mnringelbased - Mathbox for Rohan Ridenour

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Theorem mnringelbased 44183

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 2740	. . . 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 44182	. . 3 ⊢ (𝜑 → 𝐵 = (Base‘(𝑅 freeLMod 𝐴)))
8	7	eleq2d 2830	. 2 ⊢ (𝜑 → (𝑋 ∈ 𝐵 ↔ 𝑋 ∈ (Base‘(𝑅 freeLMod 𝐴))))
9	3	fvexi 6934	. . 3 ⊢ 𝐴 ∈ V
10		mnringelbased.4	. . . 4 ⊢ 𝐶 = (Base‘𝑅)
11		mnringelbased.5	. . . 4 ⊢ 0 = (0_g‘𝑅)
12		eqid 2740	. . . 4 ⊢ (Base‘(𝑅 freeLMod 𝐴)) = (Base‘(𝑅 freeLMod 𝐴))
13	4, 10, 11, 12	frlmelbas 21799	. . 3 ⊢ ((𝑅 ∈ 𝑈 ∧ 𝐴 ∈ V) → (𝑋 ∈ (Base‘(𝑅 freeLMod 𝐴)) ↔ (𝑋 ∈ (𝐶 ↑_m 𝐴) ∧ 𝑋 finSupp 0 )))
14	5, 9, 13	sylancl 585	. 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 1537 ∈ wcel 2108 Vcvv 3488 class class class wbr 5166 ‘cfv 6573 (class class class)co 7448 ↑_m cmap 8884 finSupp cfsupp 9431 Basecbs 17258 0_gc0g 17499 freeLMod cfrlm 21789 MndRing cmnring 44175

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-rep 5303 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7770 ax-cnex 11240 ax-resscn 11241 ax-1cn 11242 ax-icn 11243 ax-addcl 11244 ax-addrcl 11245 ax-mulcl 11246 ax-mulrcl 11247 ax-mulcom 11248 ax-addass 11249 ax-mulass 11250 ax-distr 11251 ax-i2m1 11252 ax-1ne0 11253 ax-1rid 11254 ax-rnegex 11255 ax-rrecex 11256 ax-cnre 11257 ax-pre-lttri 11258 ax-pre-lttrn 11259 ax-pre-ltadd 11260 ax-pre-mulgt0 11261

This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-nel 3053 df-ral 3068 df-rex 3077 df-reu 3389 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-pss 3996 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-tp 4653 df-op 4655 df-uni 4932 df-iun 5017 df-br 5167 df-opab 5229 df-mpt 5250 df-tr 5284 df-id 5593 df-eprel 5599 df-po 5607 df-so 5608 df-fr 5652 df-we 5654 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-pred 6332 df-ord 6398 df-on 6399 df-lim 6400 df-suc 6401 df-iota 6525 df-fun 6575 df-fn 6576 df-f 6577 df-f1 6578 df-fo 6579 df-f1o 6580 df-fv 6581 df-riota 7404 df-ov 7451 df-oprab 7452 df-mpo 7453 df-om 7904 df-1st 8030 df-2nd 8031 df-supp 8202 df-frecs 8322 df-wrecs 8353 df-recs 8427 df-rdg 8466 df-1o 8522 df-er 8763 df-map 8886 df-ixp 8956 df-en 9004 df-dom 9005 df-sdom 9006 df-fin 9007 df-fsupp 9432 df-sup 9511 df-pnf 11326 df-mnf 11327 df-xr 11328 df-ltxr 11329 df-le 11330 df-sub 11522 df-neg 11523 df-nn 12294 df-2 12356 df-3 12357 df-4 12358 df-5 12359 df-6 12360 df-7 12361 df-8 12362 df-9 12363 df-n0 12554 df-z 12640 df-dec 12759 df-uz 12904 df-fz 13568 df-struct 17194 df-sets 17211 df-slot 17229 df-ndx 17241 df-base 17259 df-ress 17288 df-plusg 17324 df-mulr 17325 df-sca 17327 df-vsca 17328 df-ip 17329 df-tset 17330 df-ple 17331 df-ds 17333 df-hom 17335 df-cco 17336 df-0g 17501 df-prds 17507 df-pws 17509 df-sra 21195 df-rgmod 21196 df-dsmm 21775 df-frlm 21790 df-mnring 44176

This theorem is referenced by: mnringbasefd 44184 mnringbasefsuppd 44185 mnringmulrcld 44197

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