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Theorem negcncf 24956

Description: The negative function is continuous. (Contributed by Mario Carneiro, 30-Dec-2016.) Avoid ax-mulf 11245. (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 12388	. . . . . 6 ⊢ -1 ∈ ℂ
2		ssel2 3986	. . . . . 6 ⊢ ((𝐴 ⊆ ℂ ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ ℂ)
3		ovmpot 7592	. . . . . . 7 ⊢ ((-1 ∈ ℂ ∧ 𝑥 ∈ ℂ) → (-1(𝑎 ∈ ℂ, 𝑏 ∈ ℂ ↦ (𝑎 · 𝑏))𝑥) = (-1 · 𝑥))
4	3	eqcomd 2735	. . . . . 6 ⊢ ((-1 ∈ ℂ ∧ 𝑥 ∈ ℂ) → (-1 · 𝑥) = (-1(𝑎 ∈ ℂ, 𝑏 ∈ ℂ ↦ (𝑎 · 𝑏))𝑥))
5	1, 2, 4	sylancr 585	. . . . 5 ⊢ ((𝐴 ⊆ ℂ ∧ 𝑥 ∈ 𝐴) → (-1 · 𝑥) = (-1(𝑎 ∈ ℂ, 𝑏 ∈ ℂ ↦ (𝑎 · 𝑏))𝑥))
6	2	mulm1d 11723	. . . . 5 ⊢ ((𝐴 ⊆ ℂ ∧ 𝑥 ∈ 𝐴) → (-1 · 𝑥) = -𝑥)
7	5, 6	eqtr3d 2771	. . . 4 ⊢ ((𝐴 ⊆ ℂ ∧ 𝑥 ∈ 𝐴) → (-1(𝑎 ∈ ℂ, 𝑏 ∈ ℂ ↦ (𝑎 · 𝑏))𝑥) = -𝑥)
8	7	mpteq2dva 5256	. . 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 2787	. 2 ⊢ (𝐴 ⊆ ℂ → (𝑥 ∈ 𝐴 ↦ (-1(𝑎 ∈ ℂ, 𝑏 ∈ ℂ ↦ (𝑎 · 𝑏))𝑥)) = 𝐹)
11		eqid 2729	. . 3 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
12	11	mpomulcn 24899	. . . 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 4014	. . . 4 ⊢ ℂ ⊆ ℂ
15		cncfmptc 24946	. . . 4 ⊢ ((-1 ∈ ℂ ∧ 𝐴 ⊆ ℂ ∧ ℂ ⊆ ℂ) → (𝑥 ∈ 𝐴 ↦ -1) ∈ (𝐴–cn→ℂ))
16	1, 14, 15	mp3an13 1449	. . 3 ⊢ (𝐴 ⊆ ℂ → (𝑥 ∈ 𝐴 ↦ -1) ∈ (𝐴–cn→ℂ))
17		cncfmptid 24947	. . . 4 ⊢ ((𝐴 ⊆ ℂ ∧ ℂ ⊆ ℂ) → (𝑥 ∈ 𝐴 ↦ 𝑥) ∈ (𝐴–cn→ℂ))
18	14, 17	mpan2 689	. . 3 ⊢ (𝐴 ⊆ ℂ → (𝑥 ∈ 𝐴 ↦ 𝑥) ∈ (𝐴–cn→ℂ))
19	11, 13, 16, 18	cncfmpt2f 24949	. 2 ⊢ (𝐴 ⊆ ℂ → (𝑥 ∈ 𝐴 ↦ (-1(𝑎 ∈ ℂ, 𝑏 ∈ ℂ ↦ (𝑎 · 𝑏))𝑥)) ∈ (𝐴–cn→ℂ))
20	10, 19	eqeltrrd 2830	1 ⊢ (𝐴 ⊆ ℂ → 𝐹 ∈ (𝐴–cn→ℂ))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 394 = wceq 1534 ∈ wcel 2100 ⊆ wss 3959 ↦ cmpt 5239 ‘cfv 6558 (class class class)co 7430 ∈ cmpo 7432 ℂcc 11163 1c1 11166 · cmul 11170 -cneg 11502 TopOpenctopn 17457 ℂ_fldccnfld 21365 Cn ccn 23242 ×_t ctx 23578 –cn→ccncf 24910

