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Mirrors > Home > MPE Home > Th. List > tgpmulg | Structured version Visualization version GIF version |
Description: In a topological group, the n-times group multiple function is continuous. (Contributed by Mario Carneiro, 19-Sep-2015.) |
Ref | Expression |
---|---|
tgpmulg.j | ⊢ 𝐽 = (TopOpen‘𝐺) |
tgpmulg.t | ⊢ · = (.g‘𝐺) |
tgpmulg.b | ⊢ 𝐵 = (Base‘𝐺) |
Ref | Expression |
---|---|
tgpmulg | ⊢ ((𝐺 ∈ TopGrp ∧ 𝑁 ∈ ℤ) → (𝑥 ∈ 𝐵 ↦ (𝑁 · 𝑥)) ∈ (𝐽 Cn 𝐽)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tgptmd 24102 | . . . 4 ⊢ (𝐺 ∈ TopGrp → 𝐺 ∈ TopMnd) | |
2 | tgpmulg.j | . . . . 5 ⊢ 𝐽 = (TopOpen‘𝐺) | |
3 | tgpmulg.t | . . . . 5 ⊢ · = (.g‘𝐺) | |
4 | tgpmulg.b | . . . . 5 ⊢ 𝐵 = (Base‘𝐺) | |
5 | 2, 3, 4 | tmdmulg 24115 | . . . 4 ⊢ ((𝐺 ∈ TopMnd ∧ 𝑁 ∈ ℕ0) → (𝑥 ∈ 𝐵 ↦ (𝑁 · 𝑥)) ∈ (𝐽 Cn 𝐽)) |
6 | 1, 5 | sylan 580 | . . 3 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑁 ∈ ℕ0) → (𝑥 ∈ 𝐵 ↦ (𝑁 · 𝑥)) ∈ (𝐽 Cn 𝐽)) |
7 | 6 | adantlr 715 | . 2 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑁 ∈ ℤ) ∧ 𝑁 ∈ ℕ0) → (𝑥 ∈ 𝐵 ↦ (𝑁 · 𝑥)) ∈ (𝐽 Cn 𝐽)) |
8 | simpllr 776 | . . . . . . . . 9 ⊢ ((((𝐺 ∈ TopGrp ∧ 𝑁 ∈ ℤ) ∧ -𝑁 ∈ ℕ) ∧ 𝑥 ∈ 𝐵) → 𝑁 ∈ ℤ) | |
9 | 8 | zcnd 12720 | . . . . . . . 8 ⊢ ((((𝐺 ∈ TopGrp ∧ 𝑁 ∈ ℤ) ∧ -𝑁 ∈ ℕ) ∧ 𝑥 ∈ 𝐵) → 𝑁 ∈ ℂ) |
10 | 9 | negnegd 11608 | . . . . . . 7 ⊢ ((((𝐺 ∈ TopGrp ∧ 𝑁 ∈ ℤ) ∧ -𝑁 ∈ ℕ) ∧ 𝑥 ∈ 𝐵) → --𝑁 = 𝑁) |
11 | 10 | oveq1d 7445 | . . . . . 6 ⊢ ((((𝐺 ∈ TopGrp ∧ 𝑁 ∈ ℤ) ∧ -𝑁 ∈ ℕ) ∧ 𝑥 ∈ 𝐵) → (--𝑁 · 𝑥) = (𝑁 · 𝑥)) |
12 | eqid 2734 | . . . . . . . 8 ⊢ (invg‘𝐺) = (invg‘𝐺) | |
13 | 4, 3, 12 | mulgnegnn 19114 | . . . . . . 7 ⊢ ((-𝑁 ∈ ℕ ∧ 𝑥 ∈ 𝐵) → (--𝑁 · 𝑥) = ((invg‘𝐺)‘(-𝑁 · 𝑥))) |
14 | 13 | adantll 714 | . . . . . 6 ⊢ ((((𝐺 ∈ TopGrp ∧ 𝑁 ∈ ℤ) ∧ -𝑁 ∈ ℕ) ∧ 𝑥 ∈ 𝐵) → (--𝑁 · 𝑥) = ((invg‘𝐺)‘(-𝑁 · 𝑥))) |
15 | 11, 14 | eqtr3d 2776 | . . . . 5 ⊢ ((((𝐺 ∈ TopGrp ∧ 𝑁 ∈ ℤ) ∧ -𝑁 ∈ ℕ) ∧ 𝑥 ∈ 𝐵) → (𝑁 · 𝑥) = ((invg‘𝐺)‘(-𝑁 · 𝑥))) |
16 | 15 | mpteq2dva 5247 | . . . 4 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑁 ∈ ℤ) ∧ -𝑁 ∈ ℕ) → (𝑥 ∈ 𝐵 ↦ (𝑁 · 𝑥)) = (𝑥 ∈ 𝐵 ↦ ((invg‘𝐺)‘(-𝑁 · 𝑥)))) |
17 | 2, 4 | tgptopon 24105 | . . . . . 6 ⊢ (𝐺 ∈ TopGrp → 𝐽 ∈ (TopOn‘𝐵)) |
18 | 17 | ad2antrr 726 | . . . . 5 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑁 ∈ ℤ) ∧ -𝑁 ∈ ℕ) → 𝐽 ∈ (TopOn‘𝐵)) |
19 | 1 | adantr 480 | . . . . . 6 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑁 ∈ ℤ) → 𝐺 ∈ TopMnd) |
20 | nnnn0 12530 | . . . . . 6 ⊢ (-𝑁 ∈ ℕ → -𝑁 ∈ ℕ0) | |
21 | 2, 3, 4 | tmdmulg 24115 | . . . . . 6 ⊢ ((𝐺 ∈ TopMnd ∧ -𝑁 ∈ ℕ0) → (𝑥 ∈ 𝐵 ↦ (-𝑁 · 𝑥)) ∈ (𝐽 Cn 𝐽)) |
22 | 19, 20, 21 | syl2an 596 | . . . . 5 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑁 ∈ ℤ) ∧ -𝑁 ∈ ℕ) → (𝑥 ∈ 𝐵 ↦ (-𝑁 · 𝑥)) ∈ (𝐽 Cn 𝐽)) |
23 | 2, 12 | tgpinv 24108 | . . . . . 6 ⊢ (𝐺 ∈ TopGrp → (invg‘𝐺) ∈ (𝐽 Cn 𝐽)) |
24 | 23 | ad2antrr 726 | . . . . 5 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑁 ∈ ℤ) ∧ -𝑁 ∈ ℕ) → (invg‘𝐺) ∈ (𝐽 Cn 𝐽)) |
