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Theorem voliunlem3 25076
Description: Lemma for voliun 25078. (Contributed by Mario Carneiro, 20-Mar-2014.)
Hypotheses
Ref Expression
voliunlem.3 (πœ‘ β†’ 𝐹:β„•βŸΆdom vol)
voliunlem.5 (πœ‘ β†’ Disj 𝑖 ∈ β„• (πΉβ€˜π‘–))
voliunlem.6 𝐻 = (𝑛 ∈ β„• ↦ (vol*β€˜(π‘₯ ∩ (πΉβ€˜π‘›))))
voliunlem3.1 𝑆 = seq1( + , 𝐺)
voliunlem3.2 𝐺 = (𝑛 ∈ β„• ↦ (volβ€˜(πΉβ€˜π‘›)))
voliunlem3.4 (πœ‘ β†’ βˆ€π‘– ∈ β„• (volβ€˜(πΉβ€˜π‘–)) ∈ ℝ)
Assertion
Ref Expression
voliunlem3 (πœ‘ β†’ (volβ€˜βˆͺ ran 𝐹) = sup(ran 𝑆, ℝ*, < ))
Distinct variable groups:   𝑖,𝑛,π‘₯,𝐹   π‘₯,𝑆   πœ‘,𝑛,π‘₯
Allowed substitution hints:   πœ‘(𝑖)   𝑆(𝑖,𝑛)   𝐺(π‘₯,𝑖,𝑛)   𝐻(π‘₯,𝑖,𝑛)

Proof of Theorem voliunlem3
Dummy variables π‘˜ 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 voliunlem.3 . . . 4 (πœ‘ β†’ 𝐹:β„•βŸΆdom vol)
2 voliunlem.5 . . . 4 (πœ‘ β†’ Disj 𝑖 ∈ β„• (πΉβ€˜π‘–))
3 voliunlem.6 . . . 4 𝐻 = (𝑛 ∈ β„• ↦ (vol*β€˜(π‘₯ ∩ (πΉβ€˜π‘›))))
41, 2, 3voliunlem2 25075 . . 3 (πœ‘ β†’ βˆͺ ran 𝐹 ∈ dom vol)
5 mblvol 25054 . . 3 (βˆͺ ran 𝐹 ∈ dom vol β†’ (volβ€˜βˆͺ ran 𝐹) = (vol*β€˜βˆͺ ran 𝐹))
64, 5syl 17 . 2 (πœ‘ β†’ (volβ€˜βˆͺ ran 𝐹) = (vol*β€˜βˆͺ ran 𝐹))
71frnd 6725 . . . . . 6 (πœ‘ β†’ ran 𝐹 βŠ† dom vol)
8 mblss 25055 . . . . . . . 8 (π‘₯ ∈ dom vol β†’ π‘₯ βŠ† ℝ)
9 reex 11203 . . . . . . . . 9 ℝ ∈ V
109elpw2 5345 . . . . . . . 8 (π‘₯ ∈ 𝒫 ℝ ↔ π‘₯ βŠ† ℝ)
118, 10sylibr 233 . . . . . . 7 (π‘₯ ∈ dom vol β†’ π‘₯ ∈ 𝒫 ℝ)
1211ssriv 3986 . . . . . 6 dom vol βŠ† 𝒫 ℝ
137, 12sstrdi 3994 . . . . 5 (πœ‘ β†’ ran 𝐹 βŠ† 𝒫 ℝ)
14 sspwuni 5103 . . . . 5 (ran 𝐹 βŠ† 𝒫 ℝ ↔ βˆͺ ran 𝐹 βŠ† ℝ)
1513, 14sylib 217 . . . 4 (πœ‘ β†’ βˆͺ ran 𝐹 βŠ† ℝ)
16 ovolcl 25002 . . . 4 (βˆͺ ran 𝐹 βŠ† ℝ β†’ (vol*β€˜βˆͺ ran 𝐹) ∈ ℝ*)
1715, 16syl 17 . . 3 (πœ‘ β†’ (vol*β€˜βˆͺ ran 𝐹) ∈ ℝ*)
18 nnuz 12867 . . . . . . . 8 β„• = (β„€β‰₯β€˜1)
19 1zzd 12595 . . . . . . . 8 (πœ‘ β†’ 1 ∈ β„€)
20 2fveq3 6896 . . . . . . . . . . 11 (𝑛 = π‘˜ β†’ (volβ€˜(πΉβ€˜π‘›)) = (volβ€˜(πΉβ€˜π‘˜)))
21 voliunlem3.2 . . . . . . . . . . 11 𝐺 = (𝑛 ∈ β„• ↦ (volβ€˜(πΉβ€˜π‘›)))
