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Theorem ftc2nc 36163
Description: Choice-free proof of ftc2 25411. (Contributed by Brendan Leahy, 19-Jun-2018.)
Hypotheses
Ref Expression
ftc2nc.a (πœ‘ β†’ 𝐴 ∈ ℝ)
ftc2nc.b (πœ‘ β†’ 𝐡 ∈ ℝ)
ftc2nc.le (πœ‘ β†’ 𝐴 ≀ 𝐡)
ftc2nc.c (πœ‘ β†’ (ℝ D 𝐹) ∈ ((𝐴(,)𝐡)–cnβ†’β„‚))
ftc2nc.i (πœ‘ β†’ (ℝ D 𝐹) ∈ 𝐿1)
ftc2nc.f (πœ‘ β†’ 𝐹 ∈ ((𝐴[,]𝐡)–cnβ†’β„‚))
Assertion
Ref Expression
ftc2nc (πœ‘ β†’ ∫(𝐴(,)𝐡)((ℝ D 𝐹)β€˜π‘‘) d𝑑 = ((πΉβ€˜π΅) βˆ’ (πΉβ€˜π΄)))
Distinct variable groups:   𝑑,𝐴   𝑑,𝐡   𝑑,𝐹   πœ‘,𝑑

Proof of Theorem ftc2nc
Dummy variables 𝑠 π‘₯ are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ftc2nc.a . . . . . . 7 (πœ‘ β†’ 𝐴 ∈ ℝ)
21rexrd 11206 . . . . . 6 (πœ‘ β†’ 𝐴 ∈ ℝ*)
3 ftc2nc.b . . . . . . 7 (πœ‘ β†’ 𝐡 ∈ ℝ)
43rexrd 11206 . . . . . 6 (πœ‘ β†’ 𝐡 ∈ ℝ*)
5 ftc2nc.le . . . . . 6 (πœ‘ β†’ 𝐴 ≀ 𝐡)
6 ubicc2 13383 . . . . . 6 ((𝐴 ∈ ℝ* ∧ 𝐡 ∈ ℝ* ∧ 𝐴 ≀ 𝐡) β†’ 𝐡 ∈ (𝐴[,]𝐡))
72, 4, 5, 6syl3anc 1372 . . . . 5 (πœ‘ β†’ 𝐡 ∈ (𝐴[,]𝐡))
8 fvex 6856 . . . . . 6 ((π‘₯ ∈ (𝐴[,]𝐡) ↦ (∫(𝐴(,)π‘₯)((ℝ D 𝐹)β€˜π‘‘) d𝑑 βˆ’ (πΉβ€˜π‘₯)))β€˜π΄) ∈ V
98fvconst2 7154 . . . . 5 (𝐡 ∈ (𝐴[,]𝐡) β†’ (((𝐴[,]𝐡) Γ— {((π‘₯ ∈ (𝐴[,]𝐡) ↦ (∫(𝐴(,)π‘₯)((ℝ D 𝐹)β€˜π‘‘) d𝑑 βˆ’ (πΉβ€˜π‘₯)))β€˜π΄)})β€˜π΅) = ((π‘₯ ∈ (𝐴[,]𝐡) ↦ (∫(𝐴(,)π‘₯)((ℝ D 𝐹)β€˜π‘‘) d𝑑 βˆ’ (πΉβ€˜π‘₯)))β€˜π΄))
107, 9syl 17 . . . 4 (πœ‘ β†’ (((𝐴[,]𝐡) Γ— {((π‘₯ ∈ (𝐴[,]𝐡) ↦ (∫(𝐴(,)π‘₯)((ℝ D 𝐹)β€˜π‘‘) d𝑑 βˆ’ (πΉβ€˜π‘₯)))β€˜π΄)})β€˜π΅) = ((π‘₯ ∈ (𝐴[,]𝐡) ↦ (∫(𝐴(,)π‘₯)((ℝ D 𝐹)β€˜π‘‘) d𝑑 βˆ’ (πΉβ€˜π‘₯)))β€˜π΄))
11 eqid 2737 . . . . . . . 8 (TopOpenβ€˜β„‚fld) = (TopOpenβ€˜β„‚fld)
1211subcn 24232 . . . . . . . . 9 βˆ’ ∈ (((TopOpenβ€˜β„‚fld) Γ—t (TopOpenβ€˜β„‚fld)) Cn (TopOpenβ€˜β„‚fld))
1312a1i 11 . . . . . . . 8 (πœ‘ β†’ βˆ’ ∈ (((TopOpenβ€˜β„‚fld) Γ—t (TopOpenβ€˜β„‚fld)) Cn (TopOpenβ€˜β„‚fld)))
14 eqid 2737 . . . . . . . . 9 (π‘₯ ∈ (𝐴[,]𝐡) ↦ ∫(𝐴(,)π‘₯)((ℝ D 𝐹)β€˜π‘‘) d𝑑) = (π‘₯ ∈ (𝐴[,]𝐡) ↦ ∫(𝐴(,)π‘₯)((ℝ D 𝐹)β€˜π‘‘) d𝑑)
15 ssidd 3968 . . . . . . . . 9 (πœ‘ β†’ (𝐴(,)𝐡) βŠ† (𝐴(,)𝐡))
16 ioossre 13326 . . . . . . . . . 10 (𝐴(,)𝐡) βŠ† ℝ
1716a1i 11 . . . . . . . . 9 (πœ‘ β†’ (𝐴(,)𝐡) βŠ† ℝ)
18 ftc2nc.i . . . . . . . . 9 (πœ‘ β†’ (ℝ D 𝐹) ∈ 𝐿1)
19 ftc2nc.c . . . . . . . . . 10 (πœ‘ β†’ (ℝ D 𝐹) ∈ ((𝐴(,)𝐡)–cnβ†’β„‚))
20 cncff 24259 . . . . . . . . . 10 ((ℝ D 𝐹) ∈ ((𝐴(,)𝐡)–cnβ†’β„‚) β†’ (ℝ D 𝐹):(𝐴(,)𝐡)βŸΆβ„‚)
2119, 20syl 17 . . . . . . . . 9 (πœ‘ β†’ (ℝ D 𝐹):(𝐴(,)𝐡)βŸΆβ„‚)
22 ioof 13365 . . . . . . . . . . . . 13 (,):(ℝ* Γ— ℝ*)βŸΆπ’« ℝ
23 ffun 6672 . . . . . . . . . . . . 13 ((,):(ℝ* Γ— ℝ*)βŸΆπ’« ℝ β†’ Fun (,))
2422, 23ax-mp 5 . . . . . . . . . . . 12 Fun (,)
25 fvelima 6909 . . . . . . . . . . . 12 ((Fun (,) ∧ 𝑠 ∈ ((,) β€œ ((𝐴[,]𝐡) Γ— (𝐴[,]𝐡)))) β†’ βˆƒπ‘₯ ∈ ((𝐴[,]𝐡) Γ— (𝐴[,]𝐡))((,)β€˜π‘₯) = 𝑠)
2624, 25mpan 689 . . . . . . . . . . 11 (𝑠 ∈ ((,) β€œ ((𝐴[,]𝐡) Γ— (𝐴[,]𝐡))) β†’ βˆƒπ‘₯ ∈ ((𝐴[,]𝐡) Γ— (𝐴[,]𝐡))((,)β€˜π‘₯) = 𝑠)
27 1st2nd2 7961 . . . . . . . . . . . . . . . . 17 (π‘₯ ∈ ((𝐴[,]𝐡) Γ— (𝐴[,]𝐡)) β†’ π‘₯ = ⟨(1st β€˜π‘₯), (2nd β€˜π‘₯)⟩)
2827fveq2d 6847 . . . . . . . . . . . . . . . 16 (π‘₯ ∈ ((𝐴[,]𝐡) Γ— (𝐴[,]𝐡)) β†’ ((,)β€˜π‘₯) = ((,)β€˜βŸ¨(1st β€˜π‘₯), (2nd β€˜π‘₯)⟩))
29 df-ov 7361 . . . . . . . . . . . . . . . 16 ((1st β€˜π‘₯)(,)(2nd β€˜π‘₯)) = ((,)β€˜βŸ¨(1st β€˜π‘₯), (2nd β€˜π‘₯)⟩)
3028, 29eqtr4di 2795 . . . . . . . . . . . . . . 15 (π‘₯ ∈ ((𝐴[,]𝐡) Γ— (𝐴[,]𝐡)) β†’ ((,)β€˜π‘₯) = ((1st β€˜π‘₯)(,)(2nd β€˜π‘₯)))
3130eqeq1d 2739 . . . . . . . . . . . . . 14 (π‘₯ ∈ ((𝐴[,]𝐡) Γ— (𝐴[,]𝐡)) β†’ (((,)β€˜π‘₯) = 𝑠 ↔ ((1st β€˜π‘₯)(,)(2nd β€˜π‘₯)) = 𝑠))
3231adantl 483 . . . . . . . . . . . . 13 ((πœ‘ ∧ π‘₯ ∈ ((𝐴[,]𝐡) Γ— (𝐴[,]𝐡))) β†’ (((,)β€˜π‘₯) = 𝑠 ↔ ((1st β€˜π‘₯)(,)(2nd β€˜π‘₯)) = 𝑠))
332, 4jca 513 . . . . . . . . . . . . . . . . . . . 20 (πœ‘ β†’ (𝐴 ∈ ℝ* ∧ 𝐡 ∈ ℝ*))
3433adantr 482 . . . . . . . . . . . . . . . . . . 19 ((πœ‘ ∧ π‘₯ ∈ ((𝐴[,]𝐡) Γ— (𝐴[,]𝐡))) β†’ (𝐴 ∈ ℝ* ∧ 𝐡 ∈ ℝ*))
