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Mirrors > Home > MPE Home > Th. List > Mathboxes > mapdpglem18 | Structured version Visualization version GIF version |
Description: Lemma for mapdpg 38844. Baer p. 45, line 7: "Then y =/= 0..." (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 |
---|---|
mapdpglem18 | ⊢ (𝜑 → 𝐸 ≠ (0g‘𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mapdpglem.h | . . . . . . 7 ⊢ 𝐻 = (LHyp‘𝐾) | |
2 | mapdpglem.u | . . . . . . 7 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) | |
3 | mapdpglem.k | . . . . . . 7 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
4 | 1, 2, 3 | dvhlvec 38247 | . . . . . 6 ⊢ (𝜑 → 𝑈 ∈ LVec) |
5 | mapdpglem3.a | . . . . . . 7 ⊢ 𝐴 = (Scalar‘𝑈) | |
6 | 5 | lvecdrng 19879 | . . . . . 6 ⊢ (𝑈 ∈ LVec → 𝐴 ∈ DivRing) |
7 | 4, 6 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ DivRing) |
8 | mapdpglem4.g4 | . . . . 5 ⊢ (𝜑 → 𝑔 ∈ 𝐵) | |
9 | mapdpglem.m | . . . . . 6 ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) | |
10 | mapdpglem.v | . . . . . 6 ⊢ 𝑉 = (Base‘𝑈) | |
11 | mapdpglem.s | . . . . . 6 ⊢ − = (-g‘𝑈) | |
12 | mapdpglem.n | . . . . . 6 ⊢ 𝑁 = (LSpan‘𝑈) | |
13 | mapdpglem.c | . . . . . 6 ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) | |
14 | mapdpglem.x | . . . . . 6 ⊢ (𝜑 → 𝑋 ∈ 𝑉) | |
15 | mapdpglem.y | . . . . . 6 ⊢ (𝜑 → 𝑌 ∈ 𝑉) | |
16 | mapdpglem1.p | . . . . . 6 ⊢ ⊕ = (LSSum‘𝐶) | |
17 | mapdpglem2.j | . . . . . 6 ⊢ 𝐽 = (LSpan‘𝐶) | |
18 | mapdpglem3.f | . . . . . 6 ⊢ 𝐹 = (Base‘𝐶) | |
19 | mapdpglem3.te | . . . . . 6 ⊢ (𝜑 → 𝑡 ∈ ((𝑀‘(𝑁‘{𝑋})) ⊕ (𝑀‘(𝑁‘{𝑌})))) | |
20 | mapdpglem3.b | . . . . . 6 ⊢ 𝐵 = (Base‘𝐴) | |
21 | mapdpglem3.t | . . . . . 6 ⊢ · = ( ·𝑠 ‘𝐶) | |
22 | mapdpglem3.r | . . . . . 6 ⊢ 𝑅 = (-g‘𝐶) | |
23 | mapdpglem3.g | . . . . . 6 ⊢ (𝜑 → 𝐺 ∈ 𝐹) | |
24 | mapdpglem3.e | . . . . . 6 ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐺})) | |
25 | mapdpglem4.q | . . . . . 6 ⊢ 𝑄 = (0g‘𝑈) | |
26 | mapdpglem.ne | . . . . . 6 ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌})) | |
27 | mapdpglem4.jt | . . . . . 6 ⊢ (𝜑 → (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{𝑡})) | |
28 | mapdpglem4.z | . . . . . 6 ⊢ 0 = (0g‘𝐴) | |
29 | mapdpglem4.z4 | . . . . . 6 ⊢ (𝜑 → 𝑧 ∈ (𝑀‘(𝑁‘{𝑌}))) | |
30 | mapdpglem4.t4 | . . . . . 6 ⊢ (𝜑 → 𝑡 = ((𝑔 · 𝐺)𝑅𝑧)) | |
31 | mapdpglem4.xn | . . . . . 6 ⊢ (𝜑 → 𝑋 ≠ 𝑄) | |
