volioofmpt - Mathbox for Glauco Siliprandi

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Theorem volioofmpt 45992

Description: ((vol ∘ (,)) ∘ 𝐹) expressed in maps-to notation. (Contributed by Glauco Siliprandi, 3-Mar-2021.)

**Hypotheses**
Ref	Expression
volioofmpt.x	⊢ Ⅎ𝑥𝐹
volioofmpt.f	⊢ (𝜑 → 𝐹:𝐴⟶(ℝ^* × ℝ^*))

**Assertion**
Ref	Expression
volioofmpt	⊢ (𝜑 → ((vol ∘ (,)) ∘ 𝐹) = (𝑥 ∈ 𝐴 ↦ (vol‘((1^st ‘(𝐹‘𝑥))(,)(2^nd ‘(𝐹‘𝑥))))))

Distinct variable groups: 𝑥,𝐴 𝜑,𝑥 Allowed substitution hint: 𝐹(𝑥)

**Proof of Theorem volioofmpt**
Step	Hyp	Ref	Expression
1		nfcv 2891	. . 3 ⊢ Ⅎ𝑥𝐴
2		nfcv 2891	. . . 4 ⊢ Ⅎ𝑥(vol ∘ (,))
3		volioofmpt.x	. . . 4 ⊢ Ⅎ𝑥𝐹
4	2, 3	nfco 5829	. . 3 ⊢ Ⅎ𝑥((vol ∘ (,)) ∘ 𝐹)
5		volioof 45985	. . . . 5 ⊢ (vol ∘ (,)):(ℝ^* × ℝ^*)⟶(0[,]+∞)
6	5	a1i 11	. . . 4 ⊢ (𝜑 → (vol ∘ (,)):(ℝ^* × ℝ^*)⟶(0[,]+∞))
7		volioofmpt.f	. . . 4 ⊢ (𝜑 → 𝐹:𝐴⟶(ℝ^* × ℝ^*))
8		fco 6712	. . . 4 ⊢ (((vol ∘ (,)):(ℝ^* × ℝ^)⟶(0[,]+∞) ∧ 𝐹:𝐴⟶(ℝ^ × ℝ^*)) → ((vol ∘ (,)) ∘ 𝐹):𝐴⟶(0[,]+∞))
9	6, 7, 8	syl2anc 584	. . 3 ⊢ (𝜑 → ((vol ∘ (,)) ∘ 𝐹):𝐴⟶(0[,]+∞))
10	1, 4, 9	feqmptdf 6931	. 2 ⊢ (𝜑 → ((vol ∘ (,)) ∘ 𝐹) = (𝑥 ∈ 𝐴 ↦ (((vol ∘ (,)) ∘ 𝐹)‘𝑥)))
11	7	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐹:𝐴⟶(ℝ^* × ℝ^*))
12		simpr 484	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐴)
13	11, 12	fvvolioof 45987	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (((vol ∘ (,)) ∘ 𝐹)‘𝑥) = (vol‘((1^st ‘(𝐹‘𝑥))(,)(2^nd ‘(𝐹‘𝑥)))))
14	13	mpteq2dva 5200	. 2 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (((vol ∘ (,)) ∘ 𝐹)‘𝑥)) = (𝑥 ∈ 𝐴 ↦ (vol‘((1^st ‘(𝐹‘𝑥))(,)(2^nd ‘(𝐹‘𝑥))))))
15	10, 14	eqtrd 2764	1 ⊢ (𝜑 → ((vol ∘ (,)) ∘ 𝐹) = (𝑥 ∈ 𝐴 ↦ (vol‘((1^st ‘(𝐹‘𝑥))(,)(2^nd ‘(𝐹‘𝑥))))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2109 Ⅎwnfc 2876 ↦ cmpt 5188 × cxp 5636 ∘ ccom 5642 ⟶wf 6507 ‘cfv 6511 (class class class)co 7387 1^st c1st 7966 2^nd c2nd 7967 0cc0 11068 +∞cpnf 11205 ℝ^*cxr 11207 (,)cioo 13306 [,]cicc 13309 volcvol 25364

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 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5234 ax-sep 5251 ax-nul 5261 ax-pow 5320 ax-pr 5387 ax-un 7711 ax-inf2 9594 ax-cnex 11124 ax-resscn 11125 ax-1cn 11126 ax-icn 11127 ax-addcl 11128 ax-addrcl 11129 ax-mulcl 11130 ax-mulrcl 11131 ax-mulcom 11132 ax-addass 11133 ax-mulass 11134 ax-distr 11135 ax-i2m1 11136 ax-1ne0 11137 ax-1rid 11138 ax-rnegex 11139 ax-rrecex 11140 ax-cnre 11141 ax-pre-lttri 11142 ax-pre-lttrn 11143 ax-pre-ltadd 11144 ax-pre-mulgt0 11145 ax-pre-sup 11146

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3354 df-reu 3355 df-rab 3406 df-v 3449 df-sbc 3754 df-csb 3863 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3934 df-nul 4297 df-if 4489 df-pw 4565 df-sn 4590 df-pr 4592 df-op 4596 df-uni 4872 df-int 4911 df-iun 4957 df-br 5108 df-opab 5170 df-mpt 5189 df-tr 5215 df-id 5533 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5591 df-se 5592 df-we 5593 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6274 df-ord 6335 df-on 6336 df-lim 6337 df-suc 6338 df-iota 6464 df-fun 6513 df-fn 6514 df-f 6515 df-f1 6516 df-fo 6517 df-f1o 6518 df-fv 6519 df-isom 6520 df-riota 7344 df-ov 7390 df-oprab 7391 df-mpo 7392 df-of 7653 df-om 7843 df-1st 7968 df-2nd 7969 df-frecs 8260 df-wrecs 8291 df-recs 8340 df-rdg 8378 df-1o 8434 df-2o 8435 df-er 8671 df-map 8801 df-pm 8802 df-en 8919 df-dom 8920 df-sdom 8921 df-fin 8922 df-sup 9393 df-inf 9394 df-oi 9463 df-dju 9854 df-card 9892 df-pnf 11210 df-mnf 11211 df-xr 11212 df-ltxr 11213 df-le 11214 df-sub 11407 df-neg 11408 df-div 11836 df-nn 12187 df-2 12249 df-3 12250 df-n0 12443 df-z 12530 df-uz 12794 df-q 12908 df-rp 12952 df-xadd 13073 df-ioo 13310 df-ico 13312 df-icc 13313 df-fz 13469 df-fzo 13616 df-fl 13754 df-seq 13967 df-exp 14027 df-hash 14296 df-cj 15065 df-re 15066 df-im 15067 df-sqrt 15201 df-abs 15202 df-clim 15454 df-rlim 15455 df-sum 15653 df-xmet 21257 df-met 21258 df-ovol 25365 df-vol 25366

This theorem is referenced by: ovolval5lem2 46651

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