Mathbox for Rohan Ridenour |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > Mathboxes > mnringaddgd | Structured version Visualization version GIF version |
Description: The additive operation of a monoid ring. (Contributed by Rohan Ridenour, 14-May-2024.) (Proof shortened by AV, 1-Nov-2024.) |
Ref | Expression |
---|---|
mnringaddgd.1 | ⊢ 𝐹 = (𝑅 MndRing 𝑀) |
mnringaddgd.2 | ⊢ 𝐴 = (Base‘𝑀) |
mnringaddgd.3 | ⊢ 𝑉 = (𝑅 freeLMod 𝐴) |
mnringaddgd.4 | ⊢ (𝜑 → 𝑅 ∈ 𝑈) |
mnringaddgd.5 | ⊢ (𝜑 → 𝑀 ∈ 𝑊) |
Ref | Expression |
---|---|
mnringaddgd | ⊢ (𝜑 → (+g‘𝑉) = (+g‘𝐹)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mnringaddgd.1 | . 2 ⊢ 𝐹 = (𝑅 MndRing 𝑀) | |
2 | plusgid 16979 | . 2 ⊢ +g = Slot (+g‘ndx) | |
3 | plusgndxnmulrndx 16997 | . 2 ⊢ (+g‘ndx) ≠ (.r‘ndx) | |
4 | mnringaddgd.2 | . 2 ⊢ 𝐴 = (Base‘𝑀) | |
5 | mnringaddgd.3 | . 2 ⊢ 𝑉 = (𝑅 freeLMod 𝐴) | |
6 | mnringaddgd.4 | . 2 ⊢ (𝜑 → 𝑅 ∈ 𝑈) | |
7 | mnringaddgd.5 | . 2 ⊢ (𝜑 → 𝑀 ∈ 𝑊) | |
8 | 1, 2, 3, 4, 5, 6, 7 | mnringnmulrd 41789 | 1 ⊢ (𝜑 → (+g‘𝑉) = (+g‘𝐹)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2110 ‘cfv 6431 (class class class)co 7269 Basecbs 16902 +gcplusg 16952 freeLMod cfrlm 20943 MndRing cmnring 41786 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2015 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2711 ax-sep 5227 ax-nul 5234 ax-pow 5292 ax-pr 5356 ax-un 7580 ax-cnex 10920 ax-resscn 10921 ax-1cn 10922 ax-icn 10923 ax-addcl 10924 ax-addrcl 10925 ax-mulcl 10926 ax-mulrcl 10927 ax-mulcom 10928 ax-addass 10929 ax-mulass 10930 ax-distr 10931 ax-i2m1 10932 ax-1ne0 10933 ax-1rid 10934 ax-rnegex 10935 ax-rrecex 10936 ax-cnre 10937 ax-pre-lttri 10938 ax-pre-lttrn 10939 ax-pre-ltadd 10940 ax-pre-mulgt0 10941 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2072 df-mo 2542 df-eu 2571 df-clab 2718 df-cleq 2732 df-clel 2818 df-nfc 2891 df-ne 2946 df-nel 3052 df-ral 3071 df-rex 3072 df-reu 3073 df-rab 3075 df-v 3433 df-sbc 3721 df-csb 3838 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-pss 3911 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4846 df-iun 4932 df-br 5080 df-opab 5142 df-mpt 5163 df-tr 5197 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6200 df-ord 6267 df-on 6268 df-lim 6269 df-suc 6270 df-iota 6389 df-fun 6433 df-fn 6434 df-f 6435 df-f1 6436 df-fo 6437 df-f1o 6438 df-fv 6439 df-riota 7226 df-ov 7272 df-oprab 7273 df-mpo 7274 df-om 7702 df-2nd 7819 df-frecs 8082 df-wrecs 8113 df-recs 8187 df-rdg 8226 df-er 8473 df-en 8709 df-dom 8710 df-sdom 8711 df-pnf 11004 df-mnf 11005 df-xr 11006 df-ltxr 11007 df-le 11008 df-sub 11199 df-neg 11200 df-nn 11966 df-2 12028 df-3 12029 df-sets 16855 df-slot 16873 df-ndx 16885 df-plusg 16965 df-mulr 16966 df-mnring 41787 |
This theorem is referenced by: mnring0gd 41799 mnringmulrd 41801 mnringlmodd 41806 |
Copyright terms: Public domain | W3C validator |