35 xp1st 7954 . . . . . . . . . . . . . . . . . . . 20 (π‘₯ ∈ ((𝐴[,]𝐡) Γ— (𝐴[,]𝐡)) β†’ (1st β€˜π‘₯) ∈ (𝐴[,]𝐡))
36 elicc1 13309 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝐴 ∈ ℝ* ∧ 𝐡 ∈ ℝ*) β†’ ((1st β€˜π‘₯) ∈ (𝐴[,]𝐡) ↔ ((1st β€˜π‘₯) ∈ ℝ* ∧ 𝐴 ≀ (1st β€˜π‘₯) ∧ (1st β€˜π‘₯) ≀ 𝐡)))
372, 4, 36syl2anc 585 . . . . . . . . . . . . . . . . . . . . . 22 (πœ‘ β†’ ((1st β€˜π‘₯) ∈ (𝐴[,]𝐡) ↔ ((1st β€˜π‘₯) ∈ ℝ* ∧ 𝐴 ≀ (1st β€˜π‘₯) ∧ (1st β€˜π‘₯) ≀ 𝐡)))
3837biimpa 478 . . . . . . . . . . . . . . . . . . . . 21 ((πœ‘ ∧ (1st β€˜π‘₯) ∈ (𝐴[,]𝐡)) β†’ ((1st β€˜π‘₯) ∈ ℝ* ∧ 𝐴 ≀ (1st β€˜π‘₯) ∧ (1st β€˜π‘₯) ≀ 𝐡))
3938simp2d 1144 . . . . . . . . . . . . . . . . . . . 20 ((πœ‘ ∧ (1st β€˜π‘₯) ∈ (𝐴[,]𝐡)) β†’ 𝐴 ≀ (1st β€˜π‘₯))
4035, 39sylan2 594 . . . . . . . . . . . . . . . . . . 19 ((πœ‘ ∧ π‘₯ ∈ ((𝐴[,]𝐡) Γ— (𝐴[,]𝐡))) β†’ 𝐴 ≀ (1st β€˜π‘₯))
41 xp2nd 7955 . . . . . . . . . . . . . . . . . . . 20 (π‘₯ ∈ ((𝐴[,]𝐡) Γ— (𝐴[,]𝐡)) β†’ (2nd β€˜π‘₯) ∈ (𝐴[,]𝐡))
42 iccleub 13320 . . . . . . . . . . . . . . . . . . . . 21 ((𝐴 ∈ ℝ* ∧ 𝐡 ∈ ℝ* ∧ (2nd β€˜π‘₯) ∈ (𝐴[,]𝐡)) β†’ (2nd β€˜π‘₯) ≀ 𝐡)
43423expa 1119 . . . . . . . . . . . . . . . . . . . 20 (((𝐴 ∈ ℝ* ∧ 𝐡 ∈ ℝ*) ∧ (2nd β€˜π‘₯) ∈ (𝐴[,]𝐡)) β†’ (2nd β€˜π‘₯) ≀ 𝐡)
4433, 41, 43syl2an 597 . . . . . . . . . . . . . . . . . . 19 ((πœ‘ ∧ π‘₯ ∈ ((𝐴[,]𝐡) Γ— (𝐴[,]𝐡))) β†’ (2nd β€˜π‘₯) ≀ 𝐡)
45 ioossioo 13359 . . . . . . . . . . . . . . . . . . 19 (((𝐴 ∈ ℝ* ∧ 𝐡 ∈ ℝ*) ∧ (𝐴 ≀ (1st β€˜π‘₯) ∧ (2nd β€˜π‘₯) ≀ 𝐡)) β†’ ((1st β€˜π‘₯)(,)(2nd β€˜π‘₯)) βŠ† (𝐴(,)𝐡))
4634, 40, 44, 45syl12anc 836 . . . . . . . . . . . . . . . . . 18 ((πœ‘ ∧ π‘₯ ∈ ((𝐴[,]𝐡) Γ— (𝐴[,]𝐡))) β†’ ((1st β€˜π‘₯)(,)(2nd β€˜π‘₯)) βŠ† (𝐴(,)𝐡))
4746sselda 3945 . . . . . . . . . . . . . . . . 17 (((πœ‘ ∧ π‘₯ ∈ ((𝐴[,]𝐡) Γ— (𝐴[,]𝐡))) ∧ 𝑑 ∈ ((1st β€˜π‘₯)(,)(2nd β€˜π‘₯))) β†’ 𝑑 ∈ (𝐴(,)𝐡))
4821ffvelcdmda 7036 . . . . . . . . . . . . . . . . . 18 ((πœ‘ ∧ 𝑑 ∈ (𝐴(,)𝐡)) β†’ ((ℝ D 𝐹)β€˜π‘‘) ∈ β„‚)
4948adantlr 714 . . . . . . . . . . . . . . . . 17 (((πœ‘ ∧ π‘₯ ∈ ((𝐴[,]𝐡) Γ— (𝐴[,]𝐡))) ∧ 𝑑 ∈ (𝐴(,)𝐡)) β†’ ((ℝ D 𝐹)β€˜π‘‘) ∈ β„‚)
5047, 49syldan 592 . . . . . . . . . . . . . . . 16 (((πœ‘ ∧ π‘₯ ∈ ((𝐴[,]𝐡) Γ— (𝐴[,]𝐡))) ∧ 𝑑 ∈ ((1st β€˜π‘₯)(,)(2nd β€˜π‘₯))) β†’ ((ℝ D 𝐹)β€˜π‘‘) ∈ β„‚)
51 ioombl 24932 . . . . . . . . . . . . . . . . . 18 ((1st β€˜π‘₯)(,)(2nd β€˜π‘₯)) ∈ dom vol
5251a1i 11 . . . . . . . . . . . . . . . . 17 ((πœ‘ ∧ π‘₯ ∈ ((𝐴[,]𝐡) Γ— (𝐴[,]𝐡))) β†’ ((1st β€˜π‘₯)(,)(2nd β€˜π‘₯)) ∈ dom vol)
53 fvexd 6858 . . . . . . . . . . . . . . . . 17 (((πœ‘ ∧ π‘₯ ∈ ((𝐴[,]𝐡) Γ— (𝐴[,]𝐡))) ∧ 𝑑 ∈ (𝐴(,)𝐡)) β†’ ((ℝ D 𝐹)β€˜π‘‘) ∈ V)
5421feqmptd 6911 . . . . . . . . . . . . . . . . . . 19 (πœ‘ β†’ (ℝ D 𝐹) = (𝑑 ∈ (𝐴(,)𝐡) ↦ ((ℝ D 𝐹)β€˜π‘‘)))
5554, 18eqeltrrd 2839 . . . . . . . . . . . . . . . . . 18 (πœ‘ β†’ (𝑑 ∈ (𝐴(,)𝐡) ↦ ((ℝ D 𝐹)β€˜π‘‘)) ∈ 𝐿1)
5655adantr 482 . . . . . . . . . . . . . . . . 17 ((πœ‘ ∧ π‘₯ ∈ ((𝐴[,]𝐡) Γ— (𝐴[,]𝐡))) β†’ (𝑑 ∈ (𝐴(,)𝐡) ↦ ((ℝ D 𝐹)β€˜π‘‘)) ∈ 𝐿1)
5746, 52, 53, 56iblss 25172 . . . . . . . . . . . . . . . 16 ((πœ‘ ∧ π‘₯ ∈ ((𝐴[,]𝐡) Γ— (𝐴[,]𝐡))) β†’ (𝑑 ∈ ((1st β€˜π‘₯)(,)(2nd β€˜π‘₯)) ↦ ((ℝ D 𝐹)β€˜π‘‘)) ∈ 𝐿1)
58 ax-resscn 11109 . . . . . . . . . . . . . . . . . . . . 21 ℝ βŠ† β„‚
59 ssid 3967 . . . . . . . . . . . . . . . . . . . . 21 β„‚ βŠ† β„‚
60 cncfss 24265 . . . . . . . . . . . . . . . . . . . . 21 ((ℝ βŠ† β„‚ ∧ β„‚ βŠ† β„‚) β†’ (ℂ–cn→ℝ) βŠ† (ℂ–cnβ†’β„‚))
6158, 59, 60mp2an 691 . . . . . . . . . . . . . . . . . . . 20 (ℂ–cn→ℝ) βŠ† (ℂ–cnβ†’β„‚)
62 abscncf 24267 . . . . . . . . . . . . . . . . . . . 20 abs ∈ (ℂ–cn→ℝ)
6361, 62sselii 3942 . . . . . . . . . . . . . . . . . . 19 abs ∈ (ℂ–cnβ†’β„‚)
6463a1i 11 . . . . . . . . . . . . . . . . . 18 ((πœ‘ ∧ π‘₯ ∈ ((𝐴[,]𝐡) Γ— (𝐴[,]𝐡))) β†’ abs ∈ (ℂ–cnβ†’β„‚))