22 fvex 6904 . . . . . . . . . . 11 (volβ€˜(πΉβ€˜π‘˜)) ∈ V
2320, 21, 22fvmpt 6998 . . . . . . . . . 10 (π‘˜ ∈ β„• β†’ (πΊβ€˜π‘˜) = (volβ€˜(πΉβ€˜π‘˜)))
2423adantl 482 . . . . . . . . 9 ((πœ‘ ∧ π‘˜ ∈ β„•) β†’ (πΊβ€˜π‘˜) = (volβ€˜(πΉβ€˜π‘˜)))
25 voliunlem3.4 . . . . . . . . . 10 (πœ‘ β†’ βˆ€π‘– ∈ β„• (volβ€˜(πΉβ€˜π‘–)) ∈ ℝ)
26 2fveq3 6896 . . . . . . . . . . . 12 (𝑖 = π‘˜ β†’ (volβ€˜(πΉβ€˜π‘–)) = (volβ€˜(πΉβ€˜π‘˜)))
2726eleq1d 2818 . . . . . . . . . . 11 (𝑖 = π‘˜ β†’ ((volβ€˜(πΉβ€˜π‘–)) ∈ ℝ ↔ (volβ€˜(πΉβ€˜π‘˜)) ∈ ℝ))
2827rspccva 3611 . . . . . . . . . 10 ((βˆ€π‘– ∈ β„• (volβ€˜(πΉβ€˜π‘–)) ∈ ℝ ∧ π‘˜ ∈ β„•) β†’ (volβ€˜(πΉβ€˜π‘˜)) ∈ ℝ)
2925, 28sylan 580 . . . . . . . . 9 ((πœ‘ ∧ π‘˜ ∈ β„•) β†’ (volβ€˜(πΉβ€˜π‘˜)) ∈ ℝ)
3024, 29eqeltrd 2833 . . . . . . . 8 ((πœ‘ ∧ π‘˜ ∈ β„•) β†’ (πΊβ€˜π‘˜) ∈ ℝ)
3118, 19, 30serfre 13999 . . . . . . 7 (πœ‘ β†’ seq1( + , 𝐺):β„•βŸΆβ„)
32 voliunlem3.1 . . . . . . . 8 𝑆 = seq1( + , 𝐺)
3332feq1i 6708 . . . . . . 7 (𝑆:β„•βŸΆβ„ ↔ seq1( + , 𝐺):β„•βŸΆβ„)
3431, 33sylibr 233 . . . . . 6 (πœ‘ β†’ 𝑆:β„•βŸΆβ„)
3534frnd 6725 . . . . 5 (πœ‘ β†’ ran 𝑆 βŠ† ℝ)
36 ressxr 11260 . . . . 5 ℝ βŠ† ℝ*
3735, 36sstrdi 3994 . . . 4 (πœ‘ β†’ ran 𝑆 βŠ† ℝ*)
38 supxrcl 13296 . . . 4 (ran 𝑆 βŠ† ℝ* β†’ sup(ran 𝑆, ℝ*, < ) ∈ ℝ*)
3937, 38syl 17 . . 3 (πœ‘ β†’ sup(ran 𝑆, ℝ*, < ) ∈ ℝ*)
40 eqid 2732 . . . . 5 seq1( + , (𝑛 ∈ β„• ↦ (vol*β€˜(πΉβ€˜π‘›)))) = seq1( + , (𝑛 ∈ β„• ↦ (vol*β€˜(πΉβ€˜π‘›))))
41 eqid 2732 . . . . 5 (𝑛 ∈ β„• ↦ (vol*β€˜(πΉβ€˜π‘›))) = (𝑛 ∈ β„• ↦ (vol*β€˜(πΉβ€˜π‘›)))
421ffvelcdmda 7086 . . . . . 6 ((πœ‘ ∧ 𝑛 ∈ β„•) β†’ (πΉβ€˜π‘›) ∈ dom vol)
43 mblss 25055 . . . . . 6 ((πΉβ€˜π‘›) ∈ dom vol β†’ (πΉβ€˜π‘›) βŠ† ℝ)
4442, 43syl 17 . . . . 5 ((πœ‘ ∧ 𝑛 ∈ β„•) β†’ (πΉβ€˜π‘›) βŠ† ℝ)
45 mblvol 25054 . . . . . . 7 ((πΉβ€˜π‘›) ∈ dom vol β†’ (volβ€˜(πΉβ€˜π‘›)) = (vol*β€˜(πΉβ€˜π‘›)))
4642, 45syl 17 . . . . . 6 ((πœ‘ ∧ 𝑛 ∈ β„•) β†’ (volβ€˜(πΉβ€˜π‘›)) = (vol*β€˜(πΉβ€˜π‘›)))
47 2fveq3 6896 . . . . . . . . 9 (𝑖 = 𝑛 β†’ (volβ€˜(πΉβ€˜π‘–)) = (volβ€˜(πΉβ€˜π‘›)))
