isclmi0 - Metamath Proof Explorer

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Theorem isclmi0 22898

Description: Properties that determine a subcomplex module. (Contributed by NM, 5-Nov-2006.) (Revised by AV, 4-Oct-2021.)

**Hypotheses**
Ref	Expression
isclmp.t	⊢ · = ( ·_𝑠 ‘𝑊)
isclmp.a	⊢ + = (+_g‘𝑊)
isclmp.v	⊢ 𝑉 = (Base‘𝑊)
isclmp.s	⊢ 𝑆 = (Scalar‘𝑊)
isclmp.k	⊢ 𝐾 = (Base‘𝑆)
isclmi0.1	⊢ 𝑆 = (ℂ_fld ↾_s 𝐾)
isclmi0.2	⊢ 𝑊 ∈ Grp
isclmi0.3	⊢ 𝐾 ∈ (SubRing‘ℂ_fld)
isclmi0.4	⊢ (𝑥 ∈ 𝑉 → (1 · 𝑥) = 𝑥)
isclmi0.5	⊢ ((𝑦 ∈ 𝐾 ∧ 𝑥 ∈ 𝑉) → (𝑦 · 𝑥) ∈ 𝑉)
isclmi0.6	⊢ ((𝑦 ∈ 𝐾 ∧ 𝑥 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉) → (𝑦 · (𝑥 + 𝑧)) = ((𝑦 · 𝑥) + (𝑦 · 𝑧)))
isclmi0.7	⊢ ((𝑦 ∈ 𝐾 ∧ 𝑧 ∈ 𝐾 ∧ 𝑥 ∈ 𝑉) → ((𝑧 + 𝑦) · 𝑥) = ((𝑧 · 𝑥) + (𝑦 · 𝑥)))
isclmi0.8	⊢ ((𝑦 ∈ 𝐾 ∧ 𝑧 ∈ 𝐾 ∧ 𝑥 ∈ 𝑉) → ((𝑧 · 𝑦) · 𝑥) = (𝑧 · (𝑦 · 𝑥)))