6554reseq1d 5937 . . . . . . . . . . . . . . . . . . . . 21 (πœ‘ β†’ ((ℝ D 𝐹) β†Ύ ((1st β€˜π‘₯)(,)(2nd β€˜π‘₯))) = ((𝑑 ∈ (𝐴(,)𝐡) ↦ ((ℝ D 𝐹)β€˜π‘‘)) β†Ύ ((1st β€˜π‘₯)(,)(2nd β€˜π‘₯))))
6665adantr 482 . . . . . . . . . . . . . . . . . . . 20 ((πœ‘ ∧ π‘₯ ∈ ((𝐴[,]𝐡) Γ— (𝐴[,]𝐡))) β†’ ((ℝ D 𝐹) β†Ύ ((1st β€˜π‘₯)(,)(2nd β€˜π‘₯))) = ((𝑑 ∈ (𝐴(,)𝐡) ↦ ((ℝ D 𝐹)β€˜π‘‘)) β†Ύ ((1st β€˜π‘₯)(,)(2nd β€˜π‘₯))))
6746resmptd 5995 . . . . . . . . . . . . . . . . . . . 20 ((πœ‘ ∧ π‘₯ ∈ ((𝐴[,]𝐡) Γ— (𝐴[,]𝐡))) β†’ ((𝑑 ∈ (𝐴(,)𝐡) ↦ ((ℝ D 𝐹)β€˜π‘‘)) β†Ύ ((1st β€˜π‘₯)(,)(2nd β€˜π‘₯))) = (𝑑 ∈ ((1st β€˜π‘₯)(,)(2nd β€˜π‘₯)) ↦ ((ℝ D 𝐹)β€˜π‘‘)))
6866, 67eqtrd 2777 . . . . . . . . . . . . . . . . . . 19 ((πœ‘ ∧ π‘₯ ∈ ((𝐴[,]𝐡) Γ— (𝐴[,]𝐡))) β†’ ((ℝ D 𝐹) β†Ύ ((1st β€˜π‘₯)(,)(2nd β€˜π‘₯))) = (𝑑 ∈ ((1st β€˜π‘₯)(,)(2nd β€˜π‘₯)) ↦ ((ℝ D 𝐹)β€˜π‘‘)))
6919adantr 482 . . . . . . . . . . . . . . . . . . . 20 ((πœ‘ ∧ π‘₯ ∈ ((𝐴[,]𝐡) Γ— (𝐴[,]𝐡))) β†’ (ℝ D 𝐹) ∈ ((𝐴(,)𝐡)–cnβ†’β„‚))
70 rescncf 24263 . . . . . . . . . . . . . . . . . . . 20 (((1st β€˜π‘₯)(,)(2nd β€˜π‘₯)) βŠ† (𝐴(,)𝐡) β†’ ((ℝ D 𝐹) ∈ ((𝐴(,)𝐡)–cnβ†’β„‚) β†’ ((ℝ D 𝐹) β†Ύ ((1st β€˜π‘₯)(,)(2nd β€˜π‘₯))) ∈ (((1st β€˜π‘₯)(,)(2nd β€˜π‘₯))–cnβ†’β„‚)))
7146, 69, 70sylc 65 . . . . . . . . . . . . . . . . . . 19 ((πœ‘ ∧ π‘₯ ∈ ((𝐴[,]𝐡) Γ— (𝐴[,]𝐡))) β†’ ((ℝ D 𝐹) β†Ύ ((1st β€˜π‘₯)(,)(2nd β€˜π‘₯))) ∈ (((1st β€˜π‘₯)(,)(2nd β€˜π‘₯))–cnβ†’β„‚))
7268, 71eqeltrrd 2839 . . . . . . . . . . . . . . . . . 18 ((πœ‘ ∧ π‘₯ ∈ ((𝐴[,]𝐡) Γ— (𝐴[,]𝐡))) β†’ (𝑑 ∈ ((1st β€˜π‘₯)(,)(2nd β€˜π‘₯)) ↦ ((ℝ D 𝐹)β€˜π‘‘)) ∈ (((1st β€˜π‘₯)(,)(2nd β€˜π‘₯))–cnβ†’β„‚))
7364, 72cncfmpt1f 24280 . . . . . . . . . . . . . . . . 17 ((πœ‘ ∧ π‘₯ ∈ ((𝐴[,]𝐡) Γ— (𝐴[,]𝐡))) β†’ (𝑑 ∈ ((1st β€˜π‘₯)(,)(2nd β€˜π‘₯)) ↦ (absβ€˜((ℝ D 𝐹)β€˜π‘‘))) ∈ (((1st β€˜π‘₯)(,)(2nd β€˜π‘₯))–cnβ†’β„‚))
74 cnmbf 25026 . . . . . . . . . . . . . . . . 17 ((((1st β€˜π‘₯)(,)(2nd β€˜π‘₯)) ∈ dom vol ∧ (𝑑 ∈ ((1st β€˜π‘₯)(,)(2nd β€˜π‘₯)) ↦ (absβ€˜((ℝ D 𝐹)β€˜π‘‘))) ∈ (((1st β€˜π‘₯)(,)(2nd β€˜π‘₯))–cnβ†’β„‚)) β†’ (𝑑 ∈ ((1st β€˜π‘₯)(,)(2nd β€˜π‘₯)) ↦ (absβ€˜((ℝ D 𝐹)β€˜π‘‘))) ∈ MblFn)
7551, 73, 74sylancr 588 . . . . . . . . . . . . . . . 16 ((πœ‘ ∧ π‘₯ ∈ ((𝐴[,]𝐡) Γ— (𝐴[,]𝐡))) β†’ (𝑑 ∈ ((1st β€˜π‘₯)(,)(2nd β€˜π‘₯)) ↦ (absβ€˜((ℝ D 𝐹)β€˜π‘‘))) ∈ MblFn)
7650, 57itgcl 25151 . . . . . . . . . . . . . . . . . . . 20 ((πœ‘ ∧ π‘₯ ∈ ((𝐴[,]𝐡) Γ— (𝐴[,]𝐡))) β†’ ∫((1st β€˜π‘₯)(,)(2nd β€˜π‘₯))((ℝ D 𝐹)β€˜π‘‘) d𝑑 ∈ β„‚)
7776cjcld 15082 . . . . . . . . . . . . . . . . . . 19 ((πœ‘ ∧ π‘₯ ∈ ((𝐴[,]𝐡) Γ— (𝐴[,]𝐡))) β†’ (βˆ—β€˜βˆ«((1st β€˜π‘₯)(,)(2nd β€˜π‘₯))((ℝ D 𝐹)β€˜π‘‘) d𝑑) ∈ β„‚)
78 ioossre 13326 . . . . . . . . . . . . . . . . . . . . 21 ((1st β€˜π‘₯)(,)(2nd β€˜π‘₯)) βŠ† ℝ
7978, 58sstri 3954 . . . . . . . . . . . . . . . . . . . 20 ((1st β€˜π‘₯)(,)(2nd β€˜π‘₯)) βŠ† β„‚
80 cncfmptc 24278 . . . . . . . . . . . . . . . . . . . 20 (((βˆ—β€˜βˆ«((1st β€˜π‘₯)(,)(2nd β€˜π‘₯))((ℝ D 𝐹)β€˜π‘‘) d𝑑) ∈ β„‚ ∧ ((1st β€˜π‘₯)(,)(2nd β€˜π‘₯)) βŠ† β„‚ ∧ β„‚ βŠ† β„‚) β†’ (𝑠 ∈ ((1st β€˜π‘₯)(,)(2nd β€˜π‘₯)) ↦ (βˆ—β€˜βˆ«((1st β€˜π‘₯)(,)(2nd β€˜π‘₯))((ℝ D 𝐹)β€˜π‘‘) d𝑑)) ∈ (((1st β€˜π‘₯)(,)(2nd β€˜π‘₯))–cnβ†’β„‚))
8179, 59, 80mp3an23 1454 . . . . . . . . . . . . . . . . . . 19 ((βˆ—β€˜βˆ«((1st β€˜π‘₯)(,)(2nd β€˜π‘₯))((ℝ D 𝐹)β€˜π‘‘) d𝑑) ∈ β„‚ β†’ (𝑠 ∈ ((1st β€˜π‘₯)(,)(2nd β€˜π‘₯)) ↦ (βˆ—β€˜βˆ«((1st β€˜π‘₯)(,)(2nd β€˜π‘₯))((ℝ D 𝐹)β€˜π‘‘) d𝑑)) ∈ (((1st β€˜π‘₯)(,)(2nd β€˜π‘₯))–cnβ†’β„‚))
8277, 81syl 17 . . . . . . . . . . . . . . . . . 18 ((πœ‘ ∧ π‘₯ ∈ ((𝐴[,]𝐡) Γ— (𝐴[,]𝐡))) β†’ (𝑠 ∈ ((1st β€˜π‘₯)(,)(2nd β€˜π‘₯)) ↦ (βˆ—β€˜βˆ«((1st β€˜π‘₯)(,)(2nd β€˜π‘₯))((ℝ D 𝐹)β€˜π‘‘) d𝑑)) ∈ (((1st β€˜π‘₯)(,)(2nd β€˜π‘₯))–cnβ†’β„‚))
83 nfcv 2908 . . . . . . . . . . . . . . . . . . . 20 Ⅎ𝑠((ℝ D 𝐹)β€˜π‘‘)
84 nfcsb1v 3881 . . . . . . . . . . . . . . . . . . . 20 Ⅎ𝑑⦋𝑠 / π‘‘β¦Œ((ℝ D 𝐹)β€˜π‘‘)
85 csbeq1a 3870 . . . . . . . . . . . . . . . . . . . 20 (𝑑 = 𝑠 β†’ ((ℝ D 𝐹)β€˜π‘‘) = ⦋𝑠 / π‘‘β¦Œ((ℝ D 𝐹)β€˜π‘‘))
