mnringelbased - Mathbox for Rohan Ridenour

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Theorem mnringelbased 44249

Description: Membership in the base set of a monoid ring. (Contributed by Rohan Ridenour, 14-May-2024.)

**Hypotheses**
Ref	Expression
mnringelbased.1	⊢ 𝐹 = (𝑅 MndRing 𝑀)
mnringelbased.2	⊢ 𝐵 = (Base‘𝐹)
mnringelbased.3	⊢ 𝐴 = (Base‘𝑀)
mnringelbased.4	⊢ 𝐶 = (Base‘𝑅)
mnringelbased.5	⊢ 0 = (0_g‘𝑅)
mnringelbased.6	⊢ (𝜑 → 𝑅 ∈ 𝑈)
mnringelbased.7	⊢ (𝜑 → 𝑀 ∈ 𝑊)

**Assertion**
Ref	Expression
mnringelbased	⊢ (𝜑 → (𝑋 ∈ 𝐵 ↔ (𝑋 ∈ (𝐶 ↑_m 𝐴) ∧ 𝑋 finSupp 0 )))

**Proof of Theorem mnringelbased**
Step	Hyp	Ref	Expression
1		mnringelbased.1	. . . 4 ⊢ 𝐹 = (𝑅 MndRing 𝑀)
2		mnringelbased.2	. . . 4 ⊢ 𝐵 = (Base‘𝐹)
3		mnringelbased.3	. . . 4 ⊢ 𝐴 = (Base‘𝑀)
4		eqid 2731	. . . 4 ⊢ (𝑅 freeLMod 𝐴) = (𝑅 freeLMod 𝐴)
5		mnringelbased.6	. . . 4 ⊢ (𝜑 → 𝑅 ∈ 𝑈)
6		mnringelbased.7	. . . 4 ⊢ (𝜑 → 𝑀 ∈ 𝑊)
7	1, 2, 3, 4, 5, 6	mnringbaserd 44248	. . 3 ⊢ (𝜑 → 𝐵 = (Base‘(𝑅 freeLMod 𝐴)))
8	7	eleq2d 2817	. 2 ⊢ (𝜑 → (𝑋 ∈ 𝐵 ↔ 𝑋 ∈ (Base‘(𝑅 freeLMod 𝐴))))
9	3	fvexi 6836	. . 3 ⊢ 𝐴 ∈ V
10		mnringelbased.4	. . . 4 ⊢ 𝐶 = (Base‘𝑅)
11		mnringelbased.5	. . . 4 ⊢ 0 = (0_g‘𝑅)
12		eqid 2731	. . . 4 ⊢ (Base‘(𝑅 freeLMod 𝐴)) = (Base‘(𝑅 freeLMod 𝐴))
13	4, 10, 11, 12	frlmelbas 21691	. . 3 ⊢ ((𝑅 ∈ 𝑈 ∧ 𝐴 ∈ V) → (𝑋 ∈ (Base‘(𝑅 freeLMod 𝐴)) ↔ (𝑋 ∈ (𝐶 ↑_m 𝐴) ∧ 𝑋 finSupp 0 )))
14	5, 9, 13	sylancl 586	. 2 ⊢ (𝜑 → (𝑋 ∈ (Base‘(𝑅 freeLMod 𝐴)) ↔ (𝑋 ∈ (𝐶 ↑_m 𝐴) ∧ 𝑋 finSupp 0 )))
15	8, 14	bitrd 279	1 ⊢ (𝜑 → (𝑋 ∈ 𝐵 ↔ (𝑋 ∈ (𝐶 ↑_m 𝐴) ∧ 𝑋 finSupp 0 )))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1541 ∈ wcel 2111 Vcvv 3436 class class class wbr 5091 ‘cfv 6481 (class class class)co 7346 ↑_m cmap 8750 finSupp cfsupp 9245 Basecbs 17117 0_gc0g 17340 freeLMod cfrlm 21681 MndRing cmnring 44243

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2113 ax-9 2121 ax-10 2144 ax-11 2160 ax-12 2180 ax-ext 2703 ax-rep 5217 ax-sep 5234 ax-nul 5244 ax-pow 5303 ax-pr 5370 ax-un 7668 ax-cnex 11059 ax-resscn 11060 ax-1cn 11061 ax-icn 11062 ax-addcl 11063 ax-addrcl 11064 ax-mulcl 11065 ax-mulrcl 11066 ax-mulcom 11067 ax-addass 11068 ax-mulass 11069 ax-distr 11070 ax-i2m1 11071 ax-1ne0 11072 ax-1rid 11073 ax-rnegex 11074 ax-rrecex 11075 ax-cnre 11076 ax-pre-lttri 11077 ax-pre-lttrn 11078 ax-pre-ltadd 11079 ax-pre-mulgt0 11080

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2535 df-eu 2564 df-clab 2710 df-cleq 2723 df-clel 2806 df-nfc 2881 df-ne 2929 df-nel 3033 df-ral 3048 df-rex 3057 df-reu 3347 df-rab 3396 df-v 3438 df-sbc 3742 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-pss 3922 df-nul 4284 df-if 4476 df-pw 4552 df-sn 4577 df-pr 4579 df-tp 4581 df-op 4583 df-uni 4860 df-iun 4943 df-br 5092 df-opab 5154 df-mpt 5173 df-tr 5199 df-id 5511 df-eprel 5516 df-po 5524 df-so 5525 df-fr 5569 df-we 5571 df-xp 5622 df-rel 5623 df-cnv 5624 df-co 5625 df-dm 5626 df-rn 5627 df-res 5628 df-ima 5629 df-pred 6248 df-ord 6309 df-on 6310 df-lim 6311 df-suc 6312 df-iota 6437 df-fun 6483 df-fn 6484 df-f 6485 df-f1 6486 df-fo 6487 df-f1o 6488 df-fv 6489 df-riota 7303 df-ov 7349 df-oprab 7350 df-mpo 7351 df-om 7797 df-1st 7921 df-2nd 7922 df-supp 8091 df-frecs 8211 df-wrecs 8242 df-recs 8291 df-rdg 8329 df-1o 8385 df-er 8622 df-map 8752 df-ixp 8822 df-en 8870 df-dom 8871 df-sdom 8872 df-fin 8873 df-fsupp 9246 df-sup 9326 df-pnf 11145 df-mnf 11146 df-xr 11147 df-ltxr 11148 df-le 11149 df-sub 11343 df-neg 11344 df-nn 12123 df-2 12185 df-3 12186 df-4 12187 df-5 12188 df-6 12189 df-7 12190 df-8 12191 df-9 12192 df-n0 12379 df-z 12466 df-dec 12586 df-uz 12730 df-fz 13405 df-struct 17055 df-sets 17072 df-slot 17090 df-ndx 17102 df-base 17118 df-ress 17139 df-plusg 17171 df-mulr 17172 df-sca 17174 df-vsca 17175 df-ip 17176 df-tset 17177 df-ple 17178 df-ds 17180 df-hom 17182 df-cco 17183 df-0g 17342 df-prds 17348 df-pws 17350 df-sra 21105 df-rgmod 21106 df-dsmm 21667 df-frlm 21682 df-mnring 44244

This theorem is referenced by: mnringbasefd 44250 mnringbasefsuppd 44251 mnringmulrcld 44260

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