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Mirrors > Home > MPE Home > Th. List > Mathboxes > mapdpglem31 | Structured version Visualization version GIF version |
Description: Lemma for mapdpg 38836. Baer p. 45 line 19: "...and we have consequently that y' = y'', as we claimed." (Contributed by NM, 23-Mar-2015.) |
Ref | Expression |
---|---|
mapdpg.h | ⊢ 𝐻 = (LHyp‘𝐾) |
mapdpg.m | ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) |
mapdpg.u | ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) |
mapdpg.v | ⊢ 𝑉 = (Base‘𝑈) |
mapdpg.s | ⊢ − = (-g‘𝑈) |
mapdpg.z | ⊢ 0 = (0g‘𝑈) |
mapdpg.n | ⊢ 𝑁 = (LSpan‘𝑈) |
mapdpg.c | ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) |
mapdpg.f | ⊢ 𝐹 = (Base‘𝐶) |
mapdpg.r | ⊢ 𝑅 = (-g‘𝐶) |
mapdpg.j | ⊢ 𝐽 = (LSpan‘𝐶) |
mapdpg.k | ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
mapdpg.x | ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 })) |
mapdpg.y | ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 })) |
mapdpg.g | ⊢ (𝜑 → 𝐺 ∈ 𝐹) |
mapdpg.ne | ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌})) |
mapdpg.e | ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐺})) |
mapdpgem25.h1 | ⊢ (𝜑 → (ℎ ∈ 𝐹 ∧ ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐺𝑅ℎ)})))) |
mapdpgem25.i1 | ⊢ (𝜑 → (𝑖 ∈ 𝐹 ∧ ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{𝑖}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐺𝑅𝑖)})))) |
mapdpglem26.a | ⊢ 𝐴 = (Scalar‘𝑈) |
mapdpglem26.b | ⊢ 𝐵 = (Base‘𝐴) |
mapdpglem26.t | ⊢ · = ( ·𝑠 ‘𝐶) |
mapdpglem26.o | ⊢ 𝑂 = (0g‘𝐴) |
mapdpglem28.ve | ⊢ (𝜑 → 𝑣 ∈ 𝐵) |
mapdpglem28.u1 | ⊢ (𝜑 → ℎ = (𝑢 · 𝑖)) |
mapdpglem28.u2 | ⊢ (𝜑 → (𝐺𝑅ℎ) = (𝑣 · (𝐺𝑅𝑖))) |
mapdpglem28.ue | ⊢ (𝜑 → 𝑢 ∈ 𝐵) |
Ref | Expression |
---|---|
mapdpglem31 | ⊢ (𝜑 → ℎ = 𝑖) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mapdpglem28.u1 | . 2 ⊢ (𝜑 → ℎ = (𝑢 · 𝑖)) | |
2 | mapdpg.h | . . . . 5 ⊢ 𝐻 = (LHyp‘𝐾) | |
3 | mapdpg.u | . . . . 5 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) | |
4 | mapdpglem26.a | . . . . 5 ⊢ 𝐴 = (Scalar‘𝑈) | |
5 | eqid 2821 | . . . . 5 ⊢ (1r‘𝐴) = (1r‘𝐴) | |
6 | mapdpg.c | . . . . 5 ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) | |
7 | eqid 2821 | . . . . 5 ⊢ (Scalar‘𝐶) = (Scalar‘𝐶) | |
8 | eqid 2821 | . . . . 5 ⊢ (1r‘(Scalar‘𝐶)) = (1r‘(Scalar‘𝐶)) | |
9 | mapdpg.k | . . . . 5 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
10 | 2, 3, 4, 5, 6, 7, 8, 9 | lcd1 38739 | . . . 4 ⊢ (𝜑 → (1r‘(Scalar‘𝐶)) = (1r‘𝐴)) |
