Users' Mathboxes Mathbox for Glauco Siliprandi < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  hspmbllem1 Structured version   Visualization version   GIF version

Theorem hspmbllem1 41346
Description: Any half-space of the n-dimensional Real numbers is Lebesgue measurable. This is Step (a) of Lemma 115F of [Fremlin1] p. 31. (Contributed by Glauco Siliprandi, 24-Dec-2020.)
Hypotheses
Ref Expression
hspmbllem1.x (𝜑𝑋 ∈ Fin)
hspmbllem1.k (𝜑𝐾𝑋)
hspmbllem1.y (𝜑𝑌 ∈ ℝ)
hspmbllem1.a (𝜑𝐴:𝑋⟶ℝ)
hspmbllem1.b (𝜑𝐵:𝑋⟶ℝ)
hspmbllem1.l 𝐿 = (𝑥 ∈ Fin ↦ (𝑎 ∈ (ℝ ↑𝑚 𝑥), 𝑏 ∈ (ℝ ↑𝑚 𝑥) ↦ if(𝑥 = ∅, 0, ∏𝑘𝑥 (vol‘((𝑎𝑘)[,)(𝑏𝑘))))))
hspmbllem1.t 𝑇 = (𝑦 ∈ ℝ ↦ (𝑐 ∈ (ℝ ↑𝑚 𝑋) ↦ (𝑋 ↦ if( ∈ (𝑋 ∖ {𝐾}), (𝑐), if((𝑐) ≤ 𝑦, (𝑐), 𝑦)))))
hspmbllem1.s 𝑆 = (𝑥 ∈ ℝ ↦ (𝑐 ∈ (ℝ ↑𝑚 𝑋) ↦ (𝑋 ↦ if( = 𝐾, if(𝑥 ≤ (𝑐), (𝑐), 𝑥), (𝑐)))))
Assertion
Ref Expression
hspmbllem1 (𝜑 → (𝐴(𝐿𝑋)𝐵) = ((𝐴(𝐿𝑋)((𝑇𝑌)‘𝐵)) +𝑒 (((𝑆𝑌)‘𝐴)(𝐿𝑋)𝐵)))
Distinct variable groups:   𝐴,𝑎,𝑏,𝑘   𝐴,𝑐,,𝑘   𝐵,𝑎,𝑏,𝑘   𝐵,𝑐,   𝐾,𝑐,,𝑘,𝑥   𝑦,𝐾,𝑐,,𝑘   𝑆,𝑎,𝑏,𝑘   𝑇,𝑎,𝑏,𝑘   𝑋,𝑎,𝑏,𝑘,𝑥   𝑋,𝑐,,𝑦   𝑌,𝑎,𝑏,𝑘,𝑥   𝑌,𝑐,,𝑦   𝜑,𝑎,𝑏,𝑘,𝑥   𝜑,𝑐,,𝑦
Allowed substitution hints:   𝐴(𝑥,𝑦)   𝐵(𝑥,𝑦)   𝑆(𝑥,𝑦,,𝑐)   𝑇(𝑥,𝑦,,𝑐)   𝐾(𝑎,𝑏)   𝐿(𝑥,𝑦,,𝑘,𝑎,𝑏,𝑐)

Proof of Theorem hspmbllem1
StepHypRef Expression
1 rge0ssre 12473 . . . 4 (0[,)+∞) ⊆ ℝ
2 hspmbllem1.l . . . . 5 𝐿 = (𝑥 ∈ Fin ↦ (𝑎 ∈ (ℝ ↑𝑚 𝑥), 𝑏 ∈ (ℝ ↑𝑚 𝑥) ↦ if(𝑥 = ∅, 0, ∏𝑘𝑥 (vol‘((𝑎𝑘)[,)(𝑏𝑘))))))
3 hspmbllem1.x . . . . 5 (𝜑𝑋 ∈ Fin)
4 hspmbllem1.a . . . . 5 (𝜑𝐴:𝑋⟶ℝ)
5 hspmbllem1.t . . . . . 6 𝑇 = (𝑦 ∈ ℝ ↦ (𝑐 ∈ (ℝ ↑𝑚 𝑋) ↦ (𝑋 ↦ if( ∈ (𝑋 ∖ {𝐾}), (𝑐), if((𝑐) ≤ 𝑦, (𝑐), 𝑦)))))
6 hspmbllem1.y . . . . . 6 (𝜑𝑌 ∈ ℝ)
7 hspmbllem1.b . . . . . 6 (𝜑𝐵:𝑋⟶ℝ)
85, 6, 3, 7hsphoif 41296 . . . . 5 (𝜑 → ((𝑇𝑌)‘𝐵):𝑋⟶ℝ)
92, 3, 4, 8hoidmvcl 41302 . . . 4 (𝜑 → (𝐴(𝐿𝑋)((𝑇𝑌)‘𝐵)) ∈ (0[,)+∞))
101, 9sseldi 3742 . . 3 (𝜑 → (𝐴(𝐿𝑋)((𝑇𝑌)‘𝐵)) ∈ ℝ)
11 hspmbllem1.s . . . . . 6 𝑆 = (𝑥 ∈ ℝ ↦ (𝑐 ∈ (ℝ ↑𝑚 𝑋) ↦ (𝑋 ↦ if( = 𝐾, if(𝑥 ≤ (𝑐), (𝑐), 𝑥), (𝑐)))))
1211, 6, 3, 4hoidifhspf 41338 . . . . 5 (𝜑 → ((𝑆𝑌)‘𝐴):𝑋⟶ℝ)
132, 3, 12, 7hoidmvcl 41302 . . . 4 (𝜑 → (((𝑆𝑌)‘𝐴)(𝐿𝑋)𝐵) ∈ (0[,)+∞))
141, 13sseldi 3742 . . 3 (𝜑 → (((𝑆𝑌)‘𝐴)(𝐿𝑋)𝐵) ∈ ℝ)
1510, 14rexaddd 12258 . 2 (𝜑 → ((𝐴(𝐿𝑋)((𝑇𝑌)‘𝐵)) +𝑒 (((𝑆𝑌)‘𝐴)(𝐿𝑋)𝐵)) = ((𝐴(𝐿𝑋)((𝑇𝑌)‘𝐵)) + (((𝑆𝑌)‘𝐴)(𝐿𝑋)𝐵)))
16 hspmbllem1.k . . . . . 6 (𝜑𝐾𝑋)
17 ne0i 4064 . . . . . 6 (𝐾𝑋𝑋 ≠ ∅)
1816, 17syl 17 . . . . 5 (𝜑𝑋 ≠ ∅)
192, 3, 18, 4, 8hoidmvn0val 41304 . . . 4 (𝜑 → (𝐴(𝐿𝑋)((𝑇𝑌)‘𝐵)) = ∏𝑘𝑋 (vol‘((𝐴𝑘)[,)(((𝑇𝑌)‘𝐵)‘𝑘))))
202, 3, 18, 12, 7hoidmvn0val 41304 . . . 4 (𝜑 → (((𝑆𝑌)‘𝐴)(𝐿𝑋)𝐵) = ∏𝑘𝑋 (vol‘((((𝑆𝑌)‘𝐴)‘𝑘)[,)(𝐵𝑘))))
2119, 20oveq12d 6831 . . 3 (𝜑 → ((𝐴(𝐿𝑋)((𝑇𝑌)‘𝐵)) + (((𝑆𝑌)‘𝐴)(𝐿𝑋)𝐵)) = (∏𝑘𝑋 (vol‘((𝐴𝑘)[,)(((𝑇𝑌)‘𝐵)‘𝑘))) + ∏𝑘𝑋 (vol‘((((𝑆𝑌)‘𝐴)‘𝑘)[,)(𝐵𝑘)))))
22 uncom 3900 . . . . . . . . 9 ((𝑋 ∖ {𝐾}) ∪ {𝐾}) = ({𝐾} ∪ (𝑋 ∖ {𝐾}))
2322a1i 11 . . . . . . . 8 (𝜑 → ((𝑋 ∖ {𝐾}) ∪ {𝐾}) = ({𝐾} ∪ (𝑋 ∖ {𝐾})))
2416snssd 4485 . . . . . . . . 9 (𝜑 → {𝐾} ⊆ 𝑋)
25 undif 4193 . . . . . . . . 9 ({𝐾} ⊆ 𝑋 ↔ ({𝐾} ∪ (𝑋 ∖ {𝐾})) = 𝑋)
2624, 25sylib 208 . . . . . . . 8 (𝜑 → ({𝐾} ∪ (𝑋 ∖ {𝐾})) = 𝑋)
2723, 26eqtrd 2794 . . . . . . 7 (𝜑 → ((𝑋 ∖ {𝐾}) ∪ {𝐾}) = 𝑋)
2827eqcomd 2766 . . . . . 6 (𝜑𝑋 = ((𝑋 ∖ {𝐾}) ∪ {𝐾}))
2928prodeq1d 14850 . . . . 5 (𝜑 → ∏𝑘𝑋 (vol‘((𝐴𝑘)[,)(((𝑇𝑌)‘𝐵)‘𝑘))) = ∏𝑘 ∈ ((𝑋 ∖ {𝐾}) ∪ {𝐾})(vol‘((𝐴𝑘)[,)(((𝑇𝑌)‘𝐵)‘𝑘))))
30 nfv 1992 . . . . . 6 𝑘𝜑
31 nfcv 2902 . . . . . 6 𝑘(vol‘((𝐴𝐾)[,)(((𝑇𝑌)‘𝐵)‘𝐾)))
32 difssd 3881 . . . . . . 7 (𝜑 → (𝑋 ∖ {𝐾}) ⊆ 𝑋)
333, 32ssfid 8348 . . . . . 6 (𝜑 → (𝑋 ∖ {𝐾}) ∈ Fin)
34 neldifsnd 4468 . . . . . 6 (𝜑 → ¬ 𝐾 ∈ (𝑋 ∖ {𝐾}))
354adantr 472 . . . . . . . . 9 ((𝜑𝑘 ∈ (𝑋 ∖ {𝐾})) → 𝐴:𝑋⟶ℝ)
3632sselda 3744 . . . . . . . . 9 ((𝜑𝑘 ∈ (𝑋 ∖ {𝐾})) → 𝑘𝑋)
3735, 36ffvelrnd 6523 . . . . . . . 8 ((𝜑𝑘 ∈ (𝑋 ∖ {𝐾})) → (𝐴𝑘) ∈ ℝ)
