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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 44219 | . . . 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 2783 | . . 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 2736 | . 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 6860 | . . . . . . . 8 ⊢ (𝑥 = 𝑋 → (𝑥‘𝑎) = (𝑋‘𝑎)) | |
| 14 | fveq1 6860 | . . . . . . . 8 ⊢ (𝑦 = 𝑌 → (𝑦‘𝑏) = (𝑌‘𝑏)) | |
| 15 | 13, 14 | oveqan12d 7409 | . . . . . . 7 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → ((𝑥‘𝑎) ∙ (𝑦‘𝑏)) = ((𝑋‘𝑎) ∙ (𝑌‘𝑏))) |
| 16 | 15 | ifeq1d 4511 | . . . . . 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 5204 | . . . . 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 7471 | . . . 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 7406 | . . 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 481 | . 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 7425 | . 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 7544 | 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 395 = wceq 1540 ∈ wcel 2109 Vcvv 3450 ifcif 4491 ↦ cmpt 5191 ‘cfv 6514 (class class class)co 7390 ∈ cmpo 7392 Basecbs 17186 +gcplusg 17227 .rcmulr 17228 0gc0g 17409 Σg cgsu 17410 MndRing cmnring 44207 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2702 ax-rep 5237 ax-sep 5254 ax-nul 5264 ax-pow 5323 ax-pr 5390 ax-un 7714 ax-cnex 11131 ax-resscn 11132 ax-1cn 11133 ax-icn 11134 ax-addcl 11135 ax-addrcl 11136 ax-mulcl 11137 ax-mulrcl 11138 ax-mulcom 11139 ax-addass 11140 ax-mulass 11141 ax-distr 11142 ax-i2m1 11143 ax-1ne0 11144 ax-1rid 11145 ax-rnegex 11146 ax-rrecex 11147 ax-cnre 11148 ax-pre-lttri 11149 ax-pre-lttrn 11150 ax-pre-ltadd 11151 ax-pre-mulgt0 11152 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2534 df-eu 2563 df-clab 2709 df-cleq 2722 df-clel 2804 df-nfc 2879 df-ne 2927 df-nel 3031 df-ral 3046 df-rex 3055 df-reu 3357 df-rab 3409 df-v 3452 df-sbc 3757 df-csb 3866 df-dif 3920 df-un 3922 df-in 3924 df-ss 3934 df-pss 3937 df-nul 4300 df-if 4492 df-pw 4568 df-sn 4593 df-pr 4595 df-op 4599 df-uni 4875 df-iun 4960 df-br 5111 df-opab 5173 df-mpt 5192 df-tr 5218 df-id 5536 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5594 df-we 5596 df-xp 5647 df-rel 5648 df-cnv 5649 df-co 5650 df-dm 5651 df-rn 5652 df-res 5653 df-ima 5654 df-pred 6277 df-ord 6338 df-on 6339 df-lim 6340 df-suc 6341 df-iota 6467 df-fun 6516 df-fn 6517 df-f 6518 df-f1 6519 df-fo 6520 df-f1o 6521 df-fv 6522 df-riota 7347 df-ov 7393 df-oprab 7394 df-mpo 7395 df-om 7846 df-1st 7971 df-2nd 7972 df-frecs 8263 df-wrecs 8294 df-recs 8343 df-rdg 8381 df-er 8674 df-en 8922 df-dom 8923 df-sdom 8924 df-pnf 11217 df-mnf 11218 df-xr 11219 df-ltxr 11220 df-le 11221 df-sub 11414 df-neg 11415 df-nn 12194 df-2 12256 df-3 12257 df-seq 13974 df-sets 17141 df-slot 17159 df-ndx 17171 df-base 17187 df-plusg 17240 df-mulr 17241 df-0g 17411 df-gsum 17412 df-mnring 44208 |
| This theorem is referenced by: mnringmulrcld 44224 |
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