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Theorem hsphoidmvle2 46541
Description: The dimensional volume of a half-open interval intersected with a two half-spaces. Used in the last inequality of step (c) of Lemma 115B of [Fremlin1] p. 29. (Contributed by Glauco Siliprandi, 21-Nov-2020.)
Hypotheses
Ref Expression
hsphoidmvle2.l 𝐿 = (𝑥 ∈ Fin ↦ (𝑎 ∈ (ℝ ↑m 𝑥), 𝑏 ∈ (ℝ ↑m 𝑥) ↦ if(𝑥 = ∅, 0, ∏𝑘𝑥 (vol‘((𝑎𝑘)[,)(𝑏𝑘))))))
hsphoidmvle2.x (𝜑𝑋 ∈ Fin)
hsphoidmvle2.z (𝜑𝑍 ∈ (𝑋𝑌))
hsphoidmvle2.y 𝑋 = (𝑌 ∪ {𝑍})
hsphoidmvle2.c (𝜑𝐶 ∈ ℝ)
hsphoidmvle2.d (𝜑𝐷 ∈ ℝ)
hsphoidmvle2.e (𝜑𝐶𝐷)
hsphoidmvle2.h 𝐻 = (𝑥 ∈ ℝ ↦ (𝑐 ∈ (ℝ ↑m 𝑋) ↦ (𝑗𝑋 ↦ if(𝑗𝑌, (𝑐𝑗), if((𝑐𝑗) ≤ 𝑥, (𝑐𝑗), 𝑥)))))
hsphoidmvle2.a (𝜑𝐴:𝑋⟶ℝ)
hsphoidmvle2.b (𝜑𝐵:𝑋⟶ℝ)
Assertion
Ref Expression
hsphoidmvle2 (𝜑 → (𝐴(𝐿𝑋)((𝐻𝐶)‘𝐵)) ≤ (𝐴(𝐿𝑋)((𝐻𝐷)‘𝐵)))
Distinct variable groups:   𝐴,𝑎,𝑏,𝑘   𝐵,𝑎,𝑏,𝑘   𝐵,𝑐,𝑗,𝑘   𝐶,𝑎,𝑏,𝑘,𝑥   𝐶,𝑐,𝑗,𝑥   𝐷,𝑎,𝑏,𝑘,𝑥   𝐷,𝑐,𝑗   𝐻,𝑎,𝑏,𝑘   𝑋,𝑎,𝑏,𝑘,𝑥   𝑋,𝑐,𝑗   𝑌,𝑐,𝑗,𝑥   𝑍,𝑐,𝑗,𝑘,𝑥   𝜑,𝑎,𝑏,𝑘,𝑥   𝜑,𝑐,𝑗
Allowed substitution hints:   𝐴(𝑥,𝑗,𝑐)   𝐵(𝑥)   𝐻(𝑥,𝑗,𝑐)   𝐿(𝑥,𝑗,𝑘,𝑎,𝑏,𝑐)   𝑌(𝑘,𝑎,𝑏)   𝑍(𝑎,𝑏)

Proof of Theorem hsphoidmvle2
StepHypRef Expression
1 hsphoidmvle2.a . . . . 5 (𝜑𝐴:𝑋⟶ℝ)
2 hsphoidmvle2.z . . . . . 6 (𝜑𝑍 ∈ (𝑋𝑌))
32eldifad 3975 . . . . 5 (𝜑𝑍𝑋)
41, 3ffvelcdmd 7105 . . . 4 (𝜑 → (𝐴𝑍) ∈ ℝ)
5 hsphoidmvle2.b . . . . . 6 (𝜑𝐵:𝑋⟶ℝ)
65, 3ffvelcdmd 7105 . . . . 5 (𝜑 → (𝐵𝑍) ∈ ℝ)
7 hsphoidmvle2.c . . . . 5 (𝜑𝐶 ∈ ℝ)
86, 7ifcld 4577 . . . 4 (𝜑 → if((𝐵𝑍) ≤ 𝐶, (𝐵𝑍), 𝐶) ∈ ℝ)
9 volicore 46537 . . . 4 (((𝐴𝑍) ∈ ℝ ∧ if((𝐵𝑍) ≤ 𝐶, (𝐵𝑍), 𝐶) ∈ ℝ) → (vol‘((𝐴𝑍)[,)if((𝐵𝑍) ≤ 𝐶, (𝐵𝑍), 𝐶))) ∈ ℝ)
104, 8, 9syl2anc 584 . . 3 (𝜑 → (vol‘((𝐴𝑍)[,)if((𝐵𝑍) ≤ 𝐶, (𝐵𝑍), 𝐶))) ∈ ℝ)
11 hsphoidmvle2.d . . . . 5 (𝜑𝐷 ∈ ℝ)
126, 11ifcld 4577 . . . 4 (𝜑 → if((𝐵𝑍) ≤ 𝐷, (𝐵𝑍), 𝐷) ∈ ℝ)
13 volicore 46537 . . . 4 (((𝐴𝑍) ∈ ℝ ∧ if((𝐵𝑍) ≤ 𝐷, (𝐵𝑍), 𝐷) ∈ ℝ) → (vol‘((𝐴𝑍)[,)if((𝐵𝑍) ≤ 𝐷, (𝐵𝑍), 𝐷))) ∈ ℝ)
144, 12, 13syl2anc 584 . . 3 (𝜑 → (vol‘((𝐴𝑍)[,)if((𝐵𝑍) ≤ 𝐷, (𝐵𝑍), 𝐷))) ∈ ℝ)
