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Mirrors > Home > ILE Home > Th. List > dfur2g | GIF version |
Description: The multiplicative identity is the unique element of the ring that is left- and right-neutral on all elements under multiplication. (Contributed by Mario Carneiro, 10-Jan-2015.) |
Ref | Expression |
---|---|
dfur2.b | ⊢ 𝐵 = (Base‘𝑅) |
dfur2.t | ⊢ · = (.r‘𝑅) |
dfur2.u | ⊢ 1 = (1r‘𝑅) |
Ref | Expression |
---|---|
dfur2g | ⊢ (𝑅 ∈ 𝑉 → 1 = (℩𝑒(𝑒 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐵 ((𝑒 · 𝑥) = 𝑥 ∧ (𝑥 · 𝑒) = 𝑥)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fnmgp 12956 | . . . 4 ⊢ mulGrp Fn V | |
2 | elex 2748 | . . . 4 ⊢ (𝑅 ∈ 𝑉 → 𝑅 ∈ V) | |
3 | funfvex 5528 | . . . . 5 ⊢ ((Fun mulGrp ∧ 𝑅 ∈ dom mulGrp) → (mulGrp‘𝑅) ∈ V) | |
4 | 3 | funfni 5312 | . . . 4 ⊢ ((mulGrp Fn V ∧ 𝑅 ∈ V) → (mulGrp‘𝑅) ∈ V) |
5 | 1, 2, 4 | sylancr 414 | . . 3 ⊢ (𝑅 ∈ 𝑉 → (mulGrp‘𝑅) ∈ V) |
6 | eqid 2177 | . . . 4 ⊢ (Base‘(mulGrp‘𝑅)) = (Base‘(mulGrp‘𝑅)) | |
7 | eqid 2177 | . . . 4 ⊢ (+g‘(mulGrp‘𝑅)) = (+g‘(mulGrp‘𝑅)) | |
8 | eqid 2177 | . . . 4 ⊢ (0g‘(mulGrp‘𝑅)) = (0g‘(mulGrp‘𝑅)) | |
9 | 6, 7, 8 | grpidvalg 12681 | . . 3 ⊢ ((mulGrp‘𝑅) ∈ V → (0g‘(mulGrp‘𝑅)) = (℩𝑒(𝑒 ∈ (Base‘(mulGrp‘𝑅)) ∧ ∀𝑥 ∈ (Base‘(mulGrp‘𝑅))((𝑒(+g‘(mulGrp‘𝑅))𝑥) = 𝑥 ∧ (𝑥(+g‘(mulGrp‘𝑅))𝑒) = 𝑥)))) |
10 | 5, 9 | syl 14 | . 2 ⊢ (𝑅 ∈ 𝑉 → (0g‘(mulGrp‘𝑅)) = (℩𝑒(𝑒 ∈ (Base‘(mulGrp‘𝑅)) ∧ ∀𝑥 ∈ (Base‘(mulGrp‘𝑅))((𝑒(+g‘(mulGrp‘𝑅))𝑥) = 𝑥 ∧ (𝑥(+g‘(mulGrp‘𝑅))𝑒) = 𝑥)))) |
11 | eqid 2177 | . . 3 ⊢ (mulGrp‘𝑅) = (mulGrp‘𝑅) | |
12 | dfur2.u | . . 3 ⊢ 1 = (1r‘𝑅) | |
13 | 11, 12 | ringidvalg 12967 | . 2 ⊢ (𝑅 ∈ 𝑉 → 1 = (0g‘(mulGrp‘𝑅))) |
14 | dfur2.b | . . . . . 6 ⊢ 𝐵 = (Base‘𝑅) | |
15 | 11, 14 | mgpbasg 12960 | . . . . 5 ⊢ (𝑅 ∈ 𝑉 → 𝐵 = (Base‘(mulGrp‘𝑅))) |
16 | 15 | eleq2d 2247 | . . . 4 ⊢ (𝑅 ∈ 𝑉 → (𝑒 ∈ 𝐵 ↔ 𝑒 ∈ (Base‘(mulGrp‘𝑅)))) |
17 | dfur2.t | . . . . . . . . 9 ⊢ · = (.r‘𝑅) | |
18 | 11, 17 | mgpplusgg 12958 | . . . . . . . 8 ⊢ (𝑅 ∈ 𝑉 → · = (+g‘(mulGrp‘𝑅))) |
19 | 18 | oveqd 5886 | . . . . . . 7 ⊢ (𝑅 ∈ 𝑉 → (𝑒 · 𝑥) = (𝑒(+g‘(mulGrp‘𝑅))𝑥)) |
20 | 19 | eqeq1d 2186 | . . . . . 6 ⊢ (𝑅 ∈ 𝑉 → ((𝑒 · 𝑥) = 𝑥 ↔ (𝑒(+g‘(mulGrp‘𝑅))𝑥) = 𝑥)) |
21 | 18 | oveqd 5886 | . . . . . . 7 ⊢ (𝑅 ∈ 𝑉 → (𝑥 · 𝑒) = (𝑥(+g‘(mulGrp‘𝑅))𝑒)) |
