Mathbox for Norm Megill |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > Mathboxes > lduallmodlem | Structured version Visualization version GIF version |
Description: Lemma for lduallmod 36304. (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 2821 | . . . 4 ⊢ (Base‘𝐷) = (Base‘𝐷) | |
4 | lduallmod.w | . . . 4 ⊢ (𝜑 → 𝑊 ∈ LMod) | |
5 | 1, 2, 3, 4 | ldualvbase 36277 | . . 3 ⊢ (𝜑 → (Base‘𝐷) = 𝐹) |
6 | 5 | eqcomd 2827 | . 2 ⊢ (𝜑 → 𝐹 = (Base‘𝐷)) |
7 | eqidd 2822 | . 2 ⊢ (𝜑 → (+g‘𝐷) = (+g‘𝐷)) | |
8 | lduallmod.r | . . . 4 ⊢ 𝑅 = (Scalar‘𝑊) | |
9 | lduallmod.o | . . . 4 ⊢ 𝑂 = (oppr‘𝑅) | |
10 | eqid 2821 | . . . 4 ⊢ (Scalar‘𝐷) = (Scalar‘𝐷) | |
11 | 8, 9, 2, 10, 4 | ldualsca 36283 | . . 3 ⊢ (𝜑 → (Scalar‘𝐷) = 𝑂) |
12 | 11 | eqcomd 2827 | . 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 19379 | . . 3 ⊢ 𝐾 = (Base‘𝑂) |
17 | 16 | a1i 11 | . 2 ⊢ (𝜑 → 𝐾 = (Base‘𝑂)) |
18 | eqid 2821 | . . . 4 ⊢ (+g‘𝑅) = (+g‘𝑅) | |
19 | 9, 18 | oppradd 19380 | . . 3 ⊢ (+g‘𝑅) = (+g‘𝑂) |
20 | 19 | a1i 11 | . 2 ⊢ (𝜑 → (+g‘𝑅) = (+g‘𝑂)) |
21 | 11 | fveq2d 6674 | . 2 ⊢ (𝜑 → (.r‘(Scalar‘𝐷)) = (.r‘𝑂)) |
22 | eqid 2821 | . . . 4 ⊢ (1r‘𝑅) = (1r‘𝑅) | |
23 | 9, 22 | oppr1 19384 | . . 3 ⊢ (1r‘𝑅) = (1r‘𝑂) |
24 | 23 | a1i 11 | . 2 ⊢ (𝜑 → (1r‘𝑅) = (1r‘𝑂)) |
25 | 8 | lmodring 19642 | . . 3 ⊢ (𝑊 ∈ LMod → 𝑅 ∈ Ring) |
26 | 9 | opprring 19381 | . . 3 ⊢ (𝑅 ∈ Ring → 𝑂 ∈ Ring) |
27 | 4, 25, 26 | 3syl 18 | . 2 ⊢ (𝜑 → 𝑂 ∈ Ring) |
28 | 2, 4 | ldualgrp 36297 | . 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 36290 | . 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐾 ∧ 𝑦 ∈ 𝐹) → (𝑥 · 𝑦) ∈ 𝐹) |
33 | eqid 2821 | . . 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 36294 | . 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 36295 | . 2 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐾 ∧ 𝑦 ∈ 𝐾 ∧ 𝑧 ∈ 𝐹)) → ((𝑥(+g‘𝑅)𝑦) · 𝑧) = ((𝑥 · 𝑧)(+g‘𝐷)(𝑦 · 𝑧))) |
44 | eqid 2821 | . . 3 ⊢ (.r‘(Scalar‘𝐷)) = (.r‘(Scalar‘𝐷)) | |
45 | 1, 8, 15, 2, 10, 44, 13, 39, 40, 41, 42 | ldualvsass2 36293 | . 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 19318 | . . . . . 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 36288 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐹) → ((1r‘𝑅) · 𝑥) = (𝑥 ∘f × (𝑉 × {(1r‘𝑅)}))) |
54 | 46, 8, 1, 15, 47, 22, 48, 52 | lfl1sc 36235 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐹) → (𝑥 ∘f × (𝑉 × {(1r‘𝑅)})) = 𝑥) |
55 | 53, 54 | eqtrd 2856 | . 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 19640 | 1 ⊢ (𝜑 → 𝐷 ∈ LMod) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 {csn 4567 × cxp 5553 ‘cfv 6355 (class class class)co 7156 ∘f cof 7407 Basecbs 16483 +gcplusg 16565 .rcmulr 16566 Scalarcsca 16568 ·𝑠 cvsca 16569 1rcur 19251 Ringcrg 19297 opprcoppr 19372 LModclmod 19634 LFnlclfn 36208 LDualcld 36274 |
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 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 |
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 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-of 7409 df-om 7581 df-1st 7689 df-2nd 7690 df-tpos 7892 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-oadd 8106 df-er 8289 df-map 8408 df-en 8510 df-dom 8511 df-sdom 8512 df-fin 8513 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-nn 11639 df-2 11701 df-3 11702 df-4 11703 df-5 11704 df-6 11705 df-n0 11899 df-z 11983 df-uz 12245 df-fz 12894 df-struct 16485 df-ndx 16486 df-slot 16487 df-base 16489 df-sets 16490 df-plusg 16578 df-mulr 16579 df-sca 16581 df-vsca 16582 df-0g 16715 df-mgm 17852 df-sgrp 17901 df-mnd 17912 df-grp 18106 df-minusg 18107 df-sbg 18108 df-cmn 18908 df-abl 18909 df-mgp 19240 df-ur 19252 df-ring 19299 df-oppr 19373 df-lmod 19636 df-lfl 36209 df-ldual 36275 |
This theorem is referenced by: lduallmod 36304 |
Copyright terms: Public domain | W3C validator |