![]() |
Mathbox for Glauco Siliprandi |
< Previous
Next >
Nearby theorems |
|
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 25578 | . . . 4 ⊢ vol:dom vol⟶(0[,]+∞) | |
2 | 1 | a1i 11 | . . 3 ⊢ (𝜑 → vol:dom vol⟶(0[,]+∞)) |
3 | icof 45162 | . . . . . . 7 ⊢ [,):(ℝ* × ℝ*)⟶𝒫 ℝ* | |
4 | 3 | a1i 11 | . . . . . 6 ⊢ (𝜑 → [,):(ℝ* × ℝ*)⟶𝒫 ℝ*) |
5 | ressxr 11303 | . . . . . . . 8 ⊢ ℝ ⊆ ℝ* | |
6 | xpss1 5708 | . . . . . . . 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 45153 | . . . . 5 ⊢ (𝜑 → ([,) ∘ 𝐹):𝐴⟶𝒫 ℝ*) |
11 | 10 | ffnd 6738 | . . . 4 ⊢ (𝜑 → ([,) ∘ 𝐹) Fn 𝐴) |
12 | 9 | adantr 480 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐹:𝐴⟶(ℝ × ℝ*)) |
13 | simpr 484 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐴) | |
14 | 12, 13 | fvovco 45136 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (([,) ∘ 𝐹)‘𝑥) = ((1st ‘(𝐹‘𝑥))[,)(2nd ‘(𝐹‘𝑥)))) |
15 | 9 | ffvelcdmda 7104 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) ∈ (ℝ × ℝ*)) |
16 | xp1st 8045 | . . . . . . . 8 ⊢ ((𝐹‘𝑥) ∈ (ℝ × ℝ*) → (1st ‘(𝐹‘𝑥)) ∈ ℝ) | |
17 | 15, 16 | syl 17 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (1st ‘(𝐹‘𝑥)) ∈ ℝ) |
18 | xp2nd 8046 | . . . . . . . 8 ⊢ ((𝐹‘𝑥) ∈ (ℝ × ℝ*) → (2nd ‘(𝐹‘𝑥)) ∈ ℝ*) | |
19 | 15, 18 | syl 17 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (2nd ‘(𝐹‘𝑥)) ∈ ℝ*) |
20 | icombl 25613 | . . . . . . 7 ⊢ (((1st ‘(𝐹‘𝑥)) ∈ ℝ ∧ (2nd ‘(𝐹‘𝑥)) ∈ ℝ*) → ((1st ‘(𝐹‘𝑥))[,)(2nd ‘(𝐹‘𝑥))) ∈ dom vol) | |
21 | 17, 19, 20 | syl2anc 584 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((1st ‘(𝐹‘𝑥))[,)(2nd ‘(𝐹‘𝑥))) ∈ dom vol) |
22 | 14, 21 | eqeltrd 2839 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (([,) ∘ 𝐹)‘𝑥) ∈ dom vol) |
23 | 22 | ralrimiva 3144 | . . . 4 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 (([,) ∘ 𝐹)‘𝑥) ∈ dom vol) |
24 | fnfvrnss 7141 | . . . 4 ⊢ ((([,) ∘ 𝐹) Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (([,) ∘ 𝐹)‘𝑥) ∈ dom vol) → ran ([,) ∘ 𝐹) ⊆ dom vol) | |
25 | 11, 23, 24 | syl2anc 584 | . . 3 ⊢ (𝜑 → ran ([,) ∘ 𝐹) ⊆ dom vol) |
26 | ffrn 6750 | . . . 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 45153 | . 2 ⊢ (𝜑 → (vol ∘ ([,) ∘ 𝐹)):𝐴⟶(0[,]+∞)) |
29 | coass 6287 | . . . 4 ⊢ ((vol ∘ [,)) ∘ 𝐹) = (vol ∘ ([,) ∘ 𝐹)) | |
30 | 29 | feq1i 6728 | . . 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 257 | 1 ⊢ (𝜑 → ((vol ∘ [,)) ∘ 𝐹):𝐴⟶(0[,]+∞)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∈ wcel 2106 ∀wral 3059 ⊆ wss 3963 𝒫 cpw 4605 × cxp 5687 dom cdm 5689 ran crn 5690 ∘ ccom 5693 Fn wfn 6558 ⟶wf 6559 ‘cfv 6563 (class class class)co 7431 1st c1st 8011 2nd c2nd 8012 ℝcr 11152 0cc0 11153 +∞cpnf 11290 ℝ*cxr 11292 [,)cico 13386 [,]cicc 13387 volcvol 25512 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-inf2 9679 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 ax-pre-sup 11231 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-se 5642 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-isom 6572 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-om 7888 df-1st 8013 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-2o 8506 df-er 8744 df-map 8867 df-pm 8868 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-sup 9480 df-inf 9481 df-oi 9548 df-dju 9939 df-card 9977 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-div 11919 df-nn 12265 df-2 12327 df-3 12328 df-n0 12525 df-z 12612 df-uz 12877 df-q 12989 df-rp 13033 df-xadd 13153 df-ioo 13388 df-ico 13390 df-icc 13391 df-fz 13545 df-fzo 13692 df-fl 13829 df-seq 14040 df-exp 14100 df-hash 14367 df-cj 15135 df-re 15136 df-im 15137 df-sqrt 15271 df-abs 15272 df-clim 15521 df-rlim 15522 df-sum 15720 df-xmet 21375 df-met 21376 df-ovol 25513 df-vol 25514 |
This theorem is referenced by: volicofmpt 45953 |
Copyright terms: Public domain | W3C validator |