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Mirrors > Home > ILE Home > Th. List > topgrpstrd | GIF version |
Description: A constructed topological group is a structure. (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 |
---|---|
topgrpstrd | ⊢ (𝜑 → 𝑊 Struct 〈1, 9〉) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | topgrpfn.w | . 2 ⊢ 𝑊 = {〈(Base‘ndx), 𝐵〉, 〈(+g‘ndx), + 〉, 〈(TopSet‘ndx), 𝐽〉} | |
2 | topgrpfnd.b | . . 3 ⊢ (𝜑 → 𝐵 ∈ 𝑉) | |
3 | topgrpfnd.p | . . 3 ⊢ (𝜑 → + ∈ 𝑊) | |
4 | topgrpfnd.j | . . 3 ⊢ (𝜑 → 𝐽 ∈ 𝑋) | |
5 | 1nn 8933 | . . . 4 ⊢ 1 ∈ ℕ | |
6 | basendx 12520 | . . . 4 ⊢ (Base‘ndx) = 1 | |
7 | 1lt2 9091 | . . . 4 ⊢ 1 < 2 | |
8 | 2nn 9083 | . . . 4 ⊢ 2 ∈ ℕ | |
9 | plusgndx 12571 | . . . 4 ⊢ (+g‘ndx) = 2 | |
10 | 2lt9 9125 | . . . 4 ⊢ 2 < 9 | |
11 | 9nn 9090 | . . . 4 ⊢ 9 ∈ ℕ | |
12 | tsetndx 12647 | . . . 4 ⊢ (TopSet‘ndx) = 9 | |
13 | 5, 6, 7, 8, 9, 10, 11, 12 | strle3g 12570 | . . 3 ⊢ ((𝐵 ∈ 𝑉 ∧ + ∈ 𝑊 ∧ 𝐽 ∈ 𝑋) → {〈(Base‘ndx), 𝐵〉, 〈(+g‘ndx), + 〉, 〈(TopSet‘ndx), 𝐽〉} Struct 〈1, 9〉) |
14 | 2, 3, 4, 13 | syl3anc 1238 | . 2 ⊢ (𝜑 → {〈(Base‘ndx), 𝐵〉, 〈(+g‘ndx), + 〉, 〈(TopSet‘ndx), 𝐽〉} Struct 〈1, 9〉) |
15 | 1, 14 | eqbrtrid 4040 | 1 ⊢ (𝜑 → 𝑊 Struct 〈1, 9〉) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1353 ∈ wcel 2148 {ctp 3596 〈cop 3597 class class class wbr 4005 ‘cfv 5218 1c1 7815 2c2 8973 9c9 8980 Struct cstr 12461 ndxcnx 12462 Basecbs 12465 +gcplusg 12539 TopSetcts 12545 |
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 4123 ax-pow 4176 ax-pr 4211 ax-un 4435 ax-setind 4538 ax-cnex 7905 ax-resscn 7906 ax-1cn 7907 ax-1re 7908 ax-icn 7909 ax-addcl 7910 ax-addrcl 7911 ax-mulcl 7912 ax-addcom 7914 ax-addass 7916 ax-distr 7918 ax-i2m1 7919 ax-0lt1 7920 ax-0id 7922 ax-rnegex 7923 ax-cnre 7925 ax-pre-ltirr 7926 ax-pre-ltwlin 7927 ax-pre-lttrn 7928 ax-pre-apti 7929 ax-pre-ltadd 7930 |
This theorem depends on definitions: df-bi 117 df-3or 979 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 2741 df-sbc 2965 df-dif 3133 df-un 3135 df-in 3137 df-ss 3144 df-nul 3425 df-pw 3579 df-sn 3600 df-pr 3601 df-tp 3602 df-op 3603 df-uni 3812 df-int 3847 df-br 4006 df-opab 4067 df-mpt 4068 df-id 4295 df-xp 4634 df-rel 4635 df-cnv 4636 df-co 4637 df-dm 4638 df-rn 4639 df-res 4640 df-ima 4641 df-iota 5180 df-fun 5220 df-fn 5221 df-f 5222 df-fv 5226 df-riota 5834 df-ov 5881 df-oprab 5882 df-mpo 5883 df-pnf 7997 df-mnf 7998 df-xr 7999 df-ltxr 8000 df-le 8001 df-sub 8133 df-neg 8134 df-inn 8923 df-2 8981 df-3 8982 df-4 8983 df-5 8984 df-6 8985 df-7 8986 df-8 8987 df-9 8988 df-n0 9180 df-z 9257 df-uz 9532 df-fz 10012 df-struct 12467 df-ndx 12468 df-slot 12469 df-base 12471 df-plusg 12552 df-tset 12558 |
This theorem is referenced by: topgrpbasd 12658 topgrpplusgd 12659 topgrptsetd 12660 |
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