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Mirrors > Home > ILE Home > Th. List > mgpex | GIF version |
Description: Existence of the multiplication group. If 𝑅 is known to be a semiring, see srgmgp 13104. (Contributed by Jim Kingdon, 10-Jan-2025.) |
Ref | Expression |
---|---|
mgpbas.1 | ⊢ 𝑀 = (mulGrp‘𝑅) |
Ref | Expression |
---|---|
mgpex | ⊢ (𝑅 ∈ 𝑉 → 𝑀 ∈ V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mgpbas.1 | . . 3 ⊢ 𝑀 = (mulGrp‘𝑅) | |
2 | eqid 2177 | . . 3 ⊢ (.r‘𝑅) = (.r‘𝑅) | |
3 | 1, 2 | mgpvalg 13086 | . 2 ⊢ (𝑅 ∈ 𝑉 → 𝑀 = (𝑅 sSet 〈(+g‘ndx), (.r‘𝑅)〉)) |
4 | plusgslid 12565 | . . . . 5 ⊢ (+g = Slot (+g‘ndx) ∧ (+g‘ndx) ∈ ℕ) | |
5 | 4 | simpri 113 | . . . 4 ⊢ (+g‘ndx) ∈ ℕ |
6 | 5 | a1i 9 | . . 3 ⊢ (𝑅 ∈ 𝑉 → (+g‘ndx) ∈ ℕ) |
7 | mulrslid 12584 | . . . 4 ⊢ (.r = Slot (.r‘ndx) ∧ (.r‘ndx) ∈ ℕ) | |
8 | 7 | slotex 12483 | . . 3 ⊢ (𝑅 ∈ 𝑉 → (.r‘𝑅) ∈ V) |
9 | setsex 12488 | . . 3 ⊢ ((𝑅 ∈ 𝑉 ∧ (+g‘ndx) ∈ ℕ ∧ (.r‘𝑅) ∈ V) → (𝑅 sSet 〈(+g‘ndx), (.r‘𝑅)〉) ∈ V) | |
10 | 6, 8, 9 | mpd3an23 1339 | . 2 ⊢ (𝑅 ∈ 𝑉 → (𝑅 sSet 〈(+g‘ndx), (.r‘𝑅)〉) ∈ V) |
11 | 3, 10 | eqeltrd 2254 | 1 ⊢ (𝑅 ∈ 𝑉 → 𝑀 ∈ V) |
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 +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-1re 7904 ax-addrcl 7907 |
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-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-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-inn 8918 df-2 8976 df-3 8977 df-ndx 12459 df-slot 12460 df-sets 12463 df-plusg 12543 df-mulr 12544 df-mgp 13084 |
This theorem is referenced by: mgpress 13094 ringidss 13165 oppr1g 13205 unitgrpbasd 13237 unitgrp 13238 unitlinv 13248 unitrinv 13249 rngidpropdg 13268 |
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