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Mathbox for Rohan Ridenour |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > mnringmulrvald | Structured version Visualization version GIF version |
Description: Value of multiplication in a monoid ring. (Contributed by Rohan Ridenour, 14-May-2024.) |
Ref | Expression |
---|---|
mnringmulrvald.1 | ⊢ 𝐹 = (𝑅 MndRing 𝑀) |
mnringmulrvald.2 | ⊢ 𝐵 = (Base‘𝐹) |
mnringmulrvald.3 | ⊢ ∙ = (.r‘𝑅) |
mnringmulrvald.4 | ⊢ 𝟎 = (0g‘𝑅) |
mnringmulrvald.5 | ⊢ 𝐴 = (Base‘𝑀) |
mnringmulrvald.6 | ⊢ + = (+g‘𝑀) |
mnringmulrvald.7 | ⊢ · = (.r‘𝐹) |
mnringmulrvald.8 | ⊢ (𝜑 → 𝑅 ∈ 𝑈) |
mnringmulrvald.9 | ⊢ (𝜑 → 𝑀 ∈ 𝑊) |
mnringmulrvald.10 | ⊢ (𝜑 → 𝑋 ∈ 𝐵) |
mnringmulrvald.11 | ⊢ (𝜑 → 𝑌 ∈ 𝐵) |
Ref | Expression |
---|---|
mnringmulrvald | ⊢ (𝜑 → (𝑋 · 𝑌) = (𝐹 Σg (𝑎 ∈ 𝐴, 𝑏 ∈ 𝐴 ↦ (𝑖 ∈ 𝐴 ↦ if(𝑖 = (𝑎 + 𝑏), ((𝑋‘𝑎) ∙ (𝑌‘𝑏)), 𝟎 ))))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mnringmulrvald.1 | . . . . 5 ⊢ 𝐹 = (𝑅 MndRing 𝑀) | |
2 | mnringmulrvald.2 | . . . . 5 ⊢ 𝐵 = (Base‘𝐹) | |
3 | mnringmulrvald.3 | . . . . 5 ⊢ ∙ = (.r‘𝑅) | |
4 | mnringmulrvald.4 | . . . . 5 ⊢ 𝟎 = (0g‘𝑅) | |
5 | mnringmulrvald.5 | . . . . 5 ⊢ 𝐴 = (Base‘𝑀) | |
6 | mnringmulrvald.6 | . . . . 5 ⊢ + = (+g‘𝑀) | |
7 | mnringmulrvald.8 | . . . . 5 ⊢ (𝜑 → 𝑅 ∈ 𝑈) | |
8 | mnringmulrvald.9 | . . . . 5 ⊢ (𝜑 → 𝑀 ∈ 𝑊) | |
9 | 1, 2, 3, 4, 5, 6, 7, 8 | mnringmulrd 41289 | . . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝐹 Σg (𝑎 ∈ 𝐴, 𝑏 ∈ 𝐴 ↦ (𝑖 ∈ 𝐴 ↦ if(𝑖 = (𝑎 + 𝑏), ((𝑥‘𝑎) ∙ (𝑦‘𝑏)), 𝟎 ))))) = (.r‘𝐹)) |
10 | mnringmulrvald.7 | . . . 4 ⊢ · = (.r‘𝐹) | |
11 | 9, 10 | eqtr4di 2812 | . . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝐹 Σg (𝑎 ∈ 𝐴, 𝑏 ∈ 𝐴 ↦ (𝑖 ∈ 𝐴 ↦ if(𝑖 = (𝑎 + 𝑏), ((𝑥‘𝑎) ∙ (𝑦‘𝑏)), 𝟎 ))))) = · ) |
12 | 11 | eqcomd 2765 | . 2 ⊢ (𝜑 → · = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝐹 Σg (𝑎 ∈ 𝐴, 𝑏 ∈ 𝐴 ↦ (𝑖 ∈ 𝐴 ↦ if(𝑖 = (𝑎 + 𝑏), ((𝑥‘𝑎) ∙ (𝑦‘𝑏)), 𝟎 )))))) |
