Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > raddcn | Structured version Visualization version GIF version |
Description: Addition in the real numbers is a continuous function. (Contributed by Thierry Arnoux, 23-May-2017.) |
Ref | Expression |
---|---|
raddcn.1 | ⊢ 𝐽 = (topGen‘ran (,)) |
Ref | Expression |
---|---|
raddcn | ⊢ (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ (𝑥 + 𝑦)) ∈ ((𝐽 ×t 𝐽) Cn 𝐽) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2821 | . . . . . 6 ⊢ (TopOpen‘ℂfld) = (TopOpen‘ℂfld) | |
2 | 1 | addcn 23472 | . . . . 5 ⊢ + ∈ (((TopOpen‘ℂfld) ×t (TopOpen‘ℂfld)) Cn (TopOpen‘ℂfld)) |
3 | ax-resscn 10593 | . . . . . 6 ⊢ ℝ ⊆ ℂ | |
4 | xpss12 5569 | . . . . . 6 ⊢ ((ℝ ⊆ ℂ ∧ ℝ ⊆ ℂ) → (ℝ × ℝ) ⊆ (ℂ × ℂ)) | |
5 | 3, 3, 4 | mp2an 690 | . . . . 5 ⊢ (ℝ × ℝ) ⊆ (ℂ × ℂ) |
6 | 1 | cnfldtop 23391 | . . . . . . 7 ⊢ (TopOpen‘ℂfld) ∈ Top |
7 | 1 | cnfldtopon 23390 | . . . . . . . 8 ⊢ (TopOpen‘ℂfld) ∈ (TopOn‘ℂ) |
8 | 7 | toponunii 21523 | . . . . . . 7 ⊢ ℂ = ∪ (TopOpen‘ℂfld) |
9 | 6, 6, 8, 8 | txunii 22200 | . . . . . 6 ⊢ (ℂ × ℂ) = ∪ ((TopOpen‘ℂfld) ×t (TopOpen‘ℂfld)) |
10 | 9 | cnrest 21892 | . . . . 5 ⊢ (( + ∈ (((TopOpen‘ℂfld) ×t (TopOpen‘ℂfld)) Cn (TopOpen‘ℂfld)) ∧ (ℝ × ℝ) ⊆ (ℂ × ℂ)) → ( + ↾ (ℝ × ℝ)) ∈ ((((TopOpen‘ℂfld) ×t (TopOpen‘ℂfld)) ↾t (ℝ × ℝ)) Cn (TopOpen‘ℂfld))) |
11 | 2, 5, 10 | mp2an 690 | . . . 4 ⊢ ( + ↾ (ℝ × ℝ)) ∈ ((((TopOpen‘ℂfld) ×t (TopOpen‘ℂfld)) ↾t (ℝ × ℝ)) Cn (TopOpen‘ℂfld)) |
12 | reex 10627 | . . . . . . 7 ⊢ ℝ ∈ V | |
13 | txrest 22238 | . . . . . . 7 ⊢ ((((TopOpen‘ℂfld) ∈ Top ∧ (TopOpen‘ℂfld) ∈ Top) ∧ (ℝ ∈ V ∧ ℝ ∈ V)) → (((TopOpen‘ℂfld) ×t (TopOpen‘ℂfld)) ↾t (ℝ × ℝ)) = (((TopOpen‘ℂfld) ↾t ℝ) ×t ((TopOpen‘ℂfld) ↾t ℝ))) | |
14 | 6, 6, 12, 12, 13 | mp4an 691 | . . . . . 6 ⊢ (((TopOpen‘ℂfld) ×t (TopOpen‘ℂfld)) ↾t (ℝ × ℝ)) = (((TopOpen‘ℂfld) ↾t ℝ) ×t ((TopOpen‘ℂfld) ↾t ℝ)) |
15 | raddcn.1 | . . . . . . . 8 ⊢ 𝐽 = (topGen‘ran (,)) | |
16 | 1 | tgioo2 23410 | . . . . . . . 8 ⊢ (topGen‘ran (,)) = ((TopOpen‘ℂfld) ↾t ℝ) |
17 | 15, 16 | eqtr2i 2845 | . . . . . . 7 ⊢ ((TopOpen‘ℂfld) ↾t ℝ) = 𝐽 |
18 | 17, 17 | oveq12i 7167 | . . . . . 6 ⊢ (((TopOpen‘ℂfld) ↾t ℝ) ×t ((TopOpen‘ℂfld) ↾t ℝ)) = (𝐽 ×t 𝐽) |
19 | 14, 18 | eqtri 2844 | . . . . 5 ⊢ (((TopOpen‘ℂfld) ×t (TopOpen‘ℂfld)) ↾t (ℝ × ℝ)) = (𝐽 ×t 𝐽) |
20 | 19 | oveq1i 7165 | . . . 4 ⊢ ((((TopOpen‘ℂfld) ×t (TopOpen‘ℂfld)) ↾t (ℝ × ℝ)) Cn (TopOpen‘ℂfld)) = ((𝐽 ×t 𝐽) Cn (TopOpen‘ℂfld)) |
21 | 11, 20 | eleqtri 2911 | . . 3 ⊢ ( + ↾ (ℝ × ℝ)) ∈ ((𝐽 ×t 𝐽) Cn (TopOpen‘ℂfld)) |
22 | ax-addf 10615 | . . . . . . . . . 10 ⊢ + :(ℂ × ℂ)⟶ℂ | |
23 | ffn 6513 | . . . . . . . . . 10 ⊢ ( + :(ℂ × ℂ)⟶ℂ → + Fn (ℂ × ℂ)) | |
24 | 22, 23 | ax-mp 5 | . . . . . . . . 9 ⊢ + Fn (ℂ × ℂ) |
