volioofmpt - Mathbox for Glauco Siliprandi

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Theorem volioofmpt 45287

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 2897	. . 3 ⊢ Ⅎ𝑥𝐴
2		nfcv 2897	. . . 4 ⊢ Ⅎ𝑥(vol ∘ (,))
3		volioofmpt.x	. . . 4 ⊢ Ⅎ𝑥𝐹
4	2, 3	nfco 5859	. . 3 ⊢ Ⅎ𝑥((vol ∘ (,)) ∘ 𝐹)
5		volioof 45280	. . . . 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 6735	. . . 4 ⊢ (((vol ∘ (,)):(ℝ^* × ℝ^)⟶(0[,]+∞) ∧ 𝐹:𝐴⟶(ℝ^ × ℝ^*)) → ((vol ∘ (,)) ∘ 𝐹):𝐴⟶(0[,]+∞))
9	6, 7, 8	syl2anc 583	. . 3 ⊢ (𝜑 → ((vol ∘ (,)) ∘ 𝐹):𝐴⟶(0[,]+∞))
10	1, 4, 9	feqmptdf 6956	. 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 45282	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (((vol ∘ (,)) ∘ 𝐹)‘𝑥) = (vol‘((1^st ‘(𝐹‘𝑥))(,)(2^nd ‘(𝐹‘𝑥)))))
14	13	mpteq2dva 5241	. 2 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (((vol ∘ (,)) ∘ 𝐹)‘𝑥)) = (𝑥 ∈ 𝐴 ↦ (vol‘((1^st ‘(𝐹‘𝑥))(,)(2^nd ‘(𝐹‘𝑥))))))
15	10, 14	eqtrd 2766	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 1533 ∈ wcel 2098 Ⅎwnfc 2877 ↦ cmpt 5224 × cxp 5667 ∘ ccom 5673 ⟶wf 6533 ‘cfv 6537 (class class class)co 7405 1^st c1st 7972 2^nd c2nd 7973 0cc0 11112 +∞cpnf 11249 ℝ^*cxr 11251 (,)cioo 13330 [,]cicc 13333 volcvol 25347

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 ax-inf2 9638 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 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-int 4944 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-se 5625 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6294 df-ord 6361 df-on 6362 df-lim 6363 df-suc 6364 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-isom 6546 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-of 7667 df-om 7853 df-1st 7974 df-2nd 7975 df-frecs 8267 df-wrecs 8298 df-recs 8372 df-rdg 8411 df-1o 8467 df-2o 8468 df-er 8705 df-map 8824 df-pm 8825 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-sup 9439 df-inf 9440 df-oi 9507 df-dju 9898 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-n0 12477 df-z 12563 df-uz 12827 df-q 12937 df-rp 12981 df-xadd 13099 df-ioo 13334 df-ico 13336 df-icc 13337 df-fz 13491 df-fzo 13634 df-fl 13763 df-seq 13973 df-exp 14033 df-hash 14296 df-cj 15052 df-re 15053 df-im 15054 df-sqrt 15188 df-abs 15189 df-clim 15438 df-rlim 15439 df-sum 15639 df-xmet 21233 df-met 21234 df-ovol 25348 df-vol 25349

This theorem is referenced by: ovolval5lem2 45946

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