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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.) (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 17129 | . 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 2733 | . . . 4 ⊢ (.r‘𝑅) = (.r‘𝑅) | |
6 | eqid 2733 | . . . 4 ⊢ (0g‘𝑅) = (0g‘𝑅) | |
7 | mnringnmulrd.5 | . . . 4 ⊢ 𝐴 = (Base‘𝑀) | |
8 | eqid 2733 | . . . 4 ⊢ (+g‘𝑀) = (+g‘𝑀) | |
9 | mnringnmulrd.6 | . . . 4 ⊢ 𝑉 = (𝑅 freeLMod 𝐴) | |
10 | eqid 2733 | . . . 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 42838 | . . 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 6885 | . 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 2792 | 1 ⊢ (𝜑 → (𝐸‘𝑉) = (𝐸‘𝐹)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2107 ≠ wne 2941 ifcif 4524 〈cop 4630 ↦ cmpt 5227 ‘cfv 6535 (class class class)co 7396 ∈ cmpo 7398 sSet csts 17083 Slot cslot 17101 ndxcnx 17113 Basecbs 17131 +gcplusg 17184 .rcmulr 17185 0gc0g 17372 Σg cgsu 17373 freeLMod cfrlm 21274 MndRing cmnring 42836 |
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-sep 5295 ax-nul 5302 ax-pr 5423 ax-un 7712 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 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-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-nul 4321 df-if 4525 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4905 df-br 5145 df-opab 5207 df-mpt 5228 df-id 5570 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-res 5684 df-iota 6487 df-fun 6537 df-fv 6543 df-ov 7399 df-oprab 7400 df-mpo 7401 df-sets 17084 df-slot 17102 df-mnring 42837 |
This theorem is referenced by: mnringbased 42841 mnringaddgd 42847 mnringscad 42852 mnringvscad 42854 |
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