![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > ILE Home > Th. List > mgptopng | GIF version |
Description: Topology of the multiplication group. (Contributed by Mario Carneiro, 5-Oct-2015.) |
Ref | Expression |
---|---|
mgpbas.1 | ⊢ 𝑀 = (mulGrp‘𝑅) |
mgptopn.2 | ⊢ 𝐽 = (TopOpen‘𝑅) |
Ref | Expression |
---|---|
mgptopng | ⊢ (𝑅 ∈ 𝑉 → 𝐽 = (TopOpen‘𝑀)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mgptopn.2 | . . 3 ⊢ 𝐽 = (TopOpen‘𝑅) | |
2 | eqid 2177 | . . . . 5 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
3 | eqid 2177 | . . . . 5 ⊢ (TopSet‘𝑅) = (TopSet‘𝑅) | |
4 | 2, 3 | topnvalg 12635 | . . . 4 ⊢ (𝑅 ∈ 𝑉 → ((TopSet‘𝑅) ↾t (Base‘𝑅)) = (TopOpen‘𝑅)) |
5 | mgpbas.1 | . . . . . 6 ⊢ 𝑀 = (mulGrp‘𝑅) | |
6 | 5 | mgptsetg 12952 | . . . . 5 ⊢ (𝑅 ∈ 𝑉 → (TopSet‘𝑅) = (TopSet‘𝑀)) |
7 | 5, 2 | mgpbasg 12950 | . . . . 5 ⊢ (𝑅 ∈ 𝑉 → (Base‘𝑅) = (Base‘𝑀)) |
8 | 6, 7 | oveq12d 5886 | . . . 4 ⊢ (𝑅 ∈ 𝑉 → ((TopSet‘𝑅) ↾t (Base‘𝑅)) = ((TopSet‘𝑀) ↾t (Base‘𝑀))) |
9 | 4, 8 | eqtr3d 2212 | . . 3 ⊢ (𝑅 ∈ 𝑉 → (TopOpen‘𝑅) = ((TopSet‘𝑀) ↾t (Base‘𝑀))) |
10 | 1, 9 | eqtrid 2222 | . 2 ⊢ (𝑅 ∈ 𝑉 → 𝐽 = ((TopSet‘𝑀) ↾t (Base‘𝑀))) |
11 | fnmgp 12946 | . . . . 5 ⊢ mulGrp Fn V | |
12 | elex 2748 | . . . . 5 ⊢ (𝑅 ∈ 𝑉 → 𝑅 ∈ V) | |
13 | funfvex 5527 | . . . . . 6 ⊢ ((Fun mulGrp ∧ 𝑅 ∈ dom mulGrp) → (mulGrp‘𝑅) ∈ V) | |
14 | 13 | funfni 5311 | . . . . 5 ⊢ ((mulGrp Fn V ∧ 𝑅 ∈ V) → (mulGrp‘𝑅) ∈ V) |
15 | 11, 12, 14 | sylancr 414 | . . . 4 ⊢ (𝑅 ∈ 𝑉 → (mulGrp‘𝑅) ∈ V) |
16 | 5, 15 | eqeltrid 2264 | . . 3 ⊢ (𝑅 ∈ 𝑉 → 𝑀 ∈ V) |
17 | eqid 2177 | . . . 4 ⊢ (Base‘𝑀) = (Base‘𝑀) | |
18 | eqid 2177 | . . . 4 ⊢ (TopSet‘𝑀) = (TopSet‘𝑀) | |
19 | 17, 18 | topnvalg 12635 | . . 3 ⊢ (𝑀 ∈ V → ((TopSet‘𝑀) ↾t (Base‘𝑀)) = (TopOpen‘𝑀)) |
20 | 16, 19 | syl 14 | . 2 ⊢ (𝑅 ∈ 𝑉 → ((TopSet‘𝑀) ↾t (Base‘𝑀)) = (TopOpen‘𝑀)) |
21 | 10, 20 | eqtrd 2210 | 1 ⊢ (𝑅 ∈ 𝑉 → 𝐽 = (TopOpen‘𝑀)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1353 ∈ wcel 2148 Vcvv 2737 Fn wfn 5206 ‘cfv 5211 (class class class)co 5868 Basecbs 12432 TopSetcts 12511 ↾t crest 12623 TopOpenctopn 12624 mulGrpcmgp 12944 |
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-coll 4115 ax-sep 4118 ax-pow 4171 ax-pr 4205 ax-un 4429 ax-setind 4532 ax-cnex 7880 ax-resscn 7881 ax-1cn 7882 ax-1re 7883 ax-icn 7884 ax-addcl 7885 ax-addrcl 7886 ax-mulcl 7887 ax-addcom 7889 ax-addass 7891 ax-i2m1 7894 ax-0lt1 7895 ax-0id 7897 ax-rnegex 7898 ax-pre-ltirr 7901 ax-pre-lttrn 7903 ax-pre-ltadd 7905 |
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-reu 2462 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-iun 3886 df-br 4001 df-opab 4062 df-mpt 4063 df-id 4289 df-xp 4628 df-rel 4629 df-cnv 4630 df-co 4631 df-dm 4632 df-rn 4633 df-res 4634 df-ima 4635 df-iota 5173 df-fun 5213 df-fn 5214 df-f 5215 df-f1 5216 df-fo 5217 df-f1o 5218 df-fv 5219 df-ov 5871 df-oprab 5872 df-mpo 5873 df-1st 6134 df-2nd 6135 df-pnf 7971 df-mnf 7972 df-ltxr 7974 df-inn 8896 df-2 8954 df-3 8955 df-4 8956 df-5 8957 df-6 8958 df-7 8959 df-8 8960 df-9 8961 df-ndx 12435 df-slot 12436 df-base 12438 df-sets 12439 df-plusg 12518 df-mulr 12519 df-tset 12524 df-rest 12625 df-topn 12626 df-mgp 12945 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |