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Mirrors > Home > MPE Home > Th. List > tmdtopon | Structured version Visualization version GIF version |
Description: The topology of a topological monoid. (Contributed by Mario Carneiro, 27-Jun-2014.) (Revised by Mario Carneiro, 13-Aug-2015.) |
Ref | Expression |
---|---|
tgpcn.j | ⊢ 𝐽 = (TopOpen‘𝐺) |
tgptopon.x | ⊢ 𝑋 = (Base‘𝐺) |
Ref | Expression |
---|---|
tmdtopon | ⊢ (𝐺 ∈ TopMnd → 𝐽 ∈ (TopOn‘𝑋)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tmdtps 22927 | . 2 ⊢ (𝐺 ∈ TopMnd → 𝐺 ∈ TopSp) | |
2 | tgptopon.x | . . 3 ⊢ 𝑋 = (Base‘𝐺) | |
3 | tgpcn.j | . . 3 ⊢ 𝐽 = (TopOpen‘𝐺) | |
4 | 2, 3 | istps 21785 | . 2 ⊢ (𝐺 ∈ TopSp ↔ 𝐽 ∈ (TopOn‘𝑋)) |
5 | 1, 4 | sylib 221 | 1 ⊢ (𝐺 ∈ TopMnd → 𝐽 ∈ (TopOn‘𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1543 ∈ wcel 2112 ‘cfv 6358 Basecbs 16666 TopOpenctopn 16880 TopOnctopon 21761 TopSpctps 21783 TopMndctmd 22921 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-ral 3056 df-rex 3057 df-rab 3060 df-v 3400 df-sbc 3684 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-op 4534 df-uni 4806 df-br 5040 df-opab 5102 df-mpt 5121 df-id 5440 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-iota 6316 df-fun 6360 df-fv 6366 df-ov 7194 df-top 21745 df-topon 21762 df-topsp 21784 df-tmd 22923 |
This theorem is referenced by: cnmpt1plusg 22938 cnmpt2plusg 22939 tmdcn2 22940 tmdmulg 22943 tmdgsum 22946 tmdgsum2 22947 oppgtmd 22948 tmdlactcn 22953 submtmd 22955 ghmcnp 22966 prdstgpd 22976 tsmsxp 23006 mhmhmeotmd 31545 |
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