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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 36189 | . 2 ⊢ (𝜑 → ((𝑆‘(𝐹 · 𝑋)) = (𝐺 ∙ (𝑆‘𝑋)) ↔ ∀𝑦 ∈ (𝑉 ∖ { 0 })(𝑆‘(𝐹 · 𝑦)) = (𝐺 ∙ (𝑆‘𝑦)))) |
18 | velsn 4141 | . . . . . 6 ⊢ (𝑦 ∈ { 0 } ↔ 𝑦 = 0 ) | |
19 | 1, 7, 10 | lcdlmod 35899 | . . . . . . . . 9 ⊢ (𝜑 → 𝐶 ∈ LMod) |
20 | eqid 2610 | . . . . . . . . . 10 ⊢ (0g‘𝐶) = (0g‘𝐶) | |
21 | 12, 8, 13, 20 | lmodvs0 18720 | . . . . . . . . 9 ⊢ ((𝐶 ∈ LMod ∧ 𝐺 ∈ 𝐴) → (𝐺 ∙ (0g‘𝐶)) = (0g‘𝐶)) |
22 | 19, 16, 21 | syl2anc 691 | . . . . . . . 8 ⊢ (𝜑 → (𝐺 ∙ (0g‘𝐶)) = (0g‘𝐶)) |
23 | 1, 2, 14, 7, 20, 9, 10 | hdmapval0 36143 | . . . . . . . . 9 ⊢ (𝜑 → (𝑆‘ 0 ) = (0g‘𝐶)) |
24 | 23 | oveq2d 6565 | . . . . . . . 8 ⊢ (𝜑 → (𝐺 ∙ (𝑆‘ 0 )) = (𝐺 ∙ (0g‘𝐶))) |
25 | 1, 2, 10 | dvhlmod 35417 | . . . . . . . . . . 11 ⊢ (𝜑 → 𝑈 ∈ LMod) |
26 | 5, 4, 6, 14 | lmodvs0 18720 | . . . . . . . . . . 11 ⊢ ((𝑈 ∈ LMod ∧ 𝐹 ∈ 𝐵) → (𝐹 · 0 ) = 0 ) |
27 | 25, 11, 26 | syl2anc 691 | . . . . . . . . . 10 ⊢ (𝜑 → (𝐹 · 0 ) = 0 ) |
28 | 27 | fveq2d 6107 | . . . . . . . . 9 ⊢ (𝜑 → (𝑆‘(𝐹 · 0 )) = (𝑆‘ 0 )) |
29 | 28, 23 | eqtrd 2644 | . . . . . . . 8 ⊢ (𝜑 → (𝑆‘(𝐹 · 0 )) = (0g‘𝐶)) |
30 | 22, 24, 29 | 3eqtr4rd 2655 | . . . . . . 7 ⊢ (𝜑 → (𝑆‘(𝐹 · 0 )) = (𝐺 ∙ (𝑆‘ 0 ))) |
31 | oveq2 6557 | . . . . . . . . 9 ⊢ (𝑦 = 0 → (𝐹 · 𝑦) = (𝐹 · 0 )) | |
32 | 31 | fveq2d 6107 | . . . . . . . 8 ⊢ (𝑦 = 0 → (𝑆‘(𝐹 · 𝑦)) = (𝑆‘(𝐹 · 0 ))) |
33 | fveq2 6103 | . . . . . . . . 9 ⊢ (𝑦 = 0 → (𝑆‘𝑦) = (𝑆‘ 0 )) | |
34 | 33 | oveq2d 6565 | . . . . . . . 8 ⊢ (𝑦 = 0 → (𝐺 ∙ (𝑆‘𝑦)) = (𝐺 ∙ (𝑆‘ 0 ))) |
35 | 32, 34 | eqeq12d 2625 | . . . . . . 7 ⊢ (𝑦 = 0 → ((𝑆‘(𝐹 · 𝑦)) = (𝐺 ∙ (𝑆‘𝑦)) ↔ (𝑆‘(𝐹 · 0 )) = (𝐺 ∙ (𝑆‘ 0 )))) |
36 | 30, 35 | syl5ibrcom 236 | . . . . . 6 ⊢ (𝜑 → (𝑦 = 0 → (𝑆‘(𝐹 · 𝑦)) = (𝐺 ∙ (𝑆‘𝑦)))) |
37 | 18, 36 | syl5bi 231 | . . . . 5 ⊢ (𝜑 → (𝑦 ∈ { 0 } → (𝑆‘(𝐹 · 𝑦)) = (𝐺 ∙ (𝑆‘𝑦)))) |
38 | 37 | ralrimiv 2948 | . . . 4 ⊢ (𝜑 → ∀𝑦 ∈ { 0 } (𝑆‘(𝐹 · 𝑦)) = (𝐺 ∙ (𝑆‘𝑦))) |
39 | 38 | biantrud 527 | . . 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 3756 | . . 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 277 | . 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 18715 | . . . 4 ⊢ (𝑈 ∈ LMod → 0 ∈ 𝑉) |
43 | difsnid 4282 | . . . 4 ⊢ ( 0 ∈ 𝑉 → ((𝑉 ∖ { 0 }) ∪ { 0 }) = 𝑉) | |
44 | 25, 42, 43 | 3syl 18 | . . 3 ⊢ (𝜑 → ((𝑉 ∖ { 0 }) ∪ { 0 }) = 𝑉) |