386adantr 472 . . . . . . . . . 10 ((𝜑𝑘 ∈ (𝑋 ∖ {𝐾})) → 𝑌 ∈ ℝ)
393adantr 472 . . . . . . . . . 10 ((𝜑𝑘 ∈ (𝑋 ∖ {𝐾})) → 𝑋 ∈ Fin)
407adantr 472 . . . . . . . . . 10 ((𝜑𝑘 ∈ (𝑋 ∖ {𝐾})) → 𝐵:𝑋⟶ℝ)
415, 38, 39, 40hsphoif 41296 . . . . . . . . 9 ((𝜑𝑘 ∈ (𝑋 ∖ {𝐾})) → ((𝑇𝑌)‘𝐵):𝑋⟶ℝ)
4241, 36ffvelrnd 6523 . . . . . . . 8 ((𝜑𝑘 ∈ (𝑋 ∖ {𝐾})) → (((𝑇𝑌)‘𝐵)‘𝑘) ∈ ℝ)
43 volicore 41301 . . . . . . . 8 (((𝐴𝑘) ∈ ℝ ∧ (((𝑇𝑌)‘𝐵)‘𝑘) ∈ ℝ) → (vol‘((𝐴𝑘)[,)(((𝑇𝑌)‘𝐵)‘𝑘))) ∈ ℝ)
4437, 42, 43syl2anc 696 . . . . . . 7 ((𝜑𝑘 ∈ (𝑋 ∖ {𝐾})) → (vol‘((𝐴𝑘)[,)(((𝑇𝑌)‘𝐵)‘𝑘))) ∈ ℝ)
4544recnd 10260 . . . . . 6 ((𝜑𝑘 ∈ (𝑋 ∖ {𝐾})) → (vol‘((𝐴𝑘)[,)(((𝑇𝑌)‘𝐵)‘𝑘))) ∈ ℂ)
46 fveq2 6352 . . . . . . . 8 (𝑘 = 𝐾 → (𝐴𝑘) = (𝐴𝐾))
47 fveq2 6352 . . . . . . . 8 (𝑘 = 𝐾 → (((𝑇𝑌)‘𝐵)‘𝑘) = (((𝑇𝑌)‘𝐵)‘𝐾))
4846, 47oveq12d 6831 . . . . . . 7 (𝑘 = 𝐾 → ((𝐴𝑘)[,)(((𝑇𝑌)‘𝐵)‘𝑘)) = ((𝐴𝐾)[,)(((𝑇𝑌)‘𝐵)‘𝐾)))
4948fveq2d 6356 . . . . . 6 (𝑘 = 𝐾 → (vol‘((𝐴𝑘)[,)(((𝑇𝑌)‘𝐵)‘𝑘))) = (vol‘((𝐴𝐾)[,)(((𝑇𝑌)‘𝐵)‘𝐾))))
504, 16ffvelrnd 6523 . . . . . . . 8 (𝜑 → (𝐴𝐾) ∈ ℝ)
518, 16ffvelrnd 6523 . . . . . . . 8 (𝜑 → (((𝑇𝑌)‘𝐵)‘𝐾) ∈ ℝ)
52 volicore 41301 . . . . . . . 8 (((𝐴𝐾) ∈ ℝ ∧ (((𝑇𝑌)‘𝐵)‘𝐾) ∈ ℝ) → (vol‘((𝐴𝐾)[,)(((𝑇𝑌)‘𝐵)‘𝐾))) ∈ ℝ)
5350, 51, 52syl2anc 696 . . . . . . 7 (𝜑 → (vol‘((𝐴𝐾)[,)(((𝑇𝑌)‘𝐵)‘𝐾))) ∈ ℝ)
5453recnd 10260 . . . . . 6 (𝜑 → (vol‘((𝐴𝐾)[,)(((𝑇𝑌)‘𝐵)‘𝐾))) ∈ ℂ)
5530, 31, 33, 16, 34, 45, 49, 54fprodsplitsn 14919 . . . . 5 (𝜑 → ∏𝑘 ∈ ((𝑋 ∖ {𝐾}) ∪ {𝐾})(vol‘((𝐴𝑘)[,)(((𝑇𝑌)‘𝐵)‘𝑘))) = (∏𝑘 ∈ (𝑋 ∖ {𝐾})(vol‘((𝐴𝑘)[,)(((𝑇𝑌)‘𝐵)‘𝑘))) · (vol‘((𝐴𝐾)[,)(((𝑇𝑌)‘𝐵)‘𝐾)))))
565, 38, 39, 40, 36hsphoival 41299 . . . . . . . . . 10 ((𝜑𝑘 ∈ (𝑋 ∖ {𝐾})) → (((𝑇𝑌)‘𝐵)‘𝑘) = if(𝑘 ∈ (𝑋 ∖ {𝐾}), (𝐵𝑘), if((𝐵𝑘) ≤ 𝑌, (𝐵𝑘), 𝑌)))
57 iftrue 4236 . . . . . . . . . . 11 (𝑘 ∈ (𝑋 ∖ {𝐾}) → if(𝑘 ∈ (𝑋 ∖ {𝐾}), (𝐵𝑘), if((𝐵𝑘) ≤ 𝑌, (𝐵𝑘), 𝑌)) = (𝐵𝑘))
5857adantl 473 . . . . . . . . . 10 ((𝜑𝑘 ∈ (𝑋 ∖ {𝐾})) → if(𝑘 ∈ (𝑋 ∖ {𝐾}), (𝐵𝑘), if((𝐵𝑘) ≤ 𝑌, (𝐵𝑘), 𝑌)) = (𝐵𝑘))
5956, 58eqtrd 2794 . . . . . . . . 9 ((𝜑𝑘 ∈ (𝑋 ∖ {𝐾})) → (((𝑇𝑌)‘𝐵)‘𝑘) = (𝐵𝑘))
6059oveq2d 6829 . . . . . . . 8 ((𝜑𝑘 ∈ (𝑋 ∖ {𝐾})) → ((𝐴𝑘)[,)(((𝑇𝑌)‘𝐵)‘𝑘)) = ((𝐴𝑘)[,)(𝐵𝑘)))
6160fveq2d 6356 . . . . . . 7 ((𝜑𝑘 ∈ (𝑋 ∖ {𝐾})) → (vol‘((𝐴𝑘)[,)(((𝑇𝑌)‘𝐵)‘𝑘))) = (vol‘((𝐴𝑘)[,)(𝐵𝑘))))
6261prodeq2dv 14852 . . . . . 6 (𝜑 → ∏𝑘 ∈ (𝑋 ∖ {𝐾})(vol‘((𝐴𝑘)[,)(((𝑇𝑌)‘𝐵)‘𝑘))) = ∏𝑘 ∈ (𝑋 ∖ {𝐾})(vol‘((𝐴𝑘)[,)(𝐵𝑘))))
6362oveq1d 6828 . . . . 5 (𝜑 → (∏𝑘 ∈ (𝑋 ∖ {𝐾})(vol‘((𝐴𝑘)[,)(((𝑇𝑌)‘𝐵)‘𝑘))) · (vol‘((𝐴𝐾)[,)(((𝑇𝑌)‘𝐵)‘𝐾)))) = (∏𝑘 ∈ (𝑋 ∖ {𝐾})(vol‘((𝐴𝑘)[,)(𝐵𝑘))) · (vol‘((𝐴𝐾)[,)(((𝑇𝑌)‘𝐵)‘𝐾)))))
6429, 55, 633eqtrd 2798 . . . 4 (𝜑 → ∏𝑘𝑋 (vol‘((𝐴𝑘)[,)(((𝑇𝑌)‘𝐵)‘𝑘))) = (∏𝑘 ∈ (𝑋 ∖ {𝐾})(vol‘((𝐴𝑘)[,)(𝐵𝑘))) · (vol‘((𝐴𝐾)[,)(((𝑇𝑌)‘𝐵)‘𝐾)))))
6528prodeq1d 14850 . . . . 5 (𝜑 → ∏𝑘𝑋 (vol‘((((𝑆𝑌)‘𝐴)‘𝑘)[,)(𝐵𝑘))) = ∏𝑘 ∈ ((𝑋 ∖ {𝐾}) ∪ {𝐾})(vol‘((((𝑆𝑌)‘𝐴)‘𝑘)[,)(𝐵𝑘))))
66 nfcv 2902 . . . . . 6 𝑘(vol‘((((𝑆𝑌)‘𝐴)‘𝐾)[,)(𝐵𝐾)))
6712adantr 472 . . . . . . . . 9 ((𝜑𝑘 ∈ (𝑋 ∖ {𝐾})) → ((𝑆𝑌)‘𝐴):𝑋⟶ℝ)
6867, 36ffvelrnd 6523 . . . . . . . 8 ((𝜑𝑘 ∈ (𝑋 ∖ {𝐾})) → (((𝑆𝑌)‘𝐴)‘𝑘) ∈ ℝ)
6959, 42eqeltrrd 2840 . . . . . . . 8 ((𝜑𝑘 ∈ (𝑋 ∖ {𝐾})) → (𝐵𝑘) ∈ ℝ)
70 volicore 41301 . . . . . . . 8 (((((𝑆𝑌)‘𝐴)‘𝑘) ∈ ℝ ∧ (𝐵𝑘) ∈ ℝ) → (vol‘((((𝑆𝑌)‘𝐴)‘𝑘)[,)(𝐵𝑘))) ∈ ℝ)
7168, 69, 70syl2anc 696 . . . . . . 7 ((𝜑𝑘 ∈ (𝑋 ∖ {𝐾})) → (vol‘((((𝑆𝑌)‘𝐴)‘𝑘)[,)(𝐵𝑘))) ∈ ℝ)
7271recnd 10260 . . . . . 6 ((𝜑𝑘 ∈ (𝑋 ∖ {𝐾})) → (vol‘((((𝑆𝑌)‘𝐴)‘𝑘)[,)(𝐵𝑘))) ∈ ℂ)
73 fveq2 6352 . . . . . . . 8 (𝑘 = 𝐾 → (((𝑆𝑌)‘𝐴)‘𝑘) = (((𝑆𝑌)‘𝐴)‘𝐾))
74 fveq2 6352 . . . . . . . 8 (𝑘 = 𝐾 → (𝐵𝑘) = (𝐵𝐾))
7573, 74oveq12d 6831 . . . . . . 7 (𝑘 = 𝐾 → ((((𝑆𝑌)‘𝐴)‘𝑘)[,)(𝐵𝑘)) = ((((𝑆𝑌)‘𝐴)‘𝐾)[,)(𝐵𝐾)))
7675fveq2d 6356 . . . . . 6 (𝑘 = 𝐾 → (vol‘((((𝑆𝑌)‘𝐴)‘𝑘)[,)(𝐵𝑘))) = (vol‘((((𝑆𝑌)‘𝐴)‘𝐾)[,)(𝐵𝐾))))
7712, 16ffvelrnd 6523 . . . . . . . 8 (𝜑 → (((𝑆𝑌)‘𝐴)‘𝐾) ∈ ℝ)
