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Mirrors > Home > ILE Home > Th. List > topgrptsetd | GIF version |
Description: The topology of a constructed topological group. (Contributed by Mario Carneiro, 29-Aug-2015.) (Revised by Jim Kingdon, 9-Feb-2023.) |
Ref | Expression |
---|---|
topgrpfn.w | ⊢ 𝑊 = {〈(Base‘ndx), 𝐵〉, 〈(+g‘ndx), + 〉, 〈(TopSet‘ndx), 𝐽〉} |
topgrpfnd.b | ⊢ (𝜑 → 𝐵 ∈ 𝑉) |
topgrpfnd.p | ⊢ (𝜑 → + ∈ 𝑊) |
topgrpfnd.j | ⊢ (𝜑 → 𝐽 ∈ 𝑋) |
Ref | Expression |
---|---|
topgrptsetd | ⊢ (𝜑 → 𝐽 = (TopSet‘𝑊)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tsetslid 12664 | . 2 ⊢ (TopSet = Slot (TopSet‘ndx) ∧ (TopSet‘ndx) ∈ ℕ) | |
2 | topgrpfn.w | . . 3 ⊢ 𝑊 = {〈(Base‘ndx), 𝐵〉, 〈(+g‘ndx), + 〉, 〈(TopSet‘ndx), 𝐽〉} | |
3 | topgrpfnd.b | . . 3 ⊢ (𝜑 → 𝐵 ∈ 𝑉) | |
4 | topgrpfnd.p | . . 3 ⊢ (𝜑 → + ∈ 𝑊) | |
5 | topgrpfnd.j | . . 3 ⊢ (𝜑 → 𝐽 ∈ 𝑋) | |
6 | 2, 3, 4, 5 | topgrpstrd 12672 | . 2 ⊢ (𝜑 → 𝑊 Struct 〈1, 9〉) |
7 | 1 | simpri 113 | . . . . 5 ⊢ (TopSet‘ndx) ∈ ℕ |
8 | opexg 4242 | . . . . 5 ⊢ (((TopSet‘ndx) ∈ ℕ ∧ 𝐽 ∈ 𝑋) → 〈(TopSet‘ndx), 𝐽〉 ∈ V) | |
9 | 7, 5, 8 | sylancr 414 | . . . 4 ⊢ (𝜑 → 〈(TopSet‘ndx), 𝐽〉 ∈ V) |
10 | tpid3g 3721 | . . . 4 ⊢ (〈(TopSet‘ndx), 𝐽〉 ∈ V → 〈(TopSet‘ndx), 𝐽〉 ∈ {〈(Base‘ndx), 𝐵〉, 〈(+g‘ndx), + 〉, 〈(TopSet‘ndx), 𝐽〉}) | |
11 | 9, 10 | syl 14 | . . 3 ⊢ (𝜑 → 〈(TopSet‘ndx), 𝐽〉 ∈ {〈(Base‘ndx), 𝐵〉, 〈(+g‘ndx), + 〉, 〈(TopSet‘ndx), 𝐽〉}) |
12 | 11, 2 | eleqtrrdi 2282 | . 2 ⊢ (𝜑 → 〈(TopSet‘ndx), 𝐽〉 ∈ 𝑊) |
13 | 1, 6, 5, 12 | opelstrsl 12591 | 1 ⊢ (𝜑 → 𝐽 = (TopSet‘𝑊)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1363 ∈ wcel 2159 Vcvv 2751 {ctp 3608 〈cop 3609 ‘cfv 5230 1c1 7829 ℕcn 8936 9c9 8994 ndxcnx 12476 Slot cslot 12478 Basecbs 12479 +gcplusg 12554 TopSetcts 12560 |
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 615 ax-in2 616 ax-io 710 ax-5 1457 ax-7 1458 ax-gen 1459 ax-ie1 1503 ax-ie2 1504 ax-8 1514 ax-10 1515 ax-11 1516 ax-i12 1517 ax-bndl 1519 ax-4 1520 ax-17 1536 ax-i9 1540 ax-ial 1544 ax-i5r 1545 ax-13 2161 ax-14 2162 ax-ext 2170 ax-sep 4135 ax-pow 4188 ax-pr 4223 ax-un 4447 ax-setind 4550 ax-cnex 7919 ax-resscn 7920 ax-1cn 7921 ax-1re 7922 ax-icn 7923 ax-addcl 7924 ax-addrcl 7925 ax-mulcl 7926 ax-addcom 7928 ax-addass 7930 ax-distr 7932 ax-i2m1 7933 ax-0lt1 7934 ax-0id 7936 ax-rnegex 7937 ax-cnre 7939 ax-pre-ltirr 7940 ax-pre-ltwlin 7941 ax-pre-lttrn 7942 ax-pre-apti 7943 ax-pre-ltadd 7944 |
This theorem depends on definitions: df-bi 117 df-3or 980 df-3an 981 df-tru 1366 df-fal 1369 df-nf 1471 df-sb 1773 df-eu 2040 df-mo 2041 df-clab 2175 df-cleq 2181 df-clel 2184 df-nfc 2320 df-ne 2360 df-nel 2455 df-ral 2472 df-rex 2473 df-reu 2474 df-rab 2476 df-v 2753 df-sbc 2977 df-dif 3145 df-un 3147 df-in 3149 df-ss 3156 df-nul 3437 df-pw 3591 df-sn 3612 df-pr 3613 df-tp 3614 df-op 3615 df-uni 3824 df-int 3859 df-br 4018 df-opab 4079 df-mpt 4080 df-id 4307 df-xp 4646 df-rel 4647 df-cnv 4648 df-co 4649 df-dm 4650 df-rn 4651 df-res 4652 df-ima 4653 df-iota 5192 df-fun 5232 df-fn 5233 df-f 5234 df-fv 5238 df-riota 5846 df-ov 5893 df-oprab 5894 df-mpo 5895 df-pnf 8011 df-mnf 8012 df-xr 8013 df-ltxr 8014 df-le 8015 df-sub 8147 df-neg 8148 df-inn 8937 df-2 8995 df-3 8996 df-4 8997 df-5 8998 df-6 8999 df-7 9000 df-8 9001 df-9 9002 df-n0 9194 df-z 9271 df-uz 9546 df-fz 10026 df-struct 12481 df-ndx 12482 df-slot 12483 df-base 12485 df-plusg 12567 df-tset 12573 |
This theorem is referenced by: (None) |
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