dvmptre - Metamath Proof Explorer

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Theorem dvmptre 25987

Description: Function-builder for derivative, real part. (Contributed by Mario Carneiro, 1-Sep-2014.)

**Hypotheses**
Ref	Expression
dvmptcj.a	⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐴 ∈ ℂ)
dvmptcj.b	⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐵 ∈ 𝑉)
dvmptcj.da	⊢ (𝜑 → (ℝ D (𝑥 ∈ 𝑋 ↦ 𝐴)) = (𝑥 ∈ 𝑋 ↦ 𝐵))

**Assertion**
Ref	Expression
dvmptre	⊢ (𝜑 → (ℝ D (𝑥 ∈ 𝑋 ↦ (ℜ‘𝐴))) = (𝑥 ∈ 𝑋 ↦ (ℜ‘𝐵)))

Distinct variable groups: 𝜑,𝑥 𝑥,𝑉 𝑥,𝑋 Allowed substitution hints: 𝐴(𝑥) 𝐵(𝑥)

**Proof of Theorem dvmptre**
Step	Hyp	Ref	Expression
1		reelprrecn 11239	. . . 4 ⊢ ℝ ∈ {ℝ, ℂ}
2	1	a1i 11	. . 3 ⊢ (𝜑 → ℝ ∈ {ℝ, ℂ})
3		dvmptcj.a	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐴 ∈ ℂ)
4	3	cjcld 15194	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (∗‘𝐴) ∈ ℂ)
5	3, 4	addcld 11272	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝐴 + (∗‘𝐴)) ∈ ℂ)
6		dvmptcj.b	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐵 ∈ 𝑉)
7		dvmptcj.da	. . . . 5 ⊢ (𝜑 → (ℝ D (𝑥 ∈ 𝑋 ↦ 𝐴)) = (𝑥 ∈ 𝑋 ↦ 𝐵))
8	2, 3, 6, 7	dvmptcl 25977	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐵 ∈ ℂ)
9	8	cjcld 15194	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (∗‘𝐵) ∈ ℂ)
10	8, 9	addcld 11272	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝐵 + (∗‘𝐵)) ∈ ℂ)
11	3, 6, 7	dvmptcj 25986	. . . 4 ⊢ (𝜑 → (ℝ D (𝑥 ∈ 𝑋 ↦ (∗‘𝐴))) = (𝑥 ∈ 𝑋 ↦ (∗‘𝐵)))
12	2, 3, 6, 7, 4, 9, 11	dvmptadd 25978	. . 3 ⊢ (𝜑 → (ℝ D (𝑥 ∈ 𝑋 ↦ (𝐴 + (∗‘𝐴)))) = (𝑥 ∈ 𝑋 ↦ (𝐵 + (∗‘𝐵))))
13		halfcn 12471	. . . 4 ⊢ (1 / 2) ∈ ℂ
14	13	a1i 11	. . 3 ⊢ (𝜑 → (1 / 2) ∈ ℂ)
15	2, 5, 10, 12, 14	dvmptcmul 25982	. 2 ⊢ (𝜑 → (ℝ D (𝑥 ∈ 𝑋 ↦ ((1 / 2) · (𝐴 + (∗‘𝐴))))) = (𝑥 ∈ 𝑋 ↦ ((1 / 2) · (𝐵 + (∗‘𝐵)))))
16		reval 15104	. . . . . 6 ⊢ (𝐴 ∈ ℂ → (ℜ‘𝐴) = ((𝐴 + (∗‘𝐴)) / 2))