**Assertion**
Ref	Expression
isclmi0	⊢ 𝑊 ∈ ℂMod

Distinct variable groups: 𝑥,𝐾,𝑦,𝑧 𝑥,𝑆,𝑦,𝑧 𝑥,𝑉,𝑦,𝑧 𝑥,𝑊,𝑦,𝑧 𝑥, + ,𝑦,𝑧 𝑥, · ,𝑦,𝑧

**Proof of Theorem isclmi0**
Step	Hyp	Ref	Expression
1		isclmi0.2	. . 3 ⊢ 𝑊 ∈ Grp
2		isclmi0.1	. . 3 ⊢ 𝑆 = (ℂ_fld ↾_s 𝐾)
3		isclmi0.3	. . 3 ⊢ 𝐾 ∈ (SubRing‘ℂ_fld)
4	1, 2, 3	3pm3.2i 1239	. 2 ⊢ (𝑊 ∈ Grp ∧ 𝑆 = (ℂ_fld ↾_s 𝐾) ∧ 𝐾 ∈ (SubRing‘ℂ_fld))
5		isclmi0.4	. . . 4 ⊢ (𝑥 ∈ 𝑉 → (1 · 𝑥) = 𝑥)
6		isclmi0.5	. . . . . . 7 ⊢ ((𝑦 ∈ 𝐾 ∧ 𝑥 ∈ 𝑉) → (𝑦 · 𝑥) ∈ 𝑉)
7	6	ancoms 469	. . . . . 6 ⊢ ((𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝐾) → (𝑦 · 𝑥) ∈ 𝑉)
8		isclmi0.6	. . . . . . . . 9 ⊢ ((𝑦 ∈ 𝐾 ∧ 𝑥 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉) → (𝑦 · (𝑥 + 𝑧)) = ((𝑦 · 𝑥) + (𝑦 · 𝑧)))
9	8	3com12 1269	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝐾 ∧ 𝑧 ∈ 𝑉) → (𝑦 · (𝑥 + 𝑧)) = ((𝑦 · 𝑥) + (𝑦 · 𝑧)))
10	9	3expa 1265	. . . . . . 7 ⊢ (((𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝐾) ∧ 𝑧 ∈ 𝑉) → (𝑦 · (𝑥 + 𝑧)) = ((𝑦 · 𝑥) + (𝑦 · 𝑧)))
11	10	ralrimiva 2966	. . . . . 6 ⊢ ((𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝐾) → ∀𝑧 ∈ 𝑉 (𝑦 · (𝑥 + 𝑧)) = ((𝑦 · 𝑥) + (𝑦 · 𝑧)))
12		isclmi0.7	. . . . . . . . . 10 ⊢ ((𝑦 ∈ 𝐾 ∧ 𝑧 ∈ 𝐾 ∧ 𝑥 ∈ 𝑉) → ((𝑧 + 𝑦) · 𝑥) = ((𝑧 · 𝑥) + (𝑦 · 𝑥)))
13		isclmi0.8	. . . . . . . . . 10 ⊢ ((𝑦 ∈ 𝐾 ∧ 𝑧 ∈ 𝐾 ∧ 𝑥 ∈ 𝑉) → ((𝑧 · 𝑦) · 𝑥) = (𝑧 · (𝑦 · 𝑥)))
14	12, 13	jca 554	. . . . . . . . 9 ⊢ ((𝑦 ∈ 𝐾 ∧ 𝑧 ∈ 𝐾 ∧ 𝑥 ∈ 𝑉) → (((𝑧 + 𝑦) · 𝑥) = ((𝑧 · 𝑥) + (𝑦 · 𝑥)) ∧ ((𝑧 · 𝑦) · 𝑥) = (𝑧 · (𝑦 · 𝑥))))
15	14	3comr 1273	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝐾 ∧ 𝑧 ∈ 𝐾) → (((𝑧 + 𝑦) · 𝑥) = ((𝑧 · 𝑥) + (𝑦 · 𝑥)) ∧ ((𝑧 · 𝑦) · 𝑥) = (𝑧 · (𝑦 · 𝑥))))
16	15	3expa 1265	. . . . . . 7 ⊢ (((𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝐾) ∧ 𝑧 ∈ 𝐾) → (((𝑧 + 𝑦) · 𝑥) = ((𝑧 · 𝑥) + (𝑦 · 𝑥)) ∧ ((𝑧 · 𝑦) · 𝑥) = (𝑧 · (𝑦 · 𝑥))))
17	16	ralrimiva 2966	. . . . . 6 ⊢ ((𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝐾) → ∀𝑧 ∈ 𝐾 (((𝑧 + 𝑦) · 𝑥) = ((𝑧 · 𝑥) + (𝑦 · 𝑥)) ∧ ((𝑧 · 𝑦) · 𝑥) = (𝑧 · (𝑦 · 𝑥))))
18	7, 11, 17	3jca 1242	. . . . 5 ⊢ ((𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝐾) → ((𝑦 · 𝑥) ∈ 𝑉 ∧ ∀𝑧 ∈ 𝑉 (𝑦 · (𝑥 + 𝑧)) = ((𝑦 · 𝑥) + (𝑦 · 𝑧)) ∧ ∀𝑧 ∈ 𝐾 (((𝑧 + 𝑦) · 𝑥) = ((𝑧 · 𝑥) + (𝑦 · 𝑥)) ∧ ((𝑧 · 𝑦) · 𝑥) = (𝑧 · (𝑦 · 𝑥)))))
19	18	ralrimiva 2966	. . . 4 ⊢ (𝑥 ∈ 𝑉 → ∀𝑦 ∈ 𝐾 ((𝑦 · 𝑥) ∈ 𝑉 ∧ ∀𝑧 ∈ 𝑉 (𝑦 · (𝑥 + 𝑧)) = ((𝑦 · 𝑥) + (𝑦 · 𝑧)) ∧ ∀𝑧 ∈ 𝐾 (((𝑧 + 𝑦) · 𝑥) = ((𝑧 · 𝑥) + (𝑦 · 𝑥)) ∧ ((𝑧 · 𝑦) · 𝑥) = (𝑧 · (𝑦 · 𝑥)))))
20	5, 19	jca 554	. . 3 ⊢ (𝑥 ∈ 𝑉 → ((1 · 𝑥) = 𝑥 ∧ ∀𝑦 ∈ 𝐾 ((𝑦 · 𝑥) ∈ 𝑉 ∧ ∀𝑧 ∈ 𝑉 (𝑦 · (𝑥 + 𝑧)) = ((𝑦 · 𝑥) + (𝑦 · 𝑧)) ∧ ∀𝑧 ∈ 𝐾 (((𝑧 + 𝑦) · 𝑥) = ((𝑧 · 𝑥) + (𝑦 · 𝑥)) ∧ ((𝑧 · 𝑦) · 𝑥) = (𝑧 · (𝑦 · 𝑥))))))
21	20	rgen 2922	. 2 ⊢ ∀𝑥 ∈ 𝑉 ((1 · 𝑥) = 𝑥 ∧ ∀𝑦 ∈ 𝐾 ((𝑦 · 𝑥) ∈ 𝑉 ∧ ∀𝑧 ∈ 𝑉 (𝑦 · (𝑥 + 𝑧)) = ((𝑦 · 𝑥) + (𝑦 · 𝑧)) ∧ ∀𝑧 ∈ 𝐾 (((𝑧 + 𝑦) · 𝑥) = ((𝑧 · 𝑥) + (𝑦 · 𝑥)) ∧ ((𝑧 · 𝑦) · 𝑥) = (𝑧 · (𝑦 · 𝑥)))))
22		isclmp.t	. . 3 ⊢ · = ( ·_𝑠 ‘𝑊)
23		isclmp.a	. . 3 ⊢ + = (+_g‘𝑊)
24		isclmp.v	. . 3 ⊢ 𝑉 = (Base‘𝑊)
25		isclmp.s	. . 3 ⊢ 𝑆 = (Scalar‘𝑊)
26		isclmp.k	. . 3 ⊢ 𝐾 = (Base‘𝑆)
27	22, 23, 24, 25, 26	isclmp 22897	. 2 ⊢ (𝑊 ∈ ℂMod ↔ ((𝑊 ∈ Grp ∧ 𝑆 = (ℂ_fld ↾_s 𝐾) ∧ 𝐾 ∈ (SubRing‘ℂ_fld)) ∧ ∀𝑥 ∈ 𝑉 ((1 · 𝑥) = 𝑥 ∧ ∀𝑦 ∈ 𝐾 ((𝑦 · 𝑥) ∈ 𝑉 ∧ ∀𝑧 ∈ 𝑉 (𝑦 · (𝑥 + 𝑧)) = ((𝑦 · 𝑥) + (𝑦 · 𝑧)) ∧ ∀𝑧 ∈ 𝐾 (((𝑧 + 𝑦) · 𝑥) = ((𝑧 · 𝑥) + (𝑦 · 𝑥)) ∧ ((𝑧 · 𝑦) · 𝑥) = (𝑧 · (𝑦 · 𝑥)))))))
28	4, 21, 27	mpbir2an 955	1 ⊢ 𝑊 ∈ ℂMod

