negcncf - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > negcncf		Structured version Visualization version GIF version

Theorem negcncf 25081

Description: The negative function is continuous. (Contributed by Mario Carneiro, 30-Dec-2016.) Avoid ax-mulf 11175. (Revised by GG, 16-Mar-2025.)

**Hypothesis**
Ref	Expression
negcncf.1	⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ -𝑥)

**Assertion**
Ref	Expression
negcncf	⊢ (𝐴 ⊆ ℂ → 𝐹 ∈ (𝐴–cn→ℂ))

Distinct variable group: 𝑥,𝐴 Allowed substitution hint: 𝐹(𝑥)

Proof of Theorem negcncf Dummy variables 𝑎 𝑏 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		neg1cn 12198	. . . . . 6 ⊢ -1 ∈ ℂ
2		ssel2 3932	. . . . . 6 ⊢ ((𝐴 ⊆ ℂ ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ ℂ)
3		ovmpot 7571	. . . . . . 7 ⊢ ((-1 ∈ ℂ ∧ 𝑥 ∈ ℂ) → (-1(𝑎 ∈ ℂ, 𝑏 ∈ ℂ ↦ (𝑎 · 𝑏))𝑥) = (-1 · 𝑥))
4	3	eqcomd 2769	. . . . . 6 ⊢ ((-1 ∈ ℂ ∧ 𝑥 ∈ ℂ) → (-1 · 𝑥) = (-1(𝑎 ∈ ℂ, 𝑏 ∈ ℂ ↦ (𝑎 · 𝑏))𝑥))
5	1, 2, 4	sylancr 598	. . . . 5 ⊢ ((𝐴 ⊆ ℂ ∧ 𝑥 ∈ 𝐴) → (-1 · 𝑥) = (-1(𝑎 ∈ ℂ, 𝑏 ∈ ℂ ↦ (𝑎 · 𝑏))𝑥))
6	2	mulm1d 11661	. . . . 5 ⊢ ((𝐴 ⊆ ℂ ∧ 𝑥 ∈ 𝐴) → (-1 · 𝑥) = -𝑥)
7	5, 6	eqtr3d 2800	. . . 4 ⊢ ((𝐴 ⊆ ℂ ∧ 𝑥 ∈ 𝐴) → (-1(𝑎 ∈ ℂ, 𝑏 ∈ ℂ ↦ (𝑎 · 𝑏))𝑥) = -𝑥)
8	7	mpteq2dva 5204	. . 3 ⊢ (𝐴 ⊆ ℂ → (𝑥 ∈ 𝐴 ↦ (-1(𝑎 ∈ ℂ, 𝑏 ∈ ℂ ↦ (𝑎 · 𝑏))𝑥)) = (𝑥 ∈ 𝐴 ↦ -𝑥))
9		negcncf.1	. . 3 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ -𝑥)
10	8, 9	eqtr4di 2816	. 2 ⊢ (𝐴 ⊆ ℂ → (𝑥 ∈ 𝐴 ↦ (-1(𝑎 ∈ ℂ, 𝑏 ∈ ℂ ↦ (𝑎 · 𝑏))𝑥)) = 𝐹)
11		eqid 2763	. . 3 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
12	11	mpomulcn 25026	. . . 4 ⊢ (𝑎 ∈ ℂ, 𝑏 ∈ ℂ ↦ (𝑎 · 𝑏)) ∈ (((TopOpen‘ℂ_fld) ×_t (TopOpen‘ℂ_fld)) Cn (TopOpen‘ℂ_fld))
13	12	a1i 11	. . 3 ⊢ (𝐴 ⊆ ℂ → (𝑎 ∈ ℂ, 𝑏 ∈ ℂ ↦ (𝑎 · 𝑏)) ∈ (((TopOpen‘ℂ_fld) ×_t (TopOpen‘ℂ_fld)) Cn (TopOpen‘ℂ_fld)))
14		ssid 3959	. . . 4 ⊢ ℂ ⊆ ℂ
15		cncfmptc 25071	. . . 4 ⊢ ((-1 ∈ ℂ ∧ 𝐴 ⊆ ℂ ∧ ℂ ⊆ ℂ) → (𝑥 ∈ 𝐴 ↦ -1) ∈ (𝐴–cn→ℂ))
16	1, 14, 15	mp3an13 1481	. . 3 ⊢ (𝐴 ⊆ ℂ → (𝑥 ∈ 𝐴 ↦ -1) ∈ (𝐴–cn→ℂ))
17		cncfmptid 25072	. . . 4 ⊢ ((𝐴 ⊆ ℂ ∧ ℂ ⊆ ℂ) → (𝑥 ∈ 𝐴 ↦ 𝑥) ∈ (𝐴–cn→ℂ))
18	14, 17	mpan2 703	. . 3 ⊢ (𝐴 ⊆ ℂ → (𝑥 ∈ 𝐴 ↦ 𝑥) ∈ (𝐴–cn→ℂ))
19	11, 13, 16, 18	cncfmpt2f 25074	. 2 ⊢ (𝐴 ⊆ ℂ → (𝑥 ∈ 𝐴 ↦ (-1(𝑎 ∈ ℂ, 𝑏 ∈ ℂ ↦ (𝑎 · 𝑏))𝑥)) ∈ (𝐴–cn→ℂ))
20	10, 19	eqeltrrd 2864	1 ⊢ (𝐴 ⊆ ℂ → 𝐹 ∈ (𝐴–cn→ℂ))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 400 = wceq 1570 ∈ wcel 2143 ⊆ wss 3905 ↦ cmpt 5192 ‘cfv 6536 (class class class)co 7410 ∈ cmpo 7412 ℂcc 11093 1c1 11096 · cmul 11100 -cneg 11437 TopOpenctopn 17469 ℂ_fldccnfld 21522 Cn ccn 23381 ×_t ctx 23717 –cn→ccncf 25035