4847eleq1d 2818 . . . . . . . 8 (𝑖 = 𝑛 β†’ ((volβ€˜(πΉβ€˜π‘–)) ∈ ℝ ↔ (volβ€˜(πΉβ€˜π‘›)) ∈ ℝ))
4948rspccva 3611 . . . . . . 7 ((βˆ€π‘– ∈ β„• (volβ€˜(πΉβ€˜π‘–)) ∈ ℝ ∧ 𝑛 ∈ β„•) β†’ (volβ€˜(πΉβ€˜π‘›)) ∈ ℝ)
5025, 49sylan 580 . . . . . 6 ((πœ‘ ∧ 𝑛 ∈ β„•) β†’ (volβ€˜(πΉβ€˜π‘›)) ∈ ℝ)
5146, 50eqeltrrd 2834 . . . . 5 ((πœ‘ ∧ 𝑛 ∈ β„•) β†’ (vol*β€˜(πΉβ€˜π‘›)) ∈ ℝ)
5240, 41, 44, 51ovoliun 25029 . . . 4 (πœ‘ β†’ (vol*β€˜βˆͺ 𝑛 ∈ β„• (πΉβ€˜π‘›)) ≀ sup(ran seq1( + , (𝑛 ∈ β„• ↦ (vol*β€˜(πΉβ€˜π‘›)))), ℝ*, < ))
531ffnd 6718 . . . . . 6 (πœ‘ β†’ 𝐹 Fn β„•)
54 fniunfv 7248 . . . . . 6 (𝐹 Fn β„• β†’ βˆͺ 𝑛 ∈ β„• (πΉβ€˜π‘›) = βˆͺ ran 𝐹)
5553, 54syl 17 . . . . 5 (πœ‘ β†’ βˆͺ 𝑛 ∈ β„• (πΉβ€˜π‘›) = βˆͺ ran 𝐹)
5655fveq2d 6895 . . . 4 (πœ‘ β†’ (vol*β€˜βˆͺ 𝑛 ∈ β„• (πΉβ€˜π‘›)) = (vol*β€˜βˆͺ ran 𝐹))
5746mpteq2dva 5248 . . . . . . . . 9 (πœ‘ β†’ (𝑛 ∈ β„• ↦ (volβ€˜(πΉβ€˜π‘›))) = (𝑛 ∈ β„• ↦ (vol*β€˜(πΉβ€˜π‘›))))
5821, 57eqtrid 2784 . . . . . . . 8 (πœ‘ β†’ 𝐺 = (𝑛 ∈ β„• ↦ (vol*β€˜(πΉβ€˜π‘›))))
5958seqeq3d 13976 . . . . . . 7 (πœ‘ β†’ seq1( + , 𝐺) = seq1( + , (𝑛 ∈ β„• ↦ (vol*β€˜(πΉβ€˜π‘›)))))
6032, 59eqtr2id 2785 . . . . . 6 (πœ‘ β†’ seq1( + , (𝑛 ∈ β„• ↦ (vol*β€˜(πΉβ€˜π‘›)))) = 𝑆)
6160rneqd 5937 . . . . 5 (πœ‘ β†’ ran seq1( + , (𝑛 ∈ β„• ↦ (vol*β€˜(πΉβ€˜π‘›)))) = ran 𝑆)
6261supeq1d 9443 . . . 4 (πœ‘ β†’ sup(ran seq1( + , (𝑛 ∈ β„• ↦ (vol*β€˜(πΉβ€˜π‘›)))), ℝ*, < ) = sup(ran 𝑆, ℝ*, < ))
6352, 56, 623brtr3d 5179 . . 3 (πœ‘ β†’ (vol*β€˜βˆͺ ran 𝐹) ≀ sup(ran 𝑆, ℝ*, < ))
64 ovolge0 25005 . . . . . . . . . 10 (βˆͺ ran 𝐹 βŠ† ℝ β†’ 0 ≀ (vol*β€˜βˆͺ ran 𝐹))
6515, 64syl 17 . . . . . . . . 9 (πœ‘ β†’ 0 ≀ (vol*β€˜βˆͺ ran 𝐹))
66 mnflt0 13107 . . . . . . . . . 10 -∞ < 0
67 mnfxr 11273 . . . . . . . . . . 11 -∞ ∈ ℝ*
68 0xr 11263 . . . . . . . . . . 11 0 ∈ ℝ*
69 xrltletr 13138 . . . . . . . . . . 11 ((-∞ ∈ ℝ* ∧ 0 ∈ ℝ* ∧ (vol*β€˜βˆͺ ran 𝐹) ∈ ℝ*) β†’ ((-∞ < 0 ∧ 0 ≀ (vol*β€˜βˆͺ ran 𝐹)) β†’ -∞ < (vol*β€˜βˆͺ ran 𝐹)))
7067, 68, 69mp3an12 1451 . . . . . . . . . 10 ((vol*β€˜βˆͺ ran 𝐹) ∈ ℝ* β†’ ((-∞ < 0 ∧ 0 ≀ (vol*β€˜βˆͺ ran 𝐹)) β†’ -∞ < (vol*β€˜βˆͺ ran 𝐹)))
