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Mirrors > Home > MPE Home > Th. List > Mathboxes > lduallmodlem | Structured version Visualization version GIF version |
Description: Lemma for lduallmod 36449. (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 2798 | . . . 4 ⊢ (Base‘𝐷) = (Base‘𝐷) | |
4 | lduallmod.w | . . . 4 ⊢ (𝜑 → 𝑊 ∈ LMod) | |
5 | 1, 2, 3, 4 | ldualvbase 36422 | . . 3 ⊢ (𝜑 → (Base‘𝐷) = 𝐹) |
6 | 5 | eqcomd 2804 | . 2 ⊢ (𝜑 → 𝐹 = (Base‘𝐷)) |
7 | eqidd 2799 | . 2 ⊢ (𝜑 → (+g‘𝐷) = (+g‘𝐷)) | |
8 | lduallmod.r | . . . 4 ⊢ 𝑅 = (Scalar‘𝑊) | |
9 | lduallmod.o | . . . 4 ⊢ 𝑂 = (oppr‘𝑅) | |
10 | eqid 2798 | . . . 4 ⊢ (Scalar‘𝐷) = (Scalar‘𝐷) | |
11 | 8, 9, 2, 10, 4 | ldualsca 36428 | . . 3 ⊢ (𝜑 → (Scalar‘𝐷) = 𝑂) |
12 | 11 | eqcomd 2804 | . 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 19375 | . . 3 ⊢ 𝐾 = (Base‘𝑂) |
17 | 16 | a1i 11 | . 2 ⊢ (𝜑 → 𝐾 = (Base‘𝑂)) |
18 | eqid 2798 | . . . 4 ⊢ (+g‘𝑅) = (+g‘𝑅) | |
19 | 9, 18 | oppradd 19376 | . . 3 ⊢ (+g‘𝑅) = (+g‘𝑂) |
20 | 19 | a1i 11 | . 2 ⊢ (𝜑 → (+g‘𝑅) = (+g‘𝑂)) |
21 | 11 | fveq2d 6649 | . 2 ⊢ (𝜑 → (.r‘(Scalar‘𝐷)) = (.r‘𝑂)) |
22 | eqid 2798 | . . . 4 ⊢ (1r‘𝑅) = (1r‘𝑅) | |
23 | 9, 22 | oppr1 19380 | . . 3 ⊢ (1r‘𝑅) = (1r‘𝑂) |
24 | 23 | a1i 11 | . 2 ⊢ (𝜑 → (1r‘𝑅) = (1r‘𝑂)) |
25 | 8 | lmodring 19635 | . . 3 ⊢ (𝑊 ∈ LMod → 𝑅 ∈ Ring) |
26 | 9 | opprring 19377 | . . 3 ⊢ (𝑅 ∈ Ring → 𝑂 ∈ Ring) |
27 | 4, 25, 26 | 3syl 18 | . 2 ⊢ (𝜑 → 𝑂 ∈ Ring) |
28 | 2, 4 | ldualgrp 36442 | . 2 ⊢ (𝜑 → 𝐷 ∈ Grp) |
29 | 4 | 3ad2ant1 1130 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐾 ∧ 𝑦 ∈ 𝐹) → 𝑊 ∈ LMod) |
30 | simp2 1134 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐾 ∧ 𝑦 ∈ 𝐹) → 𝑥 ∈ 𝐾) | |
31 | simp3 1135 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐾 ∧ 𝑦 ∈ 𝐹) → 𝑦 ∈ 𝐹) | |
32 | 1, 8, 15, 2, 13, 29, 30, 31 | ldualvscl 36435 | . 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐾 ∧ 𝑦 ∈ 𝐹) → (𝑥 · 𝑦) ∈ 𝐹) |
33 | eqid 2798 | . . 3 ⊢ (+g‘𝐷) = (+g‘𝐷) | |
34 | 4 | adantr 484 | . . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐾 ∧ 𝑦 ∈ 𝐹 ∧ 𝑧 ∈ 𝐹)) → 𝑊 ∈ LMod) |
35 | simpr1 1191 | . . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐾 ∧ 𝑦 ∈ 𝐹 ∧ 𝑧 ∈ 𝐹)) → 𝑥 ∈ 𝐾) | |
36 | simpr2 1192 | . . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐾 ∧ 𝑦 ∈ 𝐹 ∧ 𝑧 ∈ 𝐹)) → 𝑦 ∈ 𝐹) | |
37 | simpr3 1193 | . . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐾 ∧ 𝑦 ∈ 𝐹 ∧ 𝑧 ∈ 𝐹)) → 𝑧 ∈ 𝐹) | |
38 | 1, 8, 15, 2, 33, 13, 34, 35, 36, 37 | ldualvsdi1 36439 | . 2 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐾 ∧ 𝑦 ∈ 𝐹 ∧ 𝑧 ∈ 𝐹)) → (𝑥 · (𝑦(+g‘𝐷)𝑧)) = ((𝑥 · 𝑦)(+g‘𝐷)(𝑥 · 𝑧))) |
39 | 4 | adantr 484 | . . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐾 ∧ 𝑦 ∈ 𝐾 ∧ 𝑧 ∈ 𝐹)) → 𝑊 ∈ LMod) |
