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Mirrors > Home > MPE Home > Th. List > Mathboxes > ascl0 | Structured version Visualization version GIF version |
Description: The scalar 0 embedded into a left module corresponds to the 0 of the left module if the left module is also a ring. (Contributed by AV, 31-Jul-2019.) |
Ref | Expression |
---|---|
ascl0.a | ⊢ 𝐴 = (algSc‘𝑊) |
ascl0.f | ⊢ 𝐹 = (Scalar‘𝑊) |
ascl0.l | ⊢ (𝜑 → 𝑊 ∈ LMod) |
ascl0.r | ⊢ (𝜑 → 𝑊 ∈ Ring) |
Ref | Expression |
---|---|
ascl0 | ⊢ (𝜑 → (𝐴‘(0g‘𝐹)) = (0g‘𝑊)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ascl0.l | . . . . 5 ⊢ (𝜑 → 𝑊 ∈ LMod) | |
2 | ascl0.f | . . . . . 6 ⊢ 𝐹 = (Scalar‘𝑊) | |
3 | 2 | lmodfgrp 19264 | . . . . 5 ⊢ (𝑊 ∈ LMod → 𝐹 ∈ Grp) |
4 | 1, 3 | syl 17 | . . . 4 ⊢ (𝜑 → 𝐹 ∈ Grp) |
5 | eqid 2778 | . . . . 5 ⊢ (Base‘𝐹) = (Base‘𝐹) | |
6 | eqid 2778 | . . . . 5 ⊢ (0g‘𝐹) = (0g‘𝐹) | |
7 | 5, 6 | grpidcl 17837 | . . . 4 ⊢ (𝐹 ∈ Grp → (0g‘𝐹) ∈ (Base‘𝐹)) |
8 | 4, 7 | syl 17 | . . 3 ⊢ (𝜑 → (0g‘𝐹) ∈ (Base‘𝐹)) |
9 | ascl0.a | . . . 4 ⊢ 𝐴 = (algSc‘𝑊) | |
10 | eqid 2778 | . . . 4 ⊢ ( ·𝑠 ‘𝑊) = ( ·𝑠 ‘𝑊) | |
11 | eqid 2778 | . . . 4 ⊢ (1r‘𝑊) = (1r‘𝑊) | |
12 | 9, 2, 5, 10, 11 | asclval 19732 | . . 3 ⊢ ((0g‘𝐹) ∈ (Base‘𝐹) → (𝐴‘(0g‘𝐹)) = ((0g‘𝐹)( ·𝑠 ‘𝑊)(1r‘𝑊))) |
13 | 8, 12 | syl 17 | . 2 ⊢ (𝜑 → (𝐴‘(0g‘𝐹)) = ((0g‘𝐹)( ·𝑠 ‘𝑊)(1r‘𝑊))) |
14 | ascl0.r | . . . 4 ⊢ (𝜑 → 𝑊 ∈ Ring) | |
15 | eqid 2778 | . . . . 5 ⊢ (Base‘𝑊) = (Base‘𝑊) | |
16 | 15, 11 | ringidcl 18955 | . . . 4 ⊢ (𝑊 ∈ Ring → (1r‘𝑊) ∈ (Base‘𝑊)) |
17 | 14, 16 | syl 17 | . . 3 ⊢ (𝜑 → (1r‘𝑊) ∈ (Base‘𝑊)) |
18 | eqid 2778 | . . . 4 ⊢ (0g‘𝑊) = (0g‘𝑊) | |
19 | 15, 2, 10, 6, 18 | lmod0vs 19288 | . . 3 ⊢ ((𝑊 ∈ LMod ∧ (1r‘𝑊) ∈ (Base‘𝑊)) → ((0g‘𝐹)( ·𝑠 ‘𝑊)(1r‘𝑊)) = (0g‘𝑊)) |
20 | 1, 17, 19 | syl2anc 579 | . 2 ⊢ (𝜑 → ((0g‘𝐹)( ·𝑠 ‘𝑊)(1r‘𝑊)) = (0g‘𝑊)) |
21 | 13, 20 | eqtrd 2814 | 1 ⊢ (𝜑 → (𝐴‘(0g‘𝐹)) = (0g‘𝑊)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1601 ∈ wcel 2107 ‘cfv 6135 (class class class)co 6922 Basecbs 16255 Scalarcsca 16341 ·𝑠 cvsca 16342 0gc0g 16486 Grpcgrp 17809 1rcur 18888 Ringcrg 18934 LModclmod 19255 algSccascl 19708 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-rep 5006 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 ax-cnex 10328 ax-resscn 10329 ax-1cn 10330 ax-icn 10331 ax-addcl 10332 ax-addrcl 10333 ax-mulcl 10334 ax-mulrcl 10335 ax-mulcom 10336 ax-addass 10337 ax-mulass 10338 ax-distr 10339 ax-i2m1 10340 ax-1ne0 10341 ax-1rid 10342 ax-rnegex 10343 ax-rrecex 10344 ax-cnre 10345 ax-pre-lttri 10346 ax-pre-lttrn 10347 ax-pre-ltadd 10348 ax-pre-mulgt0 10349 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-nel 3076 df-ral 3095 df-rex 3096 df-reu 3097 df-rmo 3098 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-pss 3808 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4672 df-iun 4755 df-br 4887 df-opab 4949 df-mpt 4966 df-tr 4988 df-id 5261 df-eprel 5266 df-po 5274 df-so 5275 df-fr 5314 df-we 5316 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-pred 5933 df-ord 5979 df-on 5980 df-lim 5981 df-suc 5982 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-f1 6140 df-fo 6141 df-f1o 6142 df-fv 6143 df-riota 6883 df-ov 6925 df-oprab 6926 df-mpt2 6927 df-om 7344 df-wrecs 7689 df-recs 7751 df-rdg 7789 df-er 8026 df-en 8242 df-dom 8243 df-sdom 8244 df-pnf 10413 df-mnf 10414 df-xr 10415 df-ltxr 10416 df-le 10417 df-sub 10608 df-neg 10609 df-nn 11375 df-2 11438 df-ndx 16258 df-slot 16259 df-base 16261 df-sets 16262 df-plusg 16351 df-0g 16488 df-mgm 17628 df-sgrp 17670 df-mnd 17681 df-grp 17812 df-mgp 18877 df-ur 18889 df-ring 18936 df-lmod 19257 df-ascl 19711 |
This theorem is referenced by: assaascl0 43182 |
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