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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > volicoff | Structured version Visualization version GIF version |
Description: ((vol ∘ [,)) ∘ 𝐹) expressed in maps-to notation. (Contributed by Glauco Siliprandi, 3-Mar-2021.) |
Ref | Expression |
---|---|
volicoff.1 | ⊢ (𝜑 → 𝐹:𝐴⟶(ℝ × ℝ*)) |
Ref | Expression |
---|---|
volicoff | ⊢ (𝜑 → ((vol ∘ [,)) ∘ 𝐹):𝐴⟶(0[,]+∞)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | volf 25507 | . . . 4 ⊢ vol:dom vol⟶(0[,]+∞) | |
2 | 1 | a1i 11 | . . 3 ⊢ (𝜑 → vol:dom vol⟶(0[,]+∞)) |
3 | icof 44733 | . . . . . . 7 ⊢ [,):(ℝ* × ℝ*)⟶𝒫 ℝ* | |
4 | 3 | a1i 11 | . . . . . 6 ⊢ (𝜑 → [,):(ℝ* × ℝ*)⟶𝒫 ℝ*) |
5 | ressxr 11295 | . . . . . . . 8 ⊢ ℝ ⊆ ℝ* | |
6 | xpss1 5697 | . . . . . . . 8 ⊢ (ℝ ⊆ ℝ* → (ℝ × ℝ*) ⊆ (ℝ* × ℝ*)) | |
7 | 5, 6 | ax-mp 5 | . . . . . . 7 ⊢ (ℝ × ℝ*) ⊆ (ℝ* × ℝ*) |
8 | 7 | a1i 11 | . . . . . 6 ⊢ (𝜑 → (ℝ × ℝ*) ⊆ (ℝ* × ℝ*)) |
9 | volicoff.1 | . . . . . 6 ⊢ (𝜑 → 𝐹:𝐴⟶(ℝ × ℝ*)) | |
10 | 4, 8, 9 | fcoss 44724 | . . . . 5 ⊢ (𝜑 → ([,) ∘ 𝐹):𝐴⟶𝒫 ℝ*) |
11 | 10 | ffnd 6724 | . . . 4 ⊢ (𝜑 → ([,) ∘ 𝐹) Fn 𝐴) |
12 | 9 | adantr 479 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐹:𝐴⟶(ℝ × ℝ*)) |
13 | simpr 483 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐴) | |
14 | 12, 13 | fvovco 44707 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (([,) ∘ 𝐹)‘𝑥) = ((1st ‘(𝐹‘𝑥))[,)(2nd ‘(𝐹‘𝑥)))) |
15 | 9 | ffvelcdmda 7093 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) ∈ (ℝ × ℝ*)) |
16 | xp1st 8026 | . . . . . . . 8 ⊢ ((𝐹‘𝑥) ∈ (ℝ × ℝ*) → (1st ‘(𝐹‘𝑥)) ∈ ℝ) | |
17 | 15, 16 | syl 17 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (1st ‘(𝐹‘𝑥)) ∈ ℝ) |
18 | xp2nd 8027 | . . . . . . . 8 ⊢ ((𝐹‘𝑥) ∈ (ℝ × ℝ*) → (2nd ‘(𝐹‘𝑥)) ∈ ℝ*) | |
19 | 15, 18 | syl 17 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (2nd ‘(𝐹‘𝑥)) ∈ ℝ*) |
20 | icombl 25542 | . . . . . . 7 ⊢ (((1st ‘(𝐹‘𝑥)) ∈ ℝ ∧ (2nd ‘(𝐹‘𝑥)) ∈ ℝ*) → ((1st ‘(𝐹‘𝑥))[,)(2nd ‘(𝐹‘𝑥))) ∈ dom vol) | |
21 | 17, 19, 20 | syl2anc 582 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((1st ‘(𝐹‘𝑥))[,)(2nd ‘(𝐹‘𝑥))) ∈ dom vol) |
22 | 14, 21 | eqeltrd 2825 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (([,) ∘ 𝐹)‘𝑥) ∈ dom vol) |
23 | 22 | ralrimiva 3135 | . . . 4 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 (([,) ∘ 𝐹)‘𝑥) ∈ dom vol) |
24 | fnfvrnss 7130 | . . . 4 ⊢ ((([,) ∘ 𝐹) Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (([,) ∘ 𝐹)‘𝑥) ∈ dom vol) → ran ([,) ∘ 𝐹) ⊆ dom vol) | |
25 | 11, 23, 24 | syl2anc 582 | . . 3 ⊢ (𝜑 → ran ([,) ∘ 𝐹) ⊆ dom vol) |
26 | ffrn 6736 | . . . 4 ⊢ (([,) ∘ 𝐹):𝐴⟶𝒫 ℝ* → ([,) ∘ 𝐹):𝐴⟶ran ([,) ∘ 𝐹)) | |
27 | 10, 26 | syl 17 | . . 3 ⊢ (𝜑 → ([,) ∘ 𝐹):𝐴⟶ran ([,) ∘ 𝐹)) |