40 | simpr1 1191 | . . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐾 ∧ 𝑦 ∈ 𝐾 ∧ 𝑧 ∈ 𝐹)) → 𝑥 ∈ 𝐾) | |
41 | simpr2 1192 | . . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐾 ∧ 𝑦 ∈ 𝐾 ∧ 𝑧 ∈ 𝐹)) → 𝑦 ∈ 𝐾) | |
42 | simpr3 1193 | . . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐾 ∧ 𝑦 ∈ 𝐾 ∧ 𝑧 ∈ 𝐹)) → 𝑧 ∈ 𝐹) | |
43 | 1, 8, 18, 15, 2, 33, 13, 39, 40, 41, 42 | ldualvsdi2 36440 | . 2 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐾 ∧ 𝑦 ∈ 𝐾 ∧ 𝑧 ∈ 𝐹)) → ((𝑥(+g‘𝑅)𝑦) · 𝑧) = ((𝑥 · 𝑧)(+g‘𝐷)(𝑦 · 𝑧))) |
44 | eqid 2798 | . . 3 ⊢ (.r‘(Scalar‘𝐷)) = (.r‘(Scalar‘𝐷)) | |
45 | 1, 8, 15, 2, 10, 44, 13, 39, 40, 41, 42 | ldualvsass2 36438 | . 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 484 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐹) → 𝑊 ∈ LMod) |
49 | 15, 22 | ringidcl 19314 | . . . . . 6 ⊢ (𝑅 ∈ Ring → (1r‘𝑅) ∈ 𝐾) |
50 | 4, 25, 49 | 3syl 18 | . . . . 5 ⊢ (𝜑 → (1r‘𝑅) ∈ 𝐾) |
51 | 50 | adantr 484 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐹) → (1r‘𝑅) ∈ 𝐾) |
52 | simpr 488 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐹) → 𝑥 ∈ 𝐹) | |
53 | 1, 46, 8, 15, 47, 2, 13, 48, 51, 52 | ldualvs 36433 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐹) → ((1r‘𝑅) · 𝑥) = (𝑥 ∘f × (𝑉 × {(1r‘𝑅)}))) |
54 | 46, 8, 1, 15, 47, 22, 48, 52 | lfl1sc 36380 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐹) → (𝑥 ∘f × (𝑉 × {(1r‘𝑅)})) = 𝑥) |
55 | 53, 54 | eqtrd 2833 | . 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 19633 | 1 ⊢ (𝜑 → 𝐷 ∈ LMod) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∧ w3a 1084 = wceq 1538 ∈ wcel 2111 {csn 4525 × cxp 5517 ‘cfv 6324 (class class class)co 7135 ∘f cof 7387 Basecbs 16475 +gcplusg 16557 .rcmulr 16558 Scalarcsca 16560 ·𝑠 cvsca 16561 1rcur 19244 Ringcrg 19290 opprcoppr 19368 LModclmod 19627 LFnlclfn 36353 LDualcld 36419 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-of 7389 df-om 7561 df-1st 7671 df-2nd 7672 df-tpos 7875 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-oadd 8089 df-er 8272 df-map 8391 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-nn 11626 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-n0 11886 df-z 11970 df-uz 12232 df-fz 12886 df-struct 16477 df-ndx 16478 df-slot 16479 df-base 16481 df-sets 16482 df-plusg 16570 df-mulr 16571 df-sca 16573 df-vsca 16574 df-0g 16707 df-mgm 17844 df-sgrp 17893 df-mnd 17904 df-grp 18098 df-minusg 18099 df-sbg 18100 df-cmn 18900 df-abl 18901 df-mgp 19233 df-ur 19245 df-ring 19292 df-oppr 19369 df-lmod 19629 df-lfl 36354 df-ldual 36420 |
This theorem is referenced by: lduallmod 36449 |
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