8683, 84, 85cbvmpt 5217 . . . . . . . . . . . . . . . . . . 19 (𝑑 ∈ ((1st β€˜π‘₯)(,)(2nd β€˜π‘₯)) ↦ ((ℝ D 𝐹)β€˜π‘‘)) = (𝑠 ∈ ((1st β€˜π‘₯)(,)(2nd β€˜π‘₯)) ↦ ⦋𝑠 / π‘‘β¦Œ((ℝ D 𝐹)β€˜π‘‘))
8786, 72eqeltrrid 2843 . . . . . . . . . . . . . . . . . 18 ((πœ‘ ∧ π‘₯ ∈ ((𝐴[,]𝐡) Γ— (𝐴[,]𝐡))) β†’ (𝑠 ∈ ((1st β€˜π‘₯)(,)(2nd β€˜π‘₯)) ↦ ⦋𝑠 / π‘‘β¦Œ((ℝ D 𝐹)β€˜π‘‘)) ∈ (((1st β€˜π‘₯)(,)(2nd β€˜π‘₯))–cnβ†’β„‚))
8882, 87mulcncf 24813 . . . . . . . . . . . . . . . . 17 ((πœ‘ ∧ π‘₯ ∈ ((𝐴[,]𝐡) Γ— (𝐴[,]𝐡))) β†’ (𝑠 ∈ ((1st β€˜π‘₯)(,)(2nd β€˜π‘₯)) ↦ ((βˆ—β€˜βˆ«((1st β€˜π‘₯)(,)(2nd β€˜π‘₯))((ℝ D 𝐹)β€˜π‘‘) d𝑑) Β· ⦋𝑠 / π‘‘β¦Œ((ℝ D 𝐹)β€˜π‘‘))) ∈ (((1st β€˜π‘₯)(,)(2nd β€˜π‘₯))–cnβ†’β„‚))
89 cnmbf 25026 . . . . . . . . . . . . . . . . 17 ((((1st β€˜π‘₯)(,)(2nd β€˜π‘₯)) ∈ dom vol ∧ (𝑠 ∈ ((1st β€˜π‘₯)(,)(2nd β€˜π‘₯)) ↦ ((βˆ—β€˜βˆ«((1st β€˜π‘₯)(,)(2nd β€˜π‘₯))((ℝ D 𝐹)β€˜π‘‘) d𝑑) Β· ⦋𝑠 / π‘‘β¦Œ((ℝ D 𝐹)β€˜π‘‘))) ∈ (((1st β€˜π‘₯)(,)(2nd β€˜π‘₯))–cnβ†’β„‚)) β†’ (𝑠 ∈ ((1st β€˜π‘₯)(,)(2nd β€˜π‘₯)) ↦ ((βˆ—β€˜βˆ«((1st β€˜π‘₯)(,)(2nd β€˜π‘₯))((ℝ D 𝐹)β€˜π‘‘) d𝑑) Β· ⦋𝑠 / π‘‘β¦Œ((ℝ D 𝐹)β€˜π‘‘))) ∈ MblFn)
9051, 88, 89sylancr 588 . . . . . . . . . . . . . . . 16 ((πœ‘ ∧ π‘₯ ∈ ((𝐴[,]𝐡) Γ— (𝐴[,]𝐡))) β†’ (𝑠 ∈ ((1st β€˜π‘₯)(,)(2nd β€˜π‘₯)) ↦ ((βˆ—β€˜βˆ«((1st β€˜π‘₯)(,)(2nd β€˜π‘₯))((ℝ D 𝐹)β€˜π‘‘) d𝑑) Β· ⦋𝑠 / π‘‘β¦Œ((ℝ D 𝐹)β€˜π‘‘))) ∈ MblFn)
9150, 57, 75, 90itgabsnc 36150 . . . . . . . . . . . . . . 15 ((πœ‘ ∧ π‘₯ ∈ ((𝐴[,]𝐡) Γ— (𝐴[,]𝐡))) β†’ (absβ€˜βˆ«((1st β€˜π‘₯)(,)(2nd β€˜π‘₯))((ℝ D 𝐹)β€˜π‘‘) d𝑑) ≀ ∫((1st β€˜π‘₯)(,)(2nd β€˜π‘₯))(absβ€˜((ℝ D 𝐹)β€˜π‘‘)) d𝑑)
9250abscld 15322 . . . . . . . . . . . . . . . 16 (((πœ‘ ∧ π‘₯ ∈ ((𝐴[,]𝐡) Γ— (𝐴[,]𝐡))) ∧ 𝑑 ∈ ((1st β€˜π‘₯)(,)(2nd β€˜π‘₯))) β†’ (absβ€˜((ℝ D 𝐹)β€˜π‘‘)) ∈ ℝ)
93 fvexd 6858 . . . . . . . . . . . . . . . . 17 (((πœ‘ ∧ π‘₯ ∈ ((𝐴[,]𝐡) Γ— (𝐴[,]𝐡))) ∧ 𝑑 ∈ ((1st β€˜π‘₯)(,)(2nd β€˜π‘₯))) β†’ ((ℝ D 𝐹)β€˜π‘‘) ∈ V)
9493, 57, 75iblabsnc 36145 . . . . . . . . . . . . . . . 16 ((πœ‘ ∧ π‘₯ ∈ ((𝐴[,]𝐡) Γ— (𝐴[,]𝐡))) β†’ (𝑑 ∈ ((1st β€˜π‘₯)(,)(2nd β€˜π‘₯)) ↦ (absβ€˜((ℝ D 𝐹)β€˜π‘‘))) ∈ 𝐿1)
9550absge0d 15330 . . . . . . . . . . . . . . . 16 (((πœ‘ ∧ π‘₯ ∈ ((𝐴[,]𝐡) Γ— (𝐴[,]𝐡))) ∧ 𝑑 ∈ ((1st β€˜π‘₯)(,)(2nd β€˜π‘₯))) β†’ 0 ≀ (absβ€˜((ℝ D 𝐹)β€˜π‘‘)))
9692, 94, 95itgposval 25163 . . . . . . . . . . . . . . 15 ((πœ‘ ∧ π‘₯ ∈ ((𝐴[,]𝐡) Γ— (𝐴[,]𝐡))) β†’ ∫((1st β€˜π‘₯)(,)(2nd β€˜π‘₯))(absβ€˜((ℝ D 𝐹)β€˜π‘‘)) d𝑑 = (∫2β€˜(𝑑 ∈ ℝ ↦ if(𝑑 ∈ ((1st β€˜π‘₯)(,)(2nd β€˜π‘₯)), (absβ€˜((ℝ D 𝐹)β€˜π‘‘)), 0))))
9791, 96breqtrd 5132 . . . . . . . . . . . . . 14 ((πœ‘ ∧ π‘₯ ∈ ((𝐴[,]𝐡) Γ— (𝐴[,]𝐡))) β†’ (absβ€˜βˆ«((1st β€˜π‘₯)(,)(2nd β€˜π‘₯))((ℝ D 𝐹)β€˜π‘‘) d𝑑) ≀ (∫2β€˜(𝑑 ∈ ℝ ↦ if(𝑑 ∈ ((1st β€˜π‘₯)(,)(2nd β€˜π‘₯)), (absβ€˜((ℝ D 𝐹)β€˜π‘‘)), 0))))
98 itgeq1 25140 . . . . . . . . . . . . . . . 16 (((1st β€˜π‘₯)(,)(2nd β€˜π‘₯)) = 𝑠 β†’ ∫((1st β€˜π‘₯)(,)(2nd β€˜π‘₯))((ℝ D 𝐹)β€˜π‘‘) d𝑑 = βˆ«π‘ ((ℝ D 𝐹)β€˜π‘‘) d𝑑)
9998fveq2d 6847 . . . . . . . . . . . . . . 15 (((1st β€˜π‘₯)(,)(2nd β€˜π‘₯)) = 𝑠 β†’ (absβ€˜βˆ«((1st β€˜π‘₯)(,)(2nd β€˜π‘₯))((ℝ D 𝐹)β€˜π‘‘) d𝑑) = (absβ€˜βˆ«π‘ ((ℝ D 𝐹)β€˜π‘‘) d𝑑))
100 eleq2 2827 . . . . . . . . . . . . . . . . . 18 (((1st β€˜π‘₯)(,)(2nd β€˜π‘₯)) = 𝑠 β†’ (𝑑 ∈ ((1st β€˜π‘₯)(,)(2nd β€˜π‘₯)) ↔ 𝑑 ∈ 𝑠))
101100ifbid 4510 . . . . . . . . . . . . . . . . 17 (((1st β€˜π‘₯)(,)(2nd β€˜π‘₯)) = 𝑠 β†’ if(𝑑 ∈ ((1st β€˜π‘₯)(,)(2nd β€˜π‘₯)), (absβ€˜((ℝ D 𝐹)β€˜π‘‘)), 0) = if(𝑑 ∈ 𝑠, (absβ€˜((ℝ D 𝐹)β€˜π‘‘)), 0))
102101mpteq2dv 5208 . . . . . . . . . . . . . . . 16 (((1st β€˜π‘₯)(,)(2nd β€˜π‘₯)) = 𝑠 β†’ (𝑑 ∈ ℝ ↦ if(𝑑 ∈ ((1st β€˜π‘₯)(,)(2nd β€˜π‘₯)), (absβ€˜((ℝ D 𝐹)β€˜π‘‘)), 0)) = (𝑑 ∈ ℝ ↦ if(𝑑 ∈ 𝑠, (absβ€˜((ℝ D 𝐹)β€˜π‘‘)), 0)))
103102fveq2d 6847 . . . . . . . . . . . . . . 15 (((1st β€˜π‘₯)(,)(2nd β€˜π‘₯)) = 𝑠 β†’ (∫2β€˜(𝑑 ∈ ℝ ↦ if(𝑑 ∈ ((1st β€˜π‘₯)(,)(2nd β€˜π‘₯)), (absβ€˜((ℝ D 𝐹)β€˜π‘‘)), 0))) = (∫2β€˜(𝑑 ∈ ℝ ↦ if(𝑑 ∈ 𝑠, (absβ€˜((ℝ D 𝐹)β€˜π‘‘)), 0))))