7166, 70mpani 694 . . . . . . . . 9 ((vol*β€˜βˆͺ ran 𝐹) ∈ ℝ* β†’ (0 ≀ (vol*β€˜βˆͺ ran 𝐹) β†’ -∞ < (vol*β€˜βˆͺ ran 𝐹)))
7217, 65, 71sylc 65 . . . . . . . 8 (πœ‘ β†’ -∞ < (vol*β€˜βˆͺ ran 𝐹))
73 xrrebnd 13149 . . . . . . . . . 10 ((vol*β€˜βˆͺ ran 𝐹) ∈ ℝ* β†’ ((vol*β€˜βˆͺ ran 𝐹) ∈ ℝ ↔ (-∞ < (vol*β€˜βˆͺ ran 𝐹) ∧ (vol*β€˜βˆͺ ran 𝐹) < +∞)))
7417, 73syl 17 . . . . . . . . 9 (πœ‘ β†’ ((vol*β€˜βˆͺ ran 𝐹) ∈ ℝ ↔ (-∞ < (vol*β€˜βˆͺ ran 𝐹) ∧ (vol*β€˜βˆͺ ran 𝐹) < +∞)))
759elpw2 5345 . . . . . . . . . . . 12 (βˆͺ ran 𝐹 ∈ 𝒫 ℝ ↔ βˆͺ ran 𝐹 βŠ† ℝ)
7615, 75sylibr 233 . . . . . . . . . . 11 (πœ‘ β†’ βˆͺ ran 𝐹 ∈ 𝒫 ℝ)
77 simpl 483 . . . . . . . . . . . . . . 15 ((π‘₯ = βˆͺ ran 𝐹 ∧ πœ‘) β†’ π‘₯ = βˆͺ ran 𝐹)
7877sseq1d 4013 . . . . . . . . . . . . . 14 ((π‘₯ = βˆͺ ran 𝐹 ∧ πœ‘) β†’ (π‘₯ βŠ† ℝ ↔ βˆͺ ran 𝐹 βŠ† ℝ))
7977fveq2d 6895 . . . . . . . . . . . . . . . 16 ((π‘₯ = βˆͺ ran 𝐹 ∧ πœ‘) β†’ (vol*β€˜π‘₯) = (vol*β€˜βˆͺ ran 𝐹))
8079eleq1d 2818 . . . . . . . . . . . . . . 15 ((π‘₯ = βˆͺ ran 𝐹 ∧ πœ‘) β†’ ((vol*β€˜π‘₯) ∈ ℝ ↔ (vol*β€˜βˆͺ ran 𝐹) ∈ ℝ))
81 simpll 765 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (((π‘₯ = βˆͺ ran 𝐹 ∧ πœ‘) ∧ 𝑛 ∈ β„•) β†’ π‘₯ = βˆͺ ran 𝐹)
8281ineq1d 4211 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((π‘₯ = βˆͺ ran 𝐹 ∧ πœ‘) ∧ 𝑛 ∈ β„•) β†’ (π‘₯ ∩ (πΉβ€˜π‘›)) = (βˆͺ ran 𝐹 ∩ (πΉβ€˜π‘›)))
83 fnfvelrn 7082 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((𝐹 Fn β„• ∧ 𝑛 ∈ β„•) β†’ (πΉβ€˜π‘›) ∈ ran 𝐹)
8453, 83sylan 580 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((πœ‘ ∧ 𝑛 ∈ β„•) β†’ (πΉβ€˜π‘›) ∈ ran 𝐹)
85 elssuni 4941 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((πΉβ€˜π‘›) ∈ ran 𝐹 β†’ (πΉβ€˜π‘›) βŠ† βˆͺ ran 𝐹)
8684, 85syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((πœ‘ ∧ 𝑛 ∈ β„•) β†’ (πΉβ€˜π‘›) βŠ† βˆͺ ran 𝐹)
8786adantll 712 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (((π‘₯ = βˆͺ ran 𝐹 ∧ πœ‘) ∧ 𝑛 ∈ β„•) β†’ (πΉβ€˜π‘›) βŠ† βˆͺ ran 𝐹)
88 sseqin2 4215 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((πΉβ€˜π‘›) βŠ† βˆͺ ran 𝐹 ↔ (βˆͺ ran 𝐹 ∩ (πΉβ€˜π‘›)) = (πΉβ€˜π‘›))