25 | fnssres 6469 | . . . . . . . . 9 ⊢ (( + Fn (ℂ × ℂ) ∧ (ℝ × ℝ) ⊆ (ℂ × ℂ)) → ( + ↾ (ℝ × ℝ)) Fn (ℝ × ℝ)) | |
26 | 24, 5, 25 | mp2an 690 | . . . . . . . 8 ⊢ ( + ↾ (ℝ × ℝ)) Fn (ℝ × ℝ) |
27 | fnov 7281 | . . . . . . . 8 ⊢ (( + ↾ (ℝ × ℝ)) Fn (ℝ × ℝ) ↔ ( + ↾ (ℝ × ℝ)) = (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ (𝑥( + ↾ (ℝ × ℝ))𝑦))) | |
28 | 26, 27 | mpbi 232 | . . . . . . 7 ⊢ ( + ↾ (ℝ × ℝ)) = (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ (𝑥( + ↾ (ℝ × ℝ))𝑦)) |
29 | ovres 7313 | . . . . . . . 8 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (𝑥( + ↾ (ℝ × ℝ))𝑦) = (𝑥 + 𝑦)) | |
30 | 29 | mpoeq3ia 7231 | . . . . . . 7 ⊢ (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ (𝑥( + ↾ (ℝ × ℝ))𝑦)) = (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ (𝑥 + 𝑦)) |
31 | 28, 30 | eqtri 2844 | . . . . . 6 ⊢ ( + ↾ (ℝ × ℝ)) = (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ (𝑥 + 𝑦)) |
32 | 31 | rneqi 5806 | . . . . 5 ⊢ ran ( + ↾ (ℝ × ℝ)) = ran (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ (𝑥 + 𝑦)) |
33 | readdcl 10619 | . . . . . . 7 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (𝑥 + 𝑦) ∈ ℝ) | |
34 | 33 | rgen2 3203 | . . . . . 6 ⊢ ∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ (𝑥 + 𝑦) ∈ ℝ |
35 | eqid 2821 | . . . . . . 7 ⊢ (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ (𝑥 + 𝑦)) = (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ (𝑥 + 𝑦)) | |
36 | 35 | rnmposs 30418 | . . . . . 6 ⊢ (∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ (𝑥 + 𝑦) ∈ ℝ → ran (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ (𝑥 + 𝑦)) ⊆ ℝ) |
37 | 34, 36 | ax-mp 5 | . . . . 5 ⊢ ran (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ (𝑥 + 𝑦)) ⊆ ℝ |
38 | 32, 37 | eqsstri 4000 | . . . 4 ⊢ ran ( + ↾ (ℝ × ℝ)) ⊆ ℝ |
39 | cnrest2 21893 | . . . 4 ⊢ (((TopOpen‘ℂfld) ∈ (TopOn‘ℂ) ∧ ran ( + ↾ (ℝ × ℝ)) ⊆ ℝ ∧ ℝ ⊆ ℂ) → (( + ↾ (ℝ × ℝ)) ∈ ((𝐽 ×t 𝐽) Cn (TopOpen‘ℂfld)) ↔ ( + ↾ (ℝ × ℝ)) ∈ ((𝐽 ×t 𝐽) Cn ((TopOpen‘ℂfld) ↾t ℝ)))) | |
40 | 7, 38, 3, 39 | mp3an 1457 | . . 3 ⊢ (( + ↾ (ℝ × ℝ)) ∈ ((𝐽 ×t 𝐽) Cn (TopOpen‘ℂfld)) ↔ ( + ↾ (ℝ × ℝ)) ∈ ((𝐽 ×t 𝐽) Cn ((TopOpen‘ℂfld) ↾t ℝ))) |
41 | 21, 40 | mpbi 232 | . 2 ⊢ ( + ↾ (ℝ × ℝ)) ∈ ((𝐽 ×t 𝐽) Cn ((TopOpen‘ℂfld) ↾t ℝ)) |
42 | 17 | oveq2i 7166 | . 2 ⊢ ((𝐽 ×t 𝐽) Cn ((TopOpen‘ℂfld) ↾t ℝ)) = ((𝐽 ×t 𝐽) Cn 𝐽) |
43 | 41, 31, 42 | 3eltr3i 2925 | 1 ⊢ (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ (𝑥 + 𝑦)) ∈ ((𝐽 ×t 𝐽) Cn 𝐽) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 = wceq 1533 ∈ wcel 2110 ∀wral 3138 Vcvv 3494 ⊆ wss 3935 × cxp 5552 ran crn 5555 ↾ cres 5556 Fn wfn 6349 ⟶wf 6350 ‘cfv 6354 (class class class)co 7155 ∈ cmpo 7157 ℂcc 10534 ℝcr 10535 + caddc 10539 (,)cioo 12737 ↾t crest 16693 TopOpenctopn 16694 topGenctg 16710 ℂfldccnfld 20544 