15 hsphoidmvle2.x . . . . 5 (𝜑𝑋 ∈ Fin)
16 difssd 4147 . . . . 5 (𝜑 → (𝑋 ∖ {𝑍}) ⊆ 𝑋)
17 ssfi 9212 . . . . 5 ((𝑋 ∈ Fin ∧ (𝑋 ∖ {𝑍}) ⊆ 𝑋) → (𝑋 ∖ {𝑍}) ∈ Fin)
1815, 16, 17syl2anc 584 . . . 4 (𝜑 → (𝑋 ∖ {𝑍}) ∈ Fin)
19 eldifi 4141 . . . . . 6 (𝑘 ∈ (𝑋 ∖ {𝑍}) → 𝑘𝑋)
2019adantl 481 . . . . 5 ((𝜑𝑘 ∈ (𝑋 ∖ {𝑍})) → 𝑘𝑋)
211ffvelcdmda 7104 . . . . . 6 ((𝜑𝑘𝑋) → (𝐴𝑘) ∈ ℝ)
225ffvelcdmda 7104 . . . . . 6 ((𝜑𝑘𝑋) → (𝐵𝑘) ∈ ℝ)
23 volicore 46537 . . . . . 6 (((𝐴𝑘) ∈ ℝ ∧ (𝐵𝑘) ∈ ℝ) → (vol‘((𝐴𝑘)[,)(𝐵𝑘))) ∈ ℝ)
2421, 22, 23syl2anc 584 . . . . 5 ((𝜑𝑘𝑋) → (vol‘((𝐴𝑘)[,)(𝐵𝑘))) ∈ ℝ)
2520, 24syldan 591 . . . 4 ((𝜑𝑘 ∈ (𝑋 ∖ {𝑍})) → (vol‘((𝐴𝑘)[,)(𝐵𝑘))) ∈ ℝ)
2618, 25fprodrecl 15986 . . 3 (𝜑 → ∏𝑘 ∈ (𝑋 ∖ {𝑍})(vol‘((𝐴𝑘)[,)(𝐵𝑘))) ∈ ℝ)
27 nfv 1912 . . . 4 𝑘𝜑
2820, 21syldan 591 . . . . . 6 ((𝜑𝑘 ∈ (𝑋 ∖ {𝑍})) → (𝐴𝑘) ∈ ℝ)
2920, 22syldan 591 . . . . . . 7 ((𝜑𝑘 ∈ (𝑋 ∖ {𝑍})) → (𝐵𝑘) ∈ ℝ)
3029rexrd 11309 . . . . . 6 ((𝜑𝑘 ∈ (𝑋 ∖ {𝑍})) → (𝐵𝑘) ∈ ℝ*)
31 icombl 25613 . . . . . 6 (((𝐴𝑘) ∈ ℝ ∧ (𝐵𝑘) ∈ ℝ*) → ((𝐴𝑘)[,)(𝐵𝑘)) ∈ dom vol)
3228, 30, 31syl2anc 584 . . . . 5 ((𝜑𝑘 ∈ (𝑋 ∖ {𝑍})) → ((𝐴𝑘)[,)(𝐵𝑘)) ∈ dom vol)
33 volge0 45917 . . . . 5 (((𝐴𝑘)[,)(𝐵𝑘)) ∈ dom vol → 0 ≤ (vol‘((𝐴𝑘)[,)(𝐵𝑘))))
3432, 33syl 17 . . . 4 ((𝜑𝑘 ∈ (𝑋 ∖ {𝑍})) → 0 ≤ (vol‘((𝐴𝑘)[,)(𝐵𝑘))))
3527, 18, 25, 34fprodge0 16026 . . 3 (𝜑 → 0 ≤ ∏𝑘 ∈ (𝑋 ∖ {𝑍})(vol‘((𝐴𝑘)[,)(𝐵𝑘))))
368rexrd 11309 . . . . 5 (𝜑 → if((𝐵𝑍) ≤ 𝐶, (𝐵𝑍), 𝐶) ∈ ℝ*)
37 icombl 25613 . . . . 5 (((𝐴𝑍) ∈ ℝ ∧ if((𝐵𝑍) ≤ 𝐶, (𝐵𝑍), 𝐶) ∈ ℝ*) → ((𝐴𝑍)[,)if((𝐵𝑍) ≤ 𝐶, (𝐵𝑍), 𝐶)) ∈ dom vol)
384, 36, 37syl2anc 584 . . . 4 (𝜑 → ((𝐴𝑍)[,)if((𝐵𝑍) ≤ 𝐶, (𝐵𝑍), 𝐶)) ∈ dom vol)
3912rexrd 11309 . . . . 5 (𝜑 → if((𝐵𝑍) ≤ 𝐷, (𝐵𝑍), 𝐷) ∈ ℝ*)
40 icombl 25613 . . . . 5 (((𝐴𝑍) ∈ ℝ ∧ if((𝐵𝑍) ≤ 𝐷, (𝐵𝑍), 𝐷) ∈ ℝ*) → ((𝐴𝑍)[,)if((𝐵𝑍) ≤ 𝐷, (𝐵𝑍), 𝐷)) ∈ dom vol)
414, 39, 40syl2anc 584 . . . 4 (𝜑 → ((𝐴𝑍)[,)if((𝐵𝑍) ≤ 𝐷, (𝐵𝑍), 𝐷)) ∈ dom vol)
424rexrd 11309 . . . . 5 (𝜑 → (𝐴𝑍) ∈ ℝ*)
434leidd 11827 . . . . 5 (𝜑 → (𝐴𝑍) ≤ (𝐴𝑍))
