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Mirrors > Home > MPE Home > Th. List > istmd | Structured version Visualization version GIF version |
Description: The predicate "is a topological monoid". (Contributed by Mario Carneiro, 19-Sep-2015.) |
Ref | Expression |
---|---|
istmd.1 | ⊢ 𝐹 = (+𝑓‘𝐺) |
istmd.2 | ⊢ 𝐽 = (TopOpen‘𝐺) |
Ref | Expression |
---|---|
istmd | ⊢ (𝐺 ∈ TopMnd ↔ (𝐺 ∈ Mnd ∧ 𝐺 ∈ TopSp ∧ 𝐹 ∈ ((𝐽 ×t 𝐽) Cn 𝐽))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elin 4171 | . . 3 ⊢ (𝐺 ∈ (Mnd ∩ TopSp) ↔ (𝐺 ∈ Mnd ∧ 𝐺 ∈ TopSp)) | |
2 | 1 | anbi1i 625 | . 2 ⊢ ((𝐺 ∈ (Mnd ∩ TopSp) ∧ 𝐹 ∈ ((𝐽 ×t 𝐽) Cn 𝐽)) ↔ ((𝐺 ∈ Mnd ∧ 𝐺 ∈ TopSp) ∧ 𝐹 ∈ ((𝐽 ×t 𝐽) Cn 𝐽))) |
3 | fvexd 6687 | . . . 4 ⊢ (𝑓 = 𝐺 → (TopOpen‘𝑓) ∈ V) | |
4 | simpl 485 | . . . . . . 7 ⊢ ((𝑓 = 𝐺 ∧ 𝑗 = (TopOpen‘𝑓)) → 𝑓 = 𝐺) | |
5 | 4 | fveq2d 6676 | . . . . . 6 ⊢ ((𝑓 = 𝐺 ∧ 𝑗 = (TopOpen‘𝑓)) → (+𝑓‘𝑓) = (+𝑓‘𝐺)) |
6 | istmd.1 | . . . . . 6 ⊢ 𝐹 = (+𝑓‘𝐺) | |
7 | 5, 6 | syl6eqr 2876 | . . . . 5 ⊢ ((𝑓 = 𝐺 ∧ 𝑗 = (TopOpen‘𝑓)) → (+𝑓‘𝑓) = 𝐹) |
8 | id 22 | . . . . . . . 8 ⊢ (𝑗 = (TopOpen‘𝑓) → 𝑗 = (TopOpen‘𝑓)) | |
9 | fveq2 6672 | . . . . . . . . 9 ⊢ (𝑓 = 𝐺 → (TopOpen‘𝑓) = (TopOpen‘𝐺)) | |
10 | istmd.2 | . . . . . . . . 9 ⊢ 𝐽 = (TopOpen‘𝐺) | |
11 | 9, 10 | syl6eqr 2876 | . . . . . . . 8 ⊢ (𝑓 = 𝐺 → (TopOpen‘𝑓) = 𝐽) |
12 | 8, 11 | sylan9eqr 2880 | . . . . . . 7 ⊢ ((𝑓 = 𝐺 ∧ 𝑗 = (TopOpen‘𝑓)) → 𝑗 = 𝐽) |
13 | 12, 12 | oveq12d 7176 | . . . . . 6 ⊢ ((𝑓 = 𝐺 ∧ 𝑗 = (TopOpen‘𝑓)) → (𝑗 ×t 𝑗) = (𝐽 ×t 𝐽)) |
14 | 13, 12 | oveq12d 7176 | . . . . 5 ⊢ ((𝑓 = 𝐺 ∧ 𝑗 = (TopOpen‘𝑓)) → ((𝑗 ×t 𝑗) Cn 𝑗) = ((𝐽 ×t 𝐽) Cn 𝐽)) |
15 | 7, 14 | eleq12d 2909 | . . . 4 ⊢ ((𝑓 = 𝐺 ∧ 𝑗 = (TopOpen‘𝑓)) → ((+𝑓‘𝑓) ∈ ((𝑗 ×t 𝑗) Cn 𝑗) ↔ 𝐹 ∈ ((𝐽 ×t 𝐽) Cn 𝐽))) |
16 | 3, 15 | sbcied 3816 | . . 3 ⊢ (𝑓 = 𝐺 → ([(TopOpen‘𝑓) / 𝑗](+𝑓‘𝑓) ∈ ((𝑗 ×t 𝑗) Cn 𝑗) ↔ 𝐹 ∈ ((𝐽 ×t 𝐽) Cn 𝐽))) |
17 | df-tmd 22682 | . . 3 ⊢ TopMnd = {𝑓 ∈ (Mnd ∩ TopSp) ∣ [(TopOpen‘𝑓) / 𝑗](+𝑓‘𝑓) ∈ ((𝑗 ×t 𝑗) Cn 𝑗)} | |
18 | 16, 17 | elrab2 3685 | . 2 ⊢ (𝐺 ∈ TopMnd ↔ (𝐺 ∈ (Mnd ∩ TopSp) ∧ 𝐹 ∈ ((𝐽 ×t 𝐽) Cn 𝐽))) |
19 | df-3an 1085 | . 2 ⊢ ((𝐺 ∈ Mnd ∧ 𝐺 ∈ TopSp ∧ 𝐹 ∈ ((𝐽 ×t 𝐽) Cn 𝐽)) ↔ ((𝐺 ∈ Mnd ∧ 𝐺 ∈ TopSp) ∧ 𝐹 ∈ ((𝐽 ×t 𝐽) Cn 𝐽))) | |
20 | 2, 18, 19 | 3bitr4i 305 | 1 ⊢ (𝐺 ∈ TopMnd ↔ (𝐺 ∈ Mnd ∧ 𝐺 ∈ TopSp ∧ 𝐹 ∈ ((𝐽 ×t 𝐽) Cn 𝐽))) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 ∧ wa 398 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 Vcvv 3496 [wsbc 3774 ∩ cin 3937 ‘cfv 6357 (class class class)co 7158 TopOpenctopn 16697 +𝑓cplusf 17851 Mndcmnd 17913 TopSpctps 21542 Cn ccn 21834 ×t ctx 22170 TopMndctmd 22680 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-nul 5212 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-sbc 3775 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-br 5069 df-iota 6316 df-fv 6365 df-ov 7161 df-tmd 22682 |
This theorem is referenced by: tmdmnd 22685 tmdtps 22686 tmdcn 22693 istgp2 22701 oppgtmd 22707 efmndtmd 22711 submtmd 22714 prdstmdd 22734 nrgtrg 23301 mhmhmeotmd 31172 xrge0tmdALT 31191 |
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