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Mirrors > Home > ILE Home > Th. List > mgpbasg | GIF version |
Description: Base set of the multiplication group. (Contributed by Mario Carneiro, 21-Dec-2014.) (Revised by Mario Carneiro, 5-Oct-2015.) |
Ref | Expression |
---|---|
mgpbas.1 | ⊢ 𝑀 = (mulGrp‘𝑅) |
mgpbas.2 | ⊢ 𝐵 = (Base‘𝑅) |
Ref | Expression |
---|---|
mgpbasg | ⊢ (𝑅 ∈ 𝑉 → 𝐵 = (Base‘𝑀)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mgpbas.2 | . 2 ⊢ 𝐵 = (Base‘𝑅) | |
2 | mulrslid 12584 | . . . . 5 ⊢ (.r = Slot (.r‘ndx) ∧ (.r‘ndx) ∈ ℕ) | |
3 | 2 | slotex 12483 | . . . 4 ⊢ (𝑅 ∈ 𝑉 → (.r‘𝑅) ∈ V) |
4 | baseslid 12513 | . . . . 5 ⊢ (Base = Slot (Base‘ndx) ∧ (Base‘ndx) ∈ ℕ) | |
5 | basendxnplusgndx 12577 | . . . . 5 ⊢ (Base‘ndx) ≠ (+g‘ndx) | |
6 | plusgslid 12565 | . . . . . 6 ⊢ (+g = Slot (+g‘ndx) ∧ (+g‘ndx) ∈ ℕ) | |
7 | 6 | simpri 113 | . . . . 5 ⊢ (+g‘ndx) ∈ ℕ |
8 | 4, 5, 7 | setsslnid 12508 | . . . 4 ⊢ ((𝑅 ∈ 𝑉 ∧ (.r‘𝑅) ∈ V) → (Base‘𝑅) = (Base‘(𝑅 sSet 〈(+g‘ndx), (.r‘𝑅)〉))) |
9 | 3, 8 | mpdan 421 | . . 3 ⊢ (𝑅 ∈ 𝑉 → (Base‘𝑅) = (Base‘(𝑅 sSet 〈(+g‘ndx), (.r‘𝑅)〉))) |
10 | mgpbas.1 | . . . . 5 ⊢ 𝑀 = (mulGrp‘𝑅) | |
11 | eqid 2177 | . . . . 5 ⊢ (.r‘𝑅) = (.r‘𝑅) | |
12 | 10, 11 | mgpvalg 13086 | . . . 4 ⊢ (𝑅 ∈ 𝑉 → 𝑀 = (𝑅 sSet 〈(+g‘ndx), (.r‘𝑅)〉)) |
13 | 12 | fveq2d 5519 | . . 3 ⊢ (𝑅 ∈ 𝑉 → (Base‘𝑀) = (Base‘(𝑅 sSet 〈(+g‘ndx), (.r‘𝑅)〉))) |
14 | 9, 13 | eqtr4d 2213 | . 2 ⊢ (𝑅 ∈ 𝑉 → (Base‘𝑅) = (Base‘𝑀)) |
15 | 1, 14 | eqtrid 2222 | 1 ⊢ (𝑅 ∈ 𝑉 → 𝐵 = (Base‘𝑀)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1353 ∈ wcel 2148 Vcvv 2737 〈cop 3595 ‘cfv 5216 (class class class)co 5874 ℕcn 8917 ndxcnx 12453 sSet csts 12454 Slot cslot 12455 Basecbs 12456 +gcplusg 12530 .rcmulr 12531 mulGrpcmgp 13083 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4121 ax-pow 4174 ax-pr 4209 ax-un 4433 ax-setind 4536 ax-cnex 7901 ax-resscn 7902 ax-1cn 7903 ax-1re 7904 ax-icn 7905 ax-addcl 7906 ax-addrcl 7907 ax-mulcl 7908 ax-addcom 7910 ax-addass 7912 ax-i2m1 7915 ax-0lt1 7916 ax-0id 7918 ax-rnegex 7919 ax-pre-ltirr 7922 ax-pre-ltadd 7926 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-nel 2443 df-ral 2460 df-rex 2461 df-rab 2464 df-v 2739 df-sbc 2963 df-csb 3058 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-nul 3423 df-pw 3577 df-sn 3598 df-pr 3599 df-op 3601 df-uni 3810 df-int 3845 df-br 4004 df-opab 4065 df-mpt 4066 df-id 4293 df-xp 4632 df-rel 4633 df-cnv 4634 df-co 4635 df-dm 4636 df-rn 4637 df-res 4638 df-iota 5178 df-fun 5218 df-fn 5219 df-fv 5224 df-ov 5877 df-oprab 5878 df-mpo 5879 df-pnf 7992 df-mnf 7993 df-ltxr 7995 df-inn 8918 df-2 8976 df-3 8977 df-ndx 12459 df-slot 12460 df-base 12462 df-sets 12463 df-plusg 12543 df-mulr 12544 df-mgp 13084 |
This theorem is referenced by: mgptopng 13092 mgpress 13094 dfur2g 13098 srgcl 13106 srgass 13107 srgideu 13108 srgidcl 13112 srgidmlem 13114 issrgid 13117 srg1zr 13123 srgpcomp 13126 srgpcompp 13127 srgpcomppsc 13128 ringcl 13149 crngcom 13150 iscrng2 13151 ringass 13152 ringideu 13153 ringidcl 13156 ringidmlem 13158 isringid 13161 ringidss 13165 ringpropd 13170 crngpropd 13171 isringd 13173 iscrngd 13174 ring1 13189 oppr1g 13205 unitgrpbasd 13237 unitsubm 13241 rngidpropdg 13268 subrgsubm 13315 issubrg3 13328 cnfldexp 13362 |
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