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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 22684 | . . . 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 22697 | . . . 4 ⊢ ((𝐺 ∈ TopMnd ∧ 𝑁 ∈ ℕ0) → (𝑥 ∈ 𝐵 ↦ (𝑁 · 𝑥)) ∈ (𝐽 Cn 𝐽)) |
6 | 1, 5 | sylan 583 | . . 3 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑁 ∈ ℕ0) → (𝑥 ∈ 𝐵 ↦ (𝑁 · 𝑥)) ∈ (𝐽 Cn 𝐽)) |
7 | 6 | adantlr 714 | . 2 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑁 ∈ ℤ) ∧ 𝑁 ∈ ℕ0) → (𝑥 ∈ 𝐵 ↦ (𝑁 · 𝑥)) ∈ (𝐽 Cn 𝐽)) |
8 | simpllr 775 | . . . . . . . . 9 ⊢ ((((𝐺 ∈ TopGrp ∧ 𝑁 ∈ ℤ) ∧ -𝑁 ∈ ℕ) ∧ 𝑥 ∈ 𝐵) → 𝑁 ∈ ℤ) | |
9 | 8 | zcnd 12076 | . . . . . . . 8 ⊢ ((((𝐺 ∈ TopGrp ∧ 𝑁 ∈ ℤ) ∧ -𝑁 ∈ ℕ) ∧ 𝑥 ∈ 𝐵) → 𝑁 ∈ ℂ) |
10 | 9 | negnegd 10977 | . . . . . . 7 ⊢ ((((𝐺 ∈ TopGrp ∧ 𝑁 ∈ ℤ) ∧ -𝑁 ∈ ℕ) ∧ 𝑥 ∈ 𝐵) → --𝑁 = 𝑁) |
11 | 10 | oveq1d 7150 | . . . . . 6 ⊢ ((((𝐺 ∈ TopGrp ∧ 𝑁 ∈ ℤ) ∧ -𝑁 ∈ ℕ) ∧ 𝑥 ∈ 𝐵) → (--𝑁 · 𝑥) = (𝑁 · 𝑥)) |
12 | eqid 2798 | . . . . . . . 8 ⊢ (invg‘𝐺) = (invg‘𝐺) | |
13 | 4, 3, 12 | mulgnegnn 18230 | . . . . . . 7 ⊢ ((-𝑁 ∈ ℕ ∧ 𝑥 ∈ 𝐵) → (--𝑁 · 𝑥) = ((invg‘𝐺)‘(-𝑁 · 𝑥))) |
14 | 13 | adantll 713 | . . . . . 6 ⊢ ((((𝐺 ∈ TopGrp ∧ 𝑁 ∈ ℤ) ∧ -𝑁 ∈ ℕ) ∧ 𝑥 ∈ 𝐵) → (--𝑁 · 𝑥) = ((invg‘𝐺)‘(-𝑁 · 𝑥))) |
15 | 11, 14 | eqtr3d 2835 | . . . . 5 ⊢ ((((𝐺 ∈ TopGrp ∧ 𝑁 ∈ ℤ) ∧ -𝑁 ∈ ℕ) ∧ 𝑥 ∈ 𝐵) → (𝑁 · 𝑥) = ((invg‘𝐺)‘(-𝑁 · 𝑥))) |
16 | 15 | mpteq2dva 5125 | . . . 4 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑁 ∈ ℤ) ∧ -𝑁 ∈ ℕ) → (𝑥 ∈ 𝐵 ↦ (𝑁 · 𝑥)) = (𝑥 ∈ 𝐵 ↦ ((invg‘𝐺)‘(-𝑁 · 𝑥)))) |
17 | 2, 4 | tgptopon 22687 | . . . . . 6 ⊢ (𝐺 ∈ TopGrp → 𝐽 ∈ (TopOn‘𝐵)) |
18 | 17 | ad2antrr 725 | . . . . 5 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑁 ∈ ℤ) ∧ -𝑁 ∈ ℕ) → 𝐽 ∈ (TopOn‘𝐵)) |
19 | 1 | adantr 484 | . . . . . 6 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑁 ∈ ℤ) → 𝐺 ∈ TopMnd) |
20 | nnnn0 11892 | . . . . . 6 ⊢ (-𝑁 ∈ ℕ → -𝑁 ∈ ℕ0) | |
21 | 2, 3, 4 | tmdmulg 22697 | . . . . . 6 ⊢ ((𝐺 ∈ TopMnd ∧ -𝑁 ∈ ℕ0) → (𝑥 ∈ 𝐵 ↦ (-𝑁 · 𝑥)) ∈ (𝐽 Cn 𝐽)) |
22 | 19, 20, 21 | syl2an 598 | . . . . 5 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑁 ∈ ℤ) ∧ -𝑁 ∈ ℕ) → (𝑥 ∈ 𝐵 ↦ (-𝑁 · 𝑥)) ∈ (𝐽 Cn 𝐽)) |