13 | fveq1 6650 | . . . . . . . 8 ⊢ (𝑥 = 𝑋 → (𝑥‘𝑎) = (𝑋‘𝑎)) | |
14 | fveq1 6650 | . . . . . . . 8 ⊢ (𝑦 = 𝑌 → (𝑦‘𝑏) = (𝑌‘𝑏)) | |
15 | 13, 14 | oveqan12d 7162 | . . . . . . 7 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → ((𝑥‘𝑎) ∙ (𝑦‘𝑏)) = ((𝑋‘𝑎) ∙ (𝑌‘𝑏))) |
16 | 15 | ifeq1d 4432 | . . . . . 6 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → if(𝑖 = (𝑎 + 𝑏), ((𝑥‘𝑎) ∙ (𝑦‘𝑏)), 𝟎 ) = if(𝑖 = (𝑎 + 𝑏), ((𝑋‘𝑎) ∙ (𝑌‘𝑏)), 𝟎 )) |
17 | 16 | mpteq2dv 5121 | . . . . 5 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → (𝑖 ∈ 𝐴 ↦ if(𝑖 = (𝑎 + 𝑏), ((𝑥‘𝑎) ∙ (𝑦‘𝑏)), 𝟎 )) = (𝑖 ∈ 𝐴 ↦ if(𝑖 = (𝑎 + 𝑏), ((𝑋‘𝑎) ∙ (𝑌‘𝑏)), 𝟎 ))) |
18 | 17 | mpoeq3dv 7220 | . . . 4 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → (𝑎 ∈ 𝐴, 𝑏 ∈ 𝐴 ↦ (𝑖 ∈ 𝐴 ↦ if(𝑖 = (𝑎 + 𝑏), ((𝑥‘𝑎) ∙ (𝑦‘𝑏)), 𝟎 ))) = (𝑎 ∈ 𝐴, 𝑏 ∈ 𝐴 ↦ (𝑖 ∈ 𝐴 ↦ if(𝑖 = (𝑎 + 𝑏), ((𝑋‘𝑎) ∙ (𝑌‘𝑏)), 𝟎 )))) |
19 | 18 | oveq2d 7159 | . . 3 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → (𝐹 Σg (𝑎 ∈ 𝐴, 𝑏 ∈ 𝐴 ↦ (𝑖 ∈ 𝐴 ↦ if(𝑖 = (𝑎 + 𝑏), ((𝑥‘𝑎) ∙ (𝑦‘𝑏)), 𝟎 )))) = (𝐹 Σg (𝑎 ∈ 𝐴, 𝑏 ∈ 𝐴 ↦ (𝑖 ∈ 𝐴 ↦ if(𝑖 = (𝑎 + 𝑏), ((𝑋‘𝑎) ∙ (𝑌‘𝑏)), 𝟎 ))))) |
20 | 19 | adantl 486 | . 2 ⊢ ((𝜑 ∧ (𝑥 = 𝑋 ∧ 𝑦 = 𝑌)) → (𝐹 Σg (𝑎 ∈ 𝐴, 𝑏 ∈ 𝐴 ↦ (𝑖 ∈ 𝐴 ↦ if(𝑖 = (𝑎 + 𝑏), ((𝑥‘𝑎) ∙ (𝑦‘𝑏)), 𝟎 )))) = (𝐹 Σg (𝑎 ∈ 𝐴, 𝑏 ∈ 𝐴 ↦ (𝑖 ∈ 𝐴 ↦ if(𝑖 = (𝑎 + 𝑏), ((𝑋‘𝑎) ∙ (𝑌‘𝑏)), 𝟎 ))))) |
21 | mnringmulrvald.10 | . 2 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
22 | mnringmulrvald.11 | . 2 ⊢ (𝜑 → 𝑌 ∈ 𝐵) | |
23 | ovexd 7178 | . 2 ⊢ (𝜑 → (𝐹 Σg (𝑎 ∈ 𝐴, 𝑏 ∈ 𝐴 ↦ (𝑖 ∈ 𝐴 ↦ if(𝑖 = (𝑎 + 𝑏), ((𝑋‘𝑎) ∙ (𝑌‘𝑏)), 𝟎 )))) ∈ V) | |
24 | 12, 20, 21, 22, 23 | ovmpod 7290 | 1 ⊢ (𝜑 → (𝑋 · 𝑌) = (𝐹 Σg (𝑎 ∈ 𝐴, 𝑏 ∈ 𝐴 ↦ (𝑖 ∈ 𝐴 ↦ if(𝑖 = (𝑎 + 𝑏), ((𝑋‘𝑎) ∙ (𝑌‘𝑏)), 𝟎 ))))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 400 = wceq 1539 ∈ wcel 2112 Vcvv 3407 ifcif 4413 ↦ cmpt 5105 ‘cfv 6328 (class class class)co 7143 ∈ cmpo 7145 Basecbs 16526 +gcplusg 16608 .rcmulr 16609 0gc0g 16756 Σg cgsu 16757 MndRing cmnring 41277 |