8987, 88sylib 217 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((π‘₯ = βˆͺ ran 𝐹 ∧ πœ‘) ∧ 𝑛 ∈ β„•) β†’ (βˆͺ ran 𝐹 ∩ (πΉβ€˜π‘›)) = (πΉβ€˜π‘›))
9082, 89eqtrd 2772 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((π‘₯ = βˆͺ ran 𝐹 ∧ πœ‘) ∧ 𝑛 ∈ β„•) β†’ (π‘₯ ∩ (πΉβ€˜π‘›)) = (πΉβ€˜π‘›))
9190fveq2d 6895 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((π‘₯ = βˆͺ ran 𝐹 ∧ πœ‘) ∧ 𝑛 ∈ β„•) β†’ (vol*β€˜(π‘₯ ∩ (πΉβ€˜π‘›))) = (vol*β€˜(πΉβ€˜π‘›)))
9246adantll 712 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((π‘₯ = βˆͺ ran 𝐹 ∧ πœ‘) ∧ 𝑛 ∈ β„•) β†’ (volβ€˜(πΉβ€˜π‘›)) = (vol*β€˜(πΉβ€˜π‘›)))
9391, 92eqtr4d 2775 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((π‘₯ = βˆͺ ran 𝐹 ∧ πœ‘) ∧ 𝑛 ∈ β„•) β†’ (vol*β€˜(π‘₯ ∩ (πΉβ€˜π‘›))) = (volβ€˜(πΉβ€˜π‘›)))
9493mpteq2dva 5248 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((π‘₯ = βˆͺ ran 𝐹 ∧ πœ‘) β†’ (𝑛 ∈ β„• ↦ (vol*β€˜(π‘₯ ∩ (πΉβ€˜π‘›)))) = (𝑛 ∈ β„• ↦ (volβ€˜(πΉβ€˜π‘›))))
9594adantrr 715 . . . . . . . . . . . . . . . . . . . . . . . 24 ((π‘₯ = βˆͺ ran 𝐹 ∧ (πœ‘ ∧ π‘˜ ∈ β„•)) β†’ (𝑛 ∈ β„• ↦ (vol*β€˜(π‘₯ ∩ (πΉβ€˜π‘›)))) = (𝑛 ∈ β„• ↦ (volβ€˜(πΉβ€˜π‘›))))
9695, 3, 213eqtr4g 2797 . . . . . . . . . . . . . . . . . . . . . . 23 ((π‘₯ = βˆͺ ran 𝐹 ∧ (πœ‘ ∧ π‘˜ ∈ β„•)) β†’ 𝐻 = 𝐺)
9796seqeq3d 13976 . . . . . . . . . . . . . . . . . . . . . 22 ((π‘₯ = βˆͺ ran 𝐹 ∧ (πœ‘ ∧ π‘˜ ∈ β„•)) β†’ seq1( + , 𝐻) = seq1( + , 𝐺))
9897, 32eqtr4di 2790 . . . . . . . . . . . . . . . . . . . . 21 ((π‘₯ = βˆͺ ran 𝐹 ∧ (πœ‘ ∧ π‘˜ ∈ β„•)) β†’ seq1( + , 𝐻) = 𝑆)
9998fveq1d 6893 . . . . . . . . . . . . . . . . . . . 20 ((π‘₯ = βˆͺ ran 𝐹 ∧ (πœ‘ ∧ π‘˜ ∈ β„•)) β†’ (seq1( + , 𝐻)β€˜π‘˜) = (π‘†β€˜π‘˜))
100 difeq1 4115 . . . . . . . . . . . . . . . . . . . . . . . 24 (π‘₯ = βˆͺ ran 𝐹 β†’ (π‘₯ βˆ– βˆͺ ran 𝐹) = (βˆͺ ran 𝐹 βˆ– βˆͺ ran 𝐹))
101 difid 4370 . . . . . . . . . . . . . . . . . . . . . . . 24 (βˆͺ ran 𝐹 βˆ– βˆͺ ran 𝐹) = βˆ…
102100, 101eqtrdi 2788 . . . . . . . . . . . . . . . . . . . . . . 23 (π‘₯ = βˆͺ ran 𝐹 β†’ (π‘₯ βˆ– βˆͺ ran 𝐹) = βˆ…)