446leidd 11827 . . . . . . . 8 (𝜑 → (𝐵𝑍) ≤ (𝐵𝑍))
4544adantr 480 . . . . . . 7 ((𝜑 ∧ (𝐵𝑍) ≤ 𝐶) → (𝐵𝑍) ≤ (𝐵𝑍))
46 iftrue 4537 . . . . . . . . 9 ((𝐵𝑍) ≤ 𝐶 → if((𝐵𝑍) ≤ 𝐶, (𝐵𝑍), 𝐶) = (𝐵𝑍))
4746adantl 481 . . . . . . . 8 ((𝜑 ∧ (𝐵𝑍) ≤ 𝐶) → if((𝐵𝑍) ≤ 𝐶, (𝐵𝑍), 𝐶) = (𝐵𝑍))
486adantr 480 . . . . . . . . . 10 ((𝜑 ∧ (𝐵𝑍) ≤ 𝐶) → (𝐵𝑍) ∈ ℝ)
497adantr 480 . . . . . . . . . 10 ((𝜑 ∧ (𝐵𝑍) ≤ 𝐶) → 𝐶 ∈ ℝ)
5011adantr 480 . . . . . . . . . 10 ((𝜑 ∧ (𝐵𝑍) ≤ 𝐶) → 𝐷 ∈ ℝ)
51 simpr 484 . . . . . . . . . 10 ((𝜑 ∧ (𝐵𝑍) ≤ 𝐶) → (𝐵𝑍) ≤ 𝐶)
52 hsphoidmvle2.e . . . . . . . . . . 11 (𝜑𝐶𝐷)
5352adantr 480 . . . . . . . . . 10 ((𝜑 ∧ (𝐵𝑍) ≤ 𝐶) → 𝐶𝐷)
5448, 49, 50, 51, 53letrd 11416 . . . . . . . . 9 ((𝜑 ∧ (𝐵𝑍) ≤ 𝐶) → (𝐵𝑍) ≤ 𝐷)
5554iftrued 4539 . . . . . . . 8 ((𝜑 ∧ (𝐵𝑍) ≤ 𝐶) → if((𝐵𝑍) ≤ 𝐷, (𝐵𝑍), 𝐷) = (𝐵𝑍))
5647, 55breq12d 5161 . . . . . . 7 ((𝜑 ∧ (𝐵𝑍) ≤ 𝐶) → (if((𝐵𝑍) ≤ 𝐶, (𝐵𝑍), 𝐶) ≤ if((𝐵𝑍) ≤ 𝐷, (𝐵𝑍), 𝐷) ↔ (𝐵𝑍) ≤ (𝐵𝑍)))
5745, 56mpbird 257 . . . . . 6 ((𝜑 ∧ (𝐵𝑍) ≤ 𝐶) → if((𝐵𝑍) ≤ 𝐶, (𝐵𝑍), 𝐶) ≤ if((𝐵𝑍) ≤ 𝐷, (𝐵𝑍), 𝐷))
58 simpl 482 . . . . . . . 8 ((𝜑 ∧ ¬ (𝐵𝑍) ≤ 𝐶) → 𝜑)
59 simpr 484 . . . . . . . . 9 ((𝜑 ∧ ¬ (𝐵𝑍) ≤ 𝐶) → ¬ (𝐵𝑍) ≤ 𝐶)
6058, 7syl 17 . . . . . . . . . 10 ((𝜑 ∧ ¬ (𝐵𝑍) ≤ 𝐶) → 𝐶 ∈ ℝ)
6158, 6syl 17 . . . . . . . . . 10 ((𝜑 ∧ ¬ (𝐵𝑍) ≤ 𝐶) → (𝐵𝑍) ∈ ℝ)
6260, 61ltnled 11406 . . . . . . . . 9 ((𝜑 ∧ ¬ (𝐵𝑍) ≤ 𝐶) → (𝐶 < (𝐵𝑍) ↔ ¬ (𝐵𝑍) ≤ 𝐶))
6359, 62mpbird 257 . . . . . . . 8 ((𝜑 ∧ ¬ (𝐵𝑍) ≤ 𝐶) → 𝐶 < (𝐵𝑍))
647adantr 480 . . . . . . . . . . . 12 ((𝜑𝐶 < (𝐵𝑍)) → 𝐶 ∈ ℝ)
656adantr 480 . . . . . . . . . . . 12 ((𝜑𝐶 < (𝐵𝑍)) → (𝐵𝑍) ∈ ℝ)
66 simpr 484 . . . . . . . . . . . 12 ((𝜑𝐶 < (𝐵𝑍)) → 𝐶 < (𝐵𝑍))
6764, 65, 66ltled 11407 . . . . . . . . . . 11 ((𝜑𝐶 < (𝐵𝑍)) → 𝐶 ≤ (𝐵𝑍))
6867adantr 480 . . . . . . . . . 10 (((𝜑𝐶 < (𝐵𝑍)) ∧ (𝐵𝑍) ≤ 𝐷) → 𝐶 ≤ (𝐵𝑍))
69 iftrue 4537 . . . . . . . . . . . 12 ((𝐵𝑍) ≤ 𝐷 → if((𝐵𝑍) ≤ 𝐷, (𝐵𝑍), 𝐷) = (𝐵𝑍))
7069eqcomd 2741 . . . . . . . . . . 11 ((𝐵𝑍) ≤ 𝐷 → (𝐵𝑍) = if((𝐵𝑍) ≤ 𝐷, (𝐵𝑍), 𝐷))
