) |
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