Theorem itg2val 23495

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 11974	. . 3 ⊢ < Or ℝ*
2	1	supex 8369	. 2 ⊢ sup(𝐿, ℝ*, < ) ∈ V
3		reex 10027	. 2 ⊢ ℝ ∈ V
4		ovex 6678	. 2 ⊢ (0[,]+∞) ∈ V
5		breq2 4657	. . . . . . 7 ⊢ (𝑓 = 𝐹 → (𝑔 ∘𝑟 ≤ 𝑓 ↔ 𝑔 ∘𝑟 ≤ 𝐹))
6	5	anbi1d 741	. . . . . 6 ⊢ (𝑓 = 𝐹 → ((𝑔 ∘𝑟 ≤ 𝑓 ∧ 𝑥 = (∫1‘𝑔)) ↔ (𝑔 ∘𝑟 ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑔))))
7	6	rexbidv 3052	. . . . 5 ⊢ (𝑓 = 𝐹 → (∃𝑔 ∈ dom ∫1(𝑔 ∘𝑟 ≤ 𝑓 ∧ 𝑥 = (∫1‘𝑔)) ↔ ∃𝑔 ∈ dom ∫1(𝑔 ∘𝑟 ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑔))))
8	7	abbidv 2741	. . . 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	syl6eqr 2674	. . 3 ⊢ (𝑓 = 𝐹 → {𝑥 ∣ ∃𝑔 ∈ dom ∫1(𝑔 ∘𝑟 ≤ 𝑓 ∧ 𝑥 = (∫1‘𝑔))} = 𝐿)
11	10	supeq1d 8352	. 2 ⊢ (𝑓 = 𝐹 → sup({𝑥 ∣ ∃𝑔 ∈ dom ∫1(𝑔 ∘𝑟 ≤ 𝑓 ∧ 𝑥 = (∫1‘𝑔))}, ℝ*, < ) = sup(𝐿, ℝ*, < ))
12		df-itg2 23390	. 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 7894	1 ⊢ (𝐹:ℝ⟶(0[,]+∞) → (∫2‘𝐹) = sup(𝐿, ℝ*, < ))

Syntax hints:   → wi 4   ∧ wa 384   = wceq 1483  {cab 2608  ∃wrex 2913   class class class wbr 4653  dom cdm 5114  ⟶wf 5884  ‘cfv 5888  (class class class)co 6650   ∘_𝑟 cofr 6896  supcsup 8346  ℝcr 9935  0cc0 9936  +∞cpnf 10071  ℝ^*cxr 10073   < clt 10074   ≤ cle 10075  [,]cicc 12178  ∫₁citg1 23384  ∫₂citg2 23385

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-pre-lttri 10010  ax-pre-lttrn 10011

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-po 5035  df-so 5036  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-sup 8348  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-itg2 23390

This theorem is referenced by:  itg2cl  23499  itg2ub  23500  itg2leub  23501  itg2addnclem  33461