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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 39516 | . 2 ⊢ (𝜑 → ((𝑆‘(𝐹 · 𝑋)) = (𝐺 ∙ (𝑆‘𝑋)) ↔ ∀𝑦 ∈ (𝑉 ∖ { 0 })(𝑆‘(𝐹 · 𝑦)) = (𝐺 ∙ (𝑆‘𝑦)))) |
18 | velsn 4532 | . . . . . 6 ⊢ (𝑦 ∈ { 0 } ↔ 𝑦 = 0 ) | |
19 | 1, 7, 10 | lcdlmod 39229 | . . . . . . . . 9 ⊢ (𝜑 → 𝐶 ∈ LMod) |
20 | eqid 2738 | . . . . . . . . . 10 ⊢ (0g‘𝐶) = (0g‘𝐶) | |
21 | 12, 8, 13, 20 | lmodvs0 19787 | . . . . . . . . 9 ⊢ ((𝐶 ∈ LMod ∧ 𝐺 ∈ 𝐴) → (𝐺 ∙ (0g‘𝐶)) = (0g‘𝐶)) |
22 | 19, 16, 21 | syl2anc 587 | . . . . . . . 8 ⊢ (𝜑 → (𝐺 ∙ (0g‘𝐶)) = (0g‘𝐶)) |
23 | 1, 2, 14, 7, 20, 9, 10 | hdmapval0 39470 | . . . . . . . . 9 ⊢ (𝜑 → (𝑆‘ 0 ) = (0g‘𝐶)) |
24 | 23 | oveq2d 7186 | . . . . . . . 8 ⊢ (𝜑 → (𝐺 ∙ (𝑆‘ 0 )) = (𝐺 ∙ (0g‘𝐶))) |
25 | 1, 2, 10 | dvhlmod 38747 | . . . . . . . . . . 11 ⊢ (𝜑 → 𝑈 ∈ LMod) |
26 | 5, 4, 6, 14 | lmodvs0 19787 | . . . . . . . . . . 11 ⊢ ((𝑈 ∈ LMod ∧ 𝐹 ∈ 𝐵) → (𝐹 · 0 ) = 0 ) |
27 | 25, 11, 26 | syl2anc 587 | . . . . . . . . . 10 ⊢ (𝜑 → (𝐹 · 0 ) = 0 ) |
28 | 27 | fveq2d 6678 | . . . . . . . . 9 ⊢ (𝜑 → (𝑆‘(𝐹 · 0 )) = (𝑆‘ 0 )) |
29 | 28, 23 | eqtrd 2773 | . . . . . . . 8 ⊢ (𝜑 → (𝑆‘(𝐹 · 0 )) = (0g‘𝐶)) |
30 | 22, 24, 29 | 3eqtr4rd 2784 | . . . . . . 7 ⊢ (𝜑 → (𝑆‘(𝐹 · 0 )) = (𝐺 ∙ (𝑆‘ 0 ))) |
31 | oveq2 7178 | . . . . . . . . 9 ⊢ (𝑦 = 0 → (𝐹 · 𝑦) = (𝐹 · 0 )) | |
32 | 31 | fveq2d 6678 | . . . . . . . 8 ⊢ (𝑦 = 0 → (𝑆‘(𝐹 · 𝑦)) = (𝑆‘(𝐹 · 0 ))) |
33 | fveq2 6674 | . . . . . . . . 9 ⊢ (𝑦 = 0 → (𝑆‘𝑦) = (𝑆‘ 0 )) | |
34 | 33 | oveq2d 7186 | . . . . . . . 8 ⊢ (𝑦 = 0 → (𝐺 ∙ (𝑆‘𝑦)) = (𝐺 ∙ (𝑆‘ 0 ))) |
35 | 32, 34 | eqeq12d 2754 | . . . . . . 7 ⊢ (𝑦 = 0 → ((𝑆‘(𝐹 · 𝑦)) = (𝐺 ∙ (𝑆‘𝑦)) ↔ (𝑆‘(𝐹 · 0 )) = (𝐺 ∙ (𝑆‘ 0 )))) |
36 | 30, 35 | syl5ibrcom 250 | . . . . . 6 ⊢ (𝜑 → (𝑦 = 0 → (𝑆‘(𝐹 · 𝑦)) = (𝐺 ∙ (𝑆‘𝑦)))) |
37 | 18, 36 | syl5bi 245 | . . . . 5 ⊢ (𝜑 → (𝑦 ∈ { 0 } → (𝑆‘(𝐹 · 𝑦)) = (𝐺 ∙ (𝑆‘𝑦)))) |
38 | 37 | ralrimiv 3095 | . . . 4 ⊢ (𝜑 → ∀𝑦 ∈ { 0 } (𝑆‘(𝐹 · 𝑦)) = (𝐺 ∙ (𝑆‘𝑦))) |
39 | 38 | biantrud 535 | . . 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 4081 | . . 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 | bitr4di 292 | . 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 19782 | . . . 4 ⊢ (𝑈 ∈ LMod → 0 ∈ 𝑉) |
43 | difsnid 4698 | . . . 4 ⊢ ( 0 ∈ 𝑉 → ((𝑉 ∖ { 0 }) ∪ { 0 }) = 𝑉) | |
44 | 25, 42, 43 | 3syl 18 | . . 3 ⊢ (𝜑 → ((𝑉 ∖ { 0 }) ∪ { 0 }) = 𝑉) |
45 | 44 | raleqdv 3316 | . 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 308 | 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 209 ∧ wa 399 = wceq 1542 ∈ wcel 2114 ∀wral 3053 ∖ cdif 3840 ∪ cun 3841 {csn 4516 ‘cfv 6339 (class class class)co 7170 Basecbs 16586 Scalarcsca 16671 ·𝑠 cvsca 16672 0gc0g 16816 LModclmod 19753 HLchlt 36987 LHypclh 37621 DVecHcdvh 38715 LCDualclcd 39223 HDMapchdma 39429 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2710 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5232 ax-pr 5296 ax-un 7479 ax-cnex 10671 ax-resscn 10672 ax-1cn 