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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.) (Revised by AV, 1-Nov-2024.) | 
| Ref | Expression | 
|---|---|
| mnringnmulrd.1 | ⊢ 𝐹 = (𝑅 MndRing 𝑀) | 
| mnringnmulrd.2 | ⊢ 𝐸 = Slot (𝐸‘ndx) | 
| mnringnmulrd.4 | ⊢ (𝐸‘ndx) ≠ (.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 | . . 3 ⊢ 𝐸 = Slot (𝐸‘ndx) | |
| 2 | mnringnmulrd.4 | . . 3 ⊢ (𝐸‘ndx) ≠ (.r‘ndx) | |
| 3 | 1, 2 | setsnid 17246 | . 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))))))〉)) | 
| 4 | mnringnmulrd.1 | . . . 4 ⊢ 𝐹 = (𝑅 MndRing 𝑀) | |
| 5 | eqid 2736 | . . . 4 ⊢ (.r‘𝑅) = (.r‘𝑅) | |
| 6 | eqid 2736 | . . . 4 ⊢ (0g‘𝑅) = (0g‘𝑅) | |
| 7 | mnringnmulrd.5 | . . . 4 ⊢ 𝐴 = (Base‘𝑀) | |
| 8 | eqid 2736 | . . . 4 ⊢ (+g‘𝑀) = (+g‘𝑀) | |
| 9 | mnringnmulrd.6 | . . . 4 ⊢ 𝑉 = (𝑅 freeLMod 𝐴) | |
| 10 | eqid 2736 | . . . 4 ⊢ (Base‘𝑉) = (Base‘𝑉) | |
| 11 | mnringnmulrd.7 | . . . 4 ⊢ (𝜑 → 𝑅 ∈ 𝑈) | |
| 12 | mnringnmulrd.8 | . . . 4 ⊢ (𝜑 → 𝑀 ∈ 𝑊) | |
| 13 | 4, 5, 6, 7, 8, 9, 10, 11, 12 | mnringvald 44232 | . . 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))))))〉)) | 
| 14 | 13 | fveq2d 6909 | . 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))))))〉))) | 
| 15 | 3, 14 | eqtr4id 2795 | 1 ⊢ (𝜑 → (𝐸‘𝑉) = (𝐸‘𝐹)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 = wceq 1539 ∈ wcel 2107 ≠ wne 2939 ifcif 4524 〈cop 4631 ↦ cmpt 5224 ‘cfv 6560 (class class class)co 7432 ∈ cmpo 7434 sSet csts 17201 Slot cslot 17219 ndxcnx 17231 Basecbs 17248 +gcplusg 17298 .rcmulr 17299 0gc0g 17485 Σg cgsu 17486 freeLMod cfrlm 21767 MndRing cmnring 44230 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pr 5431 ax-un 7756 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-br 5143 df-opab 5205 df-mpt 5225 df-id 5577 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-res 5696 df-iota 6513 df-fun 6562 df-fv 6568 df-ov 7435 df-oprab 7436 df-mpo 7437 df-sets 17202 df-slot 17220 df-mnring 44231 | 
| This theorem is referenced by: mnringbased 44235 mnringaddgd 44241 mnringscad 44246 mnringvscad 44248 | 
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