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Mirrors > Home > MPE Home > Th. List > Mathboxes > mnringbasefsuppd | Structured version Visualization version GIF version |
Description: Elements of a monoid ring are finitely supported. (Contributed by Rohan Ridenour, 14-May-2024.) |
Ref | Expression |
---|---|
mnringbasefsuppd.1 | ⊢ 𝐹 = (𝑅 MndRing 𝑀) |
mnringbasefsuppd.2 | ⊢ 𝐵 = (Base‘𝐹) |
mnringbasefsuppd.3 | ⊢ 0 = (0g‘𝑅) |
mnringbasefsuppd.4 | ⊢ (𝜑 → 𝑅 ∈ 𝑈) |
mnringbasefsuppd.5 | ⊢ (𝜑 → 𝑀 ∈ 𝑊) |
mnringbasefsuppd.6 | ⊢ (𝜑 → 𝑋 ∈ 𝐵) |
Ref | Expression |
---|---|
mnringbasefsuppd | ⊢ (𝜑 → 𝑋 finSupp 0 ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mnringbasefsuppd.6 | . . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
2 | mnringbasefsuppd.1 | . . . 4 ⊢ 𝐹 = (𝑅 MndRing 𝑀) | |
3 | mnringbasefsuppd.2 | . . . 4 ⊢ 𝐵 = (Base‘𝐹) | |
4 | eqid 2738 | . . . 4 ⊢ (Base‘𝑀) = (Base‘𝑀) | |
5 | eqid 2738 | . . . 4 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
6 | mnringbasefsuppd.3 | . . . 4 ⊢ 0 = (0g‘𝑅) | |
7 | mnringbasefsuppd.4 | . . . 4 ⊢ (𝜑 → 𝑅 ∈ 𝑈) | |
8 | mnringbasefsuppd.5 | . . . 4 ⊢ (𝜑 → 𝑀 ∈ 𝑊) | |
9 | 2, 3, 4, 5, 6, 7, 8 | mnringelbased 41792 | . . 3 ⊢ (𝜑 → (𝑋 ∈ 𝐵 ↔ (𝑋 ∈ ((Base‘𝑅) ↑m (Base‘𝑀)) ∧ 𝑋 finSupp 0 ))) |
10 | 1, 9 | mpbid 231 | . 2 ⊢ (𝜑 → (𝑋 ∈ ((Base‘𝑅) ↑m (Base‘𝑀)) ∧ 𝑋 finSupp 0 )) |
11 | 10 | simprd 496 | 1 ⊢ (𝜑 → 𝑋 finSupp 0 ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1539 ∈ wcel 2106 class class class wbr 5075 ‘cfv 6428 (class class class)co 7269 ↑m cmap 8604 finSupp cfsupp 9117 Basecbs 16901 0gc0g 17139 MndRing cmnring 41784 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5210 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7580 ax-cnex 10916 ax-resscn 10917 ax-1cn 10918 ax-icn 10919 ax-addcl 10920 ax-addrcl 10921 ax-mulcl 10922 ax-mulrcl 10923 ax-mulcom 10924 ax-addass 10925 ax-mulass 10926 ax-distr 10927 ax-i2m1 10928 ax-1ne0 10929 ax-1rid 10930 ax-rnegex 10931 ax-rrecex 10932 ax-cnre 10933 ax-pre-lttri 10934 ax-pre-lttrn 10935 ax-pre-ltadd 10936 ax-pre-mulgt0 10937 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-reu 3072 df-rab 3073 df-v 3433 df-sbc 3718 df-csb 3834 df-dif 3891 df-un 3893 df-in 3895 df-ss 3905 df-pss 3907 df-nul 4259 df-if 4462 df-pw 4537 df-sn 4564 df-pr 4566 df-tp 4568 df-op 4570 df-uni 4842 df-iun 4928 df-br 5076 df-opab 5138 df-mpt 5159 df-tr 5193 df-id 5486 df-eprel 5492 df-po 5500 df-so 5501 df-fr 5541 df-we 5543 df-xp 5592 df-rel 5593 df-cnv 5594 df-co 5595 df-dm 5596 df-rn 5597 df-res 5598 df-ima 5599 df-pred 6197 df-ord 6264 df-on 6265 df-lim 6266 df-suc 6267 df-iota 6386 df-fun 6430 df-fn 6431 df-f 6432 df-f1 6433 df-fo 6434 df-f1o 6435 df-fv 6436 df-riota 7226 df-ov 7272 df-oprab 7273 df-mpo 7274 df-om 7705 df-1st 7822 df-2nd 7823 df-supp 7967 df-frecs 8086 df-wrecs 8117 df-recs 8191 df-rdg 8230 df-1o 8286 df-er 8487 df-map 8606 df-ixp 8675 df-en 8723 df-dom 8724 df-sdom 8725 df-fin 8726 df-fsupp 9118 df-sup 9190 df-pnf 11000 df-mnf 11001 df-xr 11002 df-ltxr 11003 df-le 11004 df-sub 11196 df-neg 11197 df-nn 11963 df-2 12025 df-3 12026 df-4 12027 df-5 12028 df-6 12029 df-7 12030 df-8 12031 df-9 12032 df-n0 12223 df-z 12309 df-dec 12427 df-uz 12572 df-fz 13229 df-struct 16837 df-sets 16854 df-slot 16872 df-ndx 16884 df-base 16902 df-ress 16931 df-plusg 16964 df-mulr 16965 df-sca 16967 df-vsca 16968 df-ip 16969 df-tset 16970 df-ple 16971 df-ds 16973 df-hom 16975 df-cco 16976 df-0g 17141 df-prds 17147 df-pws 17149 df-sra 20423 df-rgmod 20424 df-dsmm 20928 df-frlm 20943 df-mnring 41785 |
This theorem is referenced by: mnringmulrcld 41806 |
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