17	3, 16	syl 17	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (ℜ‘𝐴) = ((𝐴 + (∗‘𝐴)) / 2))
18		2cn 12331	. . . . . . 7 ⊢ 2 ∈ ℂ
19		2ne0 12360	. . . . . . 7 ⊢ 2 ≠ 0
20		divrec2 11929	. . . . . . 7 ⊢ (((𝐴 + (∗‘𝐴)) ∈ ℂ ∧ 2 ∈ ℂ ∧ 2 ≠ 0) → ((𝐴 + (∗‘𝐴)) / 2) = ((1 / 2) · (𝐴 + (∗‘𝐴))))
21	18, 19, 20	mp3an23 1450	. . . . . 6 ⊢ ((𝐴 + (∗‘𝐴)) ∈ ℂ → ((𝐴 + (∗‘𝐴)) / 2) = ((1 / 2) · (𝐴 + (∗‘𝐴))))
22	5, 21	syl 17	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ((𝐴 + (∗‘𝐴)) / 2) = ((1 / 2) · (𝐴 + (∗‘𝐴))))
23	17, 22	eqtrd 2766	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (ℜ‘𝐴) = ((1 / 2) · (𝐴 + (∗‘𝐴))))
24	23	mpteq2dva 5244	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (ℜ‘𝐴)) = (𝑥 ∈ 𝑋 ↦ ((1 / 2) · (𝐴 + (∗‘𝐴)))))
25	24	oveq2d 7430	. 2 ⊢ (𝜑 → (ℝ D (𝑥 ∈ 𝑋 ↦ (ℜ‘𝐴))) = (ℝ D (𝑥 ∈ 𝑋 ↦ ((1 / 2) · (𝐴 + (∗‘𝐴))))))
26		reval 15104	. . . . 5 ⊢ (𝐵 ∈ ℂ → (ℜ‘𝐵) = ((𝐵 + (∗‘𝐵)) / 2))
27	8, 26	syl 17	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (ℜ‘𝐵) = ((𝐵 + (∗‘𝐵)) / 2))
28		divrec2 11929	. . . . . 6 ⊢ (((𝐵 + (∗‘𝐵)) ∈ ℂ ∧ 2 ∈ ℂ ∧ 2 ≠ 0) → ((𝐵 + (∗‘𝐵)) / 2) = ((1 / 2) · (𝐵 + (∗‘𝐵))))
29	18, 19, 28	mp3an23 1450	. . . . 5 ⊢ ((𝐵 + (∗‘𝐵)) ∈ ℂ → ((𝐵 + (∗‘𝐵)) / 2) = ((1 / 2) · (𝐵 + (∗‘𝐵))))
30	10, 29	syl 17	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ((𝐵 + (∗‘𝐵)) / 2) = ((1 / 2) · (𝐵 + (∗‘𝐵))))
31	27, 30	eqtrd 2766	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (ℜ‘𝐵) = ((1 / 2) · (𝐵 + (∗‘𝐵))))
32	31	mpteq2dva 5244	. 2 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (ℜ‘𝐵)) = (𝑥 ∈ 𝑋 ↦ ((1 / 2) · (𝐵 + (∗‘𝐵)))))
33	15, 25, 32	3eqtr4d 2776	1 ⊢ (𝜑 → (ℝ D (𝑥 ∈ 𝑋 ↦ (ℜ‘𝐴))) = (𝑥 ∈ 𝑋 ↦ (ℜ‘𝐵)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 394 = wceq 1534 ∈ wcel 2099 ≠ wne 2930 {cpr 4626 ↦ cmpt 5227 ‘cfv 6544 (class class class)co 7414 ℂcc 11145 ℝcr 11146 0cc0 11147 1c1 11148 + caddc 11150 · cmul 11152 / cdiv 11910 2c2 12311 ∗ccj 15094 ℜcre 15095 D cdv 25878