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 384 ∧ w3a 1037 = wceq 1483 ∈ wcel 1990 ∀wral 2912 ‘cfv 5888 (class class class)co 6650 1c1 9937 + caddc 9939 · cmul 9941 Basecbs 15857 ↾_s cress 15858 +_gcplusg 15941 Scalarcsca 15944 ·_𝑠 cvsca 15945 Grpcgrp 17422 SubRingcsubrg 18776 ℂ_fldccnfld 19746 ℂModcclm 22862

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-addf 10015 ax-mulf 10016

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-oadd 7564 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-2 11079 df-3 11080 df-4 11081 df-5 11082 df-6 11083 df-7 11084 df-8 11085 df-9 11086 df-n0 11293 df-z 11378 df-dec 11494 df-uz 11688 df-fz 12327 df-struct 15859 df-ndx 15860 df-slot 15861 df-base 15863 df-sets 15864 df-ress 15865 df-plusg 15954 df-mulr 15955 df-starv 15956 df-tset 15960 df-ple 15961 df-ds 15964 df-unif 15965 df-0g 16102 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-grp 17425 df-subg 17591 df-cmn 18195 df-mgp 18490 df-ur 18502 df-ring 18549 df-cring 18550 df-subrg 18778 df-lmod 18865 df-cnfld 19747 df-clm 22863

This theorem is referenced by: (None)

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