32 | 1, 9, 2, 10, 11, 12, 13, 3, 14, 15, 16, 17, 18, 19, 5, 20, 21, 22, 23, 24, 25, 26, 27, 28, 8, 29, 30, 31 | mapdpglem11 38820 | . . . . 5 ⊢ (𝜑 → 𝑔 ≠ 0 ) |
33 | eqid 2823 | . . . . . 6 ⊢ (invr‘𝐴) = (invr‘𝐴) | |
34 | 20, 28, 33 | drnginvrn0 19522 | . . . . 5 ⊢ ((𝐴 ∈ DivRing ∧ 𝑔 ∈ 𝐵 ∧ 𝑔 ≠ 0 ) → ((invr‘𝐴)‘𝑔) ≠ 0 ) |
35 | 7, 8, 32, 34 | syl3anc 1367 | . . . 4 ⊢ (𝜑 → ((invr‘𝐴)‘𝑔) ≠ 0 ) |
36 | eqid 2823 | . . . . 5 ⊢ (Scalar‘𝐶) = (Scalar‘𝐶) | |
37 | eqid 2823 | . . . . 5 ⊢ (0g‘(Scalar‘𝐶)) = (0g‘(Scalar‘𝐶)) | |
38 | 1, 2, 5, 28, 13, 36, 37, 3 | lcd0 38746 | . . . 4 ⊢ (𝜑 → (0g‘(Scalar‘𝐶)) = 0 ) |
39 | 35, 38 | neeqtrrd 3092 | . . 3 ⊢ (𝜑 → ((invr‘𝐴)‘𝑔) ≠ (0g‘(Scalar‘𝐶))) |
40 | mapdpglem12.yn | . . . 4 ⊢ (𝜑 → 𝑌 ≠ 𝑄) | |
41 | 1, 9, 2, 10, 11, 12, 13, 3, 14, 15, 16, 17, 18, 19, 5, 20, 21, 22, 23, 24, 25, 26, 27, 28, 8, 29, 30, 31, 40 | mapdpglem16 38825 | . . 3 ⊢ (𝜑 → 𝑧 ≠ (0g‘𝐶)) |
42 | eqid 2823 | . . . 4 ⊢ (Base‘(Scalar‘𝐶)) = (Base‘(Scalar‘𝐶)) | |
43 | eqid 2823 | . . . 4 ⊢ (0g‘𝐶) = (0g‘𝐶) | |
44 | 1, 13, 3 | lcdlvec 38729 | . . . 4 ⊢ (𝜑 → 𝐶 ∈ LVec) |
45 | 20, 28, 33 | drnginvrcl 19521 | . . . . . 6 ⊢ ((𝐴 ∈ DivRing ∧ 𝑔 ∈ 𝐵 ∧ 𝑔 ≠ 0 ) → ((invr‘𝐴)‘𝑔) ∈ 𝐵) |
46 | 7, 8, 32, 45 | syl3anc 1367 | . . . . 5 ⊢ (𝜑 → ((invr‘𝐴)‘𝑔) ∈ 𝐵) |
47 | 1, 2, 5, 20, 13, 36, 42, 3 | lcdsbase 38738 | . . . . 5 ⊢ (𝜑 → (Base‘(Scalar‘𝐶)) = 𝐵) |
48 | 46, 47 | eleqtrrd 2918 | . . . 4 ⊢ (𝜑 → ((invr‘𝐴)‘𝑔) ∈ (Base‘(Scalar‘𝐶))) |
49 | eqid 2823 | . . . . . . 7 ⊢ (LSubSp‘𝑈) = (LSubSp‘𝑈) | |
50 | eqid 2823 | . . . . . . 7 ⊢ (LSubSp‘𝐶) = (LSubSp‘𝐶) | |
51 | 1, 2, 3 | dvhlmod 38248 | . . . . . . . 8 ⊢ (𝜑 → 𝑈 ∈ LMod) |
52 | 10, 49, 12 | lspsncl 19751 | . . . . . . . 8 ⊢ ((𝑈 ∈ LMod ∧ 𝑌 ∈ 𝑉) → (𝑁‘{𝑌}) ∈ (LSubSp‘𝑈)) |
53 | 51, 15, 52 | syl2anc 586 | . . . . . . 7 ⊢ (𝜑 → (𝑁‘{𝑌}) ∈ (LSubSp‘𝑈)) |
54 | 1, 9, 2, 49, 13, 50, 3, 53 | mapdcl2 38794 | . . . . . 6 ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑌})) ∈ (LSubSp‘𝐶)) |
55 | 18, 50 | lssss 19710 | . . . . . 6 ⊢ ((𝑀‘(𝑁‘{𝑌})) ∈ (LSubSp‘𝐶) → (𝑀‘(𝑁‘{𝑌})) ⊆ 𝐹) |
56 | 54, 55 | syl 17 | . . . . 5 ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑌})) ⊆ 𝐹) |
57 | 56, 29 | sseldd 3970 | . . . 4 ⊢ (𝜑 → 𝑧 ∈ 𝐹) |
58 | 18, 21, 36, 42, 37, 43, 44, 48, 57 | lvecvsn0 19883 | . . 3 ⊢ (𝜑 → ((((invr‘𝐴)‘𝑔) · 𝑧) ≠ (0g‘𝐶) ↔ (((invr‘𝐴)‘𝑔) ≠ (0g‘(Scalar‘𝐶)) ∧ 𝑧 ≠ (0g‘𝐶)))) |
59 | 39, 41, 58 | mpbir2and 711 | . 2 ⊢ (𝜑 → (((invr‘𝐴)‘𝑔) · 𝑧) ≠ (0g‘𝐶)) |