787, 16ffvelrnd 6523 . . . . . . . 8 (𝜑 → (𝐵𝐾) ∈ ℝ)
79 volicore 41301 . . . . . . . 8 (((((𝑆𝑌)‘𝐴)‘𝐾) ∈ ℝ ∧ (𝐵𝐾) ∈ ℝ) → (vol‘((((𝑆𝑌)‘𝐴)‘𝐾)[,)(𝐵𝐾))) ∈ ℝ)
8077, 78, 79syl2anc 696 . . . . . . 7 (𝜑 → (vol‘((((𝑆𝑌)‘𝐴)‘𝐾)[,)(𝐵𝐾))) ∈ ℝ)
8180recnd 10260 . . . . . 6 (𝜑 → (vol‘((((𝑆𝑌)‘𝐴)‘𝐾)[,)(𝐵𝐾))) ∈ ℂ)
8230, 66, 33, 16, 34, 72, 76, 81fprodsplitsn 14919 . . . . 5 (𝜑 → ∏𝑘 ∈ ((𝑋 ∖ {𝐾}) ∪ {𝐾})(vol‘((((𝑆𝑌)‘𝐴)‘𝑘)[,)(𝐵𝑘))) = (∏𝑘 ∈ (𝑋 ∖ {𝐾})(vol‘((((𝑆𝑌)‘𝐴)‘𝑘)[,)(𝐵𝑘))) · (vol‘((((𝑆𝑌)‘𝐴)‘𝐾)[,)(𝐵𝐾)))))
8311, 38, 39, 35, 36hoidifhspval3 41339 . . . . . . . . . 10 ((𝜑𝑘 ∈ (𝑋 ∖ {𝐾})) → (((𝑆𝑌)‘𝐴)‘𝑘) = if(𝑘 = 𝐾, if(𝑌 ≤ (𝐴𝑘), (𝐴𝑘), 𝑌), (𝐴𝑘)))
84 eldifsni 4466 . . . . . . . . . . . . 13 (𝑘 ∈ (𝑋 ∖ {𝐾}) → 𝑘𝐾)
85 neneq 2938 . . . . . . . . . . . . 13 (𝑘𝐾 → ¬ 𝑘 = 𝐾)
8684, 85syl 17 . . . . . . . . . . . 12 (𝑘 ∈ (𝑋 ∖ {𝐾}) → ¬ 𝑘 = 𝐾)
8786iffalsed 4241 . . . . . . . . . . 11 (𝑘 ∈ (𝑋 ∖ {𝐾}) → if(𝑘 = 𝐾, if(𝑌 ≤ (𝐴𝑘), (𝐴𝑘), 𝑌), (𝐴𝑘)) = (𝐴𝑘))
8887adantl 473 . . . . . . . . . 10 ((𝜑𝑘 ∈ (𝑋 ∖ {𝐾})) → if(𝑘 = 𝐾, if(𝑌 ≤ (𝐴𝑘), (𝐴𝑘), 𝑌), (𝐴𝑘)) = (𝐴𝑘))
8983, 88eqtrd 2794 . . . . . . . . 9 ((𝜑𝑘 ∈ (𝑋 ∖ {𝐾})) → (((𝑆𝑌)‘𝐴)‘𝑘) = (𝐴𝑘))
9089oveq1d 6828 . . . . . . . 8 ((𝜑𝑘 ∈ (𝑋 ∖ {𝐾})) → ((((𝑆𝑌)‘𝐴)‘𝑘)[,)(𝐵𝑘)) = ((𝐴𝑘)[,)(𝐵𝑘)))
9190fveq2d 6356 . . . . . . 7 ((𝜑𝑘 ∈ (𝑋 ∖ {𝐾})) → (vol‘((((𝑆𝑌)‘𝐴)‘𝑘)[,)(𝐵𝑘))) = (vol‘((𝐴𝑘)[,)(𝐵𝑘))))
9291prodeq2dv 14852 . . . . . 6 (𝜑 → ∏𝑘 ∈ (𝑋 ∖ {𝐾})(vol‘((((𝑆𝑌)‘𝐴)‘𝑘)[,)(𝐵𝑘))) = ∏𝑘 ∈ (𝑋 ∖ {𝐾})(vol‘((𝐴𝑘)[,)(𝐵𝑘))))
9392oveq1d 6828 . . . . 5 (𝜑 → (∏𝑘 ∈ (𝑋 ∖ {𝐾})(vol‘((((𝑆𝑌)‘𝐴)‘𝑘)[,)(𝐵𝑘))) · (vol‘((((𝑆𝑌)‘𝐴)‘𝐾)[,)(𝐵𝐾)))) = (∏𝑘 ∈ (𝑋 ∖ {𝐾})(vol‘((𝐴𝑘)[,)(𝐵𝑘))) · (vol‘((((𝑆𝑌)‘𝐴)‘𝐾)[,)(𝐵𝐾)))))
9465, 82, 933eqtrd 2798 . . . 4 (𝜑 → ∏𝑘𝑋 (vol‘((((𝑆𝑌)‘𝐴)‘𝑘)[,)(𝐵𝑘))) = (∏𝑘 ∈ (𝑋 ∖ {𝐾})(vol‘((𝐴𝑘)[,)(𝐵𝑘))) · (vol‘((((𝑆𝑌)‘𝐴)‘𝐾)[,)(𝐵𝐾)))))
9564, 94oveq12d 6831 . . 3 (𝜑 → (∏𝑘𝑋 (vol‘((𝐴𝑘)[,)(((𝑇𝑌)‘𝐵)‘𝑘))) + ∏𝑘𝑋 (vol‘((((𝑆𝑌)‘𝐴)‘𝑘)[,)(𝐵𝑘)))) = ((∏𝑘 ∈ (𝑋 ∖ {𝐾})(vol‘((𝐴𝑘)[,)(𝐵𝑘))) · (vol‘((𝐴𝐾)[,)(((𝑇𝑌)‘𝐵)‘𝐾)))) + (∏𝑘 ∈ (𝑋 ∖ {𝐾})(vol‘((𝐴𝑘)[,)(𝐵𝑘))) · (vol‘((((𝑆𝑌)‘𝐴)‘𝐾)[,)(𝐵𝐾))))))
9628prodeq1d 14850 . . . . 5 (𝜑 → ∏𝑘𝑋 (vol‘((𝐴𝑘)[,)(𝐵𝑘))) = ∏𝑘 ∈ ((𝑋 ∖ {𝐾}) ∪ {𝐾})(vol‘((𝐴𝑘)[,)(𝐵𝑘))))
97 nfcv 2902 . . . . . 6 𝑘(vol‘((𝐴𝐾)[,)(𝐵𝐾)))
9861, 45eqeltrrd 2840 . . . . . 6 ((𝜑𝑘 ∈ (𝑋 ∖ {𝐾})) → (vol‘((𝐴𝑘)[,)(𝐵𝑘))) ∈ ℂ)
9946, 74oveq12d 6831 . . . . . . 7 (𝑘 = 𝐾 → ((𝐴𝑘)[,)(𝐵𝑘)) = ((𝐴𝐾)[,)(𝐵𝐾)))
10099fveq2d 6356 . . . . . 6 (𝑘 = 𝐾 → (vol‘((𝐴𝑘)[,)(𝐵𝑘))) = (vol‘((𝐴𝐾)[,)(𝐵𝐾))))
101 volicore 41301 . . . . . . . 8 (((𝐴𝐾) ∈ ℝ ∧ (𝐵𝐾) ∈ ℝ) → (vol‘((𝐴𝐾)[,)(𝐵𝐾))) ∈ ℝ)
10250, 78, 101syl2anc 696 . . . . . . 7 (𝜑 → (vol‘((𝐴𝐾)[,)(𝐵𝐾))) ∈ ℝ)
103102recnd 10260 . . . . . 6 (𝜑 → (vol‘((𝐴𝐾)[,)(𝐵𝐾))) ∈ ℂ)
10430, 97, 33, 16, 34, 98, 100, 103fprodsplitsn 14919 . . . . 5 (𝜑 → ∏𝑘 ∈ ((𝑋 ∖ {𝐾}) ∪ {𝐾})(vol‘((𝐴𝑘)[,)(𝐵𝑘))) = (∏𝑘 ∈ (𝑋 ∖ {𝐾})(vol‘((𝐴𝑘)[,)(𝐵𝑘))) · (vol‘((𝐴𝐾)[,)(𝐵𝐾)))))
10596, 104eqtrd 2794 . . . 4 (𝜑 → ∏𝑘𝑋 (vol‘((𝐴𝑘)[,)(𝐵𝑘))) = (∏𝑘 ∈ (𝑋 ∖ {𝐾})(vol‘((𝐴𝑘)[,)(𝐵𝑘))) · (vol‘((𝐴𝐾)[,)(𝐵𝐾)))))
1065, 6, 3, 7, 16hsphoival 41299 . . . . . . . . . 10 (𝜑 → (((𝑇𝑌)‘𝐵)‘𝐾) = if(𝐾 ∈ (𝑋 ∖ {𝐾}), (𝐵𝐾), if((𝐵𝐾) ≤ 𝑌, (𝐵𝐾), 𝑌)))
10734iffalsed 4241 . . . . . . . . . 10 (𝜑 → if(𝐾 ∈ (𝑋 ∖ {𝐾}), (𝐵𝐾), if((𝐵𝐾) ≤ 𝑌, (𝐵𝐾), 𝑌)) = if((𝐵𝐾) ≤ 𝑌, (𝐵𝐾), 𝑌))
108106, 107eqtrd 2794 . . . . . . . . 9 (𝜑 → (((𝑇𝑌)‘𝐵)‘𝐾) = if((𝐵𝐾) ≤ 𝑌, (𝐵𝐾), 𝑌))
109108oveq2d 6829 . . . . . . . 8 (𝜑 → ((𝐴𝐾)[,)(((𝑇𝑌)‘𝐵)‘𝐾)) = ((𝐴𝐾)[,)if((𝐵𝐾) ≤ 𝑌, (𝐵𝐾), 𝑌)))
110109fveq2d 6356 . . . . . . 7 (𝜑 → (vol‘((𝐴𝐾)[,)(((𝑇𝑌)‘𝐵)‘𝐾))) = (vol‘((𝐴𝐾)[,)if((𝐵𝐾) ≤ 𝑌, (𝐵𝐾), 𝑌))))
11111, 6, 3, 4, 16hoidifhspval3 41339 . . . . . . . . . 10 (𝜑 → (((𝑆𝑌)‘𝐴)‘𝐾) = if(𝐾 = 𝐾, if(𝑌 ≤ (𝐴𝐾), (𝐴𝐾), 𝑌), (𝐴𝐾)))
112 eqid 2760 . . . . . . . . . . . 12 𝐾 = 𝐾
113112iftruei 4237 . . . . . . . . . . 11 if(𝐾 = 𝐾, if(𝑌 ≤ (𝐴𝐾), (𝐴𝐾), 𝑌), (𝐴𝐾)) = if(𝑌 ≤ (𝐴𝐾), (𝐴𝐾), 𝑌)