103102fveq2d 6895 . . . . . . . . . . . . . . . . . . . . . 22 (π‘₯ = βˆͺ ran 𝐹 β†’ (vol*β€˜(π‘₯ βˆ– βˆͺ ran 𝐹)) = (vol*β€˜βˆ…))
104 ovol0 25017 . . . . . . . . . . . . . . . . . . . . . 22 (vol*β€˜βˆ…) = 0
105103, 104eqtrdi 2788 . . . . . . . . . . . . . . . . . . . . 21 (π‘₯ = βˆͺ ran 𝐹 β†’ (vol*β€˜(π‘₯ βˆ– βˆͺ ran 𝐹)) = 0)
106105adantr 481 . . . . . . . . . . . . . . . . . . . 20 ((π‘₯ = βˆͺ ran 𝐹 ∧ (πœ‘ ∧ π‘˜ ∈ β„•)) β†’ (vol*β€˜(π‘₯ βˆ– βˆͺ ran 𝐹)) = 0)
10799, 106oveq12d 7429 . . . . . . . . . . . . . . . . . . 19 ((π‘₯ = βˆͺ ran 𝐹 ∧ (πœ‘ ∧ π‘˜ ∈ β„•)) β†’ ((seq1( + , 𝐻)β€˜π‘˜) + (vol*β€˜(π‘₯ βˆ– βˆͺ ran 𝐹))) = ((π‘†β€˜π‘˜) + 0))
10834ffvelcdmda 7086 . . . . . . . . . . . . . . . . . . . . . 22 ((πœ‘ ∧ π‘˜ ∈ β„•) β†’ (π‘†β€˜π‘˜) ∈ ℝ)
109108adantl 482 . . . . . . . . . . . . . . . . . . . . 21 ((π‘₯ = βˆͺ ran 𝐹 ∧ (πœ‘ ∧ π‘˜ ∈ β„•)) β†’ (π‘†β€˜π‘˜) ∈ ℝ)
110109recnd 11244 . . . . . . . . . . . . . . . . . . . 20 ((π‘₯ = βˆͺ ran 𝐹 ∧ (πœ‘ ∧ π‘˜ ∈ β„•)) β†’ (π‘†β€˜π‘˜) ∈ β„‚)
111110addridd 11416 . . . . . . . . . . . . . . . . . . 19 ((π‘₯ = βˆͺ ran 𝐹 ∧ (πœ‘ ∧ π‘˜ ∈ β„•)) β†’ ((π‘†β€˜π‘˜) + 0) = (π‘†β€˜π‘˜))
112107, 111eqtrd 2772 . . . . . . . . . . . . . . . . . 18 ((π‘₯ = βˆͺ ran 𝐹 ∧ (πœ‘ ∧ π‘˜ ∈ β„•)) β†’ ((seq1( + , 𝐻)β€˜π‘˜) + (vol*β€˜(π‘₯ βˆ– βˆͺ ran 𝐹))) = (π‘†β€˜π‘˜))
113 fveq2 6891 . . . . . . . . . . . . . . . . . . 19 (π‘₯ = βˆͺ ran 𝐹 β†’ (vol*β€˜π‘₯) = (vol*β€˜βˆͺ ran 𝐹))
114113adantr 481 . . . . . . . . . . . . . . . . . 18 ((π‘₯ = βˆͺ ran 𝐹 ∧ (πœ‘ ∧ π‘˜ ∈ β„•)) β†’ (vol*β€˜π‘₯) = (vol*β€˜βˆͺ ran 𝐹))
115112, 114breq12d 5161 . . . . . . . . . . . . . . . . 17 ((π‘₯ = βˆͺ ran 𝐹 ∧ (πœ‘ ∧ π‘˜ ∈ β„•)) β†’ (((seq1( + , 𝐻)β€˜π‘˜) + (vol*β€˜(π‘₯ βˆ– βˆͺ ran 𝐹))) ≀ (vol*β€˜π‘₯) ↔ (π‘†β€˜π‘˜) ≀ (vol*β€˜βˆͺ ran 𝐹)))
116115expr 457 . . . . . . . . . . . . . . . 16 ((π‘₯ = βˆͺ ran 𝐹 ∧ πœ‘) β†’ (π‘˜ ∈ β„• β†’ (((seq1( + , 𝐻)β€˜π‘˜) + (vol*β€˜(π‘₯ βˆ– βˆͺ ran 𝐹))) ≀ (vol*β€˜π‘₯) ↔ (π‘†β€˜π‘˜) ≀ (vol*β€˜βˆͺ ran 𝐹))))
