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Mirrors > Home > MPE Home > Th. List > Mathboxes > hdmap14lem13 | Structured version Visualization version GIF version |
Description: Lemma for proof of part 14 in [Baer] p. 50. (Contributed by NM, 6-Jun-2015.) |
Ref | Expression |
---|---|
hdmap14lem12.h | ⊢ 𝐻 = (LHyp‘𝐾) |
hdmap14lem12.u | ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) |
hdmap14lem12.v | ⊢ 𝑉 = (Base‘𝑈) |
hdmap14lem12.t | ⊢ · = ( ·𝑠 ‘𝑈) |
hdmap14lem12.r | ⊢ 𝑅 = (Scalar‘𝑈) |
hdmap14lem12.b | ⊢ 𝐵 = (Base‘𝑅) |
hdmap14lem12.c | ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) |
hdmap14lem12.e | ⊢ ∙ = ( ·𝑠 ‘𝐶) |
hdmap14lem12.s | ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊) |
hdmap14lem12.k | ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
hdmap14lem12.f | ⊢ (𝜑 → 𝐹 ∈ 𝐵) |
hdmap14lem12.p | ⊢ 𝑃 = (Scalar‘𝐶) |
hdmap14lem12.a | ⊢ 𝐴 = (Base‘𝑃) |
hdmap14lem12.o | ⊢ 0 = (0g‘𝑈) |
hdmap14lem12.x | ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 })) |
hdmap14lem12.g | ⊢ (𝜑 → 𝐺 ∈ 𝐴) |
Ref | Expression |
---|---|
hdmap14lem13 | ⊢ (𝜑 → ((𝑆‘(𝐹 · 𝑋)) = (𝐺 ∙ (𝑆‘𝑋)) ↔ ∀𝑦 ∈ 𝑉 (𝑆‘(𝐹 · 𝑦)) = (𝐺 ∙ (𝑆‘𝑦)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hdmap14lem12.h | . . 3 ⊢ 𝐻 = (LHyp‘𝐾) | |
2 | hdmap14lem12.u | . . 3 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) | |
3 | hdmap14lem12.v | . . 3 ⊢ 𝑉 = (Base‘𝑈) | |
4 | hdmap14lem12.t | . . 3 ⊢ · = ( ·𝑠 ‘𝑈) | |
5 | hdmap14lem12.r | . . 3 ⊢ 𝑅 = (Scalar‘𝑈) | |
6 | hdmap14lem12.b | . . 3 ⊢ 𝐵 = (Base‘𝑅) | |
7 | hdmap14lem12.c | . . 3 ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) | |
8 | hdmap14lem12.e | . . 3 ⊢ ∙ = ( ·𝑠 ‘𝐶) | |
9 | hdmap14lem12.s | . . 3 ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊) | |
10 | hdmap14lem12.k | . . 3 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
11 | hdmap14lem12.f | . . 3 ⊢ (𝜑 → 𝐹 ∈ 𝐵) | |
12 | hdmap14lem12.p | . . 3 ⊢ 𝑃 = (Scalar‘𝐶) | |
13 | hdmap14lem12.a | . . 3 ⊢ 𝐴 = (Base‘𝑃) | |
14 | hdmap14lem12.o | . . 3 ⊢ 0 = (0g‘𝑈) | |
15 | hdmap14lem12.x | . . 3 ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 })) | |
16 | hdmap14lem12.g | . . 3 ⊢ (𝜑 → 𝐺 ∈ 𝐴) | |
17 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 | hdmap14lem12 37171 | . 2 ⊢ (𝜑 → ((𝑆‘(𝐹 · 𝑋)) = (𝐺 ∙ (𝑆‘𝑋)) ↔ ∀𝑦 ∈ (𝑉 ∖ { 0 })(𝑆‘(𝐹 · 𝑦)) = (𝐺 ∙ (𝑆‘𝑦)))) |
18 | velsn 4193 | . . . . . 6 ⊢ (𝑦 ∈ { 0 } ↔ 𝑦 = 0 ) | |
19 | 1, 7, 10 | lcdlmod 36881 | . . . . . . . . 9 ⊢ (𝜑 → 𝐶 ∈ LMod) |