23 | 2, 12 | tgpinv 22690 | . . . . . 6 ⊢ (𝐺 ∈ TopGrp → (invg‘𝐺) ∈ (𝐽 Cn 𝐽)) |
24 | 23 | ad2antrr 725 | . . . . 5 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑁 ∈ ℤ) ∧ -𝑁 ∈ ℕ) → (invg‘𝐺) ∈ (𝐽 Cn 𝐽)) |
25 | 18, 22, 24 | cnmpt11f 22269 | . . . 4 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑁 ∈ ℤ) ∧ -𝑁 ∈ ℕ) → (𝑥 ∈ 𝐵 ↦ ((invg‘𝐺)‘(-𝑁 · 𝑥))) ∈ (𝐽 Cn 𝐽)) |
26 | 16, 25 | eqeltrd 2890 | . . 3 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑁 ∈ ℤ) ∧ -𝑁 ∈ ℕ) → (𝑥 ∈ 𝐵 ↦ (𝑁 · 𝑥)) ∈ (𝐽 Cn 𝐽)) |
27 | 26 | adantrl 715 | . 2 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑁 ∈ ℤ) ∧ (𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ)) → (𝑥 ∈ 𝐵 ↦ (𝑁 · 𝑥)) ∈ (𝐽 Cn 𝐽)) |
28 | simpr 488 | . . 3 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑁 ∈ ℤ) → 𝑁 ∈ ℤ) | |
29 | elznn0nn 11983 | . . 3 ⊢ (𝑁 ∈ ℤ ↔ (𝑁 ∈ ℕ0 ∨ (𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ))) | |
30 | 28, 29 | sylib 221 | . 2 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑁 ∈ ℤ) → (𝑁 ∈ ℕ0 ∨ (𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ))) |
31 | 7, 27, 30 | mpjaodan 956 | 1 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑁 ∈ ℤ) → (𝑥 ∈ 𝐵 ↦ (𝑁 · 𝑥)) ∈ (𝐽 Cn 𝐽)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∨ wo 844 = wceq 1538 ∈ wcel 2111 ↦ cmpt 5110 ‘cfv 6324 (class class class)co 7135 ℝcr 10525 -cneg 10860 ℕcn 11625 ℕ0cn0 11885 ℤcz 11969 Basecbs 16475 TopOpenctopn 16687 invgcminusg 18096 .gcmg 18216 TopOnctopon 21515 Cn ccn 21829 TopMndctmd 22675 TopGrpctgp 22676 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-1st 7671 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-er 8272 df-map 8391 df-en 8493 df-dom 8494 df-sdom 8495 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-nn 11626 df-n0 11886 df-z 11970 df-uz 12232 df-seq 13365 df-0g 16707 df-topgen 16709 df-plusf 17843 df-mgm 17844 df-sgrp 17893 df-mnd 17904 df-mulg 18217 df-top 21499 df-topon 21516 df-topsp 21538 df-bases 21551 df-cn 21832 df-cnp 21833 df-tx 22167 df-tmd 22677 df-tgp 22678 |
This theorem is referenced by: tgpmulg2 22699 |
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