117116pm5.74d 272 . . . . . . . . . . . . . . 15 ((π‘₯ = βˆͺ ran 𝐹 ∧ πœ‘) β†’ ((π‘˜ ∈ β„• β†’ ((seq1( + , 𝐻)β€˜π‘˜) + (vol*β€˜(π‘₯ βˆ– βˆͺ ran 𝐹))) ≀ (vol*β€˜π‘₯)) ↔ (π‘˜ ∈ β„• β†’ (π‘†β€˜π‘˜) ≀ (vol*β€˜βˆͺ ran 𝐹))))
11880, 117imbi12d 344 . . . . . . . . . . . . . 14 ((π‘₯ = βˆͺ ran 𝐹 ∧ πœ‘) β†’ (((vol*β€˜π‘₯) ∈ ℝ β†’ (π‘˜ ∈ β„• β†’ ((seq1( + , 𝐻)β€˜π‘˜) + (vol*β€˜(π‘₯ βˆ– βˆͺ ran 𝐹))) ≀ (vol*β€˜π‘₯))) ↔ ((vol*β€˜βˆͺ ran 𝐹) ∈ ℝ β†’ (π‘˜ ∈ β„• β†’ (π‘†β€˜π‘˜) ≀ (vol*β€˜βˆͺ ran 𝐹)))))
11978, 118imbi12d 344 . . . . . . . . . . . . 13 ((π‘₯ = βˆͺ ran 𝐹 ∧ πœ‘) β†’ ((π‘₯ βŠ† ℝ β†’ ((vol*β€˜π‘₯) ∈ ℝ β†’ (π‘˜ ∈ β„• β†’ ((seq1( + , 𝐻)β€˜π‘˜) + (vol*β€˜(π‘₯ βˆ– βˆͺ ran 𝐹))) ≀ (vol*β€˜π‘₯)))) ↔ (βˆͺ ran 𝐹 βŠ† ℝ β†’ ((vol*β€˜βˆͺ ran 𝐹) ∈ ℝ β†’ (π‘˜ ∈ β„• β†’ (π‘†β€˜π‘˜) ≀ (vol*β€˜βˆͺ ran 𝐹))))))
120119pm5.74da 802 . . . . . . . . . . . 12 (π‘₯ = βˆͺ ran 𝐹 β†’ ((πœ‘ β†’ (π‘₯ βŠ† ℝ β†’ ((vol*β€˜π‘₯) ∈ ℝ β†’ (π‘˜ ∈ β„• β†’ ((seq1( + , 𝐻)β€˜π‘˜) + (vol*β€˜(π‘₯ βˆ– βˆͺ ran 𝐹))) ≀ (vol*β€˜π‘₯))))) ↔ (πœ‘ β†’ (βˆͺ ran 𝐹 βŠ† ℝ β†’ ((vol*β€˜βˆͺ ran 𝐹) ∈ ℝ β†’ (π‘˜ ∈ β„• β†’ (π‘†β€˜π‘˜) ≀ (vol*β€˜βˆͺ ran 𝐹)))))))
12113ad2ant1 1133 . . . . . . . . . . . . . 14 ((πœ‘ ∧ π‘₯ βŠ† ℝ ∧ (vol*β€˜π‘₯) ∈ ℝ) β†’ 𝐹:β„•βŸΆdom vol)
12223ad2ant1 1133 . . . . . . . . . . . . . 14 ((πœ‘ ∧ π‘₯ βŠ† ℝ ∧ (vol*β€˜π‘₯) ∈ ℝ) β†’ Disj 𝑖 ∈ β„• (πΉβ€˜π‘–))
123 simp2 1137 . . . . . . . . . . . . . 14 ((πœ‘ ∧ π‘₯ βŠ† ℝ ∧ (vol*β€˜π‘₯) ∈ ℝ) β†’ π‘₯ βŠ† ℝ)
124 simp3 1138 . . . . . . . . . . . . . 14 ((πœ‘ ∧ π‘₯ βŠ† ℝ ∧ (vol*β€˜π‘₯) ∈ ℝ) β†’ (vol*β€˜π‘₯) ∈ ℝ)
125121, 122, 3, 123, 124voliunlem1 25074 . . . . . . . . . . . . 13 (((πœ‘ ∧ π‘₯ βŠ† ℝ ∧ (vol*β€˜π‘₯) ∈ ℝ) ∧ π‘˜ ∈ β„•) β†’ ((seq1( + , 𝐻)β€˜π‘˜) + (vol*β€˜(π‘₯ βˆ– βˆͺ ran 𝐹))) ≀ (vol*β€˜π‘₯))
1261253exp1 1352 . . . . . . . . . . . 12 (πœ‘ β†’ (π‘₯ βŠ† ℝ β†’ ((vol*β€˜π‘₯) ∈ ℝ β†’ (π‘˜ ∈ β„• β†’ ((seq1( + , 𝐻)β€˜π‘˜) + (vol*β€˜(π‘₯ βˆ– βˆͺ ran 𝐹))) ≀ (vol*β€˜π‘₯)))))
127120, 126vtoclg 3556 . . . . . . . . . . 11 (βˆͺ ran 𝐹 ∈ 𝒫 ℝ β†’ (πœ‘ β†’ (βˆͺ ran 𝐹 βŠ† ℝ β†’ ((vol*β€˜βˆͺ ran 𝐹) ∈ ℝ β†’ (π‘˜ ∈ β„• β†’ (π‘†β€˜π‘˜) ≀ (vol*β€˜βˆͺ ran 𝐹))))))