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1825 ax-4 1839 ax-5 1940 ax-6 1997 ax-7 2038 ax-8 2145 ax-9 2153 ax-10 2176 ax-11 2192 ax-12 2213 ax-ext 2735 ax-rep 5238 ax-sep 5257 ax-nul 5269 ax-pow 5336 ax-pr 5404 ax-un 7732 ax-cnex 11151 ax-resscn 11152 ax-1cn 11153 ax-icn 11154 ax-addcl 11155 ax-addrcl 11156 ax-mulcl 11157 ax-mulrcl 11158 ax-mulcom 11159 ax-addass 11160 ax-mulass 11161 ax-distr 11162 ax-i2m1 11163 ax-1ne0 11164 ax-1rid 11165 ax-rnegex 11166 ax-rrecex 11167 ax-cnre 11168 ax-pre-lttri 11169 ax-pre-lttrn 11170 ax-pre-ltadd 11171 ax-pre-mulgt0 11172 ax-pre-sup 11173

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1810 df-nf 1814 df-sb 2097 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2838 df-nfc 2912 df-ne 2959 df-nel 3065 df-ral 3080 df-rex 3090 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3457 df-sbc 3745 df-csb 3854 df-dif 3908 df-un 3910 df-in 3912 df-ss 3922 df-pss 3925 df-nul 4287 df-if 4488 df-pw 4564 df-sn 4590 df-pr 4592 df-tp 4594 df-op 4596 df-uni 4873 df-int 4913 df-iun 4958 df-iin 4959 df-br 5110 df-opab 5174 df-mpt 5193 df-tr 5219 df-id 5556 df-eprel 5561 df-po 5569 df-so 5570 df-fr 5614 df-se 5615 df-we 5616 df-xp 5667 df-rel 5668 df-cnv 5669 df-co 5670 df-dm 5671 df-rn 5672 df-res 5673 df-ima 5674 df-pred 6302 df-ord 6363 df-on 6364 df-lim 6365 df-suc 6366 df-iota 6492 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-isom 6545 df-riota 7367 df-ov 7413 df-oprab 7414 df-mpo 7415 df-of 7674 df-om 7859 df-1st 7982 df-2nd 7983 df-supp 8153 df-frecs 8274 df-wrecs 8305 df-recs 8354 df-rdg 8393 df-1o 8449 df-2o 8450 df-er 8690 df-map 8822 df-ixp 8892 df-en 8940 df-dom 8941 df-sdom 8942 df-fin 8943 df-fsupp 9318 df-fi 9367 df-sup 9398 df-inf 9399 df-oi 9468 df-card 9921 df-pnf 11240 df-mnf 11241 df-xr 11242 df-ltxr 11243 df-le 11244 df-sub 11438 df-neg 11439 df-div 11867 df-nn 12229 df-2 12298 df-3 12299 df-4 12300 df-5 12301 df-6 12302 df-7 12303 df-8 12304 df-9 12305 df-n0 12500 df-z 12587 df-dec 12707 df-uz 12858 df-q 12968 df-rp 13012 df-xneg 13132 df-xadd 13133 df-xmul 13134 df-icc 13374 df-fz 13531 df-fzo 13679 df-seq 14034 df-exp 14094 df-hash 14363 df-cj 15146 df-re 15147 df-im 15148 df-sqrt 15282 df-abs 15283 df-struct 17202 df-sets 17219 df-slot 17237 df-ndx 17249 df-base 17265 df-ress 17286 df-plusg 17318 df-mulr 17319 df-starv 17320 df-sca 17321 df-vsca 17322 df-ip 17323 df-tset 17324 df-ple 17325 df-ds 17327 df-unif 17328 df-hom 17329 df-cco 17330 df-rest 17470 df-topn 17471 df-0g 17489 df-gsum 17490 df-topgen 17491 df-pt 17492 df-prds 17495 df-xrs 17551 df-qtop 17556 df-imas 17557 df-xps 17559 df-mre 17633 df-mrc 17634 df-acs 17636 df-mgm 18693 df-sgrp 18772 df-mnd 18788 df-submnd 18837 df-mulg 19129 df-cntz 19382 df-cmn 19847 df-psmet 21514 df-xmet 21515 df-met 21516 df-bl 21517 df-mopn 21518 df-cnfld 21523 df-top 23051 df-topon 23068 df-topsp 23090 df-bases 23103 df-cn 23384 df-cnp 23385 df-tx 23719 df-hmeo 23912 df-xms 24477 df-ms 24478 df-tms 24479 df-cncf 25037

This theorem is referenced by: negfcncf 25082 lhop2 26174 etransclem18 46966 etransclem46 46994

W3C validator