20 | eqid 2622 | . . . . . . . . . 10 ⊢ (0g‘𝐶) = (0g‘𝐶) | |
21 | 12, 8, 13, 20 | lmodvs0 18897 | . . . . . . . . 9 ⊢ ((𝐶 ∈ LMod ∧ 𝐺 ∈ 𝐴) → (𝐺 ∙ (0g‘𝐶)) = (0g‘𝐶)) |
22 | 19, 16, 21 | syl2anc 693 | . . . . . . . 8 ⊢ (𝜑 → (𝐺 ∙ (0g‘𝐶)) = (0g‘𝐶)) |
23 | 1, 2, 14, 7, 20, 9, 10 | hdmapval0 37125 | . . . . . . . . 9 ⊢ (𝜑 → (𝑆‘ 0 ) = (0g‘𝐶)) |
24 | 23 | oveq2d 6666 | . . . . . . . 8 ⊢ (𝜑 → (𝐺 ∙ (𝑆‘ 0 )) = (𝐺 ∙ (0g‘𝐶))) |
25 | 1, 2, 10 | dvhlmod 36399 | . . . . . . . . . . 11 ⊢ (𝜑 → 𝑈 ∈ LMod) |
26 | 5, 4, 6, 14 | lmodvs0 18897 | . . . . . . . . . . 11 ⊢ ((𝑈 ∈ LMod ∧ 𝐹 ∈ 𝐵) → (𝐹 · 0 ) = 0 ) |
27 | 25, 11, 26 | syl2anc 693 | . . . . . . . . . 10 ⊢ (𝜑 → (𝐹 · 0 ) = 0 ) |
28 | 27 | fveq2d 6195 | . . . . . . . . 9 ⊢ (𝜑 → (𝑆‘(𝐹 · 0 )) = (𝑆‘ 0 )) |
29 | 28, 23 | eqtrd 2656 | . . . . . . . 8 ⊢ (𝜑 → (𝑆‘(𝐹 · 0 )) = (0g‘𝐶)) |
30 | 22, 24, 29 | 3eqtr4rd 2667 | . . . . . . 7 ⊢ (𝜑 → (𝑆‘(𝐹 · 0 )) = (𝐺 ∙ (𝑆‘ 0 ))) |
31 | oveq2 6658 | . . . . . . . . 9 ⊢ (𝑦 = 0 → (𝐹 · 𝑦) = (𝐹 · 0 )) | |
32 | 31 | fveq2d 6195 | . . . . . . . 8 ⊢ (𝑦 = 0 → (𝑆‘(𝐹 · 𝑦)) = (𝑆‘(𝐹 · 0 ))) |
33 | fveq2 6191 | . . . . . . . . 9 ⊢ (𝑦 = 0 → (𝑆‘𝑦) = (𝑆‘ 0 )) | |
34 | 33 | oveq2d 6666 | . . . . . . . 8 ⊢ (𝑦 = 0 → (𝐺 ∙ (𝑆‘𝑦)) = (𝐺 ∙ (𝑆‘ 0 ))) |
35 | 32, 34 | eqeq12d 2637 | . . . . . . 7 ⊢ (𝑦 = 0 → ((𝑆‘(𝐹 · 𝑦)) = (𝐺 ∙ (𝑆‘𝑦)) ↔ (𝑆‘(𝐹 · 0 )) = (𝐺 ∙ (𝑆‘ 0 )))) |
36 | 30, 35 | syl5ibrcom 237 | . . . . . 6 ⊢ (𝜑 → (𝑦 = 0 → (𝑆‘(𝐹 · 𝑦)) = (𝐺 ∙ (𝑆‘𝑦)))) |
37 | 18, 36 | syl5bi 232 | . . . . 5 ⊢ (𝜑 → (𝑦 ∈ { 0 } → (𝑆‘(𝐹 · 𝑦)) = (𝐺 ∙ (𝑆‘𝑦)))) |
38 | 37 | ralrimiv 2965 | . . . 4 ⊢ (𝜑 → ∀𝑦 ∈ { 0 } (𝑆‘(𝐹 · 𝑦)) = (𝐺 ∙ (𝑆‘𝑦))) |
39 | 38 | biantrud 528 | . . 3 ⊢ (𝜑 → (∀𝑦 ∈ (𝑉 ∖ { 0 })(𝑆‘(𝐹 · 𝑦)) = (𝐺 ∙ (𝑆‘𝑦)) ↔ (∀𝑦 ∈ (𝑉 ∖ { 0 })(𝑆‘(𝐹 · 𝑦)) = (𝐺 ∙ (𝑆‘𝑦)) ∧ ∀𝑦 ∈ { 0 } (𝑆‘(𝐹 · 𝑦)) = (𝐺 ∙ (𝑆‘𝑦))))) |
40 | ralunb 3794 | . . 3 ⊢ (∀𝑦 ∈ ((𝑉 ∖ { 0 }) ∪ { 0 })(𝑆‘(𝐹 · 𝑦)) = (𝐺 ∙ (𝑆‘𝑦)) ↔ (∀𝑦 ∈ (𝑉 ∖ { 0 })(𝑆‘(𝐹 · 𝑦)) = (𝐺 ∙ (𝑆‘𝑦)) ∧ ∀𝑦 ∈ { 0 } (𝑆‘(𝐹 · 𝑦)) = (𝐺 ∙ (𝑆‘𝑦)))) | |
41 | 39, 40 | syl6bbr 278 | . 2 ⊢ (𝜑 → (∀𝑦 ∈ (𝑉 ∖ { 0 })(𝑆‘(𝐹 · 𝑦)) = (𝐺 ∙ (𝑆‘𝑦)) ↔ ∀𝑦 ∈ ((𝑉 ∖ { 0 }) ∪ { 0 })(𝑆‘(𝐹 · 𝑦)) = (𝐺 ∙ (𝑆‘𝑦)))) |
42 | 3, 14 | lmod0vcl 18892 | . . . 4 ⊢ (𝑈 ∈ LMod → 0 ∈ 𝑉) |
43 | difsnid 4341 | . . . 4 ⊢ ( 0 ∈ 𝑉 → ((𝑉 ∖ { 0 }) ∪ { 0 }) = 𝑉) | |
44 | 25, 42, 43 | 3syl 18 | . . 3 ⊢ (𝜑 → ((𝑉 ∖ { 0 }) ∪ { 0 }) = 𝑉) |
45 | 44 | raleqdv 3144 | . 2 ⊢ (𝜑 → (∀𝑦 ∈ ((𝑉 ∖ { 0 }) ∪ { 0 })(𝑆‘(𝐹 · 𝑦)) = (𝐺 ∙ (𝑆‘𝑦)) ↔ ∀𝑦 ∈ 𝑉 (𝑆‘(𝐹 · 𝑦)) = (𝐺 ∙ (𝑆‘𝑦)))) |