114113a1i 11 . . . . . . . . . 10 (𝜑 → if(𝐾 = 𝐾, if(𝑌 ≤ (𝐴𝐾), (𝐴𝐾), 𝑌), (𝐴𝐾)) = if(𝑌 ≤ (𝐴𝐾), (𝐴𝐾), 𝑌))
115111, 114eqtrd 2794 . . . . . . . . 9 (𝜑 → (((𝑆𝑌)‘𝐴)‘𝐾) = if(𝑌 ≤ (𝐴𝐾), (𝐴𝐾), 𝑌))
116115oveq1d 6828 . . . . . . . 8 (𝜑 → ((((𝑆𝑌)‘𝐴)‘𝐾)[,)(𝐵𝐾)) = (if(𝑌 ≤ (𝐴𝐾), (𝐴𝐾), 𝑌)[,)(𝐵𝐾)))
117116fveq2d 6356 . . . . . . 7 (𝜑 → (vol‘((((𝑆𝑌)‘𝐴)‘𝐾)[,)(𝐵𝐾))) = (vol‘(if(𝑌 ≤ (𝐴𝐾), (𝐴𝐾), 𝑌)[,)(𝐵𝐾))))
118110, 117oveq12d 6831 . . . . . 6 (𝜑 → ((vol‘((𝐴𝐾)[,)(((𝑇𝑌)‘𝐵)‘𝐾))) + (vol‘((((𝑆𝑌)‘𝐴)‘𝐾)[,)(𝐵𝐾)))) = ((vol‘((𝐴𝐾)[,)if((𝐵𝐾) ≤ 𝑌, (𝐵𝐾), 𝑌))) + (vol‘(if(𝑌 ≤ (𝐴𝐾), (𝐴𝐾), 𝑌)[,)(𝐵𝐾)))))
119 iftrue 4236 . . . . . . . . . . . 12 ((𝐵𝐾) ≤ 𝑌 → if((𝐵𝐾) ≤ 𝑌, (𝐵𝐾), 𝑌) = (𝐵𝐾))
120119oveq2d 6829 . . . . . . . . . . 11 ((𝐵𝐾) ≤ 𝑌 → ((𝐴𝐾)[,)if((𝐵𝐾) ≤ 𝑌, (𝐵𝐾), 𝑌)) = ((𝐴𝐾)[,)(𝐵𝐾)))
121120fveq2d 6356 . . . . . . . . . 10 ((𝐵𝐾) ≤ 𝑌 → (vol‘((𝐴𝐾)[,)if((𝐵𝐾) ≤ 𝑌, (𝐵𝐾), 𝑌))) = (vol‘((𝐴𝐾)[,)(𝐵𝐾))))
122121oveq1d 6828 . . . . . . . . 9 ((𝐵𝐾) ≤ 𝑌 → ((vol‘((𝐴𝐾)[,)if((𝐵𝐾) ≤ 𝑌, (𝐵𝐾), 𝑌))) + (vol‘(if(𝑌 ≤ (𝐴𝐾), (𝐴𝐾), 𝑌)[,)(𝐵𝐾)))) = ((vol‘((𝐴𝐾)[,)(𝐵𝐾))) + (vol‘(if(𝑌 ≤ (𝐴𝐾), (𝐴𝐾), 𝑌)[,)(𝐵𝐾)))))
123122adantl 473 . . . . . . . 8 ((𝜑 ∧ (𝐵𝐾) ≤ 𝑌) → ((vol‘((𝐴𝐾)[,)if((𝐵𝐾) ≤ 𝑌, (𝐵𝐾), 𝑌))) + (vol‘(if(𝑌 ≤ (𝐴𝐾), (𝐴𝐾), 𝑌)[,)(𝐵𝐾)))) = ((vol‘((𝐴𝐾)[,)(𝐵𝐾))) + (vol‘(if(𝑌 ≤ (𝐴𝐾), (𝐴𝐾), 𝑌)[,)(𝐵𝐾)))))
124 iftrue 4236 . . . . . . . . . . . . . . 15 (𝑌 ≤ (𝐴𝐾) → if(𝑌 ≤ (𝐴𝐾), (𝐴𝐾), 𝑌) = (𝐴𝐾))
125124oveq1d 6828 . . . . . . . . . . . . . 14 (𝑌 ≤ (𝐴𝐾) → (if(𝑌 ≤ (𝐴𝐾), (𝐴𝐾), 𝑌)[,)(𝐵𝐾)) = ((𝐴𝐾)[,)(𝐵𝐾)))
126125adantl 473 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝐵𝐾) ≤ 𝑌) ∧ 𝑌 ≤ (𝐴𝐾)) → (if(𝑌 ≤ (𝐴𝐾), (𝐴𝐾), 𝑌)[,)(𝐵𝐾)) = ((𝐴𝐾)[,)(𝐵𝐾)))
12778ad2antrr 764 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝐵𝐾) ≤ 𝑌) ∧ 𝑌 ≤ (𝐴𝐾)) → (𝐵𝐾) ∈ ℝ)
1286ad2antrr 764 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝐵𝐾) ≤ 𝑌) ∧ 𝑌 ≤ (𝐴𝐾)) → 𝑌 ∈ ℝ)
12950ad2antrr 764 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝐵𝐾) ≤ 𝑌) ∧ 𝑌 ≤ (𝐴𝐾)) → (𝐴𝐾) ∈ ℝ)
130 simplr 809 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝐵𝐾) ≤ 𝑌) ∧ 𝑌 ≤ (𝐴𝐾)) → (𝐵𝐾) ≤ 𝑌)
131 simpr 479 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝐵𝐾) ≤ 𝑌) ∧ 𝑌 ≤ (𝐴𝐾)) → 𝑌 ≤ (𝐴𝐾))
132127, 128, 129, 130, 131letrd 10386 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝐵𝐾) ≤ 𝑌) ∧ 𝑌 ≤ (𝐴𝐾)) → (𝐵𝐾) ≤ (𝐴𝐾))
133129rexrd 10281 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝐵𝐾) ≤ 𝑌) ∧ 𝑌 ≤ (𝐴𝐾)) → (𝐴𝐾) ∈ ℝ*)
134127rexrd 10281 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝐵𝐾) ≤ 𝑌) ∧ 𝑌 ≤ (𝐴𝐾)) → (𝐵𝐾) ∈ ℝ*)
135 ico0 12414 . . . . . . . . . . . . . . 15 (((𝐴𝐾) ∈ ℝ* ∧ (𝐵𝐾) ∈ ℝ*) → (((𝐴𝐾)[,)(𝐵𝐾)) = ∅ ↔ (𝐵𝐾) ≤ (𝐴𝐾)))
136133, 134, 135syl2anc 696 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝐵𝐾) ≤ 𝑌) ∧ 𝑌 ≤ (𝐴𝐾)) → (((𝐴𝐾)[,)(𝐵𝐾)) = ∅ ↔ (𝐵𝐾) ≤ (𝐴𝐾)))
137132, 136mpbird 247 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝐵𝐾) ≤ 𝑌) ∧ 𝑌 ≤ (𝐴𝐾)) → ((𝐴𝐾)[,)(𝐵𝐾)) = ∅)
138126, 137eqtrd 2794 . . . . . . . . . . . 12 (((𝜑 ∧ (𝐵𝐾) ≤ 𝑌) ∧ 𝑌 ≤ (𝐴𝐾)) → (if(𝑌 ≤ (𝐴𝐾), (𝐴𝐾), 𝑌)[,)(𝐵𝐾)) = ∅)
139 iffalse 4239 . . . . . . . . . . . . . . 15 𝑌 ≤ (𝐴𝐾) → if(𝑌 ≤ (𝐴𝐾), (𝐴𝐾), 𝑌) = 𝑌)
140139oveq1d 6828 . . . . . . . . . . . . . 14 𝑌 ≤ (𝐴𝐾) → (if(𝑌 ≤ (𝐴𝐾), (𝐴𝐾), 𝑌)[,)(𝐵𝐾)) = (𝑌[,)(𝐵𝐾)))
141140adantl 473 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝐵𝐾) ≤ 𝑌) ∧ ¬ 𝑌 ≤ (𝐴𝐾)) → (if(𝑌 ≤ (𝐴𝐾), (𝐴𝐾), 𝑌)[,)(𝐵𝐾)) = (𝑌[,)(𝐵𝐾)))
142 simpr 479 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝐵𝐾) ≤ 𝑌) → (𝐵𝐾) ≤ 𝑌)
1436rexrd 10281 . . . . . . . . . . . . . . . . 17 (𝜑𝑌 ∈ ℝ*)
144143adantr 472 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝐵𝐾) ≤ 𝑌) → 𝑌 ∈ ℝ*)
