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Mirrors > Home > MPE Home > Th. List > Mathboxes > mapdpglem17N | Structured version Visualization version GIF version |
Description: Lemma for mapdpg 38846. 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 38249 | . . . . 5 ⊢ (𝜑 → 𝑈 ∈ LVec) |
11 | 4 | lvecdrng 19880 | . . . . 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 38822 | . . . 4 ⊢ (𝜑 → 𝑔 ≠ 0 ) |
34 | eqid 2824 | . . . . 5 ⊢ (invr‘𝐴) = (invr‘𝐴) | |
35 | 5, 29, 34 | drnginvrcl 19522 | . . . 4 ⊢ ((𝐴 ∈ DivRing ∧ 𝑔 ∈ 𝐵 ∧ 𝑔 ≠ 0 ) → ((invr‘𝐴)‘𝑔) ∈ 𝐵) |
36 | 12, 13, 33, 35 | syl3anc 1367 | . . 3 ⊢ (𝜑 → ((invr‘𝐴)‘𝑔) ∈ 𝐵) |
37 | eqid 2824 | . . . . . 6 ⊢ (LSubSp‘𝑈) = (LSubSp‘𝑈) | |
38 | eqid 2824 | . . . . . 6 ⊢ (LSubSp‘𝐶) = (LSubSp‘𝐶) | |
39 | 2, 3, 9 | dvhlmod 38250 | . . . . . . 7 ⊢ (𝜑 → 𝑈 ∈ LMod) |
40 | 15, 37, 17 | lspsncl 19752 | . . . . . . 7 ⊢ ((𝑈 ∈ LMod ∧ 𝑌 ∈ 𝑉) → (𝑁‘{𝑌}) ∈ (LSubSp‘𝑈)) |
41 | 39, 19, 40 | syl2anc 586 | . . . . . 6 ⊢ (𝜑 → (𝑁‘{𝑌}) ∈ (LSubSp‘𝑈)) |
42 | 2, 14, 3, 37, 6, 38, 9, 41 | mapdcl2 38796 | . . . . 5 ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑌})) ∈ (LSubSp‘𝐶)) |
43 | 7, 38 | lssss 19711 | . . . . 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 3971 | . . 3 ⊢ (𝜑 → 𝑧 ∈ 𝐹) |
46 | 2, 3, 4, 5, 6, 7, 8, 9, 36, 45 | lcdvscl 38745 | . 2 ⊢ (𝜑 → (((invr‘𝐴)‘𝑔) · 𝑧) ∈ 𝐹) |
47 | 1, 46 | eqeltrid 2920 | 1 ⊢ (𝜑 → 𝐸 ∈ 𝐹) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1536 ∈ wcel 2113 ≠ wne 3019 ⊆ wss 3939 {csn 4570 ‘cfv 6358 (class class class)co 7159 Basecbs 16486 Scalarcsca 16571 ·𝑠 cvsca 16572 0gc0g 16716 -gcsg 18108 LSSumclsm 18762 invrcinvr 19424 DivRingcdr 19505 LModclmod 19637 LSubSpclss 19706 LSpanclspn 19746 LVecclvec 19877 HLchlt 36490 LHypclh 37124 DVecHcdvh 38218 LCDualclcd 38726 mapdcmpd 38764 |
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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-rep 5193 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 ax-cnex 10596 ax-resscn 10597 ax-1cn 10598 ax-icn 10599 ax-addcl 10600 ax-addrcl 10601 ax-mulcl 10602 ax-mulrcl 10603 ax-mulcom 10604 ax-addass 10605 ax-mulass 10606 ax-distr 10607 ax-i2m1 10608 ax-1ne0 10609 ax-1rid 10610 ax-rnegex 10611 ax-rrecex 10612 ax-cnre 10613 ax-pre-lttri 10614 ax-pre-lttrn 10615 ax-pre-ltadd 10616 ax-pre-mulgt0 10617 ax-riotaBAD 36093 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1539 df-fal 1549 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-nel 3127 df-ral 3146 df-rex 3147 df-reu 3148 df-rmo 3149 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-pss 3957 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-tp 4575 df-op 4577 df-uni 4842 df-int 4880 df-iun 4924 df-iin 4925 df-br 5070 df-opab 5132 df-mpt 5150 df-tr 5176 df-id 5463 df-eprel 5468 df-po 5477 df-so 5478 df-fr 5517 df-we 5519 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-pred 6151 df-ord 6197 df-on 6198 df-lim 6199 df-suc 6200 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7117 df-ov 7162 df-oprab 7163 df-mpo 7164 df-of 7412 df-om 7584 df-1st 7692 df-2nd 7693 df-tpos 7895 df-undef 7942 df-wrecs 7950 df-recs 8011 df-rdg 8049 df-1o 8105 df-oadd 8109 df-er 8292 df-map 8411 df-en 8513 df-dom 8514 df-sdom 8515 df-fin 8516 df-pnf 10680 df-mnf 10681 df-xr 10682 df-ltxr 10683 df-le 10684 df-sub 10875 df-neg 10876 df-nn 11642 df-2 11703 df-3 11704 df-4 11705 df-5 11706 df-6 11707 df-n0 11901 df-z 11985 df-uz 12247 df-fz 12896 df-struct 16488 df-ndx 16489 df-slot 16490 df-base 16492 df-sets 16493 df-ress 16494 df-plusg 16581 df-mulr 16582 df-sca 16584 df-vsca 16585 df-0g 16718 df-mre 16860 df-mrc 16861 df-acs 16863 df-proset 17541 df-poset 17559 df-plt 17571 df-lub 17587 df-glb 17588 df-join 17589 df-meet 17590 df-p0 17652 df-p1 17653 df-lat 17659 df-clat 17721 df-mgm 17855 df-sgrp 17904 df-mnd 17915 df-submnd 17960 df-grp 18109 df-minusg 18110 df-sbg 18111 df-subg 18279 df-cntz 18450 df-oppg 18477 df-lsm 18764 df-cmn 18911 df-abl 18912 df-mgp 19243 df-ur 19255 df-ring 19302 df-oppr 19376 df-dvdsr 19394 df-unit 19395 df-invr 19425 df-dvr 19436 df-drng 19507 df-lmod 19639 df-lss 19707 df-lsp 19747 df-lvec 19878 df-lsatoms 36116 df-lshyp 36117 df-lcv 36159 df-lfl 36198 df-lkr 36226 df-ldual 36264 df-oposet 36316 df-ol 36318 df-oml 36319 df-covers 36406 df-ats 36407 df-atl 36438 df-cvlat 36462 df-hlat 36491 df-llines 36638 df-lplanes 36639 df-lvols 36640 df-lines 36641 df-psubsp 36643 df-pmap 36644 df-padd 36936 df-lhyp 37128 df-laut 37129 df-ldil 37244 df-ltrn 37245 df-trl 37299 df-tgrp 37883 df-tendo 37895 df-edring 37897 df-dveca 38143 df-disoa 38169 df-dvech 38219 df-dib 38279 df-dic 38313 df-dih 38369 df-doch 38488 df-djh 38535 df-lcdual 38727 df-mapd 38765 |
This theorem is referenced by: (None) |
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