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Theorem negcncf 24967

Description: The negative function is continuous. (Contributed by Mario Carneiro, 30-Dec-2016.) Avoid ax-mulf 11264. (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 12407	. . . . . 6 ⊢ -1 ∈ ℂ
2		ssel2 4003	. . . . . 6 ⊢ ((𝐴 ⊆ ℂ ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ ℂ)
3		ovmpot 7611	. . . . . . 7 ⊢ ((-1 ∈ ℂ ∧ 𝑥 ∈ ℂ) → (-1(𝑎 ∈ ℂ, 𝑏 ∈ ℂ ↦ (𝑎 · 𝑏))𝑥) = (-1 · 𝑥))
4	3	eqcomd 2746	. . . . . 6 ⊢ ((-1 ∈ ℂ ∧ 𝑥 ∈ ℂ) → (-1 · 𝑥) = (-1(𝑎 ∈ ℂ, 𝑏 ∈ ℂ ↦ (𝑎 · 𝑏))𝑥))
5	1, 2, 4	sylancr 586	. . . . 5 ⊢ ((𝐴 ⊆ ℂ ∧ 𝑥 ∈ 𝐴) → (-1 · 𝑥) = (-1(𝑎 ∈ ℂ, 𝑏 ∈ ℂ ↦ (𝑎 · 𝑏))𝑥))
6	2	mulm1d 11742	. . . . 5 ⊢ ((𝐴 ⊆ ℂ ∧ 𝑥 ∈ 𝐴) → (-1 · 𝑥) = -𝑥)
7	5, 6	eqtr3d 2782	. . . 4 ⊢ ((𝐴 ⊆ ℂ ∧ 𝑥 ∈ 𝐴) → (-1(𝑎 ∈ ℂ, 𝑏 ∈ ℂ ↦ (𝑎 · 𝑏))𝑥) = -𝑥)
8	7	mpteq2dva 5266	. . 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 2798	. 2 ⊢ (𝐴 ⊆ ℂ → (𝑥 ∈ 𝐴 ↦ (-1(𝑎 ∈ ℂ, 𝑏 ∈ ℂ ↦ (𝑎 · 𝑏))𝑥)) = 𝐹)
11		eqid 2740	. . 3 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
12	11	mpomulcn 24910	. . . 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 4031	. . . 4 ⊢ ℂ ⊆ ℂ
15		cncfmptc 24957	. . . 4 ⊢ ((-1 ∈ ℂ ∧ 𝐴 ⊆ ℂ ∧ ℂ ⊆ ℂ) → (𝑥 ∈ 𝐴 ↦ -1) ∈ (𝐴–cn→ℂ))
16	1, 14, 15	mp3an13 1452	. . 3 ⊢ (𝐴 ⊆ ℂ → (𝑥 ∈ 𝐴 ↦ -1) ∈ (𝐴–cn→ℂ))
17		cncfmptid 24958	. . . 4 ⊢ ((𝐴 ⊆ ℂ ∧ ℂ ⊆ ℂ) → (𝑥 ∈ 𝐴 ↦ 𝑥) ∈ (𝐴–cn→ℂ))
18	14, 17	mpan2 690	. . 3 ⊢ (𝐴 ⊆ ℂ → (𝑥 ∈ 𝐴 ↦ 𝑥) ∈ (𝐴–cn→ℂ))
19	11, 13, 16, 18	cncfmpt2f 24960	. 2 ⊢ (𝐴 ⊆ ℂ → (𝑥 ∈ 𝐴 ↦ (-1(𝑎 ∈ ℂ, 𝑏 ∈ ℂ ↦ (𝑎 · 𝑏))𝑥)) ∈ (𝐴–cn→ℂ))
20	10, 19	eqeltrrd 2845	1 ⊢ (𝐴 ⊆ ℂ → 𝐹 ∈ (𝐴–cn→ℂ))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1537 ∈ wcel 2108 ⊆ wss 3976 ↦ cmpt 5249 ‘cfv 6573 (class class class)co 7448 ∈ cmpo 7450 ℂcc 11182 1c1 11185 · cmul 11189 -cneg 11521 TopOpenctopn 17481 ℂ_fldccnfld 21387 Cn ccn 23253 ×_t ctx 23589 –cn→ccncf 24921

