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Mirrors > Home > MPE Home > Th. List > assamulgscmlem1 | Structured version Visualization version GIF version |
Description: Lemma 1 for assamulgscm 20129 (induction base). (Contributed by AV, 26-Aug-2019.) |
Ref | Expression |
---|---|
assamulgscm.v | ⊢ 𝑉 = (Base‘𝑊) |
assamulgscm.f | ⊢ 𝐹 = (Scalar‘𝑊) |
assamulgscm.b | ⊢ 𝐵 = (Base‘𝐹) |
assamulgscm.s | ⊢ · = ( ·𝑠 ‘𝑊) |
assamulgscm.g | ⊢ 𝐺 = (mulGrp‘𝐹) |
assamulgscm.p | ⊢ ↑ = (.g‘𝐺) |
assamulgscm.h | ⊢ 𝐻 = (mulGrp‘𝑊) |
assamulgscm.e | ⊢ 𝐸 = (.g‘𝐻) |
Ref | Expression |
---|---|
assamulgscmlem1 | ⊢ (((𝐴 ∈ 𝐵 ∧ 𝑋 ∈ 𝑉) ∧ 𝑊 ∈ AssAlg) → (0𝐸(𝐴 · 𝑋)) = ((0 ↑ 𝐴) · (0𝐸𝑋))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | assalmod 20091 | . . . 4 ⊢ (𝑊 ∈ AssAlg → 𝑊 ∈ LMod) | |
2 | assaring 20092 | . . . . 5 ⊢ (𝑊 ∈ AssAlg → 𝑊 ∈ Ring) | |
3 | assamulgscm.v | . . . . . 6 ⊢ 𝑉 = (Base‘𝑊) | |
4 | eqid 2821 | . . . . . 6 ⊢ (1r‘𝑊) = (1r‘𝑊) | |
5 | 3, 4 | ringidcl 19317 | . . . . 5 ⊢ (𝑊 ∈ Ring → (1r‘𝑊) ∈ 𝑉) |
6 | 2, 5 | syl 17 | . . . 4 ⊢ (𝑊 ∈ AssAlg → (1r‘𝑊) ∈ 𝑉) |
7 | assamulgscm.f | . . . . . 6 ⊢ 𝐹 = (Scalar‘𝑊) | |
8 | assamulgscm.s | . . . . . 6 ⊢ · = ( ·𝑠 ‘𝑊) | |
9 | eqid 2821 | . . . . . 6 ⊢ (1r‘𝐹) = (1r‘𝐹) | |
10 | 3, 7, 8, 9 | lmodvs1 19661 | . . . . 5 ⊢ ((𝑊 ∈ LMod ∧ (1r‘𝑊) ∈ 𝑉) → ((1r‘𝐹) · (1r‘𝑊)) = (1r‘𝑊)) |
11 | 10 | eqcomd 2827 | . . . 4 ⊢ ((𝑊 ∈ LMod ∧ (1r‘𝑊) ∈ 𝑉) → (1r‘𝑊) = ((1r‘𝐹) · (1r‘𝑊))) |
12 | 1, 6, 11 | syl2anc 586 | . . 3 ⊢ (𝑊 ∈ AssAlg → (1r‘𝑊) = ((1r‘𝐹) · (1r‘𝑊))) |
13 | 12 | adantl 484 | . 2 ⊢ (((𝐴 ∈ 𝐵 ∧ 𝑋 ∈ 𝑉) ∧ 𝑊 ∈ AssAlg) → (1r‘𝑊) = ((1r‘𝐹) · (1r‘𝑊))) |
14 | 1 | adantl 484 | . . . 4 ⊢ (((𝐴 ∈ 𝐵 ∧ 𝑋 ∈ 𝑉) ∧ 𝑊 ∈ AssAlg) → 𝑊 ∈ LMod) |
15 | simpll 765 | . . . 4 ⊢ (((𝐴 ∈ 𝐵 ∧ 𝑋 ∈ 𝑉) ∧ 𝑊 ∈ AssAlg) → 𝐴 ∈ 𝐵) | |
16 | simplr 767 | . . . 4 ⊢ (((𝐴 ∈ 𝐵 ∧ 𝑋 ∈ 𝑉) ∧ 𝑊 ∈ AssAlg) → 𝑋 ∈ 𝑉) | |
17 | assamulgscm.b | . . . . 5 ⊢ 𝐵 = (Base‘𝐹) | |
18 | 3, 7, 8, 17 | lmodvscl 19650 | . . . 4 ⊢ ((𝑊 ∈ LMod ∧ 𝐴 ∈ 𝐵 ∧ 𝑋 ∈ 𝑉) → (𝐴 · 𝑋) ∈ 𝑉) |
19 | 14, 15, 16, 18 | syl3anc 1367 | . . 3 ⊢ (((𝐴 ∈ 𝐵 ∧ 𝑋 ∈ 𝑉) ∧ 𝑊 ∈ AssAlg) → (𝐴 · 𝑋) ∈ 𝑉) |
20 | assamulgscm.h | . . . . 5 ⊢ 𝐻 = (mulGrp‘𝑊) | |
21 | 20, 3 | mgpbas 19244 | . . . 4 ⊢ 𝑉 = (Base‘𝐻) |
22 | 20, 4 | ringidval 19252 | . . . 4 ⊢ (1r‘𝑊) = (0g‘𝐻) |
23 | assamulgscm.e | . . . 4 ⊢ 𝐸 = (.g‘𝐻) | |
24 | 21, 22, 23 | mulg0 18230 | . . 3 ⊢ ((𝐴 · 𝑋) ∈ 𝑉 → (0𝐸(𝐴 · 𝑋)) = (1r‘𝑊)) |
25 | 19, 24 | syl 17 | . 2 ⊢ (((𝐴 ∈ 𝐵 ∧ 𝑋 ∈ 𝑉) ∧ 𝑊 ∈ AssAlg) → (0𝐸(𝐴 · 𝑋)) = (1r‘𝑊)) |
26 | assamulgscm.g | . . . . . 6 ⊢ 𝐺 = (mulGrp‘𝐹) | |
27 | 26, 17 | mgpbas 19244 | . . . . 5 ⊢ 𝐵 = (Base‘𝐺) |