10673 ax-icn 10674 ax-addcl 10675 ax-addrcl 10676 ax-mulcl 10677 ax-mulrcl 10678 ax-mulcom 10679 ax-addass 10680 ax-mulass 10681 ax-distr 10682 ax-i2m1 10683 ax-1ne0 10684 ax-1rid 10685 ax-rnegex 10686 ax-rrecex 10687 ax-cnre 10688 ax-pre-lttri 10689 ax-pre-lttrn 10690 ax-pre-ltadd 10691 ax-pre-mulgt0 10692 ax-riotaBAD 36590 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-nel 3039 df-ral 3058 df-rex 3059 df-reu 3060 df-rmo 3061 df-rab 3062 df-v 3400 df-sbc 3681 df-csb 3791 df-dif 3846 df-un 3848 df-in 3850 df-ss 3860 df-pss 3862 df-nul 4212 df-if 4415 df-pw 4490 df-sn 4517 df-pr 4519 df-tp 4521 df-op 4523 df-ot 4525 df-uni 4797 df-int 4837 df-iun 4883 df-iin 4884 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5429 df-eprel 5434 df-po 5442 df-so 5443 df-fr 5483 df-we 5485 df-xp 5531 df-rel 5532 df-cnv 5533 df-co 5534 df-dm 5535 df-rn 5536 df-res 5537 df-ima 5538 df-pred 6129 df-ord 6175 df-on 6176 df-lim 6177 df-suc 6178 df-iota 6297 df-fun 6341 df-fn 6342 df-f 6343 df-f1 6344 df-fo 6345 df-f1o 6346 df-fv 6347 df-riota 7127 df-ov 7173 df-oprab 7174 df-mpo 7175 df-of 7425 df-om 7600 df-1st 7714 df-2nd 7715 df-tpos 7921 df-undef 7968 df-wrecs 7976 df-recs 8037 df-rdg 8075 df-1o 8131 df-er 8320 df-map 8439 df-en 8556 df-dom 8557 df-sdom 8558 df-fin 8559 df-pnf 10755 df-mnf 10756 df-xr 10757 df-ltxr 10758 df-le 10759 df-sub 10950 df-neg 10951 df-nn 11717 df-2 11779 df-3 11780 df-4 11781 df-5 11782 df-6 11783 df-n0 11977 df-z 12063 df-uz 12325 df-fz 12982 df-struct 16588 df-ndx 16589 df-slot 16590 df-base 16592 df-sets 16593 df-ress 16594 df-plusg 16681 df-mulr 16682 df-sca 16684 df-vsca 16685 df-0g 16818 df-mre 16960 df-mrc 16961 df-acs 16963 df-proset 17654 df-poset 17672 df-plt 17684 df-lub 17700 df-glb 17701 df-join 17702 df-meet 17703 df-p0 17765 df-p1 17766 df-lat 17772 df-clat 17834 df-mgm 17968 df-sgrp 18017 df-mnd 18028 df-submnd 18073 df-grp 18222 df-minusg 18223 df-sbg 18224 df-subg 18394 df-cntz 18565 df-oppg 18592 df-lsm 18879 df-cmn 19026 df-abl 19027 df-mgp 19359 df-ur 19371 df-ring 19418 df-oppr 19495 df-dvdsr 19513 df-unit 19514 df-invr 19544 df-dvr 19555 df-drng 19623 df-lmod 19755 df-lss 19823 df-lsp 19863 df-lvec 19994 df-lsatoms 36613 df-lshyp 36614 df-lcv 36656 df-lfl 36695 df-lkr 36723 df-ldual 36761 df-oposet 36813 df-ol 36815 df-oml 36816 df-covers 36903 df-ats 36904 df-atl 36935 df-cvlat 36959 df-hlat 36988 df-llines 37135 df-lplanes 37136 df-lvols 37137 df-lines 37138 df-psubsp 37140 df-pmap 37141 df-padd 37433 df-lhyp 37625 df-laut 37626 df-ldil 37741 df-ltrn 37742 df-trl 37796 df-tgrp 38380 df-tendo 38392 df-edring 38394 df-dveca 38640 df-disoa 38666 df-dvech 38716 df-dib 38776 df-dic 38810 df-dih 38866 df-doch 38985 df-djh 39032 df-lcdual 39224 df-mapd 39262 df-hvmap 39394 df-hdmap1 39430 df-hdmap 39431 |
This theorem is referenced by: hdmap14lem14 39518 |
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