22 | 21 | eqeq1d 2186 | . . . . . 6 ⊢ (𝑅 ∈ 𝑉 → ((𝑥 · 𝑒) = 𝑥 ↔ (𝑥(+g‘(mulGrp‘𝑅))𝑒) = 𝑥)) |
23 | 20, 22 | anbi12d 473 | . . . . 5 ⊢ (𝑅 ∈ 𝑉 → (((𝑒 · 𝑥) = 𝑥 ∧ (𝑥 · 𝑒) = 𝑥) ↔ ((𝑒(+g‘(mulGrp‘𝑅))𝑥) = 𝑥 ∧ (𝑥(+g‘(mulGrp‘𝑅))𝑒) = 𝑥))) |
24 | 15, 23 | raleqbidv 2684 | . . . 4 ⊢ (𝑅 ∈ 𝑉 → (∀𝑥 ∈ 𝐵 ((𝑒 · 𝑥) = 𝑥 ∧ (𝑥 · 𝑒) = 𝑥) ↔ ∀𝑥 ∈ (Base‘(mulGrp‘𝑅))((𝑒(+g‘(mulGrp‘𝑅))𝑥) = 𝑥 ∧ (𝑥(+g‘(mulGrp‘𝑅))𝑒) = 𝑥))) |
25 | 16, 24 | anbi12d 473 | . . 3 ⊢ (𝑅 ∈ 𝑉 → ((𝑒 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐵 ((𝑒 · 𝑥) = 𝑥 ∧ (𝑥 · 𝑒) = 𝑥)) ↔ (𝑒 ∈ (Base‘(mulGrp‘𝑅)) ∧ ∀𝑥 ∈ (Base‘(mulGrp‘𝑅))((𝑒(+g‘(mulGrp‘𝑅))𝑥) = 𝑥 ∧ (𝑥(+g‘(mulGrp‘𝑅))𝑒) = 𝑥)))) |
26 | 25 | iotabidv 5195 | . 2 ⊢ (𝑅 ∈ 𝑉 → (℩𝑒(𝑒 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐵 ((𝑒 · 𝑥) = 𝑥 ∧ (𝑥 · 𝑒) = 𝑥))) = (℩𝑒(𝑒 ∈ (Base‘(mulGrp‘𝑅)) ∧ ∀𝑥 ∈ (Base‘(mulGrp‘𝑅))((𝑒(+g‘(mulGrp‘𝑅))𝑥) = 𝑥 ∧ (𝑥(+g‘(mulGrp‘𝑅))𝑒) = 𝑥)))) |
27 | 10, 13, 26 | 3eqtr4d 2220 | 1 ⊢ (𝑅 ∈ 𝑉 → 1 = (℩𝑒(𝑒 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐵 ((𝑒 · 𝑥) = 𝑥 ∧ (𝑥 · 𝑒) = 𝑥)))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 = wceq 1353 ∈ wcel 2148 ∀wral 2455 Vcvv 2737 ℩cio 5172 Fn wfn 5207 ‘cfv 5212 (class class class)co 5869 Basecbs 12442 +gcplusg 12515 .rcmulr 12516 0gc0g 12650 mulGrpcmgp 12954 1rcur 12965 |
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 4118 ax-pow 4171 ax-pr 4206 ax-un 4430 ax-setind 4533 ax-cnex 7890 ax-resscn 7891 ax-1cn 7892 ax-1re 7893 ax-icn 7894 ax-addcl 7895 ax-addrcl 7896 ax-mulcl 7897 ax-addcom 7899 ax-addass 7901 ax-i2m1 7904 ax-0lt1 7905 ax-0id 7907 ax-rnegex 7908 ax-pre-ltirr 7911 ax-pre-ltadd 7915 |
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 3576 df-sn 3597 df-pr 3598 df-op 3600 df-uni 3808 df-int 3843 df-br 4001 df-opab 4062 df-mpt 4063 df-id 4290 df-xp 4629 df-rel 4630 df-cnv 4631 df-co 4632 df-dm 4633 df-rn 4634 df-res 4635 df-ima 4636 df-iota 5174 df-fun 5214 df-fn 5215 df-fv 5220 df-riota 5825 df-ov 5872 df-oprab 5873 df-mpo 5874 df-pnf 7981 df-mnf 7982 df-ltxr 7984 df-inn 8906 df-2 8964 df-3 8965 df-ndx 12445 df-slot 12446 df-base 12448 df-sets 12449 df-plusg 12528 df-mulr 12529 df-0g 12652 df-mgp 12955 df-ur 12966 |
This theorem is referenced by: (None) |
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