28 | 26, 9 | ringidval 19252 | . . . . 5 ⊢ (1r‘𝐹) = (0g‘𝐺) |
29 | assamulgscm.p | . . . . 5 ⊢ ↑ = (.g‘𝐺) | |
30 | 27, 28, 29 | mulg0 18230 | . . . 4 ⊢ (𝐴 ∈ 𝐵 → (0 ↑ 𝐴) = (1r‘𝐹)) |
31 | 15, 30 | syl 17 | . . 3 ⊢ (((𝐴 ∈ 𝐵 ∧ 𝑋 ∈ 𝑉) ∧ 𝑊 ∈ AssAlg) → (0 ↑ 𝐴) = (1r‘𝐹)) |
32 | 21, 22, 23 | mulg0 18230 | . . . 4 ⊢ (𝑋 ∈ 𝑉 → (0𝐸𝑋) = (1r‘𝑊)) |
33 | 16, 32 | syl 17 | . . 3 ⊢ (((𝐴 ∈ 𝐵 ∧ 𝑋 ∈ 𝑉) ∧ 𝑊 ∈ AssAlg) → (0𝐸𝑋) = (1r‘𝑊)) |
34 | 31, 33 | oveq12d 7173 | . 2 ⊢ (((𝐴 ∈ 𝐵 ∧ 𝑋 ∈ 𝑉) ∧ 𝑊 ∈ AssAlg) → ((0 ↑ 𝐴) · (0𝐸𝑋)) = ((1r‘𝐹) · (1r‘𝑊))) |
35 | 13, 25, 34 | 3eqtr4d 2866 | 1 ⊢ (((𝐴 ∈ 𝐵 ∧ 𝑋 ∈ 𝑉) ∧ 𝑊 ∈ AssAlg) → (0𝐸(𝐴 · 𝑋)) = ((0 ↑ 𝐴) · (0𝐸𝑋))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1533 ∈ wcel 2110 ‘cfv 6354 (class class class)co 7155 0cc0 10536 Basecbs 16482 Scalarcsca 16567 ·𝑠 cvsca 16568 .gcmg 18223 mulGrpcmgp 19238 1rcur 19250 Ringcrg 19296 LModclmod 19633 AssAlgcasa 20081 |
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 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460 ax-cnex 10592 ax-resscn 10593 ax-1cn 10594 ax-icn 10595 ax-addcl 10596 ax-addrcl 10597 ax-mulcl 10598 ax-mulrcl 10599 ax-mulcom 10600 ax-addass 10601 ax-mulass 10602 ax-distr 10603 ax-i2m1 10604 ax-1ne0 10605 ax-1rid 10606 ax-rnegex 10607 ax-rrecex 10608 ax-cnre 10609 ax-pre-lttri 10610 ax-pre-lttrn 10611 ax-pre-ltadd 10612 ax-pre-mulgt0 10613 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 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 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4567 df-pr 4569 df-tp 4571 df-op 4573 df-uni 4838 df-iun 4920 df-br 5066 df-opab 5128 df-mpt 5146 df-tr 5172 df-id 5459 df-eprel 5464 df-po 5473 df-so 5474 df-fr 5513 df-we 5515 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-pred 6147 df-ord 6193 df-on 6194 df-lim 6195 df-suc 6196 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-f1 6359 df-fo 6360 df-f1o 6361 df-fv 6362 df-riota 7113 df-ov 7158 df-oprab 7159 df-mpo 7160 df-om 7580 df-1st 7688 df-2nd 7689 df-wrecs 7946 df-recs 8007 df-rdg 8045 df-er 8288 df-en 8509 df-dom 8510 df-sdom 8511 df-pnf 10676 df-mnf 10677 df-xr 10678 df-ltxr 10679 df-le 10680 df-sub 10871 df-neg 10872 df-nn 11638 df-2 11699 df-n0 11897 df-z 11981 df-uz 12243 df-seq 13369 df-ndx 16485 df-slot 16486 df-base 16488 df-sets 16489 df-plusg 16577 df-0g 16714 df-mgm 17851 df-sgrp 17900 df-mnd 17911 df-mulg 18224 df-mgp 19239 df-ur 19251 df-ring 19298 df-lmod 19635 df-assa 20084 |
This theorem is referenced by: assamulgscm 20129 |
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