7170adantl 481 . . . . . . . . . 10 (((𝜑𝐶 < (𝐵𝑍)) ∧ (𝐵𝑍) ≤ 𝐷) → (𝐵𝑍) = if((𝐵𝑍) ≤ 𝐷, (𝐵𝑍), 𝐷))
7268, 71breqtrd 5174 . . . . . . . . 9 (((𝜑𝐶 < (𝐵𝑍)) ∧ (𝐵𝑍) ≤ 𝐷) → 𝐶 ≤ if((𝐵𝑍) ≤ 𝐷, (𝐵𝑍), 𝐷))
7352ad2antrr 726 . . . . . . . . . 10 (((𝜑𝐶 < (𝐵𝑍)) ∧ ¬ (𝐵𝑍) ≤ 𝐷) → 𝐶𝐷)
74 iffalse 4540 . . . . . . . . . . . 12 (¬ (𝐵𝑍) ≤ 𝐷 → if((𝐵𝑍) ≤ 𝐷, (𝐵𝑍), 𝐷) = 𝐷)
7574eqcomd 2741 . . . . . . . . . . 11 (¬ (𝐵𝑍) ≤ 𝐷𝐷 = if((𝐵𝑍) ≤ 𝐷, (𝐵𝑍), 𝐷))
7675adantl 481 . . . . . . . . . 10 (((𝜑𝐶 < (𝐵𝑍)) ∧ ¬ (𝐵𝑍) ≤ 𝐷) → 𝐷 = if((𝐵𝑍) ≤ 𝐷, (𝐵𝑍), 𝐷))
7773, 76breqtrd 5174 . . . . . . . . 9 (((𝜑𝐶 < (𝐵𝑍)) ∧ ¬ (𝐵𝑍) ≤ 𝐷) → 𝐶 ≤ if((𝐵𝑍) ≤ 𝐷, (𝐵𝑍), 𝐷))
7872, 77pm2.61dan 813 . . . . . . . 8 ((𝜑𝐶 < (𝐵𝑍)) → 𝐶 ≤ if((𝐵𝑍) ≤ 𝐷, (𝐵𝑍), 𝐷))
7958, 63, 78syl2anc 584 . . . . . . 7 ((𝜑 ∧ ¬ (𝐵𝑍) ≤ 𝐶) → 𝐶 ≤ if((𝐵𝑍) ≤ 𝐷, (𝐵𝑍), 𝐷))
80 iffalse 4540 . . . . . . . . 9 (¬ (𝐵𝑍) ≤ 𝐶 → if((𝐵𝑍) ≤ 𝐶, (𝐵𝑍), 𝐶) = 𝐶)
8180adantl 481 . . . . . . . 8 ((𝜑 ∧ ¬ (𝐵𝑍) ≤ 𝐶) → if((𝐵𝑍) ≤ 𝐶, (𝐵𝑍), 𝐶) = 𝐶)
8281breq1d 5158 . . . . . . 7 ((𝜑 ∧ ¬ (𝐵𝑍) ≤ 𝐶) → (if((𝐵𝑍) ≤ 𝐶, (𝐵𝑍), 𝐶) ≤ if((𝐵𝑍) ≤ 𝐷, (𝐵𝑍), 𝐷) ↔ 𝐶 ≤ if((𝐵𝑍) ≤ 𝐷, (𝐵𝑍), 𝐷)))
8379, 82mpbird 257 . . . . . 6 ((𝜑 ∧ ¬ (𝐵𝑍) ≤ 𝐶) → if((𝐵𝑍) ≤ 𝐶, (𝐵𝑍), 𝐶) ≤ if((𝐵𝑍) ≤ 𝐷, (𝐵𝑍), 𝐷))
8457, 83pm2.61dan 813 . . . . 5 (𝜑 → if((𝐵𝑍) ≤ 𝐶, (𝐵𝑍), 𝐶) ≤ if((𝐵𝑍) ≤ 𝐷, (𝐵𝑍), 𝐷))
85 icossico 13454 . . . . 5 ((((𝐴𝑍) ∈ ℝ* ∧ if((𝐵𝑍) ≤ 𝐷, (𝐵𝑍), 𝐷) ∈ ℝ*) ∧ ((𝐴𝑍) ≤ (𝐴𝑍) ∧ if((𝐵𝑍) ≤ 𝐶, (𝐵𝑍), 𝐶) ≤ if((𝐵𝑍) ≤ 𝐷, (𝐵𝑍), 𝐷))) → ((𝐴𝑍)[,)if((𝐵𝑍) ≤ 𝐶, (𝐵𝑍), 𝐶)) ⊆ ((𝐴𝑍)[,)if((𝐵𝑍) ≤ 𝐷, (𝐵𝑍), 𝐷)))
8642, 39, 43, 84, 85syl22anc 839 . . . 4 (𝜑 → ((𝐴𝑍)[,)if((𝐵𝑍) ≤ 𝐶, (𝐵𝑍), 𝐶)) ⊆ ((𝐴𝑍)[,)if((𝐵𝑍) ≤ 𝐷, (𝐵𝑍), 𝐷)))
87 volss 25582 . . . 4 ((((𝐴𝑍)[,)if((𝐵𝑍) ≤ 𝐶, (𝐵𝑍), 𝐶)) ∈ dom vol ∧ ((𝐴𝑍)[,)if((𝐵𝑍) ≤ 𝐷, (𝐵𝑍), 𝐷)) ∈ dom vol ∧ ((𝐴𝑍)[,)if((𝐵𝑍) ≤ 𝐶, (𝐵𝑍), 𝐶)) ⊆ ((𝐴𝑍)[,)if((𝐵𝑍) ≤ 𝐷, (𝐵𝑍), 𝐷))) → (vol‘((𝐴𝑍)[,)if((𝐵𝑍) ≤ 𝐶, (𝐵𝑍), 𝐶))) ≤ (vol‘((𝐴𝑍)[,)if((𝐵𝑍) ≤ 𝐷, (𝐵𝑍), 𝐷))))
8838, 41, 86, 87syl3anc 1370 . . 3 (𝜑 → (vol‘((𝐴𝑍)[,)if((𝐵𝑍) ≤ 𝐶, (𝐵𝑍), 𝐶))) ≤ (vol‘((𝐴𝑍)[,)if((𝐵𝑍) ≤ 𝐷, (𝐵𝑍), 𝐷))))
