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Mirrors > Home > MPE Home > Th. List > assamulgscmlem1 | Structured version Visualization version GIF version |
Description: Lemma 1 for assamulgscm 20124 (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 20086 | . . . 4 ⊢ (𝑊 ∈ AssAlg → 𝑊 ∈ LMod) | |
2 | assaring 20087 | . . . . 5 ⊢ (𝑊 ∈ AssAlg → 𝑊 ∈ Ring) | |
3 | assamulgscm.v | . . . . . 6 ⊢ 𝑉 = (Base‘𝑊) | |
4 | eqid 2821 | . . . . . 6 ⊢ (1r‘𝑊) = (1r‘𝑊) | |
5 | 3, 4 | ringidcl 19312 | . . . . 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 19656 | . . . . 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 19645 | . . . 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 19239 | . . . 4 ⊢ 𝑉 = (Base‘𝐻) |
22 | 20, 4 | ringidval 19247 | . . . 4 ⊢ (1r‘𝑊) = (0g‘𝐻) |
23 | assamulgscm.e | . . . 4 ⊢ 𝐸 = (.g‘𝐻) | |
24 | 21, 22, 23 | mulg0 18225 | . . 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 19239 | . . . . 5 ⊢ 𝐵 = (Base‘𝐺) |
28 | 26, 9 | ringidval 19247 | . . . . 5 ⊢ (1r‘𝐹) = (0g‘𝐺) |
29 | assamulgscm.p | . . . . 5 ⊢ ↑ = (.g‘𝐺) | |
30 | 27, 28, 29 | mulg0 18225 | . . . 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 18225 | . . . 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 7168 | . 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 6349 (class class class)co 7150 0cc0 10531 Basecbs 16477 Scalarcsca 16562 ·𝑠 cvsca 16563 .gcmg 18218 mulGrpcmgp 19233 1rcur 19245 Ringcrg 19291 LModclmod 19628 AssAlgcasa 20076 |
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 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 |
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 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-1st 7683 df-2nd 7684 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-er 8283 df-en 8504 df-dom 8505 df-sdom 8506 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-nn 11633 df-2 11694 df-n0 11892 df-z 11976 df-uz 12238 df-seq 13364 df-ndx 16480 df-slot 16481 df-base 16483 df-sets 16484 df-plusg 16572 df-0g 16709 df-mgm 17846 df-sgrp 17895 df-mnd 17906 df-mulg 18219 df-mgp 19234 df-ur 19246 df-ring 19293 df-lmod 19630 df-assa 20079 |
This theorem is referenced by: assamulgscm 20124 |
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