28 | 2, 25, 27 | fcoss 44724 | . 2 ⊢ (𝜑 → (vol ∘ ([,) ∘ 𝐹)):𝐴⟶(0[,]+∞)) |
29 | coass 6271 | . . . 4 ⊢ ((vol ∘ [,)) ∘ 𝐹) = (vol ∘ ([,) ∘ 𝐹)) | |
30 | 29 | feq1i 6714 | . . 3 ⊢ (((vol ∘ [,)) ∘ 𝐹):𝐴⟶(0[,]+∞) ↔ (vol ∘ ([,) ∘ 𝐹)):𝐴⟶(0[,]+∞)) |
31 | 30 | a1i 11 | . 2 ⊢ (𝜑 → (((vol ∘ [,)) ∘ 𝐹):𝐴⟶(0[,]+∞) ↔ (vol ∘ ([,) ∘ 𝐹)):𝐴⟶(0[,]+∞))) |
32 | 28, 31 | mpbird 256 | 1 ⊢ (𝜑 → ((vol ∘ [,)) ∘ 𝐹):𝐴⟶(0[,]+∞)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 ∈ wcel 2098 ∀wral 3050 ⊆ wss 3944 𝒫 cpw 4604 × cxp 5676 dom cdm 5678 ran crn 5679 ∘ ccom 5682 Fn wfn 6544 ⟶wf 6545 ‘cfv 6549 (class class class)co 7419 1st c1st 7992 2nd c2nd 7993 ℝcr 11144 0cc0 11145 +∞cpnf 11282 ℝ*cxr 11284 [,)cico 13366 [,]cicc 13367 volcvol 25441 |
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 2166 ax-ext 2696 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 ax-inf2 9671 ax-cnex 11201 ax-resscn 11202 ax-1cn 11203 ax-icn 11204 ax-addcl 11205 ax-addrcl 11206 ax-mulcl 11207 ax-mulrcl 11208 ax-mulcom 11209 ax-addass 11210 ax-mulass 11211 ax-distr 11212 ax-i2m1 11213 ax-1ne0 11214 ax-1rid 11215 ax-rnegex 11216 ax-rrecex 11217 ax-cnre 11218 ax-pre-lttri 11219 ax-pre-lttrn 11220 ax-pre-ltadd 11221 ax-pre-mulgt0 11222 ax-pre-sup 11223 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 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 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-nel 3036 df-ral 3051 df-rex 3060 df-rmo 3363 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3964 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-int 4951 df-iun 4999 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-se 5634 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6307 df-ord 6374 df-on 6375 df-lim 6376 df-suc 6377 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-isom 6558 df-riota 7375 df-ov 7422 df-oprab 7423 df-mpo 7424 df-of 7685 df-om 7872 df-1st 7994 df-2nd 7995 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-1o 8487 df-2o 8488 df-er 8725 df-map 8847 df-pm 8848 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-sup 9472 df-inf 9473 df-oi 9540 df-dju 9931 df-card 9969 df-pnf 11287 df-mnf 11288 df-xr 11289 df-ltxr 11290 df-le 11291 df-sub 11483 df-neg 11484 df-div 11909 df-nn 12251 df-2 12313 df-3 12314 df-n0 12511 df-z 12597 df-uz 12861 df-q 12971 df-rp 13015 df-xadd 13133 df-ioo 13368 df-ico 13370 df-icc 13371 df-fz 13525 df-fzo 13668 df-fl 13798 df-seq 14008 df-exp 14068 df-hash 14331 df-cj 15087 df-re 15088 df-im 15089 df-sqrt 15223 df-abs 15224 df-clim 15473 df-rlim 15474 df-sum 15674 df-xmet 21294 df-met 21295 df-ovol 25442 df-vol 25443 |
This theorem is referenced by: volicofmpt 45525 |
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