Topctop 21500 TopOnctopon 21517 Cn ccn 21831 ×t ctx 22167 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5189 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460 ax-cnex 10592 ax-resscn 10593 ax-1cn 10594 ax-icn 10595 ax-addcl 10596 ax-addrcl 10597 ax-mulcl 10598 ax-mulrcl 10599 ax-mulcom 10600 ax-addass 10601 ax-mulass 10602 ax-distr 10603 ax-i2m1 10604 ax-1ne0 10605 ax-1rid 10606 ax-rnegex 10607 ax-rrecex 10608 ax-cnre 10609 ax-pre-lttri 10610 ax-pre-lttrn 10611 ax-pre-ltadd 10612 ax-pre-mulgt0 10613 ax-pre-sup 10614 ax-addf 10615 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4567 df-pr 4569 df-tp 4571 df-op 4573 df-uni 4838 df-int 4876 df-iun 4920 df-iin 4921 df-br 5066 df-opab 5128 df-mpt 5146 df-tr 5172 df-id 5459 df-eprel 5464 df-po 5473 df-so 5474 df-fr 5513 df-se 5514 df-we 5515 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-pred 6147 df-ord 6193 df-on 6194 df-lim 6195 df-suc 6196 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-f1 6359 df-fo 6360 df-f1o 6361 df-fv 6362 df-isom 6363 df-riota 7113 df-ov 7158 df-oprab 7159 df-mpo 7160 df-of 7408 df-om 7580 df-1st 7688 df-2nd 7689 df-supp 7830 df-wrecs 7946 df-recs 8007 df-rdg 8045 df-1o 8101 df-2o 8102 df-oadd 8105 df-er 8288 df-map 8407 df-ixp 8461 df-en 8509 df-dom 8510 df-sdom 8511 df-fin 8512 df-fsupp 8833 df-fi 8874 df-sup 8905 df-inf 8906 df-oi 8973 df-card 9367 df-pnf 10676 df-mnf 10677 df-xr 10678 df-ltxr 10679 df-le 10680 df-sub 10871 df-neg 10872 df-div 11297 df-nn 11638 df-2 11699 df-3 11700 df-4 11701 df-5 11702 df-6 11703 df-7 11704 df-8 11705 df-9 11706 df-n0 11897 df-z 11981 df-dec 12098 df-uz 12243 df-q 12348 df-rp 12389 df-xneg 12506 df-xadd 12507 df-xmul 12508 df-ioo 12741 df-icc 12744 df-fz 12892 df-fzo 13033 df-seq 13369 df-exp 13429 df-hash 13690 df-cj 14457 df-re 14458 df-im 14459 df-sqrt 14593 df-abs 14594 df-struct 16484 df-ndx 16485 df-slot 16486 df-base 16488 df-sets 16489 df-ress 16490 df-plusg 16577 df-mulr 16578 df-starv 16579 df-sca 16580 df-vsca 16581 df-ip 16582 df-tset 16583 df-ple 16584 df-ds 16586 df-unif 16587 df-hom 16588 df-cco 16589 df-rest 16695 df-topn 16696 df-0g 16714 df-gsum 16715 df-topgen 16716 df-pt 16717 df-prds 16720 df-xrs 16774 df-qtop 16779 df-imas 16780 df-xps 16782 df-mre 16856 df-mrc 16857 df-acs 16859 df-mgm 17851 df-sgrp 17900 df-mnd 17911 df-submnd 17956 df-mulg 18224 df-cntz 18446 df-cmn 18907 df-psmet 20536 df-xmet 20537 df-met 20538 df-bl 20539 df-mopn 20540 df-cnfld 20545 df-top 21501 df-topon 21518 df-topsp 21540 df-bases 21553 df-cn 21834 df-cnp 21835 df-tx 22169 df-hmeo 22362 df-xms 22929 df-ms 22930 df-tms 22931 |
This theorem is referenced by: rrvadd 31710 |
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