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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 40931 | . . . 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 2851 | . . 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 2804 | . 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 6644 | . . . . . . . 8 ⊢ (𝑥 = 𝑋 → (𝑥‘𝑎) = (𝑋‘𝑎)) | |
14 | fveq1 6644 | . . . . . . . 8 ⊢ (𝑦 = 𝑌 → (𝑦‘𝑏) = (𝑌‘𝑏)) | |
15 | 13, 14 | oveqan12d 7154 | . . . . . . 7 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → ((𝑥‘𝑎) ∙ (𝑦‘𝑏)) = ((𝑋‘𝑎) ∙ (𝑌‘𝑏))) |
16 | 15 | ifeq1d 4443 | . . . . . 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 5126 | . . . . 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 7212 | . . . 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 7151 | . . 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 485 | . 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 7170 | . 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 7281 | 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 399 = wceq 1538 ∈ wcel 2111 Vcvv 3441 ifcif 4425 ↦ cmpt 5110 ‘cfv 6324 (class class class)co 7135 ∈ cmpo 7137 Basecbs 16475 +gcplusg 16557 .rcmulr 16558 0gc0g 16705 Σg cgsu 16706 MndRing cmnring 40919 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-1st 7671 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-nn 11626 df-2 11688 df-3 11689 df-seq 13365 df-ndx 16478 df-slot 16479 df-base 16481 df-sets 16482 df-plusg 16570 df-mulr 16571 df-0g 16707 df-gsum 16708 df-mnring 40920 |
This theorem is referenced by: mnringmulrcld 40936 |
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