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Mirrors > Home > ILE Home > Th. List > topgrpplusgd | GIF version |
Description: The additive operation 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 |
---|---|
topgrpplusgd | ⊢ (𝜑 → + = (+g‘𝑊)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | plusgslid 12432 | . 2 ⊢ (+g = Slot (+g‘ndx) ∧ (+g‘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 12488 | . 2 ⊢ (𝜑 → 𝑊 Struct 〈1, 9〉) |
7 | 1 | simpri 112 | . . . . 5 ⊢ (+g‘ndx) ∈ ℕ |
8 | opexg 4200 | . . . . 5 ⊢ (((+g‘ndx) ∈ ℕ ∧ + ∈ 𝑊) → 〈(+g‘ndx), + 〉 ∈ V) | |
9 | 7, 4, 8 | sylancr 411 | . . . 4 ⊢ (𝜑 → 〈(+g‘ndx), + 〉 ∈ V) |
10 | tpid2g 3684 | . . . 4 ⊢ (〈(+g‘ndx), + 〉 ∈ V → 〈(+g‘ndx), + 〉 ∈ {〈(Base‘ndx), 𝐵〉, 〈(+g‘ndx), + 〉, 〈(TopSet‘ndx), 𝐽〉}) | |
11 | 9, 10 | syl 14 | . . 3 ⊢ (𝜑 → 〈(+g‘ndx), + 〉 ∈ {〈(Base‘ndx), 𝐵〉, 〈(+g‘ndx), + 〉, 〈(TopSet‘ndx), 𝐽〉}) |
12 | 11, 2 | eleqtrrdi 2258 | . 2 ⊢ (𝜑 → 〈(+g‘ndx), + 〉 ∈ 𝑊) |
13 | 1, 6, 4, 12 | opelstrsl 12433 | 1 ⊢ (𝜑 → + = (+g‘𝑊)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1342 ∈ wcel 2135 Vcvv 2721 {ctp 3572 〈cop 3573 ‘cfv 5182 1c1 7745 ℕcn 8848 9c9 8906 ndxcnx 12334 Slot cslot 12336 Basecbs 12337 +gcplusg 12399 TopSetcts 12405 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-sep 4094 ax-pow 4147 ax-pr 4181 ax-un 4405 ax-setind 4508 ax-cnex 7835 ax-resscn 7836 ax-1cn 7837 ax-1re 7838 ax-icn 7839 ax-addcl 7840 ax-addrcl 7841 ax-mulcl 7842 ax-addcom 7844 ax-addass 7846 ax-distr 7848 ax-i2m1 7849 ax-0lt1 7850 ax-0id 7852 ax-rnegex 7853 ax-cnre 7855 ax-pre-ltirr 7856 ax-pre-ltwlin 7857 ax-pre-lttrn 7858 ax-pre-apti 7859 ax-pre-ltadd 7860 |
This theorem depends on definitions: df-bi 116 df-3or 968 df-3an 969 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ne 2335 df-nel 2430 df-ral 2447 df-rex 2448 df-reu 2449 df-rab 2451 df-v 2723 df-sbc 2947 df-dif 3113 df-un 3115 df-in 3117 df-ss 3124 df-nul 3405 df-pw 3555 df-sn 3576 df-pr 3577 df-tp 3578 df-op 3579 df-uni 3784 df-int 3819 df-br 3977 df-opab 4038 df-mpt 4039 df-id 4265 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-rn 4609 df-res 4610 df-ima 4611 df-iota 5147 df-fun 5184 df-fn 5185 df-f 5186 df-fv 5190 df-riota 5792 df-ov 5839 df-oprab 5840 df-mpo 5841 df-pnf 7926 df-mnf 7927 df-xr 7928 df-ltxr 7929 df-le 7930 df-sub 8062 df-neg 8063 df-inn 8849 df-2 8907 df-3 8908 df-4 8909 df-5 8910 df-6 8911 df-7 8912 df-8 8913 df-9 8914 df-n0 9106 df-z 9183 df-uz 9458 df-fz 9936 df-struct 12339 df-ndx 12340 df-slot 12341 df-base 12343 df-plusg 12412 df-tset 12418 |
This theorem is referenced by: (None) |
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