10499, 103breq12d 5119 . . . . . . . . . . . . . 14 (((1st β€˜π‘₯)(,)(2nd β€˜π‘₯)) = 𝑠 β†’ ((absβ€˜βˆ«((1st β€˜π‘₯)(,)(2nd β€˜π‘₯))((ℝ D 𝐹)β€˜π‘‘) d𝑑) ≀ (∫2β€˜(𝑑 ∈ ℝ ↦ if(𝑑 ∈ ((1st β€˜π‘₯)(,)(2nd β€˜π‘₯)), (absβ€˜((ℝ D 𝐹)β€˜π‘‘)), 0))) ↔ (absβ€˜βˆ«π‘ ((ℝ D 𝐹)β€˜π‘‘) d𝑑) ≀ (∫2β€˜(𝑑 ∈ ℝ ↦ if(𝑑 ∈ 𝑠, (absβ€˜((ℝ D 𝐹)β€˜π‘‘)), 0)))))
10597, 104syl5ibcom 244 . . . . . . . . . . . . 13 ((πœ‘ ∧ π‘₯ ∈ ((𝐴[,]𝐡) Γ— (𝐴[,]𝐡))) β†’ (((1st β€˜π‘₯)(,)(2nd β€˜π‘₯)) = 𝑠 β†’ (absβ€˜βˆ«π‘ ((ℝ D 𝐹)β€˜π‘‘) d𝑑) ≀ (∫2β€˜(𝑑 ∈ ℝ ↦ if(𝑑 ∈ 𝑠, (absβ€˜((ℝ D 𝐹)β€˜π‘‘)), 0)))))
10632, 105sylbid 239 . . . . . . . . . . . 12 ((πœ‘ ∧ π‘₯ ∈ ((𝐴[,]𝐡) Γ— (𝐴[,]𝐡))) β†’ (((,)β€˜π‘₯) = 𝑠 β†’ (absβ€˜βˆ«π‘ ((ℝ D 𝐹)β€˜π‘‘) d𝑑) ≀ (∫2β€˜(𝑑 ∈ ℝ ↦ if(𝑑 ∈ 𝑠, (absβ€˜((ℝ D 𝐹)β€˜π‘‘)), 0)))))
107106rexlimdva 3153 . . . . . . . . . . 11 (πœ‘ β†’ (βˆƒπ‘₯ ∈ ((𝐴[,]𝐡) Γ— (𝐴[,]𝐡))((,)β€˜π‘₯) = 𝑠 β†’ (absβ€˜βˆ«π‘ ((ℝ D 𝐹)β€˜π‘‘) d𝑑) ≀ (∫2β€˜(𝑑 ∈ ℝ ↦ if(𝑑 ∈ 𝑠, (absβ€˜((ℝ D 𝐹)β€˜π‘‘)), 0)))))
10826, 107syl5 34 . . . . . . . . . 10 (πœ‘ β†’ (𝑠 ∈ ((,) β€œ ((𝐴[,]𝐡) Γ— (𝐴[,]𝐡))) β†’ (absβ€˜βˆ«π‘ ((ℝ D 𝐹)β€˜π‘‘) d𝑑) ≀ (∫2β€˜(𝑑 ∈ ℝ ↦ if(𝑑 ∈ 𝑠, (absβ€˜((ℝ D 𝐹)β€˜π‘‘)), 0)))))
109108ralrimiv 3143 . . . . . . . . 9 (πœ‘ β†’ βˆ€π‘  ∈ ((,) β€œ ((𝐴[,]𝐡) Γ— (𝐴[,]𝐡)))(absβ€˜βˆ«π‘ ((ℝ D 𝐹)β€˜π‘‘) d𝑑) ≀ (∫2β€˜(𝑑 ∈ ℝ ↦ if(𝑑 ∈ 𝑠, (absβ€˜((ℝ D 𝐹)β€˜π‘‘)), 0))))
11014, 1, 3, 5, 15, 17, 18, 21, 109ftc1anc 36162 . . . . . . . 8 (πœ‘ β†’ (π‘₯ ∈ (𝐴[,]𝐡) ↦ ∫(𝐴(,)π‘₯)((ℝ D 𝐹)β€˜π‘‘) d𝑑) ∈ ((𝐴[,]𝐡)–cnβ†’β„‚))
111 ftc2nc.f . . . . . . . . . . 11 (πœ‘ β†’ 𝐹 ∈ ((𝐴[,]𝐡)–cnβ†’β„‚))
112 cncff 24259 . . . . . . . . . . 11 (𝐹 ∈ ((𝐴[,]𝐡)–cnβ†’β„‚) β†’ 𝐹:(𝐴[,]𝐡)βŸΆβ„‚)
113111, 112syl 17 . . . . . . . . . 10 (πœ‘ β†’ 𝐹:(𝐴[,]𝐡)βŸΆβ„‚)
114113feqmptd 6911 . . . . . . . . 9 (πœ‘ β†’ 𝐹 = (π‘₯ ∈ (𝐴[,]𝐡) ↦ (πΉβ€˜π‘₯)))
115114, 111eqeltrrd 2839 . . . . . . . 8 (πœ‘ β†’ (π‘₯ ∈ (𝐴[,]𝐡) ↦ (πΉβ€˜π‘₯)) ∈ ((𝐴[,]𝐡)–cnβ†’β„‚))
11611, 13, 110, 115cncfmpt2f 24281 . . . . . . 7 (πœ‘ β†’ (π‘₯ ∈ (𝐴[,]𝐡) ↦ (∫(𝐴(,)π‘₯)((ℝ D 𝐹)β€˜π‘‘) d𝑑 βˆ’ (πΉβ€˜π‘₯))) ∈ ((𝐴[,]𝐡)–cnβ†’β„‚))
11758a1i 11 . . . . . . . . . 10 (πœ‘ β†’ ℝ βŠ† β„‚)
118 iccssre 13347 . . . . . . . . . . 11 ((𝐴 ∈ ℝ ∧ 𝐡 ∈ ℝ) β†’ (𝐴[,]𝐡) βŠ† ℝ)
1191, 3, 118syl2anc 585 . . . . . . . . . 10 (πœ‘ β†’ (𝐴[,]𝐡) βŠ† ℝ)
120 fvexd 6858 . . . . . . . . . . . 12 (((πœ‘ ∧ π‘₯ ∈ (𝐴[,]𝐡)) ∧ 𝑑 ∈ (𝐴(,)π‘₯)) β†’ ((ℝ D 𝐹)β€˜π‘‘) ∈ V)
1213adantr 482 . . . . . . . . . . . . . . 15 ((πœ‘ ∧ π‘₯ ∈ (𝐴[,]𝐡)) β†’ 𝐡 ∈ ℝ)
122121rexrd 11206 . . . . . . . . . . . . . 14 ((πœ‘ ∧ π‘₯ ∈ (𝐴[,]𝐡)) β†’ 𝐡 ∈ ℝ*)
123 elicc2 13330 . . . . . . . . . . . . . . . . 17 ((𝐴 ∈ ℝ ∧ 𝐡 ∈ ℝ) β†’ (π‘₯ ∈ (𝐴[,]𝐡) ↔ (π‘₯ ∈ ℝ ∧ 𝐴 ≀ π‘₯ ∧ π‘₯ ≀ 𝐡)))
1241, 3, 123syl2anc 585 . . . . . . . . . . . . . . . 16 (πœ‘ β†’ (π‘₯ ∈ (𝐴[,]𝐡) ↔ (π‘₯ ∈ ℝ ∧ 𝐴 ≀ π‘₯ ∧ π‘₯ ≀ 𝐡)))
125124biimpa 478 . . . . . . . . . . . . . . 15 ((πœ‘ ∧ π‘₯ ∈ (𝐴[,]𝐡)) β†’ (π‘₯ ∈ ℝ ∧ 𝐴 ≀ π‘₯ ∧ π‘₯ ≀ 𝐡))
126125simp3d 1145 . . . . . . . . . . . . . 14 ((πœ‘ ∧ π‘₯ ∈ (𝐴[,]𝐡)) β†’ π‘₯ ≀ 𝐡)
127 iooss2 13301 . . . . . . . . . . . . . 14 ((𝐡 ∈ ℝ* ∧ π‘₯ ≀ 𝐡) β†’ (𝐴(,)π‘₯) βŠ† (𝐴(,)𝐡))
128122, 126, 127syl2anc 585 . . . . . . . . . . . . 13 ((πœ‘ ∧ π‘₯ ∈ (𝐴[,]𝐡)) β†’ (𝐴(,)π‘₯) βŠ† (𝐴(,)𝐡))
129 ioombl 24932 . . . . . . . . . . . . . 14 (𝐴(,)π‘₯) ∈ dom vol
130129a1i 11 . . . . . . . . . . . . 13 ((πœ‘ ∧ π‘₯ ∈ (𝐴[,]𝐡)) β†’ (𝐴(,)π‘₯) ∈ dom vol)
131 fvexd 6858 . . . . . . . . . . . . 13 (((πœ‘ ∧ π‘₯ ∈ (𝐴[,]𝐡)) ∧ 𝑑 ∈ (𝐴(,)𝐡)) β†’ ((ℝ D 𝐹)β€˜π‘‘) ∈ V)
13255adantr 482 . . . . . . . . . . . . 13 ((πœ‘ ∧ π‘₯ ∈ (𝐴[,]𝐡)) β†’ (𝑑 ∈ (𝐴(,)𝐡) ↦ ((ℝ D 𝐹)β€˜π‘‘)) ∈ 𝐿1)
133128, 130, 131, 132iblss 25172 . . . . . . . . . . . 12 ((πœ‘ ∧ π‘₯ ∈ (𝐴[,]𝐡)) β†’ (𝑑 ∈ (𝐴(,)π‘₯) ↦ ((ℝ D 𝐹)β€˜π‘‘)) ∈ 𝐿1)
