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Mirrors > Home > MPE Home > Th. List > Mathboxes > lduallmodlem | Structured version Visualization version GIF version |
Description: Lemma for lduallmod 36316. (Contributed by NM, 22-Oct-2014.) |
Ref | Expression |
---|---|
lduallmod.d | ⊢ 𝐷 = (LDual‘𝑊) |
lduallmod.w | ⊢ (𝜑 → 𝑊 ∈ LMod) |
lduallmod.v | ⊢ 𝑉 = (Base‘𝑊) |
lduallmod.p | ⊢ + = ∘f (+g‘𝑊) |
lduallmod.f | ⊢ 𝐹 = (LFnl‘𝑊) |
lduallmod.r | ⊢ 𝑅 = (Scalar‘𝑊) |
lduallmod.k | ⊢ 𝐾 = (Base‘𝑅) |
lduallmod.t | ⊢ × = (.r‘𝑅) |
lduallmod.o | ⊢ 𝑂 = (oppr‘𝑅) |
lduallmod.s | ⊢ · = ( ·𝑠 ‘𝐷) |
Ref | Expression |
---|---|
lduallmodlem | ⊢ (𝜑 → 𝐷 ∈ LMod) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lduallmod.f | . . . 4 ⊢ 𝐹 = (LFnl‘𝑊) | |
2 | lduallmod.d | . . . 4 ⊢ 𝐷 = (LDual‘𝑊) | |
3 | eqid 2820 | . . . 4 ⊢ (Base‘𝐷) = (Base‘𝐷) | |
4 | lduallmod.w | . . . 4 ⊢ (𝜑 → 𝑊 ∈ LMod) | |
5 | 1, 2, 3, 4 | ldualvbase 36289 | . . 3 ⊢ (𝜑 → (Base‘𝐷) = 𝐹) |
6 | 5 | eqcomd 2826 | . 2 ⊢ (𝜑 → 𝐹 = (Base‘𝐷)) |
7 | eqidd 2821 | . 2 ⊢ (𝜑 → (+g‘𝐷) = (+g‘𝐷)) | |
8 | lduallmod.r | . . . 4 ⊢ 𝑅 = (Scalar‘𝑊) | |
9 | lduallmod.o | . . . 4 ⊢ 𝑂 = (oppr‘𝑅) | |
10 | eqid 2820 | . . . 4 ⊢ (Scalar‘𝐷) = (Scalar‘𝐷) | |
11 | 8, 9, 2, 10, 4 | ldualsca 36295 | . . 3 ⊢ (𝜑 → (Scalar‘𝐷) = 𝑂) |
12 | 11 | eqcomd 2826 | . 2 ⊢ (𝜑 → 𝑂 = (Scalar‘𝐷)) |
13 | lduallmod.s | . . 3 ⊢ · = ( ·𝑠 ‘𝐷) | |
14 | 13 | a1i 11 | . 2 ⊢ (𝜑 → · = ( ·𝑠 ‘𝐷)) |
15 | lduallmod.k | . . . 4 ⊢ 𝐾 = (Base‘𝑅) | |
16 | 9, 15 | opprbas 19357 | . . 3 ⊢ 𝐾 = (Base‘𝑂) |
17 | 16 | a1i 11 | . 2 ⊢ (𝜑 → 𝐾 = (Base‘𝑂)) |
18 | eqid 2820 | . . . 4 ⊢ (+g‘𝑅) = (+g‘𝑅) | |
19 | 9, 18 | oppradd 19358 | . . 3 ⊢ (+g‘𝑅) = (+g‘𝑂) |
20 | 19 | a1i 11 | . 2 ⊢ (𝜑 → (+g‘𝑅) = (+g‘𝑂)) |
21 | 11 | fveq2d 6655 | . 2 ⊢ (𝜑 → (.r‘(Scalar‘𝐷)) = (.r‘𝑂)) |
22 | eqid 2820 | . . . 4 ⊢ (1r‘𝑅) = (1r‘𝑅) | |
23 | 9, 22 | oppr1 19362 | . . 3 ⊢ (1r‘𝑅) = (1r‘𝑂) |
24 | 23 | a1i 11 | . 2 ⊢ (𝜑 → (1r‘𝑅) = (1r‘𝑂)) |
25 | 8 | lmodring 19620 | . . 3 ⊢ (𝑊 ∈ LMod → 𝑅 ∈ Ring) |
26 | 9 | opprring 19359 | . . 3 ⊢ (𝑅 ∈ Ring → 𝑂 ∈ Ring) |
27 | 4, 25, 26 | 3syl 18 | . 2 ⊢ (𝜑 → 𝑂 ∈ Ring) |
28 | 2, 4 | ldualgrp 36309 | . 2 ⊢ (𝜑 → 𝐷 ∈ Grp) |
29 | 4 | 3ad2ant1 1129 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐾 ∧ 𝑦 ∈ 𝐹) → 𝑊 ∈ LMod) |
30 | simp2 1133 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐾 ∧ 𝑦 ∈ 𝐹) → 𝑥 ∈ 𝐾) | |
31 | simp3 1134 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐾 ∧ 𝑦 ∈ 𝐹) → 𝑦 ∈ 𝐹) | |
32 | 1, 8, 15, 2, 13, 29, 30, 31 | ldualvscl 36302 | . 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐾 ∧ 𝑦 ∈ 𝐹) → (𝑥 · 𝑦) ∈ 𝐹) |