14578rexrd 10281 . . . . . . . . . . . . . . . . 17 (𝜑 → (𝐵𝐾) ∈ ℝ*)
146145adantr 472 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝐵𝐾) ≤ 𝑌) → (𝐵𝐾) ∈ ℝ*)
147 ico0 12414 . . . . . . . . . . . . . . . 16 ((𝑌 ∈ ℝ* ∧ (𝐵𝐾) ∈ ℝ*) → ((𝑌[,)(𝐵𝐾)) = ∅ ↔ (𝐵𝐾) ≤ 𝑌))
148144, 146, 147syl2anc 696 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝐵𝐾) ≤ 𝑌) → ((𝑌[,)(𝐵𝐾)) = ∅ ↔ (𝐵𝐾) ≤ 𝑌))
149142, 148mpbird 247 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝐵𝐾) ≤ 𝑌) → (𝑌[,)(𝐵𝐾)) = ∅)
150149adantr 472 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝐵𝐾) ≤ 𝑌) ∧ ¬ 𝑌 ≤ (𝐴𝐾)) → (𝑌[,)(𝐵𝐾)) = ∅)
151141, 150eqtrd 2794 . . . . . . . . . . . 12 (((𝜑 ∧ (𝐵𝐾) ≤ 𝑌) ∧ ¬ 𝑌 ≤ (𝐴𝐾)) → (if(𝑌 ≤ (𝐴𝐾), (𝐴𝐾), 𝑌)[,)(𝐵𝐾)) = ∅)
152138, 151pm2.61dan 867 . . . . . . . . . . 11 ((𝜑 ∧ (𝐵𝐾) ≤ 𝑌) → (if(𝑌 ≤ (𝐴𝐾), (𝐴𝐾), 𝑌)[,)(𝐵𝐾)) = ∅)
153152fveq2d 6356 . . . . . . . . . 10 ((𝜑 ∧ (𝐵𝐾) ≤ 𝑌) → (vol‘(if(𝑌 ≤ (𝐴𝐾), (𝐴𝐾), 𝑌)[,)(𝐵𝐾))) = (vol‘∅))
154 vol0 40678 . . . . . . . . . . 11 (vol‘∅) = 0
155154a1i 11 . . . . . . . . . 10 ((𝜑 ∧ (𝐵𝐾) ≤ 𝑌) → (vol‘∅) = 0)
156153, 155eqtrd 2794 . . . . . . . . 9 ((𝜑 ∧ (𝐵𝐾) ≤ 𝑌) → (vol‘(if(𝑌 ≤ (𝐴𝐾), (𝐴𝐾), 𝑌)[,)(𝐵𝐾))) = 0)
157156oveq2d 6829 . . . . . . . 8 ((𝜑 ∧ (𝐵𝐾) ≤ 𝑌) → ((vol‘((𝐴𝐾)[,)(𝐵𝐾))) + (vol‘(if(𝑌 ≤ (𝐴𝐾), (𝐴𝐾), 𝑌)[,)(𝐵𝐾)))) = ((vol‘((𝐴𝐾)[,)(𝐵𝐾))) + 0))
158103addid1d 10428 . . . . . . . . 9 (𝜑 → ((vol‘((𝐴𝐾)[,)(𝐵𝐾))) + 0) = (vol‘((𝐴𝐾)[,)(𝐵𝐾))))
159158adantr 472 . . . . . . . 8 ((𝜑 ∧ (𝐵𝐾) ≤ 𝑌) → ((vol‘((𝐴𝐾)[,)(𝐵𝐾))) + 0) = (vol‘((𝐴𝐾)[,)(𝐵𝐾))))
160123, 157, 1593eqtrd 2798 . . . . . . 7 ((𝜑 ∧ (𝐵𝐾) ≤ 𝑌) → ((vol‘((𝐴𝐾)[,)if((𝐵𝐾) ≤ 𝑌, (𝐵𝐾), 𝑌))) + (vol‘(if(𝑌 ≤ (𝐴𝐾), (𝐴𝐾), 𝑌)[,)(𝐵𝐾)))) = (vol‘((𝐴𝐾)[,)(𝐵𝐾))))
161 iffalse 4239 . . . . . . . . . . . 12 (¬ (𝐵𝐾) ≤ 𝑌 → if((𝐵𝐾) ≤ 𝑌, (𝐵𝐾), 𝑌) = 𝑌)
162161oveq2d 6829 . . . . . . . . . . 11 (¬ (𝐵𝐾) ≤ 𝑌 → ((𝐴𝐾)[,)if((𝐵𝐾) ≤ 𝑌, (𝐵𝐾), 𝑌)) = ((𝐴𝐾)[,)𝑌))
163162fveq2d 6356 . . . . . . . . . 10 (¬ (𝐵𝐾) ≤ 𝑌 → (vol‘((𝐴𝐾)[,)if((𝐵𝐾) ≤ 𝑌, (𝐵𝐾), 𝑌))) = (vol‘((𝐴𝐾)[,)𝑌)))
164163adantl 473 . . . . . . . . 9 ((𝜑 ∧ ¬ (𝐵𝐾) ≤ 𝑌) → (vol‘((𝐴𝐾)[,)if((𝐵𝐾) ≤ 𝑌, (𝐵𝐾), 𝑌))) = (vol‘((𝐴𝐾)[,)𝑌)))
165164oveq1d 6828 . . . . . . . 8 ((𝜑 ∧ ¬ (𝐵𝐾) ≤ 𝑌) → ((vol‘((𝐴𝐾)[,)if((𝐵𝐾) ≤ 𝑌, (𝐵𝐾), 𝑌))) + (vol‘(if(𝑌 ≤ (𝐴𝐾), (𝐴𝐾), 𝑌)[,)(𝐵𝐾)))) = ((vol‘((𝐴𝐾)[,)𝑌)) + (vol‘(if(𝑌 ≤ (𝐴𝐾), (𝐴𝐾), 𝑌)[,)(𝐵𝐾)))))
166 simpl 474 . . . . . . . . 9 ((𝜑 ∧ ¬ (𝐵𝐾) ≤ 𝑌) → 𝜑)
167 simpr 479 . . . . . . . . . 10 ((𝜑 ∧ ¬ (𝐵𝐾) ≤ 𝑌) → ¬ (𝐵𝐾) ≤ 𝑌)
168166, 6syl 17 . . . . . . . . . . 11 ((𝜑 ∧ ¬ (𝐵𝐾) ≤ 𝑌) → 𝑌 ∈ ℝ)
169166, 78syl 17 . . . . . . . . . . 11 ((𝜑 ∧ ¬ (𝐵𝐾) ≤ 𝑌) → (𝐵𝐾) ∈ ℝ)
170168, 169ltnled 10376 . . . . . . . . . 10 ((𝜑 ∧ ¬ (𝐵𝐾) ≤ 𝑌) → (𝑌 < (𝐵𝐾) ↔ ¬ (𝐵𝐾) ≤ 𝑌))
171167, 170mpbird 247 . . . . . . . . 9 ((𝜑 ∧ ¬ (𝐵𝐾) ≤ 𝑌) → 𝑌 < (𝐵𝐾))
172125fveq2d 6356 . . . . . . . . . . . . 13 (𝑌 ≤ (𝐴𝐾) → (vol‘(if(𝑌 ≤ (𝐴𝐾), (𝐴𝐾), 𝑌)[,)(𝐵𝐾))) = (vol‘((𝐴𝐾)[,)(𝐵𝐾))))
173172oveq2d 6829 . . . . . . . . . . . 12 (𝑌 ≤ (𝐴𝐾) → ((vol‘((𝐴𝐾)[,)𝑌)) + (vol‘(if(𝑌 ≤ (𝐴𝐾), (𝐴𝐾), 𝑌)[,)(𝐵𝐾)))) = ((vol‘((𝐴𝐾)[,)𝑌)) + (vol‘((𝐴𝐾)[,)(𝐵𝐾)))))
174173adantl 473 . . . . . . . . . . 11 (((𝜑𝑌 < (𝐵𝐾)) ∧ 𝑌 ≤ (𝐴𝐾)) → ((vol‘((𝐴𝐾)[,)𝑌)) + (vol‘(if(𝑌 ≤ (𝐴𝐾), (𝐴𝐾), 𝑌)[,)(𝐵𝐾)))) = ((vol‘((𝐴𝐾)[,)𝑌)) + (vol‘((𝐴𝐾)[,)(𝐵𝐾)))))
175 simpr 479 . . . . . . . . . . . . . . . 16 ((𝜑𝑌 ≤ (𝐴𝐾)) → 𝑌 ≤ (𝐴𝐾))
17650rexrd 10281 . . . . . . . . . . . . . . . . . 18 (𝜑 → (𝐴𝐾) ∈ ℝ*)
177176adantr 472 . . . . . . . . . . . . . . . . 17 ((𝜑𝑌 ≤ (𝐴𝐾)) → (𝐴𝐾) ∈ ℝ*)
178143adantr 472 . . . . . . . . . . . . . . . . 17 ((𝜑𝑌 ≤ (𝐴𝐾)) → 𝑌 ∈ ℝ*)
179 ico0 12414 . . . . . . . . . . . . . . . . 17 (((𝐴𝐾) ∈ ℝ*𝑌 ∈ ℝ*) → (((𝐴𝐾)[,)𝑌) = ∅ ↔ 𝑌 ≤ (𝐴𝐾)))
180177, 178, 179syl2anc 696 . . . . . . . . . . . . . . . 16 ((𝜑𝑌 ≤ (𝐴𝐾)) → (((𝐴𝐾)[,)𝑌) = ∅ ↔ 𝑌 ≤ (𝐴𝐾)))