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-rep 5303 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7770 ax-cnex 11240 ax-resscn 11241 ax-1cn 11242 ax-icn 11243 ax-addcl 11244 ax-addrcl 11245 ax-mulcl 11246 ax-mulrcl 11247 ax-mulcom 11248 ax-addass 11249 ax-mulass 11250 ax-distr 11251 ax-i2m1 11252 ax-1ne0 11253 ax-1rid 11254 ax-rnegex 11255 ax-rrecex 11256 ax-cnre 11257 ax-pre-lttri 11258 ax-pre-lttrn 11259 ax-pre-ltadd 11260 ax-pre-mulgt0 11261 ax-pre-sup 11262

This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-nel 3053 df-ral 3068 df-rex 3077 df-rmo 3388 df-reu 3389 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-pss 3996 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-tp 4653 df-op 4655 df-uni 4932 df-int 4971 df-iun 5017 df-iin 5018 df-br 5167 df-opab 5229 df-mpt 5250 df-tr 5284 df-id 5593 df-eprel 5599 df-po 5607 df-so 5608 df-fr 5652 df-se 5653 df-we 5654 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-pred 6332 df-ord 6398 df-on 6399 df-lim 6400 df-suc 6401 df-iota 6525 df-fun 6575 df-fn 6576 df-f 6577 df-f1 6578 df-fo 6579 df-f1o 6580 df-fv 6581 df-isom 6582 df-riota 7404 df-ov 7451 df-oprab 7452 df-mpo 7453 df-of 7714 df-om 7904 df-1st 8030 df-2nd 8031 df-supp 8202 df-frecs 8322 df-wrecs 8353 df-recs 8427 df-rdg 8466 df-1o 8522 df-2o 8523 df-er 8763 df-map 8886 df-ixp 8956 df-en 9004 df-dom 9005 df-sdom 9006 df-fin 9007 df-fsupp 9432 df-fi 9480 df-sup 9511 df-inf 9512 df-oi 9579 df-card 10008 df-pnf 11326 df-mnf 11327 df-xr 11328 df-ltxr 11329 df-le 11330 df-sub 11522 df-neg 11523 df-div 11948 df-nn 12294 df-2 12356 df-3 12357 df-4 12358 df-5 12359 df-6 12360 df-7 12361 df-8 12362 df-9 12363 df-n0 12554 df-z 12640 df-dec 12759 df-uz 12904 df-q 13014 df-rp 13058 df-xneg 13175 df-xadd 13176 df-xmul 13177 df-icc 13414 df-fz 13568 df-fzo 13712 df-seq 14053 df-exp 14113 df-hash 14380 df-cj 15148 df-re 15149 df-im 15150 df-sqrt 15284 df-abs 15285 df-struct 17194 df-sets 17211 df-slot 17229 df-ndx 17241 df-base 17259 df-ress 17288 df-plusg 17324 df-mulr 17325 df-starv 17326 df-sca 17327 df-vsca 17328 df-ip 17329 df-tset 17330 df-ple 17331 df-ds 17333 df-unif 17334 df-hom 17335 df-cco 17336 df-rest 17482 df-topn 17483 df-0g 17501 df-gsum 17502 df-topgen 17503 df-pt 17504 df-prds 17507 df-xrs 17562 df-qtop 17567 df-imas 17568 df-xps 17570 df-mre 17644 df-mrc 17645 df-acs 17647 df-mgm 18678 df-sgrp 18757 df-mnd 18773 df-submnd 18819 df-mulg 19108 df-cntz 19357 df-cmn 19824 df-psmet 21379 df-xmet 21380 df-met 21381 df-bl 21382 df-mopn 21383 df-cnfld 21388 df-top 22921 df-topon 22938 df-topsp 22960 df-bases 22974 df-cn 23256 df-cnp 23257 df-tx 23591 df-hmeo 23784 df-xms 24351 df-ms 24352 df-tms 24353 df-cncf 24923

This theorem is referenced by: negfcncf 24969 lhop2 26074 etransclem18 46173 etransclem46 46201

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