33 | eqid 2820 | . . 3 ⊢ (+g‘𝐷) = (+g‘𝐷) | |
34 | 4 | adantr 483 | . . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐾 ∧ 𝑦 ∈ 𝐹 ∧ 𝑧 ∈ 𝐹)) → 𝑊 ∈ LMod) |
35 | simpr1 1190 | . . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐾 ∧ 𝑦 ∈ 𝐹 ∧ 𝑧 ∈ 𝐹)) → 𝑥 ∈ 𝐾) | |
36 | simpr2 1191 | . . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐾 ∧ 𝑦 ∈ 𝐹 ∧ 𝑧 ∈ 𝐹)) → 𝑦 ∈ 𝐹) | |
37 | simpr3 1192 | . . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐾 ∧ 𝑦 ∈ 𝐹 ∧ 𝑧 ∈ 𝐹)) → 𝑧 ∈ 𝐹) | |
38 | 1, 8, 15, 2, 33, 13, 34, 35, 36, 37 | ldualvsdi1 36306 | . 2 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐾 ∧ 𝑦 ∈ 𝐹 ∧ 𝑧 ∈ 𝐹)) → (𝑥 · (𝑦(+g‘𝐷)𝑧)) = ((𝑥 · 𝑦)(+g‘𝐷)(𝑥 · 𝑧))) |
39 | 4 | adantr 483 | . . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐾 ∧ 𝑦 ∈ 𝐾 ∧ 𝑧 ∈ 𝐹)) → 𝑊 ∈ LMod) |
40 | simpr1 1190 | . . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐾 ∧ 𝑦 ∈ 𝐾 ∧ 𝑧 ∈ 𝐹)) → 𝑥 ∈ 𝐾) | |
41 | simpr2 1191 | . . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐾 ∧ 𝑦 ∈ 𝐾 ∧ 𝑧 ∈ 𝐹)) → 𝑦 ∈ 𝐾) | |
42 | simpr3 1192 | . . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐾 ∧ 𝑦 ∈ 𝐾 ∧ 𝑧 ∈ 𝐹)) → 𝑧 ∈ 𝐹) | |
43 | 1, 8, 18, 15, 2, 33, 13, 39, 40, 41, 42 | ldualvsdi2 36307 | . 2 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐾 ∧ 𝑦 ∈ 𝐾 ∧ 𝑧 ∈ 𝐹)) → ((𝑥(+g‘𝑅)𝑦) · 𝑧) = ((𝑥 · 𝑧)(+g‘𝐷)(𝑦 · 𝑧))) |
44 | eqid 2820 | . . 3 ⊢ (.r‘(Scalar‘𝐷)) = (.r‘(Scalar‘𝐷)) | |
45 | 1, 8, 15, 2, 10, 44, 13, 39, 40, 41, 42 | ldualvsass2 36305 | . 2 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐾 ∧ 𝑦 ∈ 𝐾 ∧ 𝑧 ∈ 𝐹)) → ((𝑥(.r‘(Scalar‘𝐷))𝑦) · 𝑧) = (𝑥 · (𝑦 · 𝑧))) |
46 | lduallmod.v | . . . 4 ⊢ 𝑉 = (Base‘𝑊) | |
47 | lduallmod.t | . . . 4 ⊢ × = (.r‘𝑅) | |
48 | 4 | adantr 483 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐹) → 𝑊 ∈ LMod) |
49 | 15, 22 | ringidcl 19296 | . . . . . 6 ⊢ (𝑅 ∈ Ring → (1r‘𝑅) ∈ 𝐾) |
50 | 4, 25, 49 | 3syl 18 | . . . . 5 ⊢ (𝜑 → (1r‘𝑅) ∈ 𝐾) |
51 | 50 | adantr 483 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐹) → (1r‘𝑅) ∈ 𝐾) |
52 | simpr 487 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐹) → 𝑥 ∈ 𝐹) | |
53 | 1, 46, 8, 15, 47, 2, 13, 48, 51, 52 | ldualvs 36300 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐹) → ((1r‘𝑅) · 𝑥) = (𝑥 ∘f × (𝑉 × {(1r‘𝑅)}))) |
54 | 46, 8, 1, 15, 47, 22, 48, 52 | lfl1sc 36247 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐹) → (𝑥 ∘f × (𝑉 × {(1r‘𝑅)})) = 𝑥) |
55 | 53, 54 | eqtrd 2855 | . 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐹) → ((1r‘𝑅) · 𝑥) = 𝑥) |