8910, 14, 26, 35, 88lemul1ad 12205 . 2 (𝜑 → ((vol‘((𝐴𝑍)[,)if((𝐵𝑍) ≤ 𝐶, (𝐵𝑍), 𝐶))) · ∏𝑘 ∈ (𝑋 ∖ {𝑍})(vol‘((𝐴𝑘)[,)(𝐵𝑘)))) ≤ ((vol‘((𝐴𝑍)[,)if((𝐵𝑍) ≤ 𝐷, (𝐵𝑍), 𝐷))) · ∏𝑘 ∈ (𝑋 ∖ {𝑍})(vol‘((𝐴𝑘)[,)(𝐵𝑘)))))
90 hsphoidmvle2.l . . . . 5 𝐿 = (𝑥 ∈ Fin ↦ (𝑎 ∈ (ℝ ↑m 𝑥), 𝑏 ∈ (ℝ ↑m 𝑥) ↦ if(𝑥 = ∅, 0, ∏𝑘𝑥 (vol‘((𝑎𝑘)[,)(𝑏𝑘))))))
913ne0d 4348 . . . . 5 (𝜑𝑋 ≠ ∅)
92 hsphoidmvle2.h . . . . . 6 𝐻 = (𝑥 ∈ ℝ ↦ (𝑐 ∈ (ℝ ↑m 𝑋) ↦ (𝑗𝑋 ↦ if(𝑗𝑌, (𝑐𝑗), if((𝑐𝑗) ≤ 𝑥, (𝑐𝑗), 𝑥)))))
9392, 7, 15, 5hsphoif 46532 . . . . 5 (𝜑 → ((𝐻𝐶)‘𝐵):𝑋⟶ℝ)
9490, 15, 91, 1, 93hoidmvn0val 46540 . . . 4 (𝜑 → (𝐴(𝐿𝑋)((𝐻𝐶)‘𝐵)) = ∏𝑘𝑋 (vol‘((𝐴𝑘)[,)(((𝐻𝐶)‘𝐵)‘𝑘))))
9593ffvelcdmda 7104 . . . . . . 7 ((𝜑𝑘𝑋) → (((𝐻𝐶)‘𝐵)‘𝑘) ∈ ℝ)
96 volicore 46537 . . . . . . 7 (((𝐴𝑘) ∈ ℝ ∧ (((𝐻𝐶)‘𝐵)‘𝑘) ∈ ℝ) → (vol‘((𝐴𝑘)[,)(((𝐻𝐶)‘𝐵)‘𝑘))) ∈ ℝ)
9721, 95, 96syl2anc 584 . . . . . 6 ((𝜑𝑘𝑋) → (vol‘((𝐴𝑘)[,)(((𝐻𝐶)‘𝐵)‘𝑘))) ∈ ℝ)
9897recnd 11287 . . . . 5 ((𝜑𝑘𝑋) → (vol‘((𝐴𝑘)[,)(((𝐻𝐶)‘𝐵)‘𝑘))) ∈ ℂ)
99 fveq2 6907 . . . . . . . . 9 (𝑘 = 𝑍 → (𝐴𝑘) = (𝐴𝑍))
100 fveq2 6907 . . . . . . . . 9 (𝑘 = 𝑍 → (((𝐻𝐶)‘𝐵)‘𝑘) = (((𝐻𝐶)‘𝐵)‘𝑍))
10199, 100oveq12d 7449 . . . . . . . 8 (𝑘 = 𝑍 → ((𝐴𝑘)[,)(((𝐻𝐶)‘𝐵)‘𝑘)) = ((𝐴𝑍)[,)(((𝐻𝐶)‘𝐵)‘𝑍)))
102101fveq2d 6911 . . . . . . 7 (𝑘 = 𝑍 → (vol‘((𝐴𝑘)[,)(((𝐻𝐶)‘𝐵)‘𝑘))) = (vol‘((𝐴𝑍)[,)(((𝐻𝐶)‘𝐵)‘𝑍))))
103102adantl 481 . . . . . 6 ((𝜑𝑘 = 𝑍) → (vol‘((𝐴𝑘)[,)(((𝐻𝐶)‘𝐵)‘𝑘))) = (vol‘((𝐴𝑍)[,)(((𝐻𝐶)‘𝐵)‘𝑍))))
10492, 7, 15, 5, 3hsphoival 46535 . . . . . . . . . 10 (𝜑 → (((𝐻𝐶)‘𝐵)‘𝑍) = if(𝑍𝑌, (𝐵𝑍), if((𝐵𝑍) ≤ 𝐶, (𝐵𝑍), 𝐶)))
1052eldifbd 3976 . . . . . . . . . . 11 (𝜑 → ¬ 𝑍𝑌)
106105iffalsed 4542 . . . . . . . . . 10 (𝜑 → if(𝑍𝑌, (𝐵𝑍), if((𝐵𝑍) ≤ 𝐶, (𝐵𝑍), 𝐶)) = if((𝐵𝑍) ≤ 𝐶, (𝐵𝑍), 𝐶))
107104, 106eqtrd 2775 . . . . . . . . 9 (𝜑 → (((𝐻𝐶)‘𝐵)‘𝑍) = if((𝐵𝑍) ≤ 𝐶, (𝐵𝑍), 𝐶))
108107oveq2d 7447 . . . . . . . 8 (𝜑 → ((𝐴𝑍)[,)(((𝐻𝐶)‘𝐵)‘𝑍)) = ((𝐴𝑍)[,)if((𝐵𝑍) ≤ 𝐶, (𝐵𝑍), 𝐶)))
109108fveq2d 6911 . . . . . . 7 (𝜑 → (vol‘((𝐴𝑍)[,)(((𝐻𝐶)‘𝐵)‘𝑍))) = (vol‘((𝐴𝑍)[,)if((𝐵𝑍) ≤ 𝐶, (𝐵𝑍), 𝐶))))