45 | 44 | raleqdv 3121 | . 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 293 | 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 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ∀wral 2896 ∖ cdif 3537 ∪ cun 3538 {csn 4125 ‘cfv 5804 (class class class)co 6549 Basecbs 15695 Scalarcsca 15771 ·𝑠 cvsca 15772 0gc0g 15923 LModclmod 18686 HLchlt 33655 LHypclh 34288 DVecHcdvh 35385 LCDualclcd 35893 HDMapchdma 36100 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-riotaBAD 33257 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-fal 1481 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-ot 4134 df-uni 4373 df-int 4411 df-iun 4457 df-iin 4458 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-of 6795 df-om 6958 df-1st 7059 df-2nd 7060 df-tpos 7239 df-undef 7286 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-oadd 7451 df-er 7629 df-map 7746 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-2 10956 df-3 10957 df-4 10958 df-5 10959 df-6 10960 df-n0 11170 df-z 11255 df-uz 11564 df-fz 12198 df-struct 15697 df-ndx 15698 df-slot 15699 df-base 15700 df-sets 15701 df-ress 15702 df-plusg 15781 df-mulr 15782 df-sca 15784 df-vsca 15785 df-0g 15925 df-mre 16069 df-mrc 16070 df-acs 16072 df-preset 16751 df-poset 16769 df-plt 16781 df-lub 16797 df-glb 16798 df-join 16799 df-meet 16800 df-p0 16862 df-p1 16863 df-lat 16869 df-clat 16931 df-mgm 17065 df-sgrp 17107 df-mnd 17118 df-submnd 17159 df-grp 17248 df-minusg 17249 df-sbg 17250 df-subg 17414 df-cntz 17573 df-oppg 17599 df-lsm 17874 df-cmn 18018 df-abl 18019 df-mgp 18313 df-ur 18325 df-ring 18372 df-oppr 18446 df-dvdsr 18464 df-unit 18465 df-invr 18495 df-dvr 18506 df-drng 18572 df-lmod 18688 df-lss 18754 df-lsp 18793 df-lvec 18924 df-lsatoms 33281 df-lshyp 33282 df-lcv 33324 df-lfl 33363 df-lkr 33391 df-ldual 33429 df-oposet 33481 df-ol 33483 df-oml 33484 df-covers 33571 df-ats 33572 df-atl 33603 df-cvlat 33627 df-hlat 33656 df-llines 33802 df-lplanes 33803 df-lvols 33804 df-lines 33805 df-psubsp 33807 df-pmap 33808 df-padd 34100 df-lhyp 34292 df-laut 34293 df-ldil 34408 df-ltrn 34409 df-trl 34464 df-tgrp 35049 df-tendo 35061 df-edring 35063 df-dveca 35309 df-disoa 35336 df-dvech 35386 df-dib 35446 df-dic 35480 df-dih 35536 df-doch 35655 df-djh 35702 df-lcdual 35894 df-mapd 35932 df-hvmap 36064 df-hdmap1 36101 df-hdmap 36102 |
This theorem is referenced by: hdmap14lem14 36191 |
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