Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > stoweidlem47 | Structured version Visualization version GIF version |
Description: Subtracting a constant from a real continuous function gives another continuous function. (Contributed by Glauco Siliprandi, 20-Apr-2017.) |
Ref | Expression |
---|---|
stoweidlem47.1 | ⊢ Ⅎ𝑡𝐹 |
stoweidlem47.2 | ⊢ Ⅎ𝑡𝑆 |
stoweidlem47.3 | ⊢ Ⅎ𝑡𝜑 |
stoweidlem47.4 | ⊢ 𝑇 = ∪ 𝐽 |
stoweidlem47.5 | ⊢ 𝐺 = (𝑇 × {-𝑆}) |
stoweidlem47.6 | ⊢ 𝐾 = (topGen‘ran (,)) |
stoweidlem47.7 | ⊢ (𝜑 → 𝐽 ∈ Top) |
stoweidlem47.8 | ⊢ 𝐶 = (𝐽 Cn 𝐾) |
stoweidlem47.9 | ⊢ (𝜑 → 𝐹 ∈ 𝐶) |
stoweidlem47.10 | ⊢ (𝜑 → 𝑆 ∈ ℝ) |
Ref | Expression |
---|---|
stoweidlem47 | ⊢ (𝜑 → (𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡) − 𝑆)) ∈ 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | stoweidlem47.3 | . . 3 ⊢ Ⅎ𝑡𝜑 | |
2 | stoweidlem47.5 | . . . . . . 7 ⊢ 𝐺 = (𝑇 × {-𝑆}) | |
3 | 2 | fveq1i 6673 | . . . . . 6 ⊢ (𝐺‘𝑡) = ((𝑇 × {-𝑆})‘𝑡) |
4 | stoweidlem47.10 | . . . . . . . 8 ⊢ (𝜑 → 𝑆 ∈ ℝ) | |
5 | 4 | renegcld 11069 | . . . . . . 7 ⊢ (𝜑 → -𝑆 ∈ ℝ) |
6 | fvconst2g 6966 | . . . . . . 7 ⊢ ((-𝑆 ∈ ℝ ∧ 𝑡 ∈ 𝑇) → ((𝑇 × {-𝑆})‘𝑡) = -𝑆) | |
7 | 5, 6 | sylan 582 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → ((𝑇 × {-𝑆})‘𝑡) = -𝑆) |
8 | 3, 7 | syl5eq 2870 | . . . . 5 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (𝐺‘𝑡) = -𝑆) |
9 | 8 | oveq2d 7174 | . . . 4 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → ((𝐹‘𝑡) + (𝐺‘𝑡)) = ((𝐹‘𝑡) + -𝑆)) |
10 | stoweidlem47.6 | . . . . . . . 8 ⊢ 𝐾 = (topGen‘ran (,)) | |
11 | stoweidlem47.4 | . . . . . . . 8 ⊢ 𝑇 = ∪ 𝐽 | |
12 | stoweidlem47.8 | . . . . . . . 8 ⊢ 𝐶 = (𝐽 Cn 𝐾) | |
13 | stoweidlem47.9 | . . . . . . . 8 ⊢ (𝜑 → 𝐹 ∈ 𝐶) | |
14 | 10, 11, 12, 13 | fcnre 41289 | . . . . . . 7 ⊢ (𝜑 → 𝐹:𝑇⟶ℝ) |
15 | 14 | ffvelrnda 6853 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (𝐹‘𝑡) ∈ ℝ) |
16 | 15 | recnd 10671 | . . . . 5 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (𝐹‘𝑡) ∈ ℂ) |
17 | 4 | recnd 10671 | . . . . . 6 ⊢ (𝜑 → 𝑆 ∈ ℂ) |
18 | 17 | adantr 483 | . . . . 5 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → 𝑆 ∈ ℂ) |
19 | 16, 18 | negsubd 11005 | . . . 4 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → ((𝐹‘𝑡) + -𝑆) = ((𝐹‘𝑡) − 𝑆)) |
20 | 9, 19 | eqtrd 2858 | . . 3 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → ((𝐹‘𝑡) + (𝐺‘𝑡)) = ((𝐹‘𝑡) − 𝑆)) |
21 | 1, 20 | mpteq2da 5162 | . 2 ⊢ (𝜑 → (𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡) + (𝐺‘𝑡))) = (𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡) − 𝑆))) |
22 | stoweidlem47.1 | . . . 4 ⊢ Ⅎ𝑡𝐹 | |
23 | nfcv 2979 | . . . . . 6 ⊢ Ⅎ𝑡𝑇 | |
24 | stoweidlem47.2 | . . . . . . . 8 ⊢ Ⅎ𝑡𝑆 | |
25 | 24 | nfneg 10884 | . . . . . . 7 ⊢ Ⅎ𝑡-𝑆 |
26 | 25 | nfsn 4645 | . . . . . 6 ⊢ Ⅎ𝑡{-𝑆} |
27 | 23, 26 | nfxp 5590 | . . . . 5 ⊢ Ⅎ𝑡(𝑇 × {-𝑆}) |
28 | 2, 27 | nfcxfr 2977 | . . . 4 ⊢ Ⅎ𝑡𝐺 |
29 | stoweidlem47.7 | . . . . 5 ⊢ (𝜑 → 𝐽 ∈ Top) | |
30 | 11 | a1i 11 | . . . . 5 ⊢ (𝜑 → 𝑇 = ∪ 𝐽) |
31 | istopon 21522 | . . . . 5 ⊢ (𝐽 ∈ (TopOn‘𝑇) ↔ (𝐽 ∈ Top ∧ 𝑇 = ∪ 𝐽)) | |
32 | 29, 30, 31 | sylanbrc 585 | . . . 4 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑇)) |
33 | 13, 12 | eleqtrdi 2925 | . . . 4 ⊢ (𝜑 → 𝐹 ∈ (𝐽 Cn 𝐾)) |
34 | retopon 23374 | . . . . . . . 8 ⊢ (topGen‘ran (,)) ∈ (TopOn‘ℝ) | |
