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Mirrors > Home > MPE Home > Th. List > Mathboxes > lduallmodlem | Structured version Visualization version GIF version |
Description: Lemma for lduallmod 39135. (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 2735 | . . . 4 ⊢ (Base‘𝐷) = (Base‘𝐷) | |
4 | lduallmod.w | . . . 4 ⊢ (𝜑 → 𝑊 ∈ LMod) | |
5 | 1, 2, 3, 4 | ldualvbase 39108 | . . 3 ⊢ (𝜑 → (Base‘𝐷) = 𝐹) |
6 | 5 | eqcomd 2741 | . 2 ⊢ (𝜑 → 𝐹 = (Base‘𝐷)) |
7 | eqidd 2736 | . 2 ⊢ (𝜑 → (+g‘𝐷) = (+g‘𝐷)) | |
8 | lduallmod.r | . . . 4 ⊢ 𝑅 = (Scalar‘𝑊) | |
9 | lduallmod.o | . . . 4 ⊢ 𝑂 = (oppr‘𝑅) | |
10 | eqid 2735 | . . . 4 ⊢ (Scalar‘𝐷) = (Scalar‘𝐷) | |
11 | 8, 9, 2, 10, 4 | ldualsca 39114 | . . 3 ⊢ (𝜑 → (Scalar‘𝐷) = 𝑂) |
12 | 11 | eqcomd 2741 | . 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 20358 | . . 3 ⊢ 𝐾 = (Base‘𝑂) |
17 | 16 | a1i 11 | . 2 ⊢ (𝜑 → 𝐾 = (Base‘𝑂)) |
18 | eqid 2735 | . . . 4 ⊢ (+g‘𝑅) = (+g‘𝑅) | |
19 | 9, 18 | oppradd 20360 | . . 3 ⊢ (+g‘𝑅) = (+g‘𝑂) |
20 | 19 | a1i 11 | . 2 ⊢ (𝜑 → (+g‘𝑅) = (+g‘𝑂)) |
21 | 11 | fveq2d 6911 | . 2 ⊢ (𝜑 → (.r‘(Scalar‘𝐷)) = (.r‘𝑂)) |
22 | eqid 2735 | . . . 4 ⊢ (1r‘𝑅) = (1r‘𝑅) | |
23 | 9, 22 | oppr1 20367 | . . 3 ⊢ (1r‘𝑅) = (1r‘𝑂) |
24 | 23 | a1i 11 | . 2 ⊢ (𝜑 → (1r‘𝑅) = (1r‘𝑂)) |
25 | 8 | lmodring 20883 | . . 3 ⊢ (𝑊 ∈ LMod → 𝑅 ∈ Ring) |
26 | 9 | opprring 20364 | . . 3 ⊢ (𝑅 ∈ Ring → 𝑂 ∈ Ring) |
27 | 4, 25, 26 | 3syl 18 | . 2 ⊢ (𝜑 → 𝑂 ∈ Ring) |
28 | 2, 4 | ldualgrp 39128 | . 2 ⊢ (𝜑 → 𝐷 ∈ Grp) |
29 | 4 | 3ad2ant1 1132 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐾 ∧ 𝑦 ∈ 𝐹) → 𝑊 ∈ LMod) |
30 | simp2 1136 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐾 ∧ 𝑦 ∈ 𝐹) → 𝑥 ∈ 𝐾) | |
31 | simp3 1137 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐾 ∧ 𝑦 ∈ 𝐹) → 𝑦 ∈ 𝐹) | |
32 | 1, 8, 15, 2, 13, 29, 30, 31 | ldualvscl 39121 | . 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐾 ∧ 𝑦 ∈ 𝐹) → (𝑥 · 𝑦) ∈ 𝐹) |
33 | eqid 2735 | . . 3 ⊢ (+g‘𝐷) = (+g‘𝐷) | |
34 | 4 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐾 ∧ 𝑦 ∈ 𝐹 ∧ 𝑧 ∈ 𝐹)) → 𝑊 ∈ LMod) |
35 | simpr1 1193 | . . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐾 ∧ 𝑦 ∈ 𝐹 ∧ 𝑧 ∈ 𝐹)) → 𝑥 ∈ 𝐾) | |
36 | simpr2 1194 | . . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐾 ∧ 𝑦 ∈ 𝐹 ∧ 𝑧 ∈ 𝐹)) → 𝑦 ∈ 𝐹) | |
37 | simpr3 1195 | . . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐾 ∧ 𝑦 ∈ 𝐹 ∧ 𝑧 ∈ 𝐹)) → 𝑧 ∈ 𝐹) | |