181175, 180mpbird 247 . . . . . . . . . . . . . . 15 ((𝜑𝑌 ≤ (𝐴𝐾)) → ((𝐴𝐾)[,)𝑌) = ∅)
182181fveq2d 6356 . . . . . . . . . . . . . 14 ((𝜑𝑌 ≤ (𝐴𝐾)) → (vol‘((𝐴𝐾)[,)𝑌)) = (vol‘∅))
183154a1i 11 . . . . . . . . . . . . . 14 ((𝜑𝑌 ≤ (𝐴𝐾)) → (vol‘∅) = 0)
184182, 183eqtrd 2794 . . . . . . . . . . . . 13 ((𝜑𝑌 ≤ (𝐴𝐾)) → (vol‘((𝐴𝐾)[,)𝑌)) = 0)
185184oveq1d 6828 . . . . . . . . . . . 12 ((𝜑𝑌 ≤ (𝐴𝐾)) → ((vol‘((𝐴𝐾)[,)𝑌)) + (vol‘((𝐴𝐾)[,)(𝐵𝐾)))) = (0 + (vol‘((𝐴𝐾)[,)(𝐵𝐾)))))
186185adantlr 753 . . . . . . . . . . 11 (((𝜑𝑌 < (𝐵𝐾)) ∧ 𝑌 ≤ (𝐴𝐾)) → ((vol‘((𝐴𝐾)[,)𝑌)) + (vol‘((𝐴𝐾)[,)(𝐵𝐾)))) = (0 + (vol‘((𝐴𝐾)[,)(𝐵𝐾)))))
187103addid2d 10429 . . . . . . . . . . . 12 (𝜑 → (0 + (vol‘((𝐴𝐾)[,)(𝐵𝐾)))) = (vol‘((𝐴𝐾)[,)(𝐵𝐾))))
188187ad2antrr 764 . . . . . . . . . . 11 (((𝜑𝑌 < (𝐵𝐾)) ∧ 𝑌 ≤ (𝐴𝐾)) → (0 + (vol‘((𝐴𝐾)[,)(𝐵𝐾)))) = (vol‘((𝐴𝐾)[,)(𝐵𝐾))))
189174, 186, 1883eqtrd 2798 . . . . . . . . . 10 (((𝜑𝑌 < (𝐵𝐾)) ∧ 𝑌 ≤ (𝐴𝐾)) → ((vol‘((𝐴𝐾)[,)𝑌)) + (vol‘(if(𝑌 ≤ (𝐴𝐾), (𝐴𝐾), 𝑌)[,)(𝐵𝐾)))) = (vol‘((𝐴𝐾)[,)(𝐵𝐾))))
190140fveq2d 6356 . . . . . . . . . . . . 13 𝑌 ≤ (𝐴𝐾) → (vol‘(if(𝑌 ≤ (𝐴𝐾), (𝐴𝐾), 𝑌)[,)(𝐵𝐾))) = (vol‘(𝑌[,)(𝐵𝐾))))
191190oveq2d 6829 . . . . . . . . . . . 12 𝑌 ≤ (𝐴𝐾) → ((vol‘((𝐴𝐾)[,)𝑌)) + (vol‘(if(𝑌 ≤ (𝐴𝐾), (𝐴𝐾), 𝑌)[,)(𝐵𝐾)))) = ((vol‘((𝐴𝐾)[,)𝑌)) + (vol‘(𝑌[,)(𝐵𝐾)))))
192191adantl 473 . . . . . . . . . . 11 (((𝜑𝑌 < (𝐵𝐾)) ∧ ¬ 𝑌 ≤ (𝐴𝐾)) → ((vol‘((𝐴𝐾)[,)𝑌)) + (vol‘(if(𝑌 ≤ (𝐴𝐾), (𝐴𝐾), 𝑌)[,)(𝐵𝐾)))) = ((vol‘((𝐴𝐾)[,)𝑌)) + (vol‘(𝑌[,)(𝐵𝐾)))))
193 simpl 474 . . . . . . . . . . . 12 (((𝜑𝑌 < (𝐵𝐾)) ∧ ¬ 𝑌 ≤ (𝐴𝐾)) → (𝜑𝑌 < (𝐵𝐾)))
194 simpr 479 . . . . . . . . . . . . . 14 ((𝜑 ∧ ¬ 𝑌 ≤ (𝐴𝐾)) → ¬ 𝑌 ≤ (𝐴𝐾))
19550adantr 472 . . . . . . . . . . . . . . 15 ((𝜑 ∧ ¬ 𝑌 ≤ (𝐴𝐾)) → (𝐴𝐾) ∈ ℝ)
1966adantr 472 . . . . . . . . . . . . . . 15 ((𝜑 ∧ ¬ 𝑌 ≤ (𝐴𝐾)) → 𝑌 ∈ ℝ)
197195, 196ltnled 10376 . . . . . . . . . . . . . 14 ((𝜑 ∧ ¬ 𝑌 ≤ (𝐴𝐾)) → ((𝐴𝐾) < 𝑌 ↔ ¬ 𝑌 ≤ (𝐴𝐾)))
198194, 197mpbird 247 . . . . . . . . . . . . 13 ((𝜑 ∧ ¬ 𝑌 ≤ (𝐴𝐾)) → (𝐴𝐾) < 𝑌)
199198adantlr 753 . . . . . . . . . . . 12 (((𝜑𝑌 < (𝐵𝐾)) ∧ ¬ 𝑌 ≤ (𝐴𝐾)) → (𝐴𝐾) < 𝑌)
20050adantr 472 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝐴𝐾) < 𝑌) → (𝐴𝐾) ∈ ℝ)
2016adantr 472 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝐴𝐾) < 𝑌) → 𝑌 ∈ ℝ)
202 volico 40703 . . . . . . . . . . . . . . . 16 (((𝐴𝐾) ∈ ℝ ∧ 𝑌 ∈ ℝ) → (vol‘((𝐴𝐾)[,)𝑌)) = if((𝐴𝐾) < 𝑌, (𝑌 − (𝐴𝐾)), 0))
203200, 201, 202syl2anc 696 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝐴𝐾) < 𝑌) → (vol‘((𝐴𝐾)[,)𝑌)) = if((𝐴𝐾) < 𝑌, (𝑌 − (𝐴𝐾)), 0))
204 iftrue 4236 . . . . . . . . . . . . . . . 16 ((𝐴𝐾) < 𝑌 → if((𝐴𝐾) < 𝑌, (𝑌 − (𝐴𝐾)), 0) = (𝑌 − (𝐴𝐾)))
205204adantl 473 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝐴𝐾) < 𝑌) → if((𝐴𝐾) < 𝑌, (𝑌 − (𝐴𝐾)), 0) = (𝑌 − (𝐴𝐾)))
206203, 205eqtrd 2794 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝐴𝐾) < 𝑌) → (vol‘((𝐴𝐾)[,)𝑌)) = (𝑌 − (𝐴𝐾)))
207206adantlr 753 . . . . . . . . . . . . 13 (((𝜑𝑌 < (𝐵𝐾)) ∧ (𝐴𝐾) < 𝑌) → (vol‘((𝐴𝐾)[,)𝑌)) = (𝑌 − (𝐴𝐾)))
2086adantr 472 . . . . . . . . . . . . . . . 16 ((𝜑𝑌 < (𝐵𝐾)) → 𝑌 ∈ ℝ)
20978adantr 472 . . . . . . . . . . . . . . . 16 ((𝜑𝑌 < (𝐵𝐾)) → (𝐵𝐾) ∈ ℝ)
210 volico 40703 . . . . . . . . . . . . . . . 16 ((𝑌 ∈ ℝ ∧ (𝐵𝐾) ∈ ℝ) → (vol‘(𝑌[,)(𝐵𝐾))) = if(𝑌 < (𝐵𝐾), ((𝐵𝐾) − 𝑌), 0))
211208, 209, 210syl2anc 696 . . . . . . . . . . . . . . 15 ((𝜑𝑌 < (𝐵𝐾)) → (vol‘(𝑌[,)(𝐵𝐾))) = if(𝑌 < (𝐵𝐾), ((𝐵𝐾) − 𝑌), 0))
212 iftrue 4236 . . . . . . . . . . . . . . . 16 (𝑌 < (𝐵𝐾) → if(𝑌 < (𝐵𝐾), ((𝐵𝐾) − 𝑌), 0) = ((𝐵𝐾) − 𝑌))
213212adantl 473 . . . . . . . . . . . . . . 15 ((𝜑𝑌 < (𝐵𝐾)) → if(𝑌 < (𝐵𝐾), ((𝐵𝐾) − 𝑌), 0) = ((𝐵𝐾) − 𝑌))
214211, 213eqtrd 2794 . . . . . . . . . . . . . 14 ((𝜑𝑌 < (𝐵𝐾)) → (vol‘(𝑌[,)(𝐵𝐾))) = ((𝐵𝐾) − 𝑌))
