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Mathbox for Rohan Ridenour |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > mnringnmulrd | Structured version Visualization version GIF version |
Description: Components of a monoid ring other than its ring product match its underlying free module. (Contributed by Rohan Ridenour, 14-May-2024.) |
Ref | Expression |
---|---|
mnringnmulrd.1 | ⊢ 𝐹 = (𝑅 MndRing 𝑀) |
mnringnmulrd.2 | ⊢ 𝐸 = Slot 𝑁 |
mnringnmulrd.3 | ⊢ 𝑁 ∈ ℕ |
mnringnmulrd.4 | ⊢ 𝑁 ≠ (.r‘ndx) |
mnringnmulrd.5 | ⊢ 𝐴 = (Base‘𝑀) |
mnringnmulrd.6 | ⊢ 𝑉 = (𝑅 freeLMod 𝐴) |
mnringnmulrd.7 | ⊢ (𝜑 → 𝑅 ∈ 𝑈) |
mnringnmulrd.8 | ⊢ (𝜑 → 𝑀 ∈ 𝑊) |
Ref | Expression |
---|---|
mnringnmulrd | ⊢ (𝜑 → (𝐸‘𝑉) = (𝐸‘𝐹)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mnringnmulrd.2 | . . . 4 ⊢ 𝐸 = Slot 𝑁 | |
2 | mnringnmulrd.3 | . . . 4 ⊢ 𝑁 ∈ ℕ | |
3 | 1, 2 | ndxid 16552 | . . 3 ⊢ 𝐸 = Slot (𝐸‘ndx) |
4 | 1, 2 | ndxarg 16551 | . . . 4 ⊢ (𝐸‘ndx) = 𝑁 |
5 | mnringnmulrd.4 | . . . 4 ⊢ 𝑁 ≠ (.r‘ndx) | |
6 | 4, 5 | eqnetri 3019 | . . 3 ⊢ (𝐸‘ndx) ≠ (.r‘ndx) |
7 | 3, 6 | setsnid 16582 | . 2 ⊢ (𝐸‘𝑉) = (𝐸‘(𝑉 sSet 〈(.r‘ndx), (𝑥 ∈ (Base‘𝑉), 𝑦 ∈ (Base‘𝑉) ↦ (𝑉 Σg (𝑎 ∈ 𝐴, 𝑏 ∈ 𝐴 ↦ (𝑖 ∈ 𝐴 ↦ if(𝑖 = (𝑎(+g‘𝑀)𝑏), ((𝑥‘𝑎)(.r‘𝑅)(𝑦‘𝑏)), (0g‘𝑅))))))〉)) |
8 | mnringnmulrd.1 | . . . 4 ⊢ 𝐹 = (𝑅 MndRing 𝑀) | |
9 | eqid 2759 | . . . 4 ⊢ (.r‘𝑅) = (.r‘𝑅) | |
10 | eqid 2759 | . . . 4 ⊢ (0g‘𝑅) = (0g‘𝑅) | |
11 | mnringnmulrd.5 | . . . 4 ⊢ 𝐴 = (Base‘𝑀) | |
12 | eqid 2759 | . . . 4 ⊢ (+g‘𝑀) = (+g‘𝑀) | |
13 | mnringnmulrd.6 | . . . 4 ⊢ 𝑉 = (𝑅 freeLMod 𝐴) | |
14 | eqid 2759 | . . . 4 ⊢ (Base‘𝑉) = (Base‘𝑉) | |
15 | mnringnmulrd.7 | . . . 4 ⊢ (𝜑 → 𝑅 ∈ 𝑈) | |
16 | mnringnmulrd.8 | . . . 4 ⊢ (𝜑 → 𝑀 ∈ 𝑊) | |
17 | 8, 9, 10, 11, 12, 13, 14, 15, 16 | mnringvald 41279 | . . 3 ⊢ (𝜑 → 𝐹 = (𝑉 sSet 〈(.r‘ndx), (𝑥 ∈ (Base‘𝑉), 𝑦 ∈ (Base‘𝑉) ↦ (𝑉 Σg (𝑎 ∈ 𝐴, 𝑏 ∈ 𝐴 ↦ (𝑖 ∈ 𝐴 ↦ if(𝑖 = (𝑎(+g‘𝑀)𝑏), ((𝑥‘𝑎)(.r‘𝑅)(𝑦‘𝑏)), (0g‘𝑅))))))〉)) |
18 | 17 | fveq2d 6655 | . 2 ⊢ (𝜑 → (𝐸‘𝐹) = (𝐸‘(𝑉 sSet 〈(.r‘ndx), (𝑥 ∈ (Base‘𝑉), 𝑦 ∈ (Base‘𝑉) ↦ (𝑉 Σg (𝑎 ∈ 𝐴, 𝑏 ∈ 𝐴 ↦ (𝑖 ∈ 𝐴 ↦ if(𝑖 = (𝑎(+g‘𝑀)𝑏), ((𝑥‘𝑎)(.r‘𝑅)(𝑦‘𝑏)), (0g‘𝑅))))))〉))) |
19 | 7, 18 | eqtr4id 2813 | 1 ⊢ (𝜑 → (𝐸‘𝑉) = (𝐸‘𝐹)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1539 ∈ wcel 2112 ≠ wne 2949 ifcif 4413 〈cop 4521 ↦ cmpt 5105 ‘cfv 6328 (class class class)co 7143 ∈ cmpo 7145 ℕcn 11659 ndxcnx 16523 sSet csts 16524 Slot cslot 16525 Basecbs 16526 +gcplusg 16608 .rcmulr 16609 0gc0g 16756 Σg cgsu 16757 freeLMod cfrlm 20496 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-sep 5162 ax-nul 5169 ax-pow 5227 ax-pr 5291 ax-un 7452 ax-cnex 10616 ax-1cn 10618 ax-addcl 10620 |
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-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-ov 7146 df-oprab 7147 df-mpo 7148 df-om 7573 df-wrecs 7950 df-recs 8011 df-rdg 8049 df-nn 11660 df-ndx 16529 df-slot 16530 df-sets 16533 df-mnring 41278 |
This theorem is referenced by: mnringbased 41281 mnringaddgd 41286 mnringscad 41290 mnringvscad 41291 |
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