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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 16501 | . . 3 ⊢ 𝐸 = Slot (𝐸‘ndx) |
4 | 1, 2 | ndxarg 16500 | . . . 4 ⊢ (𝐸‘ndx) = 𝑁 |
5 | mnringnmulrd.4 | . . . 4 ⊢ 𝑁 ≠ (.r‘ndx) | |
6 | 4, 5 | eqnetri 3057 | . . 3 ⊢ (𝐸‘ndx) ≠ (.r‘ndx) |
7 | 3, 6 | setsnid 16531 | . 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 2798 | . . . 4 ⊢ (.r‘𝑅) = (.r‘𝑅) | |
10 | eqid 2798 | . . . 4 ⊢ (0g‘𝑅) = (0g‘𝑅) | |
11 | mnringnmulrd.5 | . . . 4 ⊢ 𝐴 = (Base‘𝑀) | |
12 | eqid 2798 | . . . 4 ⊢ (+g‘𝑀) = (+g‘𝑀) | |
13 | mnringnmulrd.6 | . . . 4 ⊢ 𝑉 = (𝑅 freeLMod 𝐴) | |
14 | eqid 2798 | . . . 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 40921 | . . 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 6649 | . 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 2852 | 1 ⊢ (𝜑 → (𝐸‘𝑉) = (𝐸‘𝐹)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1538 ∈ wcel 2111 ≠ wne 2987 ifcif 4425 〈cop 4531 ↦ cmpt 5110 ‘cfv 6324 (class class class)co 7135 ∈ cmpo 7137 ℕcn 11625 ndxcnx 16472 sSet csts 16473 Slot cslot 16474 Basecbs 16475 +gcplusg 16557 .rcmulr 16558 0gc0g 16705 Σg cgsu 16706 freeLMod cfrlm 20435 MndRing cmnring 40919 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-1cn 10584 ax-addcl 10586 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-nn 11626 df-ndx 16478 df-slot 16479 df-sets 16482 df-mnring 40920 |
This theorem is referenced by: mnringbased 40923 mnringaddgd 40928 mnringscad 40932 mnringvscad 40933 |
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