110109adantr 480 . . . . . 6 ((𝜑𝑘 = 𝑍) → (vol‘((𝐴𝑍)[,)(((𝐻𝐶)‘𝐵)‘𝑍))) = (vol‘((𝐴𝑍)[,)if((𝐵𝑍) ≤ 𝐶, (𝐵𝑍), 𝐶))))
111103, 110eqtrd 2775 . . . . 5 ((𝜑𝑘 = 𝑍) → (vol‘((𝐴𝑘)[,)(((𝐻𝐶)‘𝐵)‘𝑘))) = (vol‘((𝐴𝑍)[,)if((𝐵𝑍) ≤ 𝐶, (𝐵𝑍), 𝐶))))
11215, 98, 3, 111fprodsplit1 45549 . . . 4 (𝜑 → ∏𝑘𝑋 (vol‘((𝐴𝑘)[,)(((𝐻𝐶)‘𝐵)‘𝑘))) = ((vol‘((𝐴𝑍)[,)if((𝐵𝑍) ≤ 𝐶, (𝐵𝑍), 𝐶))) · ∏𝑘 ∈ (𝑋 ∖ {𝑍})(vol‘((𝐴𝑘)[,)(((𝐻𝐶)‘𝐵)‘𝑘)))))
1137adantr 480 . . . . . . . . . 10 ((𝜑𝑘 ∈ (𝑋 ∖ {𝑍})) → 𝐶 ∈ ℝ)
11415adantr 480 . . . . . . . . . 10 ((𝜑𝑘 ∈ (𝑋 ∖ {𝑍})) → 𝑋 ∈ Fin)
1155adantr 480 . . . . . . . . . 10 ((𝜑𝑘 ∈ (𝑋 ∖ {𝑍})) → 𝐵:𝑋⟶ℝ)
11692, 113, 114, 115, 20hsphoival 46535 . . . . . . . . 9 ((𝜑𝑘 ∈ (𝑋 ∖ {𝑍})) → (((𝐻𝐶)‘𝐵)‘𝑘) = if(𝑘𝑌, (𝐵𝑘), if((𝐵𝑘) ≤ 𝐶, (𝐵𝑘), 𝐶)))
117 hsphoidmvle2.y . . . . . . . . . . . . 13 𝑋 = (𝑌 ∪ {𝑍})
11819, 117eleqtrdi 2849 . . . . . . . . . . . 12 (𝑘 ∈ (𝑋 ∖ {𝑍}) → 𝑘 ∈ (𝑌 ∪ {𝑍}))
119 eldifn 4142 . . . . . . . . . . . 12 (𝑘 ∈ (𝑋 ∖ {𝑍}) → ¬ 𝑘 ∈ {𝑍})
120 elunnel2 4165 . . . . . . . . . . . 12 ((𝑘 ∈ (𝑌 ∪ {𝑍}) ∧ ¬ 𝑘 ∈ {𝑍}) → 𝑘𝑌)
121118, 119, 120syl2anc 584 . . . . . . . . . . 11 (𝑘 ∈ (𝑋 ∖ {𝑍}) → 𝑘𝑌)
122121adantl 481 . . . . . . . . . 10 ((𝜑𝑘 ∈ (𝑋 ∖ {𝑍})) → 𝑘𝑌)
123122iftrued 4539 . . . . . . . . 9 ((𝜑𝑘 ∈ (𝑋 ∖ {𝑍})) → if(𝑘𝑌, (𝐵𝑘), if((𝐵𝑘) ≤ 𝐶, (𝐵𝑘), 𝐶)) = (𝐵𝑘))
124116, 123eqtrd 2775 . . . . . . . 8 ((𝜑𝑘 ∈ (𝑋 ∖ {𝑍})) → (((𝐻𝐶)‘𝐵)‘𝑘) = (𝐵𝑘))
125124oveq2d 7447 . . . . . . 7 ((𝜑𝑘 ∈ (𝑋 ∖ {𝑍})) → ((𝐴𝑘)[,)(((𝐻𝐶)‘𝐵)‘𝑘)) = ((𝐴𝑘)[,)(𝐵𝑘)))
126125fveq2d 6911 . . . . . 6 ((𝜑𝑘 ∈ (𝑋 ∖ {𝑍})) → (vol‘((𝐴𝑘)[,)(((𝐻𝐶)‘𝐵)‘𝑘))) = (vol‘((𝐴𝑘)[,)(𝐵𝑘))))
127126prodeq2dv 15955 . . . . 5 (𝜑 → ∏𝑘 ∈ (𝑋 ∖ {𝑍})(vol‘((𝐴𝑘)[,)(((𝐻𝐶)‘𝐵)‘𝑘))) = ∏𝑘 ∈ (𝑋 ∖ {𝑍})(vol‘((𝐴𝑘)[,)(𝐵𝑘))))
128127oveq2d 7447 . . . 4 (𝜑 → ((vol‘((𝐴𝑍)[,)if((𝐵𝑍) ≤ 𝐶, (𝐵𝑍), 𝐶))) · ∏𝑘 ∈ (𝑋 ∖ {𝑍})(vol‘((𝐴𝑘)[,)(((𝐻𝐶)‘𝐵)‘𝑘)))) = ((vol‘((𝐴𝑍)[,)if((𝐵𝑍) ≤ 𝐶, (𝐵𝑍), 𝐶))) · ∏𝑘 ∈ (𝑋 ∖ {𝑍})(vol‘((𝐴𝑘)[,)(𝐵𝑘)))))