25 | 18, 22, 24 | cnmpt11f 23687 | . . . 4 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑁 ∈ ℤ) ∧ -𝑁 ∈ ℕ) → (𝑥 ∈ 𝐵 ↦ ((invg‘𝐺)‘(-𝑁 · 𝑥))) ∈ (𝐽 Cn 𝐽)) |
26 | 16, 25 | eqeltrd 2838 | . . 3 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑁 ∈ ℤ) ∧ -𝑁 ∈ ℕ) → (𝑥 ∈ 𝐵 ↦ (𝑁 · 𝑥)) ∈ (𝐽 Cn 𝐽)) |
27 | 26 | adantrl 716 | . 2 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑁 ∈ ℤ) ∧ (𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ)) → (𝑥 ∈ 𝐵 ↦ (𝑁 · 𝑥)) ∈ (𝐽 Cn 𝐽)) |
28 | simpr 484 | . . 3 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑁 ∈ ℤ) → 𝑁 ∈ ℤ) | |
29 | elznn0nn 12624 | . . 3 ⊢ (𝑁 ∈ ℤ ↔ (𝑁 ∈ ℕ0 ∨ (𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ))) | |
30 | 28, 29 | sylib 218 | . 2 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑁 ∈ ℤ) → (𝑁 ∈ ℕ0 ∨ (𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ))) |
31 | 7, 27, 30 | mpjaodan 960 | 1 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑁 ∈ ℤ) → (𝑥 ∈ 𝐵 ↦ (𝑁 · 𝑥)) ∈ (𝐽 Cn 𝐽)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∨ wo 847 = wceq 1536 ∈ wcel 2105 ↦ cmpt 5230 ‘cfv 6562 (class class class)co 7430 ℝcr 11151 -cneg 11490 ℕcn 12263 ℕ0cn0 12523 ℤcz 12610 Basecbs 17244 TopOpenctopn 17467 invgcminusg 18964 .gcmg 19097 TopOnctopon 22931 Cn ccn 23247 TopMndctmd 24093 TopGrpctgp 24094 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 ax-cnex 11208 ax-resscn 11209 ax-1cn 11210 ax-icn 11211 ax-addcl 11212 ax-addrcl 11213 ax-mulcl 11214 ax-mulrcl 11215 ax-mulcom 11216 ax-addass 11217 ax-mulass 11218 ax-distr 11219 ax-i2m1 11220 ax-1ne0 11221 ax-1rid 11222 ax-rnegex 11223 ax-rrecex 11224 ax-cnre 11225 ax-pre-lttri 11226 ax-pre-lttrn 11227 ax-pre-ltadd 11228 ax-pre-mulgt0 11229 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3377 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-pss 3982 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-iun 4997 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5640 df-we 5642 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-pred 6322 df-ord 6388 df-on 6389 df-lim 6390 df-suc 6391 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-riota 7387 df-ov 7433 df-oprab 7434 df-mpo 7435 df-om 7887 df-1st 8012 df-2nd 8013 df-frecs 8304 df-wrecs 8335 df-recs 8409 df-rdg 8448 df-er 8743 df-map 8866 df-en 8984 df-dom 8985 df-sdom 8986 df-pnf 11294 df-mnf 11295 df-xr 11296 df-ltxr 11297 df-le 11298 df-sub 11491 df-neg 11492 df-nn 12264 df-n0 12524 df-z 12611 df-uz 12876 df-seq 14039 df-0g 17487 df-topgen 17489 df-plusf 18664 df-mgm 18665 df-sgrp 18744 df-mnd 18760 df-mulg 19098 df-top 22915 df-topon 22932 df-topsp 22954 df-bases 22968 df-cn 23250 df-cnp 23251 df-tx 23585 df-tmd 24095 df-tgp 24096 |
This theorem is referenced by: tgpmulg2 24117 |
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