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Theorem negcncf 25112

Description: The negative function is continuous. (Contributed by Mario Carneiro, 30-Dec-2016.) Avoid ax-mulf 11191. (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 12214	. . . . . 6 ⊢ -1 ∈ ℂ
2		ssel2 3933	. . . . . 6 ⊢ ((𝐴 ⊆ ℂ ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ ℂ)
3		ovmpot 7577	. . . . . . 7 ⊢ ((-1 ∈ ℂ ∧ 𝑥 ∈ ℂ) → (-1(𝑎 ∈ ℂ, 𝑏 ∈ ℂ ↦ (𝑎 · 𝑏))𝑥) = (-1 · 𝑥))
4	3	eqcomd 2771	. . . . . 6 ⊢ ((-1 ∈ ℂ ∧ 𝑥 ∈ ℂ) → (-1 · 𝑥) = (-1(𝑎 ∈ ℂ, 𝑏 ∈ ℂ ↦ (𝑎 · 𝑏))𝑥))
5	1, 2, 4	sylancr 599	. . . . 5 ⊢ ((𝐴 ⊆ ℂ ∧ 𝑥 ∈ 𝐴) → (-1 · 𝑥) = (-1(𝑎 ∈ ℂ, 𝑏 ∈ ℂ ↦ (𝑎 · 𝑏))𝑥))
6	2	mulm1d 11677	. . . . 5 ⊢ ((𝐴 ⊆ ℂ ∧ 𝑥 ∈ 𝐴) → (-1 · 𝑥) = -𝑥)
7	5, 6	eqtr3d 2802	. . . 4 ⊢ ((𝐴 ⊆ ℂ ∧ 𝑥 ∈ 𝐴) → (-1(𝑎 ∈ ℂ, 𝑏 ∈ ℂ ↦ (𝑎 · 𝑏))𝑥) = -𝑥)
8	7	mpteq2dva 5206	. . 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 2818	. 2 ⊢ (𝐴 ⊆ ℂ → (𝑥 ∈ 𝐴 ↦ (-1(𝑎 ∈ ℂ, 𝑏 ∈ ℂ ↦ (𝑎 · 𝑏))𝑥)) = 𝐹)
11		eqid 2765	. . 3 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
12	11	mpomulcn 25057	. . . 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 3960	. . . 4 ⊢ ℂ ⊆ ℂ
15		cncfmptc 25102	. . . 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 25103	. . . 4 ⊢ ((𝐴 ⊆ ℂ ∧ ℂ ⊆ ℂ) → (𝑥 ∈ 𝐴 ↦ 𝑥) ∈ (𝐴–cn→ℂ))
18	14, 17	mpan2 704	. . 3 ⊢ (𝐴 ⊆ ℂ → (𝑥 ∈ 𝐴 ↦ 𝑥) ∈ (𝐴–cn→ℂ))
19	11, 13, 16, 18	cncfmpt2f 25105	. 2 ⊢ (𝐴 ⊆ ℂ → (𝑥 ∈ 𝐴 ↦ (-1(𝑎 ∈ ℂ, 𝑏 ∈ ℂ ↦ (𝑎 · 𝑏))𝑥)) ∈ (𝐴–cn→ℂ))
20	10, 19	eqeltrrd 2866	1 ⊢ (𝐴 ⊆ ℂ → 𝐹 ∈ (𝐴–cn→ℂ))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∧ wa 401 = wceq 1570 ∈ wcel 2146 ⊆ wss 3906 ↦ cmpt 5194 ‘cfv 6540 (class class class)co 7416 ∈ cmpo 7418 ℂcc 11109 1c1 11112 · cmul 11116 -cneg 11453 TopOpenctopn 17492 ℂ_fldccnfld 21552 Cn ccn 23411 ×_t ctx 23748 –cn→ccncf 25066

