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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 12944 | . 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 13028 | . 2 ⊢ (𝜑 → 𝑊 Struct 〈1, 9〉) |
| 7 | 1 | simpri 113 | . . . . 5 ⊢ (+g‘ndx) ∈ ℕ |
| 8 | opexg 4272 | . . . . 5 ⊢ (((+g‘ndx) ∈ ℕ ∧ + ∈ 𝑊) → 〈(+g‘ndx), + 〉 ∈ V) | |
| 9 | 7, 4, 8 | sylancr 414 | . . . 4 ⊢ (𝜑 → 〈(+g‘ndx), + 〉 ∈ V) |
| 10 | tpid2g 3747 | . . . 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 2299 | . 2 ⊢ (𝜑 → 〈(+g‘ndx), + 〉 ∈ 𝑊) |
| 13 | 1, 6, 4, 12 | opelstrsl 12946 | 1 ⊢ (𝜑 → + = (+g‘𝑊)) |
| Colors of variables: wff set class |
| Syntax hints: → wi 4 = wceq 1373 ∈ wcel 2176 Vcvv 2772 {ctp 3635 〈cop 3636 ‘cfv 5271 1c1 7926 ℕcn 9036 9c9 9094 ndxcnx 12829 Slot cslot 12831 Basecbs 12832 +gcplusg 12909 TopSetcts 12915 |
| 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 711 ax-5 1470 ax-7 1471 ax-gen 1472 ax-ie1 1516 ax-ie2 1517 ax-8 1527 ax-10 1528 ax-11 1529 ax-i12 1530 ax-bndl 1532 ax-4 1533 ax-17 1549 ax-i9 1553 ax-ial 1557 ax-i5r 1558 ax-13 2178 ax-14 2179 ax-ext 2187 ax-sep 4162 ax-pow 4218 ax-pr 4253 ax-un 4480 ax-setind 4585 ax-cnex 8016 ax-resscn 8017 ax-1cn 8018 ax-1re 8019 ax-icn 8020 ax-addcl 8021 ax-addrcl 8022 ax-mulcl 8023 ax-addcom 8025 ax-addass 8027 ax-distr 8029 ax-i2m1 8030 ax-0lt1 8031 ax-0id 8033 ax-rnegex 8034 ax-cnre 8036 ax-pre-ltirr 8037 ax-pre-ltwlin 8038 ax-pre-lttrn 8039 ax-pre-apti 8040 ax-pre-ltadd 8041 |
| This theorem depends on definitions: df-bi 117 df-3or 982 df-3an 983 df-tru 1376 df-fal 1379 df-nf 1484 df-sb 1786 df-eu 2057 df-mo 2058 df-clab 2192 df-cleq 2198 df-clel 2201 df-nfc 2337 df-ne 2377 df-nel 2472 df-ral 2489 df-rex 2490 df-reu 2491 df-rab 2493 df-v 2774 df-sbc 2999 df-dif 3168 df-un 3170 df-in 3172 df-ss 3179 df-nul 3461 df-pw 3618 df-sn 3639 df-pr 3640 df-tp 3641 df-op 3642 df-uni 3851 df-int 3886 df-br 4045 df-opab 4106 df-mpt 4107 df-id 4340 df-xp 4681 df-rel 4682 df-cnv 4683 df-co 4684 df-dm 4685 df-rn 4686 df-res 4687 df-ima 4688 df-iota 5232 df-fun 5273 df-fn 5274 df-f 5275 df-fv 5279 df-riota 5899 df-ov 5947 df-oprab 5948 df-mpo 5949 df-pnf 8109 df-mnf 8110 df-xr 8111 df-ltxr 8112 df-le 8113 df-sub 8245 df-neg 8246 df-inn 9037 df-2 9095 df-3 9096 df-4 9097 df-5 9098 df-6 9099 df-7 9100 df-8 9101 df-9 9102 df-n0 9296 df-z 9373 df-uz 9649 df-fz 10131 df-struct 12834 df-ndx 12835 df-slot 12836 df-base 12838 df-plusg 12922 df-tset 12928 |
| This theorem is referenced by: (None) |
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