12994, 112, 1283eqtrd 2779 . . 3 (𝜑 → (𝐴(𝐿𝑋)((𝐻𝐶)‘𝐵)) = ((vol‘((𝐴𝑍)[,)if((𝐵𝑍) ≤ 𝐶, (𝐵𝑍), 𝐶))) · ∏𝑘 ∈ (𝑋 ∖ {𝑍})(vol‘((𝐴𝑘)[,)(𝐵𝑘)))))
13092, 11, 15, 5hsphoif 46532 . . . . 5 (𝜑 → ((𝐻𝐷)‘𝐵):𝑋⟶ℝ)
13190, 15, 91, 1, 130hoidmvn0val 46540 . . . 4 (𝜑 → (𝐴(𝐿𝑋)((𝐻𝐷)‘𝐵)) = ∏𝑘𝑋 (vol‘((𝐴𝑘)[,)(((𝐻𝐷)‘𝐵)‘𝑘))))
132130ffvelcdmda 7104 . . . . . . 7 ((𝜑𝑘𝑋) → (((𝐻𝐷)‘𝐵)‘𝑘) ∈ ℝ)
133 volicore 46537 . . . . . . 7 (((𝐴𝑘) ∈ ℝ ∧ (((𝐻𝐷)‘𝐵)‘𝑘) ∈ ℝ) → (vol‘((𝐴𝑘)[,)(((𝐻𝐷)‘𝐵)‘𝑘))) ∈ ℝ)
13421, 132, 133syl2anc 584 . . . . . 6 ((𝜑𝑘𝑋) → (vol‘((𝐴𝑘)[,)(((𝐻𝐷)‘𝐵)‘𝑘))) ∈ ℝ)
135134recnd 11287 . . . . 5 ((𝜑𝑘𝑋) → (vol‘((𝐴𝑘)[,)(((𝐻𝐷)‘𝐵)‘𝑘))) ∈ ℂ)
136 fveq2 6907 . . . . . . . 8 (𝑘 = 𝑍 → (((𝐻𝐷)‘𝐵)‘𝑘) = (((𝐻𝐷)‘𝐵)‘𝑍))
13799, 136oveq12d 7449 . . . . . . 7 (𝑘 = 𝑍 → ((𝐴𝑘)[,)(((𝐻𝐷)‘𝐵)‘𝑘)) = ((𝐴𝑍)[,)(((𝐻𝐷)‘𝐵)‘𝑍)))
138137fveq2d 6911 . . . . . 6 (𝑘 = 𝑍 → (vol‘((𝐴𝑘)[,)(((𝐻𝐷)‘𝐵)‘𝑘))) = (vol‘((𝐴𝑍)[,)(((𝐻𝐷)‘𝐵)‘𝑍))))
139138adantl 481 . . . . 5 ((𝜑𝑘 = 𝑍) → (vol‘((𝐴𝑘)[,)(((𝐻𝐷)‘𝐵)‘𝑘))) = (vol‘((𝐴𝑍)[,)(((𝐻𝐷)‘𝐵)‘𝑍))))
14015, 135, 3, 139fprodsplit1 45549 . . . 4 (𝜑 → ∏𝑘𝑋 (vol‘((𝐴𝑘)[,)(((𝐻𝐷)‘𝐵)‘𝑘))) = ((vol‘((𝐴𝑍)[,)(((𝐻𝐷)‘𝐵)‘𝑍))) · ∏𝑘 ∈ (𝑋 ∖ {𝑍})(vol‘((𝐴𝑘)[,)(((𝐻𝐷)‘𝐵)‘𝑘)))))
14192, 11, 15, 5, 3hsphoival 46535 . . . . . . . 8 (𝜑 → (((𝐻𝐷)‘𝐵)‘𝑍) = if(𝑍𝑌, (𝐵𝑍), if((𝐵𝑍) ≤ 𝐷, (𝐵𝑍), 𝐷)))
142105iffalsed 4542 . . . . . . . 8 (𝜑 → if(𝑍𝑌, (𝐵𝑍), if((𝐵𝑍) ≤ 𝐷, (𝐵𝑍), 𝐷)) = if((𝐵𝑍) ≤ 𝐷, (𝐵𝑍), 𝐷))
143141, 142eqtrd 2775 . . . . . . 7 (𝜑 → (((𝐻𝐷)‘𝐵)‘𝑍) = if((𝐵𝑍) ≤ 𝐷, (𝐵𝑍), 𝐷))
144143oveq2d 7447 . . . . . 6 (𝜑 → ((𝐴𝑍)[,)(((𝐻𝐷)‘𝐵)‘𝑍)) = ((𝐴𝑍)[,)if((𝐵𝑍) ≤ 𝐷, (𝐵𝑍), 𝐷)))
145144fveq2d 6911 . . . . 5 (𝜑 → (vol‘((𝐴𝑍)[,)(((𝐻𝐷)‘𝐵)‘𝑍))) = (vol‘((𝐴𝑍)[,)if((𝐵𝑍) ≤ 𝐷, (𝐵𝑍), 𝐷))))
14611adantr 480 . . . . . . . . . 10 ((𝜑𝑘 ∈ (𝑋 ∖ {𝑍})) → 𝐷 ∈ ℝ)
14792, 146, 114, 115, 20hsphoival 46535 . . . . . . . . 9 ((𝜑𝑘 ∈ (𝑋 ∖ {𝑍})) → (((𝐻𝐷)‘𝐵)‘𝑘) = if(𝑘𝑌, (𝐵𝑘), if((𝐵𝑘) ≤ 𝐷, (𝐵𝑘), 𝐷)))
148122iftrued 4539 . . . . . . . . 9 ((𝜑𝑘 ∈ (𝑋 ∖ {𝑍})) → if(𝑘𝑌, (𝐵𝑘), if((𝐵𝑘) ≤ 𝐷, (𝐵𝑘), 𝐷)) = (𝐵𝑘))