134120, 133itgcl 25151 . . . . . . . . . . 11 ((πœ‘ ∧ π‘₯ ∈ (𝐴[,]𝐡)) β†’ ∫(𝐴(,)π‘₯)((ℝ D 𝐹)β€˜π‘‘) d𝑑 ∈ β„‚)
135113ffvelcdmda 7036 . . . . . . . . . . 11 ((πœ‘ ∧ π‘₯ ∈ (𝐴[,]𝐡)) β†’ (πΉβ€˜π‘₯) ∈ β„‚)
136134, 135subcld 11513 . . . . . . . . . 10 ((πœ‘ ∧ π‘₯ ∈ (𝐴[,]𝐡)) β†’ (∫(𝐴(,)π‘₯)((ℝ D 𝐹)β€˜π‘‘) d𝑑 βˆ’ (πΉβ€˜π‘₯)) ∈ β„‚)
13711tgioo2 24169 . . . . . . . . . 10 (topGenβ€˜ran (,)) = ((TopOpenβ€˜β„‚fld) β†Ύt ℝ)
138 iccntr 24187 . . . . . . . . . . 11 ((𝐴 ∈ ℝ ∧ 𝐡 ∈ ℝ) β†’ ((intβ€˜(topGenβ€˜ran (,)))β€˜(𝐴[,]𝐡)) = (𝐴(,)𝐡))
1391, 3, 138syl2anc 585 . . . . . . . . . 10 (πœ‘ β†’ ((intβ€˜(topGenβ€˜ran (,)))β€˜(𝐴[,]𝐡)) = (𝐴(,)𝐡))
140117, 119, 136, 137, 11, 139dvmptntr 25338 . . . . . . . . 9 (πœ‘ β†’ (ℝ D (π‘₯ ∈ (𝐴[,]𝐡) ↦ (∫(𝐴(,)π‘₯)((ℝ D 𝐹)β€˜π‘‘) d𝑑 βˆ’ (πΉβ€˜π‘₯)))) = (ℝ D (π‘₯ ∈ (𝐴(,)𝐡) ↦ (∫(𝐴(,)π‘₯)((ℝ D 𝐹)β€˜π‘‘) d𝑑 βˆ’ (πΉβ€˜π‘₯)))))
141 reelprrecn 11144 . . . . . . . . . . 11 ℝ ∈ {ℝ, β„‚}
142141a1i 11 . . . . . . . . . 10 (πœ‘ β†’ ℝ ∈ {ℝ, β„‚})
143 ioossicc 13351 . . . . . . . . . . . 12 (𝐴(,)𝐡) βŠ† (𝐴[,]𝐡)
144143sseli 3941 . . . . . . . . . . 11 (π‘₯ ∈ (𝐴(,)𝐡) β†’ π‘₯ ∈ (𝐴[,]𝐡))
145144, 134sylan2 594 . . . . . . . . . 10 ((πœ‘ ∧ π‘₯ ∈ (𝐴(,)𝐡)) β†’ ∫(𝐴(,)π‘₯)((ℝ D 𝐹)β€˜π‘‘) d𝑑 ∈ β„‚)
14621ffvelcdmda 7036 . . . . . . . . . 10 ((πœ‘ ∧ π‘₯ ∈ (𝐴(,)𝐡)) β†’ ((ℝ D 𝐹)β€˜π‘₯) ∈ β„‚)
14714, 1, 3, 5, 19, 18ftc1cnnc 36153 . . . . . . . . . . 11 (πœ‘ β†’ (ℝ D (π‘₯ ∈ (𝐴[,]𝐡) ↦ ∫(𝐴(,)π‘₯)((ℝ D 𝐹)β€˜π‘‘) d𝑑)) = (ℝ D 𝐹))
148117, 119, 134, 137, 11, 139dvmptntr 25338 . . . . . . . . . . 11 (πœ‘ β†’ (ℝ D (π‘₯ ∈ (𝐴[,]𝐡) ↦ ∫(𝐴(,)π‘₯)((ℝ D 𝐹)β€˜π‘‘) d𝑑)) = (ℝ D (π‘₯ ∈ (𝐴(,)𝐡) ↦ ∫(𝐴(,)π‘₯)((ℝ D 𝐹)β€˜π‘‘) d𝑑)))
14921feqmptd 6911 . . . . . . . . . . 11 (πœ‘ β†’ (ℝ D 𝐹) = (π‘₯ ∈ (𝐴(,)𝐡) ↦ ((ℝ D 𝐹)β€˜π‘₯)))
150147, 148, 1493eqtr3d 2785 . . . . . . . . . 10 (πœ‘ β†’ (ℝ D (π‘₯ ∈ (𝐴(,)𝐡) ↦ ∫(𝐴(,)π‘₯)((ℝ D 𝐹)β€˜π‘‘) d𝑑)) = (π‘₯ ∈ (𝐴(,)𝐡) ↦ ((ℝ D 𝐹)β€˜π‘₯)))
151144, 135sylan2 594 . . . . . . . . . 10 ((πœ‘ ∧ π‘₯ ∈ (𝐴(,)𝐡)) β†’ (πΉβ€˜π‘₯) ∈ β„‚)
152114oveq2d 7374 . . . . . . . . . . 11 (πœ‘ β†’ (ℝ D 𝐹) = (ℝ D (π‘₯ ∈ (𝐴[,]𝐡) ↦ (πΉβ€˜π‘₯))))
153117, 119, 135, 137, 11, 139dvmptntr 25338 . . . . . . . . . . 11 (πœ‘ β†’ (ℝ D (π‘₯ ∈ (𝐴[,]𝐡) ↦ (πΉβ€˜π‘₯))) = (ℝ D (π‘₯ ∈ (𝐴(,)𝐡) ↦ (πΉβ€˜π‘₯))))
154152, 149, 1533eqtr3rd 2786 . . . . . . . . . 10 (πœ‘ β†’ (ℝ D (π‘₯ ∈ (𝐴(,)𝐡) ↦ (πΉβ€˜π‘₯))) = (π‘₯ ∈ (𝐴(,)𝐡) ↦ ((ℝ D 𝐹)β€˜π‘₯)))
155142, 145, 146, 150, 151, 146, 154dvmptsub 25334 . . . . . . . . 9 (πœ‘ β†’ (ℝ D (π‘₯ ∈ (𝐴(,)𝐡) ↦ (∫(𝐴(,)π‘₯)((ℝ D 𝐹)β€˜π‘‘) d𝑑 βˆ’ (πΉβ€˜π‘₯)))) = (π‘₯ ∈ (𝐴(,)𝐡) ↦ (((ℝ D 𝐹)β€˜π‘₯) βˆ’ ((ℝ D 𝐹)β€˜π‘₯))))
156146subidd 11501 . . . . . . . . . 10 ((πœ‘ ∧ π‘₯ ∈ (𝐴(,)𝐡)) β†’ (((ℝ D 𝐹)β€˜π‘₯) βˆ’ ((ℝ D 𝐹)β€˜π‘₯)) = 0)
157156mpteq2dva 5206 . . . . . . . . 9 (πœ‘ β†’ (π‘₯ ∈ (𝐴(,)𝐡) ↦ (((ℝ D 𝐹)β€˜π‘₯) βˆ’ ((ℝ D 𝐹)β€˜π‘₯))) = (π‘₯ ∈ (𝐴(,)𝐡) ↦ 0))
158140, 155, 1573eqtrd 2781 . . . . . . . 8 (πœ‘ β†’ (ℝ D (π‘₯ ∈ (𝐴[,]𝐡) ↦ (∫(𝐴(,)π‘₯)((ℝ D 𝐹)β€˜π‘‘) d𝑑 βˆ’ (πΉβ€˜π‘₯)))) = (π‘₯ ∈ (𝐴(,)𝐡) ↦ 0))
159 fconstmpt 5695 . . . . . . . 8 ((𝐴(,)𝐡) Γ— {0}) = (π‘₯ ∈ (𝐴(,)𝐡) ↦ 0)
160158, 159eqtr4di 2795 . . . . . . 7 (πœ‘ β†’ (ℝ D (π‘₯ ∈ (𝐴[,]𝐡) ↦ (∫(𝐴(,)π‘₯)((ℝ D 𝐹)β€˜π‘‘) d𝑑 βˆ’ (πΉβ€˜π‘₯)))) = ((𝐴(,)𝐡) Γ— {0}))
1611, 3, 116, 160dveq0 25367 . . . . . 6 (πœ‘ β†’ (π‘₯ ∈ (𝐴[,]𝐡) ↦ (∫(𝐴(,)π‘₯)((ℝ D 𝐹)β€˜π‘‘) d𝑑 βˆ’ (πΉβ€˜π‘₯))) = ((𝐴[,]𝐡) Γ— {((π‘₯ ∈ (𝐴[,]𝐡) ↦ (∫(𝐴(,)π‘₯)((ℝ D 𝐹)β€˜π‘‘) d𝑑 βˆ’ (πΉβ€˜π‘₯)))β€˜π΄)}))
162161fveq1d 6845 . . . . 5 (πœ‘ β†’ ((π‘₯ ∈ (𝐴[,]𝐡) ↦ (∫(𝐴(,)π‘₯)((ℝ D 𝐹)β€˜π‘‘) d𝑑 βˆ’ (πΉβ€˜π‘₯)))β€˜π΅) = (((𝐴[,]𝐡) Γ— {((π‘₯ ∈ (𝐴[,]𝐡) ↦ (∫(𝐴(,)π‘₯)((ℝ D 𝐹)β€˜π‘‘) d𝑑 βˆ’ (πΉβ€˜π‘₯)))β€˜π΄)})β€˜π΅))
163 oveq2 7366 . . . . . . . . 9 (π‘₯ = 𝐡 β†’ (𝐴(,)π‘₯) = (𝐴(,)𝐡))
164 itgeq1 25140 . . . . . . . . 9 ((𝐴(,)π‘₯) = (𝐴(,)𝐡) β†’ ∫(𝐴(,)π‘₯)((ℝ D 𝐹)β€˜π‘‘) d𝑑 = ∫(𝐴(,)𝐡)((ℝ D 𝐹)β€˜π‘‘) d𝑑)
165163, 164syl 17 . . . . . . . 8 (π‘₯ = 𝐡 β†’ ∫(𝐴(,)π‘₯)((ℝ D 𝐹)β€˜π‘‘) d𝑑 = ∫(𝐴(,)𝐡)((ℝ D 𝐹)β€˜π‘‘) d𝑑)
