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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 42913 | . . . 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 2791 | . . 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 2739 | . 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 6887 | . . . . . . . 8 ⊢ (𝑥 = 𝑋 → (𝑥‘𝑎) = (𝑋‘𝑎)) | |
14 | fveq1 6887 | . . . . . . . 8 ⊢ (𝑦 = 𝑌 → (𝑦‘𝑏) = (𝑌‘𝑏)) | |
15 | 13, 14 | oveqan12d 7423 | . . . . . . 7 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → ((𝑥‘𝑎) ∙ (𝑦‘𝑏)) = ((𝑋‘𝑎) ∙ (𝑌‘𝑏))) |
16 | 15 | ifeq1d 4546 | . . . . . 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 5249 | . . . . 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 7483 | . . . 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 7420 | . . 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 483 | . 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 7439 | . 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 7555 | 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 397 = wceq 1542 ∈ wcel 2107 Vcvv 3475 ifcif 4527 ↦ cmpt 5230 ‘cfv 6540 (class class class)co 7404 ∈ cmpo 7406 Basecbs 17140 +gcplusg 17193 .rcmulr 17194 0gc0g 17381 Σg cgsu 17382 MndRing cmnring 42898 |
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 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7720 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 |
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 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7360 df-ov 7407 df-oprab 7408 df-mpo 7409 df-om 7851 df-1st 7970 df-2nd 7971 df-frecs 8261 df-wrecs 8292 df-recs 8366 df-rdg 8405 df-er 8699 df-en 8936 df-dom 8937 df-sdom 8938 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-nn 12209 df-2 12271 df-3 12272 df-seq 13963 df-sets 17093 df-slot 17111 df-ndx 17123 df-base 17141 df-plusg 17206 df-mulr 17207 df-0g 17383 df-gsum 17384 df-mnring 42899 |
This theorem is referenced by: mnringmulrcld 42920 |
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