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| Description: Lemma for lduallmod 39154. (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 2737 | . . . 4 ⊢ (Base‘𝐷) = (Base‘𝐷) | |
| 4 | lduallmod.w | . . . 4 ⊢ (𝜑 → 𝑊 ∈ LMod) | |
| 5 | 1, 2, 3, 4 | ldualvbase 39127 | . . 3 ⊢ (𝜑 → (Base‘𝐷) = 𝐹) | 
| 6 | 5 | eqcomd 2743 | . 2 ⊢ (𝜑 → 𝐹 = (Base‘𝐷)) | 
| 7 | eqidd 2738 | . 2 ⊢ (𝜑 → (+g‘𝐷) = (+g‘𝐷)) | |
| 8 | lduallmod.r | . . . 4 ⊢ 𝑅 = (Scalar‘𝑊) | |
| 9 | lduallmod.o | . . . 4 ⊢ 𝑂 = (oppr‘𝑅) | |
| 10 | eqid 2737 | . . . 4 ⊢ (Scalar‘𝐷) = (Scalar‘𝐷) | |
| 11 | 8, 9, 2, 10, 4 | ldualsca 39133 | . . 3 ⊢ (𝜑 → (Scalar‘𝐷) = 𝑂) | 
| 12 | 11 | eqcomd 2743 | . 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 20341 | . . 3 ⊢ 𝐾 = (Base‘𝑂) | 
| 17 | 16 | a1i 11 | . 2 ⊢ (𝜑 → 𝐾 = (Base‘𝑂)) | 
| 18 | eqid 2737 | . . . 4 ⊢ (+g‘𝑅) = (+g‘𝑅) | |
| 19 | 9, 18 | oppradd 20343 | . . 3 ⊢ (+g‘𝑅) = (+g‘𝑂) | 
| 20 | 19 | a1i 11 | . 2 ⊢ (𝜑 → (+g‘𝑅) = (+g‘𝑂)) | 
| 21 | 11 | fveq2d 6910 | . 2 ⊢ (𝜑 → (.r‘(Scalar‘𝐷)) = (.r‘𝑂)) | 
| 22 | eqid 2737 | . . . 4 ⊢ (1r‘𝑅) = (1r‘𝑅) | |
| 23 | 9, 22 | oppr1 20350 | . . 3 ⊢ (1r‘𝑅) = (1r‘𝑂) | 
| 24 | 23 | a1i 11 | . 2 ⊢ (𝜑 → (1r‘𝑅) = (1r‘𝑂)) | 
| 25 | 8 | lmodring 20866 | . . 3 ⊢ (𝑊 ∈ LMod → 𝑅 ∈ Ring) | 
| 26 | 9 | opprring 20347 | . . 3 ⊢ (𝑅 ∈ Ring → 𝑂 ∈ Ring) | 
| 27 | 4, 25, 26 | 3syl 18 | . 2 ⊢ (𝜑 → 𝑂 ∈ Ring) | 
| 28 | 2, 4 | ldualgrp 39147 | . 2 ⊢ (𝜑 → 𝐷 ∈ Grp) | 
| 29 | 4 | 3ad2ant1 1134 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐾 ∧ 𝑦 ∈ 𝐹) → 𝑊 ∈ LMod) | 
| 30 | simp2 1138 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐾 ∧ 𝑦 ∈ 𝐹) → 𝑥 ∈ 𝐾) | |
| 31 | simp3 1139 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐾 ∧ 𝑦 ∈ 𝐹) → 𝑦 ∈ 𝐹) | |
| 32 | 1, 8, 15, 2, 13, 29, 30, 31 | ldualvscl 39140 | . 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐾 ∧ 𝑦 ∈ 𝐹) → (𝑥 · 𝑦) ∈ 𝐹) | 
| 33 | eqid 2737 | . . 3 ⊢ (+g‘𝐷) = (+g‘𝐷) | |
| 34 | 4 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐾 ∧ 𝑦 ∈ 𝐹 ∧ 𝑧 ∈ 𝐹)) → 𝑊 ∈ LMod) | 
| 35 | simpr1 1195 | . . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐾 ∧ 𝑦 ∈ 𝐹 ∧ 𝑧 ∈ 𝐹)) → 𝑥 ∈ 𝐾) | |
| 36 | simpr2 1196 | . . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐾 ∧ 𝑦 ∈ 𝐹 ∧ 𝑧 ∈ 𝐹)) → 𝑦 ∈ 𝐹) | |
| 37 | simpr3 1197 | . . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐾 ∧ 𝑦 ∈ 𝐹 ∧ 𝑧 ∈ 𝐹)) → 𝑧 ∈ 𝐹) | |
| 38 | 1, 8, 15, 2, 33, 13, 34, 35, 36, 37 | ldualvsdi1 39144 | . 2 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐾 ∧ 𝑦 ∈ 𝐹 ∧ 𝑧 ∈ 𝐹)) → (𝑥 · (𝑦(+g‘𝐷)𝑧)) = ((𝑥 · 𝑦)(+g‘𝐷)(𝑥 · 𝑧))) | 
| 39 | 4 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐾 ∧ 𝑦 ∈ 𝐾 ∧ 𝑧 ∈ 𝐹)) → 𝑊 ∈ LMod) | 