11 | 10 | oveq1d 7165 | . . 3 ⊢ (𝜑 → ((1r‘(Scalar‘𝐶)) · 𝑖) = ((1r‘𝐴) · 𝑖)) |
12 | 2, 6, 9 | lcdlmod 38722 | . . . 4 ⊢ (𝜑 → 𝐶 ∈ LMod) |
13 | mapdpgem25.i1 | . . . . 5 ⊢ (𝜑 → (𝑖 ∈ 𝐹 ∧ ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{𝑖}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐺𝑅𝑖)})))) | |
14 | 13 | simpld 497 | . . . 4 ⊢ (𝜑 → 𝑖 ∈ 𝐹) |
15 | mapdpg.f | . . . . 5 ⊢ 𝐹 = (Base‘𝐶) | |
16 | mapdpglem26.t | . . . . 5 ⊢ · = ( ·𝑠 ‘𝐶) | |
17 | 15, 7, 16, 8 | lmodvs1 19656 | . . . 4 ⊢ ((𝐶 ∈ LMod ∧ 𝑖 ∈ 𝐹) → ((1r‘(Scalar‘𝐶)) · 𝑖) = 𝑖) |
18 | 12, 14, 17 | syl2anc 586 | . . 3 ⊢ (𝜑 → ((1r‘(Scalar‘𝐶)) · 𝑖) = 𝑖) |
19 | mapdpg.m | . . . . . 6 ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) | |
20 | mapdpg.v | . . . . . 6 ⊢ 𝑉 = (Base‘𝑈) | |
21 | mapdpg.s | . . . . . 6 ⊢ − = (-g‘𝑈) | |
22 | mapdpg.z | . . . . . 6 ⊢ 0 = (0g‘𝑈) | |
23 | mapdpg.n | . . . . . 6 ⊢ 𝑁 = (LSpan‘𝑈) | |
24 | mapdpg.r | . . . . . 6 ⊢ 𝑅 = (-g‘𝐶) | |
25 | mapdpg.j | . . . . . 6 ⊢ 𝐽 = (LSpan‘𝐶) | |
26 | mapdpg.x | . . . . . 6 ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 })) | |
27 | mapdpg.y | . . . . . 6 ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 })) | |
28 | mapdpg.g | . . . . . 6 ⊢ (𝜑 → 𝐺 ∈ 𝐹) | |
29 | mapdpg.ne | . . . . . 6 ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌})) | |
30 | mapdpg.e | . . . . . 6 ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐺})) | |
31 | mapdpgem25.h1 | . . . . . 6 ⊢ (𝜑 → (ℎ ∈ 𝐹 ∧ ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐺𝑅ℎ)})))) | |
32 | mapdpglem26.b | . . . . . 6 ⊢ 𝐵 = (Base‘𝐴) | |
33 | mapdpglem26.o | . . . . . 6 ⊢ 𝑂 = (0g‘𝐴) | |
34 | mapdpglem28.ve | . . . . . 6 ⊢ (𝜑 → 𝑣 ∈ 𝐵) | |
35 | mapdpglem28.u2 | . . . . . 6 ⊢ (𝜑 → (𝐺𝑅ℎ) = (𝑣 · (𝐺𝑅𝑖))) | |
36 | mapdpglem28.ue | . . . . . 6 ⊢ (𝜑 → 𝑢 ∈ 𝐵) | |
37 | 2, 19, 3, 20, 21, 22, 23, 6, 15, 24, 25, 9, 26, 27, 28, 29, 30, 31, 13, 4, 32, 16, 33, 34, 1, 35, 36 | mapdpglem30 38832 | . . . . 5 ⊢ (𝜑 → (𝑣 = (1r‘𝐴) ∧ 𝑣 = 𝑢)) |
38 | eqtr2 2842 | . . . . 5 ⊢ ((𝑣 = (1r‘𝐴) ∧ 𝑣 = 𝑢) → (1r‘𝐴) = 𝑢) | |
39 | 37, 38 | syl 17 | . . . 4 ⊢ (𝜑 → (1r‘𝐴) = 𝑢) |
40 | 39 | oveq1d 7165 | . . 3 ⊢ (𝜑 → ((1r‘𝐴) · 𝑖) = (𝑢 · 𝑖)) |
41 | 11, 18, 40 | 3eqtr3rd 2865 | . 2 ⊢ (𝜑 → (𝑢 · 𝑖) = 𝑖) |
