mnringelbased - Mathbox for Rohan Ridenour

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Theorem mnringelbased 44190

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 2729	. . . 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 44189	. . 3 ⊢ (𝜑 → 𝐵 = (Base‘(𝑅 freeLMod 𝐴)))
8	7	eleq2d 2814	. 2 ⊢ (𝜑 → (𝑋 ∈ 𝐵 ↔ 𝑋 ∈ (Base‘(𝑅 freeLMod 𝐴))))
9	3	fvexi 6840	. . 3 ⊢ 𝐴 ∈ V
10		mnringelbased.4	. . . 4 ⊢ 𝐶 = (Base‘𝑅)
11		mnringelbased.5	. . . 4 ⊢ 0 = (0_g‘𝑅)
12		eqid 2729	. . . 4 ⊢ (Base‘(𝑅 freeLMod 𝐴)) = (Base‘(𝑅 freeLMod 𝐴))
13	4, 10, 11, 12	frlmelbas 21681	. . 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 1540 ∈ wcel 2109 Vcvv 3438 class class class wbr 5095 ‘cfv 6486 (class class class)co 7353 ↑_m cmap 8760 finSupp cfsupp 9270 Basecbs 17138 0_gc0g 17361 freeLMod cfrlm 21671 MndRing cmnring 44184

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5221 ax-sep 5238 ax-nul 5248 ax-pow 5307 ax-pr 5374 ax-un 7675 ax-cnex 11084 ax-resscn 11085 ax-1cn 11086 ax-icn 11087 ax-addcl 11088 ax-addrcl 11089 ax-mulcl 11090 ax-mulrcl 11091 ax-mulcom 11092 ax-addass 11093 ax-mulass 11094 ax-distr 11095 ax-i2m1 11096 ax-1ne0 11097 ax-1rid 11098 ax-rnegex 11099 ax-rrecex 11100 ax-cnre 11101 ax-pre-lttri 11102 ax-pre-lttrn 11103 ax-pre-ltadd 11104 ax-pre-mulgt0 11105

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-reu 3346 df-rab 3397 df-v 3440 df-sbc 3745 df-csb 3854 df-dif 3908 df-un 3910 df-in 3912 df-ss 3922 df-pss 3925 df-nul 4287 df-if 4479 df-pw 4555 df-sn 4580 df-pr 4582 df-tp 4584 df-op 4586 df-uni 4862 df-iun 4946 df-br 5096 df-opab 5158 df-mpt 5177 df-tr 5203 df-id 5518 df-eprel 5523 df-po 5531 df-so 5532 df-fr 5576 df-we 5578 df-xp 5629 df-rel 5630 df-cnv 5631 df-co 5632 df-dm 5633 df-rn 5634 df-res 5635 df-ima 5636 df-pred 6253 df-ord 6314 df-on 6315 df-lim 6316 df-suc 6317 df-iota 6442 df-fun 6488 df-fn 6489 df-f 6490 df-f1 6491 df-fo 6492 df-f1o 6493 df-fv 6494 df-riota 7310 df-ov 7356 df-oprab 7357 df-mpo 7358 df-om 7807 df-1st 7931 df-2nd 7932 df-supp 8101 df-frecs 8221 df-wrecs 8252 df-recs 8301 df-rdg 8339 df-1o 8395 df-er 8632 df-map 8762 df-ixp 8832 df-en 8880 df-dom 8881 df-sdom 8882 df-fin 8883 df-fsupp 9271 df-sup 9351 df-pnf 11170 df-mnf 11171 df-xr 11172 df-ltxr 11173 df-le 11174 df-sub 11367 df-neg 11368 df-nn 12147 df-2 12209 df-3 12210 df-4 12211 df-5 12212 df-6 12213 df-7 12214 df-8 12215 df-9 12216 df-n0 12403 df-z 12490 df-dec 12610 df-uz 12754 df-fz 13429 df-struct 17076 df-sets 17093 df-slot 17111 df-ndx 17123 df-base 17139 df-ress 17160 df-plusg 17192 df-mulr 17193 df-sca 17195 df-vsca 17196 df-ip 17197 df-tset 17198 df-ple 17199 df-ds 17201 df-hom 17203 df-cco 17204 df-0g 17363 df-prds 17369 df-pws 17371 df-sra 21095 df-rgmod 21096 df-dsmm 21657 df-frlm 21672 df-mnring 44185

This theorem is referenced by: mnringbasefd 44191 mnringbasefsuppd 44192 mnringmulrcld 44201

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