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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 42479 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 17069 | . . 3 ⊢ 𝐸 = Slot (𝐸‘ndx) |
4 | 1, 2 | ndxarg 17068 | . . . 4 ⊢ (𝐸‘ndx) = 𝑁 |
5 | mnringnmulrdOLD.4 | . . . 4 ⊢ 𝑁 ≠ (.r‘ndx) | |
6 | 4, 5 | eqnetri 3014 | . . 3 ⊢ (𝐸‘ndx) ≠ (.r‘ndx) |
7 | 3, 6 | setsnid 17081 | . 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 2736 | . . . 4 ⊢ (.r‘𝑅) = (.r‘𝑅) | |
10 | eqid 2736 | . . . 4 ⊢ (0g‘𝑅) = (0g‘𝑅) | |
11 | mnringnmulrdOLD.5 | . . . 4 ⊢ 𝐴 = (Base‘𝑀) | |
12 | eqid 2736 | . . . 4 ⊢ (+g‘𝑀) = (+g‘𝑀) | |
13 | mnringnmulrdOLD.6 | . . . 4 ⊢ 𝑉 = (𝑅 freeLMod 𝐴) | |
14 | eqid 2736 | . . . 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 42478 | . . 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 6846 | . 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 2795 | 1 ⊢ (𝜑 → (𝐸‘𝑉) = (𝐸‘𝐹)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1541 ∈ wcel 2106 ≠ wne 2943 ifcif 4486 〈cop 4592 ↦ cmpt 5188 ‘cfv 6496 (class class class)co 7357 ∈ cmpo 7359 ℕcn 12153 sSet csts 17035 Slot cslot 17053 ndxcnx 17065 Basecbs 17083 +gcplusg 17133 .rcmulr 17134 0gc0g 17321 Σg cgsu 17322 freeLMod cfrlm 21152 MndRing cmnring 42476 |
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 2707 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7672 ax-cnex 11107 ax-1cn 11109 ax-addcl 11111 |
This theorem depends on definitions: df-bi 206 df-an 397 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 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-ral 3065 df-rex 3074 df-reu 3354 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-pss 3929 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-op 4593 df-uni 4866 df-iun 4956 df-br 5106 df-opab 5168 df-mpt 5189 df-tr 5223 df-id 5531 df-eprel 5537 df-po 5545 df-so 5546 df-fr 5588 df-we 5590 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-pred 6253 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-ov 7360 df-oprab 7361 df-mpo 7362 df-om 7803 df-2nd 7922 df-frecs 8212 df-wrecs 8243 df-recs 8317 df-rdg 8356 df-nn 12154 df-sets 17036 df-slot 17054 df-ndx 17066 df-mnring 42477 |
This theorem is referenced by: mnringbasedOLD 42482 mnringaddgdOLD 42488 mnringscadOLD 42493 mnringvscadOLD 42495 |
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