215214adantr 472 . . . . . . . . . . . . 13 (((𝜑𝑌 < (𝐵𝐾)) ∧ (𝐴𝐾) < 𝑌) → (vol‘(𝑌[,)(𝐵𝐾))) = ((𝐵𝐾) − 𝑌))
216207, 215oveq12d 6831 . . . . . . . . . . . 12 (((𝜑𝑌 < (𝐵𝐾)) ∧ (𝐴𝐾) < 𝑌) → ((vol‘((𝐴𝐾)[,)𝑌)) + (vol‘(𝑌[,)(𝐵𝐾)))) = ((𝑌 − (𝐴𝐾)) + ((𝐵𝐾) − 𝑌)))
217193, 199, 216syl2anc 696 . . . . . . . . . . 11 (((𝜑𝑌 < (𝐵𝐾)) ∧ ¬ 𝑌 ≤ (𝐴𝐾)) → ((vol‘((𝐴𝐾)[,)𝑌)) + (vol‘(𝑌[,)(𝐵𝐾)))) = ((𝑌 − (𝐴𝐾)) + ((𝐵𝐾) − 𝑌)))
218200adantlr 753 . . . . . . . . . . . . . . . 16 (((𝜑𝑌 < (𝐵𝐾)) ∧ (𝐴𝐾) < 𝑌) → (𝐴𝐾) ∈ ℝ)
219208adantr 472 . . . . . . . . . . . . . . . 16 (((𝜑𝑌 < (𝐵𝐾)) ∧ (𝐴𝐾) < 𝑌) → 𝑌 ∈ ℝ)
220209adantr 472 . . . . . . . . . . . . . . . 16 (((𝜑𝑌 < (𝐵𝐾)) ∧ (𝐴𝐾) < 𝑌) → (𝐵𝐾) ∈ ℝ)
221 simpr 479 . . . . . . . . . . . . . . . 16 (((𝜑𝑌 < (𝐵𝐾)) ∧ (𝐴𝐾) < 𝑌) → (𝐴𝐾) < 𝑌)
222 simplr 809 . . . . . . . . . . . . . . . 16 (((𝜑𝑌 < (𝐵𝐾)) ∧ (𝐴𝐾) < 𝑌) → 𝑌 < (𝐵𝐾))
223218, 219, 220, 221, 222lttrd 10390 . . . . . . . . . . . . . . 15 (((𝜑𝑌 < (𝐵𝐾)) ∧ (𝐴𝐾) < 𝑌) → (𝐴𝐾) < (𝐵𝐾))
224223iftrued 4238 . . . . . . . . . . . . . 14 (((𝜑𝑌 < (𝐵𝐾)) ∧ (𝐴𝐾) < 𝑌) → if((𝐴𝐾) < (𝐵𝐾), ((𝐵𝐾) − (𝐴𝐾)), 0) = ((𝐵𝐾) − (𝐴𝐾)))
225224eqcomd 2766 . . . . . . . . . . . . 13 (((𝜑𝑌 < (𝐵𝐾)) ∧ (𝐴𝐾) < 𝑌) → ((𝐵𝐾) − (𝐴𝐾)) = if((𝐴𝐾) < (𝐵𝐾), ((𝐵𝐾) − (𝐴𝐾)), 0))
2266, 50resubcld 10650 . . . . . . . . . . . . . . . . 17 (𝜑 → (𝑌 − (𝐴𝐾)) ∈ ℝ)
227226recnd 10260 . . . . . . . . . . . . . . . 16 (𝜑 → (𝑌 − (𝐴𝐾)) ∈ ℂ)
22878, 6resubcld 10650 . . . . . . . . . . . . . . . . 17 (𝜑 → ((𝐵𝐾) − 𝑌) ∈ ℝ)
229228recnd 10260 . . . . . . . . . . . . . . . 16 (𝜑 → ((𝐵𝐾) − 𝑌) ∈ ℂ)
230227, 229addcomd 10430 . . . . . . . . . . . . . . 15 (𝜑 → ((𝑌 − (𝐴𝐾)) + ((𝐵𝐾) − 𝑌)) = (((𝐵𝐾) − 𝑌) + (𝑌 − (𝐴𝐾))))
23178recnd 10260 . . . . . . . . . . . . . . . 16 (𝜑 → (𝐵𝐾) ∈ ℂ)
2326recnd 10260 . . . . . . . . . . . . . . . 16 (𝜑𝑌 ∈ ℂ)
23350recnd 10260 . . . . . . . . . . . . . . . 16 (𝜑 → (𝐴𝐾) ∈ ℂ)
234231, 232, 233npncand 10608 . . . . . . . . . . . . . . 15 (𝜑 → (((𝐵𝐾) − 𝑌) + (𝑌 − (𝐴𝐾))) = ((𝐵𝐾) − (𝐴𝐾)))
235230, 234eqtrd 2794 . . . . . . . . . . . . . 14 (𝜑 → ((𝑌 − (𝐴𝐾)) + ((𝐵𝐾) − 𝑌)) = ((𝐵𝐾) − (𝐴𝐾)))
236235ad2antrr 764 . . . . . . . . . . . . 13 (((𝜑𝑌 < (𝐵𝐾)) ∧ (𝐴𝐾) < 𝑌) → ((𝑌 − (𝐴𝐾)) + ((𝐵𝐾) − 𝑌)) = ((𝐵𝐾) − (𝐴𝐾)))
237 volico 40703 . . . . . . . . . . . . . 14 (((𝐴𝐾) ∈ ℝ ∧ (𝐵𝐾) ∈ ℝ) → (vol‘((𝐴𝐾)[,)(𝐵𝐾))) = if((𝐴𝐾) < (𝐵𝐾), ((𝐵𝐾) − (𝐴𝐾)), 0))
238218, 220, 237syl2anc 696 . . . . . . . . . . . . 13 (((𝜑𝑌 < (𝐵𝐾)) ∧ (𝐴𝐾) < 𝑌) → (vol‘((𝐴𝐾)[,)(𝐵𝐾))) = if((𝐴𝐾) < (𝐵𝐾), ((𝐵𝐾) − (𝐴𝐾)), 0))
239225, 236, 2383eqtr4d 2804 . . . . . . . . . . . 12 (((𝜑𝑌 < (𝐵𝐾)) ∧ (𝐴𝐾) < 𝑌) → ((𝑌 − (𝐴𝐾)) + ((𝐵𝐾) − 𝑌)) = (vol‘((𝐴𝐾)[,)(𝐵𝐾))))
240193, 199, 239syl2anc 696 . . . . . . . . . . 11 (((𝜑𝑌 < (𝐵𝐾)) ∧ ¬ 𝑌 ≤ (𝐴𝐾)) → ((𝑌 − (𝐴𝐾)) + ((𝐵𝐾) − 𝑌)) = (vol‘((𝐴𝐾)[,)(𝐵𝐾))))
241192, 217, 2403eqtrd 2798 . . . . . . . . . 10 (((𝜑𝑌 < (𝐵𝐾)) ∧ ¬ 𝑌 ≤ (𝐴𝐾)) → ((vol‘((𝐴𝐾)[,)𝑌)) + (vol‘(if(𝑌 ≤ (𝐴𝐾), (𝐴𝐾), 𝑌)[,)(𝐵𝐾)))) = (vol‘((𝐴𝐾)[,)(𝐵𝐾))))
242189, 241pm2.61dan 867 . . . . . . . . 9 ((𝜑𝑌 < (𝐵𝐾)) → ((vol‘((𝐴𝐾)[,)𝑌)) + (vol‘(if(𝑌 ≤ (𝐴𝐾), (𝐴𝐾), 𝑌)[,)(𝐵𝐾)))) = (vol‘((𝐴𝐾)[,)(𝐵𝐾))))
243166, 171, 242syl2anc 696 . . . . . . . 8 ((𝜑 ∧ ¬ (𝐵𝐾) ≤ 𝑌) → ((vol‘((𝐴𝐾)[,)𝑌)) + (vol‘(if(𝑌 ≤ (𝐴𝐾), (𝐴𝐾), 𝑌)[,)(𝐵𝐾)))) = (vol‘((𝐴𝐾)[,)(𝐵𝐾))))
244165, 243eqtrd 2794 . . . . . . 7 ((𝜑 ∧ ¬ (𝐵𝐾) ≤ 𝑌) → ((vol‘((𝐴𝐾)[,)if((𝐵𝐾) ≤ 𝑌, (𝐵𝐾), 𝑌))) + (vol‘(if(𝑌 ≤ (𝐴𝐾), (𝐴𝐾), 𝑌)[,)(𝐵𝐾)))) = (vol‘((𝐴𝐾)[,)(𝐵𝐾))))
245160, 244pm2.61dan 867 . . . . . 6 (𝜑 → ((vol‘((𝐴𝐾)[,)if((𝐵𝐾) ≤ 𝑌, (𝐵𝐾), 𝑌))) + (vol‘(if(𝑌 ≤ (𝐴𝐾), (𝐴𝐾), 𝑌)[,)(𝐵𝐾)))) = (vol‘((𝐴𝐾)[,)(𝐵𝐾))))
246118, 245eqtr2d 2795 . . . . 5 (𝜑 → (vol‘((𝐴𝐾)[,)(𝐵𝐾))) = ((vol‘((𝐴𝐾)[,)(((𝑇𝑌)‘𝐵)‘𝐾))) + (vol‘((((𝑆𝑌)‘𝐴)‘𝐾)[,)(𝐵𝐾)))))