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2148 ax-9 2156 ax-10 2179 ax-11 2195 ax-12 2216 ax-ext 2737 ax-rep 5240 ax-sep 5259 ax-nul 5271 ax-pow 5338 ax-pr 5406 ax-un 7738 ax-cnex 11167 ax-resscn 11168 ax-1cn 11169 ax-icn 11170 ax-addcl 11171 ax-addrcl 11172 ax-mulcl 11173 ax-mulrcl 11174 ax-mulcom 11175 ax-addass 11176 ax-mulass 11177 ax-distr 11178 ax-i2m1 11179 ax-1ne0 11180 ax-1rid 11181 ax-rnegex 11182 ax-rrecex 11183 ax-cnre 11184 ax-pre-lttri 11185 ax-pre-lttrn 11186 ax-pre-ltadd 11187 ax-pre-mulgt0 11188 ax-pre-sup 11189

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2569 df-eu 2599 df-clab 2744 df-cleq 2757 df-clel 2840 df-nfc 2914 df-ne 2961 df-nel 3067 df-ral 3082 df-rex 3092 df-rmo 3371 df-reu 3372 df-rab 3419 df-v 3459 df-sbc 3747 df-csb 3855 df-dif 3909 df-un 3911 df-in 3913 df-ss 3923 df-pss 3926 df-nul 4287 df-if 4490 df-pw 4566 df-sn 4592 df-pr 4594 df-tp 4596 df-op 4598 df-uni 4875 df-int 4915 df-iun 4960 df-iin 4961 df-br 5112 df-opab 5176 df-mpt 5195 df-tr 5221 df-id 5558 df-eprel 5563 df-po 5571 df-so 5572 df-fr 5616 df-se 5617 df-we 5618 df-xp 5669 df-rel 5670 df-cnv 5671 df-co 5672 df-dm 5673 df-rn 5674 df-res 5675 df-ima 5676 df-pred 6306 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6496 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-isom 6549 df-riota 7373 df-ov 7419 df-oprab 7420 df-mpo 7421 df-of 7680 df-om 7865 df-1st 7988 df-2nd 7989 df-supp 8159 df-frecs 8280 df-wrecs 8311 df-recs 8360 df-rdg 8399 df-1o 8455 df-2o 8456 df-er 8696 df-map 8828 df-ixp 8898 df-en 8946 df-dom 8947 df-sdom 8948 df-fin 8949 df-fsupp 9325 df-fi 9374 df-sup 9405 df-inf 9406 df-oi 9475 df-card 9937 df-pnf 11256 df-mnf 11257 df-xr 11258 df-ltxr 11259 df-le 11260 df-sub 11454 df-neg 11455 df-div 11883 df-nn 12245 df-2 12314 df-3 12315 df-4 12316 df-5 12317 df-6 12318 df-7 12319 df-8 12320 df-9 12321 df-n0 12516 df-z 12603 df-dec 12724 df-uz 12875 df-q 12985 df-rp 13029 df-xneg 13149 df-xadd 13150 df-xmul 13151 df-icc 13391 df-fz 13548 df-fzo 13696 df-seq 14052 df-exp 14112 df-hash 14381 df-cj 15170 df-re 15171 df-im 15172 df-sqrt 15306 df-abs 15307 df-struct 17225 df-sets 17242 df-slot 17260 df-ndx 17272 df-base 17288 df-ress 17309 df-plusg 17341 df-mulr 17342 df-starv 17343 df-sca 17344 df-vsca 17345 df-ip 17346 df-tset 17347 df-ple 17348 df-ds 17350 df-unif 17351 df-hom 17352 df-cco 17353 df-rest 17493 df-topn 17494 df-0g 17512 df-gsum 17513 df-topgen 17514 df-pt 17515 df-prds 17518 df-xrs 17574 df-qtop 17579 df-imas 17580 df-xps 17582 df-mre 17656 df-mrc 17657 df-acs 17659 df-mgm 18716 df-sgrp 18799 df-mnd 18815 df-submnd 18866 df-mulg 19158 df-cntz 19411 df-cmn 19876 df-psmet 21544 df-xmet 21545 df-met 21546 df-bl 21547 df-mopn 21548 df-cnfld 21553 df-top 23081 df-topon 23098 df-topsp 23120 df-bases 23133 df-cn 23414 df-cnp 23415 df-tx 23750 df-hmeo 23943 df-xms 24508 df-ms 24509 df-tms 24510 df-cncf 25068

This theorem is used by: negfcncf 25113 lhop2 26205 etransclem18 46999 etransclem46 47027

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