42 | 1, 41 | eqtrd 2856 | 1 ⊢ (𝜑 → ℎ = 𝑖) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1533 ∈ wcel 2110 ≠ wne 3016 ∖ cdif 3933 {csn 4561 ‘cfv 6350 (class class class)co 7150 Basecbs 16477 Scalarcsca 16562 ·𝑠 cvsca 16563 0gc0g 16707 -gcsg 18099 1rcur 19245 LModclmod 19628 LSpanclspn 19737 HLchlt 36480 LHypclh 37114 DVecHcdvh 38208 LCDualclcd 38716 mapdcmpd 38754 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-rep 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 ax-riotaBAD 36083 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-fal 1546 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3497 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4833 df-int 4870 df-iun 4914 df-iin 4915 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5455 df-eprel 5460 df-po 5469 df-so 5470 df-fr 5509 df-we 5511 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-pred 6143 df-ord 6189 df-on 6190 df-lim 6191 df-suc 6192 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-of 7403 df-om 7575 df-1st 7683 df-2nd 7684 df-tpos 7886 df-undef 7933 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-1o 8096 df-oadd 8100 df-er 8283 df-map 8402 df-en 8504 df-dom 8505 df-sdom 8506 df-fin 8507 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-nn 11633 df-2 11694 df-3 11695 df-4 11696 df-5 11697 df-6 11698 df-n0 11892 df-z 11976 df-uz 12238 df-fz 12887 df-struct 16479 df-ndx 16480 df-slot 16481 df-base 16483 df-sets 16484 df-ress 16485 df-plusg 16572 df-mulr 16573 df-sca 16575 df-vsca 16576 df-0g 16709 df-mre 16851 df-mrc 16852 df-acs 16854 df-proset 17532 df-poset 17550 df-plt 17562 df-lub 17578 df-glb 17579 df-join 17580 df-meet 17581 df-p0 17643 df-p1 17644 df-lat 17650 df-clat 17712 df-mgm 17846 df-sgrp 17895 df-mnd 17906 df-submnd 17951 df-grp 18100 df-minusg 18101 df-sbg 18102 df-subg 18270 df-cntz 18441 df-oppg 18468 df-lsm 18755 df-cmn 18902 df-abl 18903 df-mgp 19234 df-ur 19246 df-ring 19293 df-oppr 19367 df-dvdsr 19385 df-unit 19386 df-invr 19416 df-dvr 19427 df-drng 19498 df-lmod 19630 df-lss 19698 df-lsp 19738 df-lvec 19869 df-lsatoms 36106 df-lshyp 36107 df-lcv 36149 df-lfl 36188 df-lkr 36216 df-ldual 36254 df-oposet 36306 df-ol 36308 df-oml 36309 df-covers 36396 df-ats 36397 df-atl 36428 df-cvlat 36452 df-hlat 36481 df-llines 36628 df-lplanes 36629 df-lvols 36630 df-lines 36631 df-psubsp 36633 df-pmap 36634 df-padd 36926 df-lhyp 37118 df-laut 37119 df-ldil 37234 df-ltrn 37235 df-trl 37289 df-tgrp 37873 df-tendo 37885 df-edring 37887 df-dveca 38133 df-disoa 38159 df-dvech 38209 df-dib 38269 df-dic 38303 df-dih 38359 df-doch 38478 df-djh 38525 df-lcdual 38717 df-mapd 38755 |
This theorem is referenced by: mapdpglem32 38835 |
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