166 fveq2 6843 . . . . . . . 8 (π‘₯ = 𝐡 β†’ (πΉβ€˜π‘₯) = (πΉβ€˜π΅))
167165, 166oveq12d 7376 . . . . . . 7 (π‘₯ = 𝐡 β†’ (∫(𝐴(,)π‘₯)((ℝ D 𝐹)β€˜π‘‘) d𝑑 βˆ’ (πΉβ€˜π‘₯)) = (∫(𝐴(,)𝐡)((ℝ D 𝐹)β€˜π‘‘) d𝑑 βˆ’ (πΉβ€˜π΅)))
168 eqid 2737 . . . . . . 7 (π‘₯ ∈ (𝐴[,]𝐡) ↦ (∫(𝐴(,)π‘₯)((ℝ D 𝐹)β€˜π‘‘) d𝑑 βˆ’ (πΉβ€˜π‘₯))) = (π‘₯ ∈ (𝐴[,]𝐡) ↦ (∫(𝐴(,)π‘₯)((ℝ D 𝐹)β€˜π‘‘) d𝑑 βˆ’ (πΉβ€˜π‘₯)))
169 ovex 7391 . . . . . . 7 (∫(𝐴(,)𝐡)((ℝ D 𝐹)β€˜π‘‘) d𝑑 βˆ’ (πΉβ€˜π΅)) ∈ V
170167, 168, 169fvmpt 6949 . . . . . 6 (𝐡 ∈ (𝐴[,]𝐡) β†’ ((π‘₯ ∈ (𝐴[,]𝐡) ↦ (∫(𝐴(,)π‘₯)((ℝ D 𝐹)β€˜π‘‘) d𝑑 βˆ’ (πΉβ€˜π‘₯)))β€˜π΅) = (∫(𝐴(,)𝐡)((ℝ D 𝐹)β€˜π‘‘) d𝑑 βˆ’ (πΉβ€˜π΅)))
1717, 170syl 17 . . . . 5 (πœ‘ β†’ ((π‘₯ ∈ (𝐴[,]𝐡) ↦ (∫(𝐴(,)π‘₯)((ℝ D 𝐹)β€˜π‘‘) d𝑑 βˆ’ (πΉβ€˜π‘₯)))β€˜π΅) = (∫(𝐴(,)𝐡)((ℝ D 𝐹)β€˜π‘‘) d𝑑 βˆ’ (πΉβ€˜π΅)))
172162, 171eqtr3d 2779 . . . 4 (πœ‘ β†’ (((𝐴[,]𝐡) Γ— {((π‘₯ ∈ (𝐴[,]𝐡) ↦ (∫(𝐴(,)π‘₯)((ℝ D 𝐹)β€˜π‘‘) d𝑑 βˆ’ (πΉβ€˜π‘₯)))β€˜π΄)})β€˜π΅) = (∫(𝐴(,)𝐡)((ℝ D 𝐹)β€˜π‘‘) d𝑑 βˆ’ (πΉβ€˜π΅)))
173 lbicc2 13382 . . . . . 6 ((𝐴 ∈ ℝ* ∧ 𝐡 ∈ ℝ* ∧ 𝐴 ≀ 𝐡) β†’ 𝐴 ∈ (𝐴[,]𝐡))
1742, 4, 5, 173syl3anc 1372 . . . . 5 (πœ‘ β†’ 𝐴 ∈ (𝐴[,]𝐡))
175 oveq2 7366 . . . . . . . . . . 11 (π‘₯ = 𝐴 β†’ (𝐴(,)π‘₯) = (𝐴(,)𝐴))
176 iooid 13293 . . . . . . . . . . 11 (𝐴(,)𝐴) = βˆ…
177175, 176eqtrdi 2793 . . . . . . . . . 10 (π‘₯ = 𝐴 β†’ (𝐴(,)π‘₯) = βˆ…)
178 itgeq1 25140 . . . . . . . . . 10 ((𝐴(,)π‘₯) = βˆ… β†’ ∫(𝐴(,)π‘₯)((ℝ D 𝐹)β€˜π‘‘) d𝑑 = βˆ«βˆ…((ℝ D 𝐹)β€˜π‘‘) d𝑑)
179177, 178syl 17 . . . . . . . . 9 (π‘₯ = 𝐴 β†’ ∫(𝐴(,)π‘₯)((ℝ D 𝐹)β€˜π‘‘) d𝑑 = βˆ«βˆ…((ℝ D 𝐹)β€˜π‘‘) d𝑑)
180 itg0 25147 . . . . . . . . 9 βˆ«βˆ…((ℝ D 𝐹)β€˜π‘‘) d𝑑 = 0
181179, 180eqtrdi 2793 . . . . . . . 8 (π‘₯ = 𝐴 β†’ ∫(𝐴(,)π‘₯)((ℝ D 𝐹)β€˜π‘‘) d𝑑 = 0)
182 fveq2 6843 . . . . . . . 8 (π‘₯ = 𝐴 β†’ (πΉβ€˜π‘₯) = (πΉβ€˜π΄))
183181, 182oveq12d 7376 . . . . . . 7 (π‘₯ = 𝐴 β†’ (∫(𝐴(,)π‘₯)((ℝ D 𝐹)β€˜π‘‘) d𝑑 βˆ’ (πΉβ€˜π‘₯)) = (0 βˆ’ (πΉβ€˜π΄)))
184 df-neg 11389 . . . . . . 7 -(πΉβ€˜π΄) = (0 βˆ’ (πΉβ€˜π΄))
185183, 184eqtr4di 2795 . . . . . 6 (π‘₯ = 𝐴 β†’ (∫(𝐴(,)π‘₯)((ℝ D 𝐹)β€˜π‘‘) d𝑑 βˆ’ (πΉβ€˜π‘₯)) = -(πΉβ€˜π΄))
186 negex 11400 . . . . . 6 -(πΉβ€˜π΄) ∈ V
187185, 168, 186fvmpt 6949 . . . . 5 (𝐴 ∈ (𝐴[,]𝐡) β†’ ((π‘₯ ∈ (𝐴[,]𝐡) ↦ (∫(𝐴(,)π‘₯)((ℝ D 𝐹)β€˜π‘‘) d𝑑 βˆ’ (πΉβ€˜π‘₯)))β€˜π΄) = -(πΉβ€˜π΄))
188174, 187syl 17 . . . 4 (πœ‘ β†’ ((π‘₯ ∈ (𝐴[,]𝐡) ↦ (∫(𝐴(,)π‘₯)((ℝ D 𝐹)β€˜π‘‘) d𝑑 βˆ’ (πΉβ€˜π‘₯)))β€˜π΄) = -(πΉβ€˜π΄))
18910, 172, 1883eqtr3d 2785 . . 3 (πœ‘ β†’ (∫(𝐴(,)𝐡)((ℝ D 𝐹)β€˜π‘‘) d𝑑 βˆ’ (πΉβ€˜π΅)) = -(πΉβ€˜π΄))
190189oveq2d 7374 . 2 (πœ‘ β†’ ((πΉβ€˜π΅) + (∫(𝐴(,)𝐡)((ℝ D 𝐹)β€˜π‘‘) d𝑑 βˆ’ (πΉβ€˜π΅))) = ((πΉβ€˜π΅) + -(πΉβ€˜π΄)))
191113, 7ffvelcdmd 7037 . . 3 (πœ‘ β†’ (πΉβ€˜π΅) ∈ β„‚)
192 fvexd 6858 . . . 4 ((πœ‘ ∧ 𝑑 ∈ (𝐴(,)𝐡)) β†’ ((ℝ D 𝐹)β€˜π‘‘) ∈ V)
193192, 55itgcl 25151 . . 3 (πœ‘ β†’ ∫(𝐴(,)𝐡)((ℝ D 𝐹)β€˜π‘‘) d𝑑 ∈ β„‚)
194191, 193pncan3d 11516 . 2 (πœ‘ β†’ ((πΉβ€˜π΅) + (∫(𝐴(,)𝐡)((ℝ D 𝐹)β€˜π‘‘) d𝑑 βˆ’ (πΉβ€˜π΅))) = ∫(𝐴(,)𝐡)((ℝ D 𝐹)β€˜π‘‘) d𝑑)
195113, 174ffvelcdmd 7037 . . 3 (πœ‘ β†’ (πΉβ€˜π΄) ∈ β„‚)
196191, 195negsubd 11519 . 2 (πœ‘ β†’ ((πΉβ€˜π΅) + -(πΉβ€˜π΄)) = ((πΉβ€˜π΅) βˆ’ (πΉβ€˜π΄)))
197190, 194, 1963eqtr3d 2785 1 (πœ‘ β†’ ∫(𝐴(,)𝐡)((ℝ D 𝐹)β€˜π‘‘) d𝑑 = ((πΉβ€˜π΅) βˆ’ (πΉβ€˜π΄)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107  βˆƒwrex 3074  Vcvv 3446  β¦‹csb 3856   βŠ† wss 3911  βˆ…c0 4283  ifcif 4487  π’« cpw 4561  {csn 4587  {cpr 4589  βŸ¨cop 4593   class class class wbr 5106   ↦ cmpt 5189   Γ— cxp 5632  dom cdm 5634  ran crn 5635   β†Ύ cres 5636   β€œ cima 5637  Fun wfun 6491  βŸΆwf 6493  β€˜cfv 6497  (class class class)co 7358  1st c1st 7920  2nd c2nd 7921  β„‚cc 11050  β„cr 11051  0cc0 11052   + caddc 11055   Β· cmul 11057  β„*cxr 11189   ≀ cle 11191   βˆ’ cmin 11386  -cneg 11387  (,)cioo 13265  [,]cicc 13268  βˆ—ccj 14982  abscabs 15120  TopOpenctopn 17304  topGenctg 17320  β„‚fldccnfld 20799  intcnt 22371   Cn ccn 22578   Γ—t ctx 22914  β€“cnβ†’ccncf 24242  volcvol 24830  MblFncmbf 24981  βˆ«2citg2 24983  πΏ1cibl 24984  βˆ«citg 24985   D cdv 25230