149147, 148eqtrd 2775 . . . . . . . 8 ((𝜑𝑘 ∈ (𝑋 ∖ {𝑍})) → (((𝐻𝐷)‘𝐵)‘𝑘) = (𝐵𝑘))
150149oveq2d 7447 . . . . . . 7 ((𝜑𝑘 ∈ (𝑋 ∖ {𝑍})) → ((𝐴𝑘)[,)(((𝐻𝐷)‘𝐵)‘𝑘)) = ((𝐴𝑘)[,)(𝐵𝑘)))
151150fveq2d 6911 . . . . . 6 ((𝜑𝑘 ∈ (𝑋 ∖ {𝑍})) → (vol‘((𝐴𝑘)[,)(((𝐻𝐷)‘𝐵)‘𝑘))) = (vol‘((𝐴𝑘)[,)(𝐵𝑘))))
152151prodeq2dv 15955 . . . . 5 (𝜑 → ∏𝑘 ∈ (𝑋 ∖ {𝑍})(vol‘((𝐴𝑘)[,)(((𝐻𝐷)‘𝐵)‘𝑘))) = ∏𝑘 ∈ (𝑋 ∖ {𝑍})(vol‘((𝐴𝑘)[,)(𝐵𝑘))))
153145, 152oveq12d 7449 . . . 4 (𝜑 → ((vol‘((𝐴𝑍)[,)(((𝐻𝐷)‘𝐵)‘𝑍))) · ∏𝑘 ∈ (𝑋 ∖ {𝑍})(vol‘((𝐴𝑘)[,)(((𝐻𝐷)‘𝐵)‘𝑘)))) = ((vol‘((𝐴𝑍)[,)if((𝐵𝑍) ≤ 𝐷, (𝐵𝑍), 𝐷))) · ∏𝑘 ∈ (𝑋 ∖ {𝑍})(vol‘((𝐴𝑘)[,)(𝐵𝑘)))))
154131, 140, 1533eqtrd 2779 . . 3 (𝜑 → (𝐴(𝐿𝑋)((𝐻𝐷)‘𝐵)) = ((vol‘((𝐴𝑍)[,)if((𝐵𝑍) ≤ 𝐷, (𝐵𝑍), 𝐷))) · ∏𝑘 ∈ (𝑋 ∖ {𝑍})(vol‘((𝐴𝑘)[,)(𝐵𝑘)))))
155129, 154breq12d 5161 . 2 (𝜑 → ((𝐴(𝐿𝑋)((𝐻𝐶)‘𝐵)) ≤ (𝐴(𝐿𝑋)((𝐻𝐷)‘𝐵)) ↔ ((vol‘((𝐴𝑍)[,)if((𝐵𝑍) ≤ 𝐶, (𝐵𝑍), 𝐶))) · ∏𝑘 ∈ (𝑋 ∖ {𝑍})(vol‘((𝐴𝑘)[,)(𝐵𝑘)))) ≤ ((vol‘((𝐴𝑍)[,)if((𝐵𝑍) ≤ 𝐷, (𝐵𝑍), 𝐷))) · ∏𝑘 ∈ (𝑋 ∖ {𝑍})(vol‘((𝐴𝑘)[,)(𝐵𝑘))))))
15689, 155mpbird 257 1 (𝜑 → (𝐴(𝐿𝑋)((𝐻𝐶)‘𝐵)) ≤ (𝐴(𝐿𝑋)((𝐻𝐷)‘𝐵)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395   = wceq 1537  wcel 2106  cdif 3960  cun 3961  wss 3963  c0 4339  ifcif 4531  {csn 4631   class class class wbr 5148  cmpt 5231  dom cdm 5689  wf 6559  cfv 6563  (class class class)co 7431  cmpo 7433  m cmap 8865  Fincfn 8984  cr 11152  0cc0 11153   · cmul 11158  *cxr 11292   < clt 11293  cle 11294  [,)cico 13386  cprod 15936  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-fi 9449  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-xneg 13152  df-xadd 13153  df-xmul 13154  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-prod 15937  df-rest 17469  df-topgen 17490  df-psmet 21374  df-xmet 21375  df-met 21376  df-bl 21377  df-mopn 21378  df-top 22916  df-topon 22933  df-bases 22969  df-cmp 23411  df-ovol 25513  df-vol 25514
This theorem is referenced by:  hoidmvlelem1  46551  hoidmvlelem2  46552
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