56 | 6, 7, 12, 14, 17, 20, 21, 24, 27, 28, 32, 38, 43, 45, 55 | islmodd 19618 | 1 ⊢ (𝜑 → 𝐷 ∈ LMod) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 {csn 4548 × cxp 5534 ‘cfv 6336 (class class class)co 7137 ∘f cof 7388 Basecbs 16461 +gcplusg 16543 .rcmulr 16544 Scalarcsca 16546 ·𝑠 cvsca 16547 1rcur 19229 Ringcrg 19275 opprcoppr 19350 LModclmod 19612 LFnlclfn 36220 LDualcld 36286 |
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 2792 ax-rep 5171 ax-sep 5184 ax-nul 5191 ax-pow 5247 ax-pr 5311 ax-un 7442 ax-cnex 10574 ax-resscn 10575 ax-1cn 10576 ax-icn 10577 ax-addcl 10578 ax-addrcl 10579 ax-mulcl 10580 ax-mulrcl 10581 ax-mulcom 10582 ax-addass 10583 ax-mulass 10584 ax-distr 10585 ax-i2m1 10586 ax-1ne0 10587 ax-1rid 10588 ax-rnegex 10589 ax-rrecex 10590 ax-cnre 10591 ax-pre-lttri 10592 ax-pre-lttrn 10593 ax-pre-ltadd 10594 ax-pre-mulgt0 10595 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2653 df-clab 2799 df-cleq 2813 df-clel 2891 df-nfc 2959 df-ne 3012 df-nel 3119 df-ral 3138 df-rex 3139 df-reu 3140 df-rmo 3141 df-rab 3142 df-v 3483 df-sbc 3759 df-csb 3867 df-dif 3922 df-un 3924 df-in 3926 df-ss 3935 df-pss 3937 df-nul 4275 df-if 4449 df-pw 4522 df-sn 4549 df-pr 4551 df-tp 4553 df-op 4555 df-uni 4820 df-int 4858 df-iun 4902 df-br 5048 df-opab 5110 df-mpt 5128 df-tr 5154 df-id 5441 df-eprel 5446 df-po 5455 df-so 5456 df-fr 5495 df-we 5497 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-pred 6129 df-ord 6175 df-on 6176 df-lim 6177 df-suc 6178 df-iota 6295 df-fun 6338 df-fn 6339 df-f 6340 df-f1 6341 df-fo 6342 df-f1o 6343 df-fv 6344 df-riota 7095 df-ov 7140 df-oprab 7141 df-mpo 7142 df-of 7390 df-om 7562 df-1st 7670 df-2nd 7671 df-tpos 7873 df-wrecs 7928 df-recs 7989 df-rdg 8027 df-1o 8083 df-oadd 8087 df-er 8270 df-map 8389 df-en 8491 df-dom 8492 df-sdom 8493 df-fin 8494 df-pnf 10658 df-mnf 10659 df-xr 10660 df-ltxr 10661 df-le 10662 df-sub 10853 df-neg 10854 df-nn 11620 df-2 11682 df-3 11683 df-4 11684 df-5 11685 df-6 11686 df-n0 11880 df-z 11964 df-uz 12226 df-fz 12878 df-struct 16463 df-ndx 16464 df-slot 16465 df-base 16467 df-sets 16468 df-plusg 16556 df-mulr 16557 df-sca 16559 df-vsca 16560 df-0g 16693 df-mgm 17830 df-sgrp 17879 df-mnd 17890 df-grp 18084 df-minusg 18085 df-sbg 18086 df-cmn 18886 df-abl 18887 df-mgp 19218 df-ur 19230 df-ring 19277 df-oppr 19351 df-lmod 19614 df-lfl 36221 df-ldual 36287 |
This theorem is referenced by: lduallmod 36316 |
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