| 40 | simpr1 1195 | . . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐾 ∧ 𝑦 ∈ 𝐾 ∧ 𝑧 ∈ 𝐹)) → 𝑥 ∈ 𝐾) | |
| 41 | simpr2 1196 | . . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐾 ∧ 𝑦 ∈ 𝐾 ∧ 𝑧 ∈ 𝐹)) → 𝑦 ∈ 𝐾) | |
| 42 | simpr3 1197 | . . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐾 ∧ 𝑦 ∈ 𝐾 ∧ 𝑧 ∈ 𝐹)) → 𝑧 ∈ 𝐹) | |
| 43 | 1, 8, 18, 15, 2, 33, 13, 39, 40, 41, 42 | ldualvsdi2 39145 | . 2 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐾 ∧ 𝑦 ∈ 𝐾 ∧ 𝑧 ∈ 𝐹)) → ((𝑥(+g‘𝑅)𝑦) · 𝑧) = ((𝑥 · 𝑧)(+g‘𝐷)(𝑦 · 𝑧))) | 
| 44 | eqid 2737 | . . 3 ⊢ (.r‘(Scalar‘𝐷)) = (.r‘(Scalar‘𝐷)) | |
| 45 | 1, 8, 15, 2, 10, 44, 13, 39, 40, 41, 42 | ldualvsass2 39143 | . 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 480 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐹) → 𝑊 ∈ LMod) | 
| 49 | 15, 22 | ringidcl 20262 | . . . . . 6 ⊢ (𝑅 ∈ Ring → (1r‘𝑅) ∈ 𝐾) | 
| 50 | 4, 25, 49 | 3syl 18 | . . . . 5 ⊢ (𝜑 → (1r‘𝑅) ∈ 𝐾) | 
| 51 | 50 | adantr 480 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐹) → (1r‘𝑅) ∈ 𝐾) | 
| 52 | simpr 484 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐹) → 𝑥 ∈ 𝐹) | |
| 53 | 1, 46, 8, 15, 47, 2, 13, 48, 51, 52 | ldualvs 39138 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐹) → ((1r‘𝑅) · 𝑥) = (𝑥 ∘f × (𝑉 × {(1r‘𝑅)}))) | 
| 54 | 46, 8, 1, 15, 47, 22, 48, 52 | lfl1sc 39085 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐹) → (𝑥 ∘f × (𝑉 × {(1r‘𝑅)})) = 𝑥) | 
| 55 | 53, 54 | eqtrd 2777 | . 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 20864 | 1 ⊢ (𝜑 → 𝐷 ∈ LMod) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 {csn 4626 × cxp 5683 ‘cfv 6561 (class class class)co 7431 ∘f cof 7695 Basecbs 17247 +gcplusg 17297 .rcmulr 17298 Scalarcsca 17300 ·𝑠 cvsca 17301 1rcur 20178 Ringcrg 20230 opprcoppr 20333 LModclmod 20858 LFnlclfn 39058 LDualcld 39124 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-tp 4631 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-om 7888 df-1st 8014 df-2nd 8015 df-tpos 8251 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-er 8745 df-map 8868 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-5 12332 df-6 12333 df-n0 12527 df-z 12614 df-uz 12879 df-fz 13548 df-struct 17184 df-sets 17201 df-slot 17219 df-ndx 17231 df-base 17248 df-plusg 17310 df-mulr 17311 df-sca 17313 df-vsca 17314 df-0g 17486 df-mgm 18653 df-sgrp 18732 df-mnd 18748 df-grp 18954 df-minusg 18955 df-sbg 18956 df-cmn 19800 df-abl 19801 df-mgp 20138 df-rng 20150 df-ur 20179 df-ring 20232 df-oppr 20334 df-lmod 20860 df-lfl 39059 df-ldual 39125 | 
| This theorem is referenced by: lduallmod 39154 | 
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