Theorem itg2val 25609

Description: Value of the integral on nonnegative real functions. (Contributed by Mario Carneiro, 28-Jun-2014.)

Hypothesis

Ref	Expression
itg2val.1	⊢ 𝐿 = {𝑥 ∣ ∃𝑔 ∈ dom ∫1(𝑔 ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑔))}

Assertion

Ref	Expression
itg2val	⊢ (𝐹:ℝ⟶(0[,]+∞) → (∫2‘𝐹) = sup(𝐿, ℝ*, < ))

Distinct variable group:   𝑥,𝑔,𝐹

Allowed substitution hints:   𝐿(𝑥,𝑔)

Proof of Theorem itg2val

Dummy variable 𝑓 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		xrltso 13123	. . 3 ⊢ < Or ℝ*
2	1	supex 9457	. 2 ⊢ sup(𝐿, ℝ*, < ) ∈ V
3		reex 11200	. 2 ⊢ ℝ ∈ V
4		ovex 7437	. 2 ⊢ (0[,]+∞) ∈ V
5		breq2 5145	. . . . . . 7 ⊢ (𝑓 = 𝐹 → (𝑔 ∘r ≤ 𝑓 ↔ 𝑔 ∘r ≤ 𝐹))
6	5	anbi1d 629	. . . . . 6 ⊢ (𝑓 = 𝐹 → ((𝑔 ∘r ≤ 𝑓 ∧ 𝑥 = (∫1‘𝑔)) ↔ (𝑔 ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑔))))
7	6	rexbidv 3172	. . . . 5 ⊢ (𝑓 = 𝐹 → (∃𝑔 ∈ dom ∫1(𝑔 ∘r ≤ 𝑓 ∧ 𝑥 = (∫1‘𝑔)) ↔ ∃𝑔 ∈ dom ∫1(𝑔 ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑔))))
8	7	abbidv 2795	. . . 4 ⊢ (𝑓 = 𝐹 → {𝑥 ∣ ∃𝑔 ∈ dom ∫1(𝑔 ∘r ≤ 𝑓 ∧ 𝑥 = (∫1‘𝑔))} = {𝑥 ∣ ∃𝑔 ∈ dom ∫1(𝑔 ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑔))})
9		itg2val.1	. . . 4 ⊢ 𝐿 = {𝑥 ∣ ∃𝑔 ∈ dom ∫1(𝑔 ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑔))}
10	8, 9	eqtr4di 2784	. . 3 ⊢ (𝑓 = 𝐹 → {𝑥 ∣ ∃𝑔 ∈ dom ∫1(𝑔 ∘r ≤ 𝑓 ∧ 𝑥 = (∫1‘𝑔))} = 𝐿)
11	10	supeq1d 9440	. 2 ⊢ (𝑓 = 𝐹 → sup({𝑥 ∣ ∃𝑔 ∈ dom ∫1(𝑔 ∘r ≤ 𝑓 ∧ 𝑥 = (∫1‘𝑔))}, ℝ*, < ) = sup(𝐿, ℝ*, < ))
12		df-itg2 25501	. 2 ⊢ ∫2 = (𝑓 ∈ ((0[,]+∞) ↑m ℝ) ↦ sup({𝑥 ∣ ∃𝑔 ∈ dom ∫1(𝑔 ∘r ≤ 𝑓 ∧ 𝑥 = (∫1‘𝑔))}, ℝ*, < ))
13	2, 3, 4, 11, 12	fvmptmap 8874	1 ⊢ (𝐹:ℝ⟶(0[,]+∞) → (∫2‘𝐹) = sup(𝐿, ℝ*, < ))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1533  {cab 2703  ∃wrex 3064   class class class wbr 5141  dom cdm 5669  ⟶wf 6532  ‘cfv 6536  (class class class)co 7404   ∘_r cofr 7665  supcsup 9434  ℝcr 11108  0cc0 11109  +∞cpnf 11246  ℝ^*cxr 11248   < clt 11249   ≤ cle 11250  [,]cicc 13330  ∫₁citg1 25495  ∫₂citg2 25496

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-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7721  ax-cnex 11165  ax-resscn 11166  ax-pre-lttri 11183  ax-pre-lttrn 11184

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-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-po 5581  df-so 5582  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-iota 6488  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-ov 7407  df-oprab 7408  df-mpo 7409  df-er 8702  df-map 8821  df-en 8939  df-dom 8940  df-sdom 8941  df-sup 9436  df-pnf 11251  df-mnf 11252  df-xr 11253  df-ltxr 11254  df-itg2 25501

This theorem is referenced by:  itg2cl  25613  itg2ub  25614  itg2leub  25615  itg2addnclem  37050