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 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-rep 5281 ax-sep 5295 ax-nul 5302 ax-pow 5360 ax-pr 5424 ax-un 7736 ax-cnex 11203 ax-resscn 11204 ax-1cn 11205 ax-icn 11206 ax-addcl 11207 ax-addrcl 11208 ax-mulcl 11209 ax-mulrcl 11210 ax-mulcom 11211 ax-addass 11212 ax-mulass 11213 ax-distr 11214 ax-i2m1 11215 ax-1ne0 11216 ax-1rid 11217 ax-rnegex 11218 ax-rrecex 11219 ax-cnre 11220 ax-pre-lttri 11221 ax-pre-lttrn 11222 ax-pre-ltadd 11223 ax-pre-mulgt0 11224 ax-pre-sup 11225 ax-addf 11226

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 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3365 df-reu 3366 df-rab 3421 df-v 3465 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3967 df-nul 4324 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-tp 4629 df-op 4631 df-uni 4907 df-int 4948 df-iun 4996 df-iin 4997 df-br 5145 df-opab 5207 df-mpt 5228 df-tr 5262 df-id 5571 df-eprel 5577 df-po 5585 df-so 5586 df-fr 5628 df-se 5629 df-we 5630 df-xp 5679 df-rel 5680 df-cnv 5681 df-co 5682 df-dm 5683 df-rn 5684 df-res 5685 df-ima 5686 df-pred 6303 df-ord 6369 df-on 6370 df-lim 6371 df-suc 6372 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-isom 6553 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-of 7680 df-om 7867 df-1st 7993 df-2nd 7994 df-supp 8165 df-frecs 8286 df-wrecs 8317 df-recs 8391 df-rdg 8430 df-1o 8486 df-2o 8487 df-er 8724 df-map 8847 df-pm 8848 df-ixp 8917 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-fsupp 9397 df-fi 9445 df-sup 9476 df-inf 9477 df-oi 9544 df-card 9973 df-pnf 11289 df-mnf 11290 df-xr 11291 df-ltxr 11292 df-le 11293 df-sub 11485 df-neg 11486 df-div 11911 df-nn 12257 df-2 12319 df-3 12320 df-4 12321 df-5 12322 df-6 12323 df-7 12324 df-8 12325 df-9 12326 df-n0 12517 df-z 12603 df-dec 12722 df-uz 12867 df-q 12977 df-rp 13021 df-xneg 13138 df-xadd 13139 df-xmul 13140 df-ioo 13374 df-icc 13377 df-fz 13531 df-fzo 13674 df-seq 14014 df-exp 14074 df-hash 14341 df-cj 15097 df-re 15098 df-im 15099 df-sqrt 15233 df-abs 15234 df-struct 17142 df-sets 17159 df-slot 17177 df-ndx 17189 df-base 17207 df-ress 17236 df-plusg 17272 df-mulr 17273 df-starv 17274 df-sca 17275 df-vsca 17276 df-ip 17277 df-tset 17278 df-ple 17279 df-ds 17281 df-unif 17282 df-hom 17283 df-cco 17284 df-rest 17430 df-topn 17431 df-0g 17449 df-gsum 17450 df-topgen 17451 df-pt 17452 df-prds 17455 df-xrs 17510 df-qtop 17515 df-imas 17516 df-xps 17518 df-mre 17592 df-mrc 17593 df-acs 17595 df-mgm 18626 df-sgrp 18705 df-mnd 18721 df-submnd 18767 df-mulg 19056 df-cntz 19305 df-cmn 19774 df-psmet 21329 df-xmet 21330 df-met 21331 df-bl 21332 df-mopn 21333 df-fbas 21334 df-fg 21335 df-cnfld 21338 df-top 22882 df-topon 22899 df-topsp 22921 df-bases 22935 df-cld 23009 df-ntr 23010 df-cls 23011 df-nei 23088 df-lp 23126 df-perf 23127 df-cn 23217 df-cnp 23218 df-haus 23305 df-tx 23552 df-hmeo 23745 df-fil 23836 df-fm 23928 df-flim 23929 df-flf 23930 df-xms 24312 df-ms 24313 df-tms 24314 df-cncf 24884 df-limc 25881 df-dv 25882

This theorem is referenced by: dvlip 26012

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