247246oveq2d 6829 . . . 4 (𝜑 → (∏𝑘 ∈ (𝑋 ∖ {𝐾})(vol‘((𝐴𝑘)[,)(𝐵𝑘))) · (vol‘((𝐴𝐾)[,)(𝐵𝐾)))) = (∏𝑘 ∈ (𝑋 ∖ {𝐾})(vol‘((𝐴𝑘)[,)(𝐵𝑘))) · ((vol‘((𝐴𝐾)[,)(((𝑇𝑌)‘𝐵)‘𝐾))) + (vol‘((((𝑆𝑌)‘𝐴)‘𝐾)[,)(𝐵𝐾))))))
24833, 98fprodcl 14881 . . . . 5 (𝜑 → ∏𝑘 ∈ (𝑋 ∖ {𝐾})(vol‘((𝐴𝑘)[,)(𝐵𝑘))) ∈ ℂ)
249248, 54, 81adddid 10256 . . . 4 (𝜑 → (∏𝑘 ∈ (𝑋 ∖ {𝐾})(vol‘((𝐴𝑘)[,)(𝐵𝑘))) · ((vol‘((𝐴𝐾)[,)(((𝑇𝑌)‘𝐵)‘𝐾))) + (vol‘((((𝑆𝑌)‘𝐴)‘𝐾)[,)(𝐵𝐾))))) = ((∏𝑘 ∈ (𝑋 ∖ {𝐾})(vol‘((𝐴𝑘)[,)(𝐵𝑘))) · (vol‘((𝐴𝐾)[,)(((𝑇𝑌)‘𝐵)‘𝐾)))) + (∏𝑘 ∈ (𝑋 ∖ {𝐾})(vol‘((𝐴𝑘)[,)(𝐵𝑘))) · (vol‘((((𝑆𝑌)‘𝐴)‘𝐾)[,)(𝐵𝐾))))))
250105, 247, 2493eqtrrd 2799 . . 3 (𝜑 → ((∏𝑘 ∈ (𝑋 ∖ {𝐾})(vol‘((𝐴𝑘)[,)(𝐵𝑘))) · (vol‘((𝐴𝐾)[,)(((𝑇𝑌)‘𝐵)‘𝐾)))) + (∏𝑘 ∈ (𝑋 ∖ {𝐾})(vol‘((𝐴𝑘)[,)(𝐵𝑘))) · (vol‘((((𝑆𝑌)‘𝐴)‘𝐾)[,)(𝐵𝐾))))) = ∏𝑘𝑋 (vol‘((𝐴𝑘)[,)(𝐵𝑘))))
25121, 95, 2503eqtrd 2798 . 2 (𝜑 → ((𝐴(𝐿𝑋)((𝑇𝑌)‘𝐵)) + (((𝑆𝑌)‘𝐴)(𝐿𝑋)𝐵)) = ∏𝑘𝑋 (vol‘((𝐴𝑘)[,)(𝐵𝑘))))
2522, 3, 18, 4, 7hoidmvn0val 41304 . . 3 (𝜑 → (𝐴(𝐿𝑋)𝐵) = ∏𝑘𝑋 (vol‘((𝐴𝑘)[,)(𝐵𝑘))))
253252eqcomd 2766 . 2 (𝜑 → ∏𝑘𝑋 (vol‘((𝐴𝑘)[,)(𝐵𝑘))) = (𝐴(𝐿𝑋)𝐵))
25415, 251, 2533eqtrrd 2799 1 (𝜑 → (𝐴(𝐿𝑋)𝐵) = ((𝐴(𝐿𝑋)((𝑇𝑌)‘𝐵)) +𝑒 (((𝑆𝑌)‘𝐴)(𝐿𝑋)𝐵)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 383   = wceq 1632  wcel 2139  wne 2932  cdif 3712  cun 3713  wss 3715  c0 4058  ifcif 4230  {csn 4321   class class class wbr 4804  cmpt 4881  wf 6045  cfv 6049  (class class class)co 6813  cmpt2 6815  𝑚 cmap 8023  Fincfn 8121  cc 10126  cr 10127  0cc0 10128   + caddc 10131   · cmul 10133  +∞cpnf 10263  *cxr 10265   < clt 10266  cle 10267  cmin 10458   +𝑒 cxad 12137  [,)cico 12370  cprod 14834  volcvol 23432
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114  ax-inf2 8711  ax-cnex 10184  ax-resscn 10185  ax-1cn 10186  ax-icn 10187  ax-addcl 10188  ax-addrcl 10189  ax-mulcl 10190  ax-mulrcl 10191  ax-mulcom 10192  ax-addass 10193  ax-mulass 10194  ax-distr 10195  ax-i2m1 10196  ax-1ne0 10197  ax-1rid 10198  ax-rnegex 10199  ax-rrecex 10200  ax-cnre 10201  ax-pre-lttri 10202  ax-pre-lttrn 10203  ax-pre-ltadd 10204  ax-pre-mulgt0 10205  ax-pre-sup 10206
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-fal 1638  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-nel 3036  df-ral 3055  df-rex 3056  df-reu 3057  df-rmo 3058  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-int 4628  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-se 5226  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-pred 5841  df-ord 5887  df-on 5888  df-lim 5889  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-isom 6058  df-riota 6774  df-ov 6816  df-oprab 6817  df-mpt2 6818  df-of 7062  df-om 7231  df-1st 7333  df-2nd 7334  df-wrecs 7576  df-recs 7637  df-rdg 7675  df-1o 7729  df-2o 7730  df-oadd 7733  df-er 7911  df-map 8025  df-pm 8026  df-en 8122  df-dom 8123  df-sdom 8124  df-fin 8125  df-fi 8482  df-sup 8513  df-inf 8514  df-oi 8580  df-card 8955  df-cda 9182  df-pnf 10268  df-mnf 10269  df-xr 10270  df-ltxr 10271  df-le 10272  df-sub 10460  df-neg 10461  df-div 10877  df-nn 11213  df-2 11271  df-3 11272  df-n0 11485  df-z 11570  df-uz 11880  df-q 11982  df-rp 12026  df-xneg 12139  df-xadd 12140  df-xmul 12141  df-ioo 12372  df-ico 12374  df-icc 12375  df-fz 12520  df-fzo 12660  df-fl 12787  df-seq 12996  df-exp 13055  df-hash 13312  df-cj 14038  df-re 14039  df-im 14040  df-sqrt 14174  df-abs 14175  df-clim 14418  df-rlim 14419  df-sum 14616  df-prod 14835  df-rest 16285  df-topgen 16306  df-psmet 19940  df-xmet 19941  df-met 19942  df-bl 19943  df-mopn 19944  df-top 20901  df-topon 20918  df-bases 20952  df-cmp 21392  df-ovol 23433  df-vol 23434
This theorem is referenced by:  hspmbllem2  41347
  Copyright terms: Public domain W3C validator