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 1912 ax-6 1971 ax-7 2016 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2159 ax-12 2176 ax-ext 2730 ax-rep 5149 ax-sep 5162 ax-nul 5169 ax-pow 5227 ax-pr 5291 ax-un 7452 ax-cnex 10616 ax-resscn 10617 ax-1cn 10618 ax-icn 10619 ax-addcl 10620 ax-addrcl 10621 ax-mulcl 10622 ax-mulrcl 10623 ax-mulcom 10624 ax-addass 10625 ax-mulass 10626 ax-distr 10627 ax-i2m1 10628 ax-1ne0 10629 ax-1rid 10630 ax-rnegex 10631 ax-rrecex 10632 ax-cnre 10633 ax-pre-lttri 10634 ax-pre-lttrn 10635 ax-pre-ltadd 10636 ax-pre-mulgt0 10637 |
This theorem depends on definitions: df-bi 210 df-an 401 df-or 846 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2071 df-mo 2558 df-eu 2589 df-clab 2737 df-cleq 2751 df-clel 2831 df-nfc 2899 df-ne 2950 df-nel 3054 df-ral 3073 df-rex 3074 df-reu 3075 df-rab 3077 df-v 3409 df-sbc 3694 df-csb 3802 df-dif 3857 df-un 3859 df-in 3861 df-ss 3871 df-pss 3873 df-nul 4222 df-if 4414 df-pw 4489 df-sn 4516 df-pr 4518 df-tp 4520 df-op 4522 df-uni 4792 df-iun 4878 df-br 5026 df-opab 5088 df-mpt 5106 df-tr 5132 df-id 5423 df-eprel 5428 df-po 5436 df-so 5437 df-fr 5476 df-we 5478 df-xp 5523 df-rel 5524 df-cnv 5525 df-co 5526 df-dm 5527 df-rn 5528 df-res 5529 df-ima 5530 df-pred 6119 df-ord 6165 df-on 6166 df-lim 6167 df-suc 6168 df-iota 6287 df-fun 6330 df-fn 6331 df-f 6332 df-f1 6333 df-fo 6334 df-f1o 6335 df-fv 6336 df-riota 7101 df-ov 7146 df-oprab 7147 df-mpo 7148 df-om 7573 df-1st 7686 df-2nd 7687 df-wrecs 7950 df-recs 8011 df-rdg 8049 df-er 8292 df-en 8521 df-dom 8522 df-sdom 8523 df-pnf 10700 df-mnf 10701 df-xr 10702 df-ltxr 10703 df-le 10704 df-sub 10895 df-neg 10896 df-nn 11660 df-2 11722 df-3 11723 df-seq 13404 df-ndx 16529 df-slot 16530 df-base 16532 df-sets 16533 df-plusg 16621 df-mulr 16622 df-0g 16758 df-gsum 16759 df-mnring 41278 |
This theorem is referenced by: mnringmulrcld 41294 |
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