38 | 1, 8, 15, 2, 33, 13, 34, 35, 36, 37 | ldualvsdi1 39125 | . 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 1193 | . . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐾 ∧ 𝑦 ∈ 𝐾 ∧ 𝑧 ∈ 𝐹)) → 𝑥 ∈ 𝐾) | |
41 | simpr2 1194 | . . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐾 ∧ 𝑦 ∈ 𝐾 ∧ 𝑧 ∈ 𝐹)) → 𝑦 ∈ 𝐾) | |
42 | simpr3 1195 | . . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐾 ∧ 𝑦 ∈ 𝐾 ∧ 𝑧 ∈ 𝐹)) → 𝑧 ∈ 𝐹) | |
43 | 1, 8, 18, 15, 2, 33, 13, 39, 40, 41, 42 | ldualvsdi2 39126 | . 2 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐾 ∧ 𝑦 ∈ 𝐾 ∧ 𝑧 ∈ 𝐹)) → ((𝑥(+g‘𝑅)𝑦) · 𝑧) = ((𝑥 · 𝑧)(+g‘𝐷)(𝑦 · 𝑧))) |
44 | eqid 2735 | . . 3 ⊢ (.r‘(Scalar‘𝐷)) = (.r‘(Scalar‘𝐷)) | |
45 | 1, 8, 15, 2, 10, 44, 13, 39, 40, 41, 42 | ldualvsass2 39124 | . 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 20280 | . . . . . 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 39119 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐹) → ((1r‘𝑅) · 𝑥) = (𝑥 ∘f × (𝑉 × {(1r‘𝑅)}))) |
54 | 46, 8, 1, 15, 47, 22, 48, 52 | lfl1sc 39066 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐹) → (𝑥 ∘f × (𝑉 × {(1r‘𝑅)})) = 𝑥) |
55 | 53, 54 | eqtrd 2775 | . 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 20881 | 1 ⊢ (𝜑 → 𝐷 ∈ LMod) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1086 = wceq 1537 ∈ wcel 2106 {csn 4631 × cxp 5687 ‘cfv 6563 (class class class)co 7431 ∘f cof 7695 Basecbs 17245 +gcplusg 17298 .rcmulr 17299 Scalarcsca 17301 ·𝑠 cvsca 17302 1rcur 20199 Ringcrg 20251 opprcoppr 20350 LModclmod 20875 LFnlclfn 39039 LDualcld 39105 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-tp 4636 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-om 7888 df-1st 8013 df-2nd 8014 df-tpos 8250 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-er 8744 df-map 8867 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-nn 12265 df-2 12327 df-3 12328 df-4 12329 df-5 12330 df-6 12331 df-n0 12525 df-z 12612 df-uz 12877 df-fz 13545 df-struct 17181 df-sets 17198 df-slot 17216 df-ndx 17228 df-base 17246 df-plusg 17311 df-mulr 17312 df-sca 17314 df-vsca 17315 df-0g 17488 df-mgm 18666 df-sgrp 18745 df-mnd 18761 df-grp 18967 df-minusg 18968 df-sbg 18969 df-cmn 19815 df-abl 19816 df-mgp 20153 df-rng 20171 df-ur 20200 df-ring 20253 df-oppr 20351 df-lmod 20877 df-lfl 39040 df-ldual 39106 |
This theorem is referenced by: lduallmod 39135 |
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