46 | 17, 41, 45 | 3bitrd 294 | 1 ⊢ (𝜑 → ((𝑆‘(𝐹 · 𝑋)) = (𝐺 ∙ (𝑆‘𝑋)) ↔ ∀𝑦 ∈ 𝑉 (𝑆‘(𝐹 · 𝑦)) = (𝐺 ∙ (𝑆‘𝑦)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ∀wral 2912 ∖ cdif 3571 ∪ cun 3572 {csn 4177 ‘cfv 5888 (class class class)co 6650 Basecbs 15857 Scalarcsca 15944 ·𝑠 cvsca 15945 0gc0g 16100 LModclmod 18863 HLchlt 34637 LHypclh 35270 DVecHcdvh 36367 LCDualclcd 36875 HDMapchdma 37082 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-riotaBAD 34239 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-fal 1489 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-ot 4186 df-uni 4437 df-int 4476 df-iun 4522 df-iin 4523 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-of 6897 df-om 7066 df-1st 7168 df-2nd 7169 df-tpos 7352 df-undef 7399 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-oadd 7564 df-er 7742 df-map 7859 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-2 11079 df-3 11080 df-4 11081 df-5 11082 df-6 11083 df-n0 11293 df-z 11378 df-uz 11688 df-fz 12327 df-struct 15859 df-ndx 15860 df-slot 15861 df-base 15863 df-sets 15864 df-ress 15865 df-plusg 15954 df-mulr 15955 df-sca 15957 df-vsca 15958 df-0g 16102 df-mre 16246 df-mrc 16247 df-acs 16249 df-preset 16928 df-poset 16946 df-plt 16958 df-lub 16974 df-glb 16975 df-join 16976 df-meet 16977 df-p0 17039 df-p1 17040 df-lat 17046 df-clat 17108 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-submnd 17336 df-grp 17425 df-minusg 17426 df-sbg 17427 df-subg 17591 df-cntz 17750 df-oppg 17776 df-lsm 18051 df-cmn 18195 df-abl 18196 df-mgp 18490 df-ur 18502 df-ring 18549 df-oppr 18623 df-dvdsr 18641 df-unit 18642 df-invr 18672 df-dvr 18683 df-drng 18749 df-lmod 18865 df-lss 18933 df-lsp 18972 df-lvec 19103 df-lsatoms 34263 df-lshyp 34264 df-lcv 34306 df-lfl 34345 df-lkr 34373 df-ldual 34411 df-oposet 34463 df-ol 34465 df-oml 34466 df-covers 34553 df-ats 34554 df-atl 34585 df-cvlat 34609 df-hlat 34638 df-llines 34784 df-lplanes 34785 df-lvols 34786 df-lines 34787 df-psubsp 34789 df-pmap 34790 df-padd 35082 df-lhyp 35274 df-laut 35275 df-ldil 35390 df-ltrn 35391 df-trl 35446 df-tgrp 36031 df-tendo 36043 df-edring 36045 df-dveca 36291 df-disoa 36318 df-dvech 36368 df-dib 36428 df-dic 36462 df-dih 36518 df-doch 36637 df-djh 36684 df-lcdual 36876 df-mapd 36914 df-hvmap 37046 df-hdmap1 37083 df-hdmap 37084 |
This theorem is referenced by: hdmap14lem14 37173 |
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