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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 17243 | . 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 2735 | . . . 4 ⊢ (.r‘𝑅) = (.r‘𝑅) | |
6 | eqid 2735 | . . . 4 ⊢ (0g‘𝑅) = (0g‘𝑅) | |
7 | mnringnmulrd.5 | . . . 4 ⊢ 𝐴 = (Base‘𝑀) | |
8 | eqid 2735 | . . . 4 ⊢ (+g‘𝑀) = (+g‘𝑀) | |
9 | mnringnmulrd.6 | . . . 4 ⊢ 𝑉 = (𝑅 freeLMod 𝐴) | |
10 | eqid 2735 | . . . 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 44204 | . . 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 6911 | . 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 2794 | 1 ⊢ (𝜑 → (𝐸‘𝑉) = (𝐸‘𝐹)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2106 ≠ wne 2938 ifcif 4531 〈cop 4637 ↦ cmpt 5231 ‘cfv 6563 (class class class)co 7431 ∈ cmpo 7433 sSet csts 17197 Slot cslot 17215 ndxcnx 17227 Basecbs 17245 +gcplusg 17298 .rcmulr 17299 0gc0g 17486 Σg cgsu 17487 freeLMod cfrlm 21784 MndRing cmnring 44202 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-res 5701 df-iota 6516 df-fun 6565 df-fv 6571 df-ov 7434 df-oprab 7435 df-mpo 7436 df-sets 17198 df-slot 17216 df-mnring 44203 |
This theorem is referenced by: mnringbased 44207 mnringaddgd 44213 mnringscad 44218 mnringvscad 44220 |
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