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Theorem negcncf 24662

Description: The negative function is continuous. (Contributed by Mario Carneiro, 30-Dec-2016.) Avoid ax-mulf 11192. (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 12330	. . . . . 6 ⊢ -1 ∈ ℂ
2		ssel2 3976	. . . . . 6 ⊢ ((𝐴 ⊆ ℂ ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ ℂ)
3		ovmul 11207	. . . . . . 7 ⊢ ((-1 ∈ ℂ ∧ 𝑥 ∈ ℂ) → (-1(𝑎 ∈ ℂ, 𝑏 ∈ ℂ ↦ (𝑎 · 𝑏))𝑥) = (-1 · 𝑥))
4	3	eqcomd 2736	. . . . . 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 11670	. . . . 5 ⊢ ((𝐴 ⊆ ℂ ∧ 𝑥 ∈ 𝐴) → (-1 · 𝑥) = -𝑥)
7	5, 6	eqtr3d 2772	. . . 4 ⊢ ((𝐴 ⊆ ℂ ∧ 𝑥 ∈ 𝐴) → (-1(𝑎 ∈ ℂ, 𝑏 ∈ ℂ ↦ (𝑎 · 𝑏))𝑥) = -𝑥)
8	7	mpteq2dva 5247	. . 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 2788	. 2 ⊢ (𝐴 ⊆ ℂ → (𝑥 ∈ 𝐴 ↦ (-1(𝑎 ∈ ℂ, 𝑏 ∈ ℂ ↦ (𝑎 · 𝑏))𝑥)) = 𝐹)
11		eqid 2730	. . 3 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
12	11	mpomulcn 24605	. . . 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 4003	. . . 4 ⊢ ℂ ⊆ ℂ
15		cncfmptc 24652	. . . 4 ⊢ ((-1 ∈ ℂ ∧ 𝐴 ⊆ ℂ ∧ ℂ ⊆ ℂ) → (𝑥 ∈ 𝐴 ↦ -1) ∈ (𝐴–cn→ℂ))
16	1, 14, 15	mp3an13 1450	. . 3 ⊢ (𝐴 ⊆ ℂ → (𝑥 ∈ 𝐴 ↦ -1) ∈ (𝐴–cn→ℂ))
17		cncfmptid 24653	. . . 4 ⊢ ((𝐴 ⊆ ℂ ∧ ℂ ⊆ ℂ) → (𝑥 ∈ 𝐴 ↦ 𝑥) ∈ (𝐴–cn→ℂ))
18	14, 17	mpan2 687	. . 3 ⊢ (𝐴 ⊆ ℂ → (𝑥 ∈ 𝐴 ↦ 𝑥) ∈ (𝐴–cn→ℂ))
19	11, 13, 16, 18	cncfmpt2f 24655	. 2 ⊢ (𝐴 ⊆ ℂ → (𝑥 ∈ 𝐴 ↦ (-1(𝑎 ∈ ℂ, 𝑏 ∈ ℂ ↦ (𝑎 · 𝑏))𝑥)) ∈ (𝐴–cn→ℂ))
20	10, 19	eqeltrrd 2832	1 ⊢ (𝐴 ⊆ ℂ → 𝐹 ∈ (𝐴–cn→ℂ))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 394 = wceq 1539 ∈ wcel 2104 ⊆ wss 3947 ↦ cmpt 5230 ‘cfv 6542 (class class class)co 7411 ∈ cmpo 7413 ℂcc 11110 1c1 11113 · cmul 11117 -cneg 11449 TopOpenctopn 17371 ℂ_fldccnfld 21144 Cn ccn 22948 ×_t ctx 23284 –cn→ccncf 24616

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2701 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7727 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 ax-pre-sup 11190

This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2532 df-eu 2561 df-clab 2708 df-cleq 2722 df-clel 2808 df-nfc 2883 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3474 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-tp 4632 df-op 4634 df-uni 4908 df-int 4950 df-iun 4998 df-iin 4999 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-se 5631 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-isom 6551 df-riota 7367 df-ov 7414 df-oprab 7415 df-mpo 7416 df-of 7672 df-om 7858 df-1st 7977 df-2nd 7978 df-supp 8149 df-frecs 8268 df-wrecs 8299 df-recs 8373 df-rdg 8412 df-1o 8468 df-2o 8469 df-er 8705 df-map 8824 df-ixp 8894 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-fsupp 9364 df-fi 9408 df-sup 9439 df-inf 9440 df-oi 9507 df-card 9936 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-div 11876 df-nn 12217 df-2 12279 df-3 12280 df-4 12281 df-5 12282 df-6 12283 df-7 12284 df-8 12285 df-9 12286 df-n0 12477 df-z 12563 df-dec 12682 df-uz 12827 df-q 12937 df-rp 12979 df-xneg 13096 df-xadd 13097 df-xmul 13098 df-icc 13335 df-fz 13489 df-fzo 13632 df-seq 13971 df-exp 14032 df-hash 14295 df-cj 15050 df-re 15051 df-im 15052 df-sqrt 15186 df-abs 15187 df-struct 17084 df-sets 17101 df-slot 17119 df-ndx 17131 df-base 17149 df-ress 17178 df-plusg 17214 df-mulr 17215 df-starv 17216 df-sca 17217 df-vsca 17218 df-ip 17219 df-tset 17220 df-ple 17221 df-ds 17223 df-unif 17224 df-hom 17225 df-cco 17226 df-rest 17372 df-topn 17373 df-0g 17391 df-gsum 17392 df-topgen 17393 df-pt 17394 df-prds 17397 df-xrs 17452 df-qtop 17457 df-imas 17458 df-xps 17460 df-mre 17534 df-mrc 17535 df-acs 17537 df-mgm 18565 df-sgrp 18644 df-mnd 18660 df-submnd 18706 df-mulg 18987 df-cntz 19222 df-cmn 19691 df-psmet 21136 df-xmet 21137 df-met 21138 df-bl 21139 df-mopn 21140 df-cnfld 21145 df-top 22616 df-topon 22633 df-topsp 22655 df-bases 22669 df-cn 22951 df-cnp 22952 df-tx 23286 df-hmeo 23479 df-xms 24046 df-ms 24047 df-tms 24048 df-cncf 24618

This theorem is referenced by: negfcncf 24664 lhop2 25767 etransclem18 45266 etransclem46 45294

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