35 | 10, 34 | eqeltri 2911 | . . . . . . 7 ⊢ 𝐾 ∈ (TopOn‘ℝ) |
36 | 35 | a1i 11 | . . . . . 6 ⊢ (𝜑 → 𝐾 ∈ (TopOn‘ℝ)) |
37 | cnconst2 21893 | . . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑇) ∧ 𝐾 ∈ (TopOn‘ℝ) ∧ -𝑆 ∈ ℝ) → (𝑇 × {-𝑆}) ∈ (𝐽 Cn 𝐾)) | |
38 | 32, 36, 5, 37 | syl3anc 1367 | . . . . 5 ⊢ (𝜑 → (𝑇 × {-𝑆}) ∈ (𝐽 Cn 𝐾)) |
39 | 2, 38 | eqeltrid 2919 | . . . 4 ⊢ (𝜑 → 𝐺 ∈ (𝐽 Cn 𝐾)) |
40 | 22, 28, 1, 10, 32, 33, 39 | refsum2cn 41302 | . . 3 ⊢ (𝜑 → (𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡) + (𝐺‘𝑡))) ∈ (𝐽 Cn 𝐾)) |
41 | 40, 12 | eleqtrrdi 2926 | . 2 ⊢ (𝜑 → (𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡) + (𝐺‘𝑡))) ∈ 𝐶) |
42 | 21, 41 | eqeltrrd 2916 | 1 ⊢ (𝜑 → (𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡) − 𝑆)) ∈ 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 Ⅎwnf 1784 ∈ wcel 2114 Ⅎwnfc 2963 {csn 4569 ∪ cuni 4840 ↦ cmpt 5148 × cxp 5555 ran crn 5558 ‘cfv 6357 (class class class)co 7158 ℂcc 10537 ℝcr 10538 + caddc 10542 − cmin 10872 -cneg 10873 (,)cioo 12741 topGenctg 16713 Topctop 21503 TopOnctopon 21520 Cn ccn 21834 |
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-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-inf2 9106 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 ax-pre-sup 10617 ax-addf 10618 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-fal 1550 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-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-iin 4924 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-se 5517 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-isom 6366 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-of 7411 df-om 7583 df-1st 7691 df-2nd 7692 df-supp 7833 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-1o 8104 df-2o 8105 df-oadd 8108 df-er 8291 df-map 8410 df-ixp 8464 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 df-fsupp 8836 df-fi 8877 df-sup 8908 df-inf 8909 df-oi 8976 df-card 9370 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-div 11300 df-nn 11641 df-2 11703 df-3 11704 df-4 11705 df-5 11706 df-6 11707 df-7 11708 df-8 11709 df-9 11710 df-n0 11901 df-z 11985 df-dec 12102 df-uz 12247 df-q 12352 df-rp 12393 df-xneg 12510 df-xadd 12511 df-xmul 12512 df-ioo 12745 df-icc 12748 df-fz 12896 df-fzo 13037 df-seq 13373 df-exp 13433 df-hash 13694 df-cj 14460 df-re 14461 df-im 14462 df-sqrt 14596 df-abs 14597 df-clim 14847 df-sum 15045 df-struct 16487 df-ndx 16488 df-slot 16489 df-base 16491 df-sets 16492 df-ress 16493 df-plusg 16580 df-mulr 16581 df-starv 16582 df-sca 16583 df-vsca 16584 df-ip 16585 df-tset 16586 df-ple 16587 df-ds 16589 df-unif 16590 df-hom 16591 df-cco 16592 df-rest 16698 df-topn 16699 df-0g 16717 df-gsum 16718 df-topgen 16719 df-pt 16720 df-prds 16723 df-xrs 16777 df-qtop 16782 df-imas 16783 df-xps 16785 df-mre 16859 df-mrc 16860 df-acs 16862 df-mgm 17854 df-sgrp 17903 df-mnd 17914 df-submnd 17959 df-mulg 18227 df-cntz 18449 df-cmn 18910 df-psmet 20539 df-xmet 20540 df-met 20541 df-bl 20542 df-mopn 20543 df-cnfld 20548 df-top 21504 df-topon 21521 df-topsp 21543 df-bases 21556 df-cn 21837 df-cnp 21838 df-tx 22172 df-hmeo 22365 df-xms 22932 df-ms 22933 df-tms 22934 |
This theorem is referenced by: stoweidlem62 42354 |
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