60 | mapdpglem17.ep | . 2 ⊢ 𝐸 = (((invr‘𝐴)‘𝑔) · 𝑧) | |
61 | 59, 60, 43 | 3netr4g 3097 | 1 ⊢ (𝜑 → 𝐸 ≠ (0g‘𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ≠ wne 3018 ⊆ wss 3938 {csn 4569 ‘cfv 6357 (class class class)co 7158 Basecbs 16485 Scalarcsca 16570 ·𝑠 cvsca 16571 0gc0g 16715 -gcsg 18107 LSSumclsm 18761 invrcinvr 19423 DivRingcdr 19504 LModclmod 19636 LSubSpclss 19705 LSpanclspn 19745 LVecclvec 19876 HLchlt 36488 LHypclh 37122 DVecHcdvh 38216 LCDualclcd 38724 mapdcmpd 38762 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 ax-riotaBAD 36091 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-fal 1550 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-iin 4924 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-of 7411 df-om 7583 df-1st 7691 df-2nd 7692 df-tpos 7894 df-undef 7941 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-1o 8104 df-oadd 8108 df-er 8291 df-map 8410 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-nn 11641 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 16487 df-ndx 16488 df-slot 16489 df-base 16491 df-sets 16492 df-ress 16493 df-plusg 16580 df-mulr 16581 df-sca 16583 df-vsca 16584 df-0g 16717 df-mre 16859 df-mrc 16860 df-acs 16862 df-proset 17540 df-poset 17558 df-plt 17570 df-lub 17586 df-glb 17587 df-join 17588 df-meet 17589 df-p0 17651 df-p1 17652 df-lat 17658 df-clat 17720 df-mgm 17854 df-sgrp 17903 df-mnd 17914 df-submnd 17959 df-grp 18108 df-minusg 18109 df-sbg 18110 df-subg 18278 df-cntz 18449 df-oppg 18476 df-lsm 18763 df-cmn 18910 df-abl 18911 df-mgp 19242 df-ur 19254 df-ring 19301 df-oppr 19375 df-dvdsr 19393 df-unit 19394 df-invr 19424 df-dvr 19435 df-drng 19506 df-lmod 19638 df-lss 19706 df-lsp 19746 df-lvec 19877 df-lsatoms 36114 df-lshyp 36115 df-lcv 36157 df-lfl 36196 df-lkr 36224 df-ldual 36262 df-oposet 36314 df-ol 36316 df-oml 36317 df-covers 36404 df-ats 36405 df-atl 36436 df-cvlat 36460 df-hlat 36489 df-llines 36636 df-lplanes 36637 df-lvols 36638 df-lines 36639 df-psubsp 36641 df-pmap 36642 df-padd 36934 df-lhyp 37126 df-laut 37127 df-ldil 37242 df-ltrn 37243 df-trl 37297 df-tgrp 37881 df-tendo 37893 df-edring 37895 df-dveca 38141 df-disoa 38167 df-dvech 38217 df-dib 38277 df-dic 38311 df-dih 38367 df-doch 38486 df-djh 38533 df-lcdual 38725 df-mapd 38763 |
This theorem is referenced by: mapdpglem20 38829 |
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