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2102 ax-9 2110 ax-10 2133 ax-11 2150 ax-12 2170 ax-ext 2700 ax-rep 5293 ax-sep 5307 ax-nul 5314 ax-pow 5373 ax-pr 5437 ax-un 7751 ax-cnex 11221 ax-resscn 11222 ax-1cn 11223 ax-icn 11224 ax-addcl 11225 ax-addrcl 11226 ax-mulcl 11227 ax-mulrcl 11228 ax-mulcom 11229 ax-addass 11230 ax-mulass 11231 ax-distr 11232 ax-i2m1 11233 ax-1ne0 11234 ax-1rid 11235 ax-rnegex 11236 ax-rrecex 11237 ax-cnre 11238 ax-pre-lttri 11239 ax-pre-lttrn 11240 ax-pre-ltadd 11241 ax-pre-mulgt0 11242 ax-pre-sup 11243

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2062 df-mo 2532 df-eu 2561 df-clab 2707 df-cleq 2721 df-clel 2806 df-nfc 2881 df-ne 2934 df-nel 3040 df-ral 3055 df-rex 3064 df-rmo 3373 df-reu 3374 df-rab 3429 df-v 3474 df-sbc 3789 df-csb 3905 df-dif 3962 df-un 3964 df-in 3966 df-ss 3976 df-pss 3979 df-nul 4336 df-if 4537 df-pw 4612 df-sn 4637 df-pr 4639 df-tp 4641 df-op 4643 df-uni 4919 df-int 4960 df-iun 5008 df-iin 5009 df-br 5157 df-opab 5219 df-mpt 5240 df-tr 5274 df-id 5584 df-eprel 5590 df-po 5598 df-so 5599 df-fr 5641 df-se 5642 df-we 5643 df-xp 5692 df-rel 5693 df-cnv 5694 df-co 5695 df-dm 5696 df-rn 5697 df-res 5698 df-ima 5699 df-pred 6317 df-ord 6383 df-on 6384 df-lim 6385 df-suc 6386 df-iota 6510 df-fun 6560 df-fn 6561 df-f 6562 df-f1 6563 df-fo 6564 df-f1o 6565 df-fv 6566 df-isom 6567 df-riota 7386 df-ov 7433 df-oprab 7434 df-mpo 7435 df-of 7695 df-om 7885 df-1st 8011 df-2nd 8012 df-supp 8183 df-frecs 8304 df-wrecs 8335 df-recs 8409 df-rdg 8448 df-1o 8504 df-2o 8505 df-er 8742 df-map 8865 df-ixp 8935 df-en 8983 df-dom 8984 df-sdom 8985 df-fin 8986 df-fsupp 9413 df-fi 9461 df-sup 9492 df-inf 9493 df-oi 9560 df-card 9989 df-pnf 11307 df-mnf 11308 df-xr 11309 df-ltxr 11310 df-le 11311 df-sub 11503 df-neg 11504 df-div 11929 df-nn 12275 df-2 12337 df-3 12338 df-4 12339 df-5 12340 df-6 12341 df-7 12342 df-8 12343 df-9 12344 df-n0 12535 df-z 12621 df-dec 12740 df-uz 12885 df-q 12995 df-rp 13039 df-xneg 13156 df-xadd 13157 df-xmul 13158 df-icc 13395 df-fz 13549 df-fzo 13692 df-seq 14033 df-exp 14093 df-hash 14360 df-cj 15125 df-re 15126 df-im 15127 df-sqrt 15261 df-abs 15262 df-struct 17170 df-sets 17187 df-slot 17205 df-ndx 17217 df-base 17235 df-ress 17264 df-plusg 17300 df-mulr 17301 df-starv 17302 df-sca 17303 df-vsca 17304 df-ip 17305 df-tset 17306 df-ple 17307 df-ds 17309 df-unif 17310 df-hom 17311 df-cco 17312 df-rest 17458 df-topn 17459 df-0g 17477 df-gsum 17478 df-topgen 17479 df-pt 17480 df-prds 17483 df-xrs 17538 df-qtop 17543 df-imas 17544 df-xps 17546 df-mre 17620 df-mrc 17621 df-acs 17623 df-mgm 18654 df-sgrp 18733 df-mnd 18749 df-submnd 18795 df-mulg 19084 df-cntz 19333 df-cmn 19802 df-psmet 21357 df-xmet 21358 df-met 21359 df-bl 21360 df-mopn 21361 df-cnfld 21366 df-top 22910 df-topon 22927 df-topsp 22949 df-bases 22963 df-cn 23245 df-cnp 23246 df-tx 23580 df-hmeo 23773 df-xms 24340 df-ms 24341 df-tms 24342 df-cncf 24912

This theorem is referenced by: negfcncf 24958 lhop2 26062 etransclem18 45934 etransclem46 45962

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