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Mathbox for Rohan Ridenour |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > mnringelbased | Structured version Visualization version GIF version |
Description: Membership in the base set of a monoid ring. (Contributed by Rohan Ridenour, 14-May-2024.) |
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 = (0g‘𝑅) |
mnringelbased.6 | ⊢ (𝜑 → 𝑅 ∈ 𝑈) |
mnringelbased.7 | ⊢ (𝜑 → 𝑀 ∈ 𝑊) |
Ref | Expression |
---|---|
mnringelbased | ⊢ (𝜑 → (𝑋 ∈ 𝐵 ↔ (𝑋 ∈ (𝐶 ↑m 𝐴) ∧ 𝑋 finSupp 0 ))) |
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 2733 | . . . 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 42585 | . . 3 ⊢ (𝜑 → 𝐵 = (Base‘(𝑅 freeLMod 𝐴))) |
8 | 7 | eleq2d 2820 | . 2 ⊢ (𝜑 → (𝑋 ∈ 𝐵 ↔ 𝑋 ∈ (Base‘(𝑅 freeLMod 𝐴)))) |
9 | 3 | fvexi 6860 | . . 3 ⊢ 𝐴 ∈ V |
10 | mnringelbased.4 | . . . 4 ⊢ 𝐶 = (Base‘𝑅) | |
11 | mnringelbased.5 | . . . 4 ⊢ 0 = (0g‘𝑅) | |
12 | eqid 2733 | . . . 4 ⊢ (Base‘(𝑅 freeLMod 𝐴)) = (Base‘(𝑅 freeLMod 𝐴)) | |
13 | 4, 10, 11, 12 | frlmelbas 21185 | . . 3 ⊢ ((𝑅 ∈ 𝑈 ∧ 𝐴 ∈ V) → (𝑋 ∈ (Base‘(𝑅 freeLMod 𝐴)) ↔ (𝑋 ∈ (𝐶 ↑m 𝐴) ∧ 𝑋 finSupp 0 ))) |
14 | 5, 9, 13 | sylancl 587 | . 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 205 ∧ wa 397 = wceq 1542 ∈ wcel 2107 Vcvv 3447 class class class wbr 5109 ‘cfv 6500 (class class class)co 7361 ↑m cmap 8771 finSupp cfsupp 9311 Basecbs 17091 0gc0g 17329 freeLMod cfrlm 21175 MndRing cmnring 42578 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5246 ax-sep 5260 ax-nul 5267 ax-pow 5324 ax-pr 5388 ax-un 7676 ax-cnex 11115 ax-resscn 11116 ax-1cn 11117 ax-icn 11118 ax-addcl 11119 ax-addrcl 11120 ax-mulcl 11121 ax-mulrcl 11122 ax-mulcom 11123 ax-addass 11124 ax-mulass 11125 ax-distr 11126 ax-i2m1 11127 ax-1ne0 11128 ax-1rid 11129 ax-rnegex 11130 ax-rrecex 11131 ax-cnre 11132 ax-pre-lttri 11133 ax-pre-lttrn 11134 ax-pre-ltadd 11135 ax-pre-mulgt0 11136 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3353 df-rab 3407 df-v 3449 df-sbc 3744 df-csb 3860 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3933 df-nul 4287 df-if 4491 df-pw 4566 df-sn 4591 df-pr 4593 df-tp 4595 df-op 4597 df-uni 4870 df-iun 4960 df-br 5110 df-opab 5172 df-mpt 5193 df-tr 5227 df-id 5535 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5592 df-we 5594 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-rn 5648 df-res 5649 df-ima 5650 df-pred 6257 df-ord 6324 df-on 6325 df-lim 6326 df-suc 6327 df-iota 6452 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-riota 7317 df-ov 7364 df-oprab 7365 df-mpo 7366 df-om 7807 df-1st 7925 df-2nd 7926 df-supp 8097 df-frecs 8216 df-wrecs 8247 df-recs 8321 df-rdg 8360 df-1o 8416 df-er 8654 df-map 8773 df-ixp 8842 df-en 8890 df-dom 8891 df-sdom 8892 df-fin 8893 df-fsupp 9312 df-sup 9386 df-pnf 11199 df-mnf 11200 df-xr 11201 df-ltxr 11202 df-le 11203 df-sub 11395 df-neg 11396 df-nn 12162 df-2 12224 df-3 12225 df-4 12226 df-5 12227 df-6 12228 df-7 12229 df-8 12230 df-9 12231 df-n0 12422 df-z 12508 df-dec 12627 df-uz 12772 df-fz 13434 df-struct 17027 df-sets 17044 df-slot 17062 df-ndx 17074 df-base 17092 df-ress 17121 df-plusg 17154 df-mulr 17155 df-sca 17157 df-vsca 17158 df-ip 17159 df-tset 17160 df-ple 17161 df-ds 17163 df-hom 17165 df-cco 17166 df-0g 17331 df-prds 17337 df-pws 17339 df-sra 20678 df-rgmod 20679 df-dsmm 21161 df-frlm 21176 df-mnring 42579 |
This theorem is referenced by: mnringbasefd 42587 mnringbasefsuppd 42588 mnringmulrcld 42600 |
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