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Mathbox for Rohan Ridenour |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > mnringnmulrdOLD | Structured version Visualization version GIF version |
Description: Obsolete version of mnringnmulrd 42200 as of 1-Nov-2024. Components of a monoid ring other than its ring product match its underlying free module. (Contributed by Rohan Ridenour, 14-May-2024.) (New usage is discouraged.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
mnringnmulrdOLD.1 | ⊢ 𝐹 = (𝑅 MndRing 𝑀) |
mnringnmulrdOLD.2 | ⊢ 𝐸 = Slot 𝑁 |
mnringnmulrdOLD.3 | ⊢ 𝑁 ∈ ℕ |
mnringnmulrdOLD.4 | ⊢ 𝑁 ≠ (.r‘ndx) |
mnringnmulrdOLD.5 | ⊢ 𝐴 = (Base‘𝑀) |
mnringnmulrdOLD.6 | ⊢ 𝑉 = (𝑅 freeLMod 𝐴) |
mnringnmulrdOLD.7 | ⊢ (𝜑 → 𝑅 ∈ 𝑈) |
mnringnmulrdOLD.8 | ⊢ (𝜑 → 𝑀 ∈ 𝑊) |
Ref | Expression |
---|---|
mnringnmulrdOLD | ⊢ (𝜑 → (𝐸‘𝑉) = (𝐸‘𝐹)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mnringnmulrdOLD.2 | . . . 4 ⊢ 𝐸 = Slot 𝑁 | |
2 | mnringnmulrdOLD.3 | . . . 4 ⊢ 𝑁 ∈ ℕ | |
3 | 1, 2 | ndxid 16995 | . . 3 ⊢ 𝐸 = Slot (𝐸‘ndx) |
4 | 1, 2 | ndxarg 16994 | . . . 4 ⊢ (𝐸‘ndx) = 𝑁 |
5 | mnringnmulrdOLD.4 | . . . 4 ⊢ 𝑁 ≠ (.r‘ndx) | |
6 | 4, 5 | eqnetri 3012 | . . 3 ⊢ (𝐸‘ndx) ≠ (.r‘ndx) |
7 | 3, 6 | setsnid 17007 | . 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 | mnringnmulrdOLD.1 | . . . 4 ⊢ 𝐹 = (𝑅 MndRing 𝑀) | |
9 | eqid 2737 | . . . 4 ⊢ (.r‘𝑅) = (.r‘𝑅) | |
10 | eqid 2737 | . . . 4 ⊢ (0g‘𝑅) = (0g‘𝑅) | |
11 | mnringnmulrdOLD.5 | . . . 4 ⊢ 𝐴 = (Base‘𝑀) | |
12 | eqid 2737 | . . . 4 ⊢ (+g‘𝑀) = (+g‘𝑀) | |
13 | mnringnmulrdOLD.6 | . . . 4 ⊢ 𝑉 = (𝑅 freeLMod 𝐴) | |
14 | eqid 2737 | . . . 4 ⊢ (Base‘𝑉) = (Base‘𝑉) | |
15 | mnringnmulrdOLD.7 | . . . 4 ⊢ (𝜑 → 𝑅 ∈ 𝑈) | |
16 | mnringnmulrdOLD.8 | . . . 4 ⊢ (𝜑 → 𝑀 ∈ 𝑊) | |
17 | 8, 9, 10, 11, 12, 13, 14, 15, 16 | mnringvald 42199 | . . 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 6833 | . 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 2796 | 1 ⊢ (𝜑 → (𝐸‘𝑉) = (𝐸‘𝐹)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1541 ∈ wcel 2106 ≠ wne 2941 ifcif 4477 〈cop 4583 ↦ cmpt 5179 ‘cfv 6483 (class class class)co 7341 ∈ cmpo 7343 ℕcn 12078 sSet csts 16961 Slot cslot 16979 ndxcnx 16991 Basecbs 17009 +gcplusg 17059 .rcmulr 17060 0gc0g 17247 Σg cgsu 17248 freeLMod cfrlm 21058 MndRing cmnring 42197 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2708 ax-sep 5247 ax-nul 5254 ax-pow 5312 ax-pr 5376 ax-un 7654 ax-cnex 11032 ax-1cn 11034 ax-addcl 11036 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2887 df-ne 2942 df-ral 3063 df-rex 3072 df-reu 3351 df-rab 3405 df-v 3444 df-sbc 3731 df-csb 3847 df-dif 3904 df-un 3906 df-in 3908 df-ss 3918 df-pss 3920 df-nul 4274 df-if 4478 df-pw 4553 df-sn 4578 df-pr 4580 df-op 4584 df-uni 4857 df-iun 4947 df-br 5097 df-opab 5159 df-mpt 5180 df-tr 5214 df-id 5522 df-eprel 5528 df-po 5536 df-so 5537 df-fr 5579 df-we 5581 df-xp 5630 df-rel 5631 df-cnv 5632 df-co 5633 df-dm 5634 df-rn 5635 df-res 5636 df-ima 5637 df-pred 6242 df-ord 6309 df-on 6310 df-lim 6311 df-suc 6312 df-iota 6435 df-fun 6485 df-fn 6486 df-f 6487 df-f1 6488 df-fo 6489 df-f1o 6490 df-fv 6491 df-ov 7344 df-oprab 7345 df-mpo 7346 df-om 7785 df-2nd 7904 df-frecs 8171 df-wrecs 8202 df-recs 8276 df-rdg 8315 df-nn 12079 df-sets 16962 df-slot 16980 df-ndx 16992 df-mnring 42198 |
This theorem is referenced by: mnringbasedOLD 42203 mnringaddgdOLD 42209 mnringscadOLD 42214 mnringvscadOLD 42216 |
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