isclmi0 - Metamath Proof Explorer

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Theorem isclmi0 25399

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 1358	. 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 464	. . . . . 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 1141	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝐾 ∧ 𝑧 ∈ 𝑉) → (𝑦 · (𝑥 + 𝑧)) = ((𝑦 · 𝑥) + (𝑦 · 𝑧)))
10	9	3expa 1136	. . . . . . 7 ⊢ (((𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝐾) ∧ 𝑧 ∈ 𝑉) → (𝑦 · (𝑥 + 𝑧)) = ((𝑦 · 𝑥) + (𝑦 · 𝑧)))
11	10	ralrimiva 3155	. . . . . 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 521	. . . . . . . . 9 ⊢ ((𝑦 ∈ 𝐾 ∧ 𝑧 ∈ 𝐾 ∧ 𝑥 ∈ 𝑉) → (((𝑧 + 𝑦) · 𝑥) = ((𝑧 · 𝑥) + (𝑦 · 𝑥)) ∧ ((𝑧 · 𝑦) · 𝑥) = (𝑧 · (𝑦 · 𝑥))))
15	14	3comr 1143	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝐾 ∧ 𝑧 ∈ 𝐾) → (((𝑧 + 𝑦) · 𝑥) = ((𝑧 · 𝑥) + (𝑦 · 𝑥)) ∧ ((𝑧 · 𝑦) · 𝑥) = (𝑧 · (𝑦 · 𝑥))))
16	15	3expa 1136	. . . . . . 7 ⊢ (((𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝐾) ∧ 𝑧 ∈ 𝐾) → (((𝑧 + 𝑦) · 𝑥) = ((𝑧 · 𝑥) + (𝑦 · 𝑥)) ∧ ((𝑧 · 𝑦) · 𝑥) = (𝑧 · (𝑦 · 𝑥))))
17	16	ralrimiva 3155	. . . . . 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 1146	. . . . 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 3155	. . . 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 521	. . 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 3079	. 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 25398	. 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 724	1 ⊢ 𝑊 ∈ ℂMod

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∧ wa 401 ∧ w3a 1103 = wceq 1570 ∈ wcel 2145 ∀wral 3077 ‘cfv 6531 (class class class)co 7412 1c1 11182 + caddc 11184 · cmul 11186 Basecbs 17367 ↾_s cress 17388 +_gcplusg 17408 Scalarcsca 17411 ·_𝑠 cvsca 17412 Grpcgrp 19124 SubRingcsubrg 20801 ℂ_fldccnfld 21658 ℂModcclm 25363

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2733 ax-sep 5249 ax-nul 5260 ax-pow 5327 ax-pr 5391 ax-un 7740 ax-cnex 11237 ax-resscn 11238 ax-1cn 11239 ax-icn 11240 ax-addcl 11241 ax-addrcl 11242 ax-mulcl 11243 ax-mulrcl 11244 ax-mulcom 11245 ax-addass 11246 ax-mulass 11247 ax-distr 11248 ax-i2m1 11249 ax-1ne0 11250 ax-1rid 11251 ax-rnegex 11252 ax-rrecex 11253 ax-cnre 11254 ax-pre-lttri 11255 ax-pre-lttrn 11256 ax-pre-ltadd 11257 ax-pre-mulgt0 11258 ax-addf 11260 ax-mulf 11261

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-nel 3063 df-ral 3078 df-rex 3088 df-rmo 3366 df-reu 3367 df-rab 3414 df-v 3453 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4591 df-uni 4868 df-iun 4953 df-br 5104 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5546 df-eprel 5551 df-po 5559 df-so 5560 df-fr 5604 df-we 5606 df-xp 5657 df-rel 5658 df-cnv 5659 df-co 5660 df-dm 5661 df-rn 5662 df-res 5663 df-ima 5664 df-pred 6297 df-ord 6358 df-on 6359 df-lim 6360 df-suc 6361 df-iota 6487 df-fun 6533 df-fn 6534 df-f 6535 df-f1 6536 df-fo 6537 df-f1o 6538 df-fv 6539 df-riota 7369 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7867 df-1st 7990 df-2nd 7991 df-frecs 8283 df-wrecs 8314 df-recs 8363 df-rdg 8402 df-1o 8460 df-er 8701 df-en 8958 df-dom 8959 df-sdom 8960 df-fin 8961 df-pnf 11326 df-mnf 11327 df-xr 11328 df-ltxr 11329 df-le 11330 df-sub 11524 df-neg 11525 df-nn 12317 df-2 12386 df-3 12387 df-4 12388 df-5 12389 df-6 12390 df-7 12391 df-8 12392 df-9 12393 df-n0 12588 df-z 12675 df-dec 12796 df-uz 12947 df-fz 13621 df-struct 17305 df-sets 17322 df-slot 17340 df-ndx 17352 df-base 17368 df-ress 17389 df-plusg 17421 df-mulr 17422 df-starv 17423 df-tset 17427 df-ple 17428 df-ds 17430 df-unif 17431 df-0g 17592 df-mgm 18796 df-sgrp 18888 df-mnd 18904 df-grp 19127 df-subg 19313 df-cmn 19976 df-mgp 20341 df-ur 20388 df-ring 20441 df-cring 20442 df-subrg 20802 df-lmod 21117 df-cnfld 21659 df-clm 25364

This theorem is used by: (None)

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