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-rep 5243  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673  ax-inf2 9578  ax-cnex 11108  ax-resscn 11109  ax-1cn 11110  ax-icn 11111  ax-addcl 11112  ax-addrcl 11113  ax-mulcl 11114  ax-mulrcl 11115  ax-mulcom 11116  ax-addass 11117  ax-mulass 11118  ax-distr 11119  ax-i2m1 11120  ax-1ne0 11121  ax-1rid 11122  ax-rnegex 11123  ax-rrecex 11124  ax-cnre 11125  ax-pre-lttri 11126  ax-pre-lttrn 11127  ax-pre-ltadd 11128  ax-pre-mulgt0 11129  ax-pre-sup 11130  ax-addf 11131  ax-mulf 11132
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-nel 3051  df-ral 3066  df-rex 3075  df-rmo 3354  df-reu 3355  df-rab 3409  df-v 3448  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3930  df-symdif 4203  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-tp 4592  df-op 4594  df-uni 4867  df-int 4909  df-iun 4957  df-iin 4958  df-disj 5072  df-br 5107  df-opab 5169  df-mpt 5190  df-tr 5224  df-id 5532  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5589  df-se 5590  df-we 5591  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-pred 6254  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-isom 6506  df-riota 7314  df-ov 7361  df-oprab 7362  df-mpo 7363  df-of 7618  df-ofr 7619  df-om 7804  df-1st 7922  df-2nd 7923  df-supp 8094  df-frecs 8213  df-wrecs 8244  df-recs 8318  df-rdg 8357  df-1o 8413  df-2o 8414  df-oadd 8417  df-omul 8418  df-er 8649  df-map 8768  df-pm 8769  df-ixp 8837  df-en 8885  df-dom 8886  df-sdom 8887  df-fin 8888  df-fsupp 9307  df-fi 9348  df-sup 9379  df-inf 9380  df-oi 9447  df-dju 9838  df-card 9876  df-acn 9879  df-pnf 11192  df-mnf 11193  df-xr 11194  df-ltxr 11195  df-le 11196  df-sub 11388  df-neg 11389  df-div 11814  df-nn 12155  df-2 12217  df-3 12218  df-4 12219  df-5 12220  df-6 12221  df-7 12222  df-8 12223  df-9 12224  df-n0 12415  df-z 12501  df-dec 12620  df-uz 12765  df-q 12875  df-rp 12917  df-xneg 13034  df-xadd 13035  df-xmul 13036  df-ioo 13269  df-ico 13271  df-icc 13272  df-fz 13426  df-fzo 13569  df-fl 13698  df-mod 13776  df-seq 13908  df-exp 13969  df-hash 14232  df-cj 14985  df-re 14986  df-im 14987  df-sqrt 15121  df-abs 15122  df-clim 15371  df-rlim 15372  df-sum 15572  df-struct 17020  df-sets 17037  df-slot 17055  df-ndx 17067  df-base 17085  df-ress 17114  df-plusg 17147  df-mulr 17148  df-starv 17149  df-sca 17150  df-vsca 17151  df-ip 17152  df-tset 17153  df-ple 17154  df-ds 17156  df-unif 17157  df-hom 17158  df-cco 17159  df-rest 17305  df-topn 17306  df-0g 17324  df-gsum 17325  df-topgen 17326  df-pt 17327  df-prds 17330  df-xrs 17385  df-qtop 17390  df-imas 17391  df-xps 17393  df-mre 17467  df-mrc 17468  df-acs 17470  df-mgm 18498  df-sgrp 18547  df-mnd 18558  df-submnd 18603  df-mulg 18874  df-cntz 19098  df-cmn 19565  df-psmet 20791  df-xmet 20792  df-met 20793  df-bl 20794  df-mopn 20795  df-fbas 20796  df-fg 20797  df-cnfld 20800  df-top 22246  df-topon 22263  df-topsp 22285  df-bases 22299  df-cld 22373  df-ntr 22374  df-cls 22375  df-nei 22452  df-lp 22490  df-perf 22491  df-cn 22581  df-cnp 22582  df-haus 22669  df-cmp 22741  df-tx 22916  df-hmeo 23109  df-fil 23200  df-fm 23292  df-flim 23293  df-flf 23294  df-xms 23676  df-ms 23677  df-tms 23678  df-cncf 24244  df-ovol 24831  df-vol 24832  df-mbf 24986  df-itg1 24987  df-itg2 24988  df-ibl 24989  df-itg 24990  df-0p 25037  df-limc 25233  df-dv 25234
This theorem is referenced by:  areacirc  36174
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