12876, 127mpcom 38 . . . . . . . . . 10 (πœ‘ β†’ (βˆͺ ran 𝐹 βŠ† ℝ β†’ ((vol*β€˜βˆͺ ran 𝐹) ∈ ℝ β†’ (π‘˜ ∈ β„• β†’ (π‘†β€˜π‘˜) ≀ (vol*β€˜βˆͺ ran 𝐹)))))
12915, 128mpd 15 . . . . . . . . 9 (πœ‘ β†’ ((vol*β€˜βˆͺ ran 𝐹) ∈ ℝ β†’ (π‘˜ ∈ β„• β†’ (π‘†β€˜π‘˜) ≀ (vol*β€˜βˆͺ ran 𝐹))))
13074, 129sylbird 259 . . . . . . . 8 (πœ‘ β†’ ((-∞ < (vol*β€˜βˆͺ ran 𝐹) ∧ (vol*β€˜βˆͺ ran 𝐹) < +∞) β†’ (π‘˜ ∈ β„• β†’ (π‘†β€˜π‘˜) ≀ (vol*β€˜βˆͺ ran 𝐹))))
13172, 130mpand 693 . . . . . . 7 (πœ‘ β†’ ((vol*β€˜βˆͺ ran 𝐹) < +∞ β†’ (π‘˜ ∈ β„• β†’ (π‘†β€˜π‘˜) ≀ (vol*β€˜βˆͺ ran 𝐹))))
132 nltpnft 13145 . . . . . . . . 9 ((vol*β€˜βˆͺ ran 𝐹) ∈ ℝ* β†’ ((vol*β€˜βˆͺ ran 𝐹) = +∞ ↔ Β¬ (vol*β€˜βˆͺ ran 𝐹) < +∞))
13317, 132syl 17 . . . . . . . 8 (πœ‘ β†’ ((vol*β€˜βˆͺ ran 𝐹) = +∞ ↔ Β¬ (vol*β€˜βˆͺ ran 𝐹) < +∞))
134 rexr 11262 . . . . . . . . . . 11 ((π‘†β€˜π‘˜) ∈ ℝ β†’ (π‘†β€˜π‘˜) ∈ ℝ*)
135 pnfge 13112 . . . . . . . . . . 11 ((π‘†β€˜π‘˜) ∈ ℝ* β†’ (π‘†β€˜π‘˜) ≀ +∞)
136108, 134, 1353syl 18 . . . . . . . . . 10 ((πœ‘ ∧ π‘˜ ∈ β„•) β†’ (π‘†β€˜π‘˜) ≀ +∞)
137136ex 413 . . . . . . . . 9 (πœ‘ β†’ (π‘˜ ∈ β„• β†’ (π‘†β€˜π‘˜) ≀ +∞))
138 breq2 5152 . . . . . . . . . 10 ((vol*β€˜βˆͺ ran 𝐹) = +∞ β†’ ((π‘†β€˜π‘˜) ≀ (vol*β€˜βˆͺ ran 𝐹) ↔ (π‘†β€˜π‘˜) ≀ +∞))
139138imbi2d 340 . . . . . . . . 9 ((vol*β€˜βˆͺ ran 𝐹) = +∞ β†’ ((π‘˜ ∈ β„• β†’ (π‘†β€˜π‘˜) ≀ (vol*β€˜βˆͺ ran 𝐹)) ↔ (π‘˜ ∈ β„• β†’ (π‘†β€˜π‘˜) ≀ +∞)))
140137, 139syl5ibrcom 246 . . . . . . . 8 (πœ‘ β†’ ((vol*β€˜βˆͺ ran 𝐹) = +∞ β†’ (π‘˜ ∈ β„• β†’ (π‘†β€˜π‘˜) ≀ (vol*β€˜βˆͺ ran 𝐹))))
141133, 140sylbird 259 . . . . . . 7 (πœ‘ β†’ (Β¬ (vol*β€˜βˆͺ ran 𝐹) < +∞ β†’ (π‘˜ ∈ β„• β†’ (π‘†β€˜π‘˜) ≀ (vol*β€˜βˆͺ ran 𝐹))))
142131, 141pm2.61d 179 . . . . . 6 (πœ‘ β†’ (π‘˜ ∈ β„• β†’ (π‘†β€˜π‘˜) ≀ (vol*β€˜βˆͺ ran 𝐹)))
143142ralrimiv 3145 . . . . 5 (πœ‘ β†’ βˆ€π‘˜ ∈ β„• (π‘†β€˜π‘˜) ≀ (vol*β€˜βˆͺ ran 𝐹))
14434ffnd 6718 . . . . . 6 (πœ‘ β†’ 𝑆 Fn β„•)
145 breq1 5151 . . . . . . 7 (𝑧 = (π‘†β€˜π‘˜) β†’ (𝑧 ≀ (vol*β€˜βˆͺ ran 𝐹) ↔ (π‘†β€˜π‘˜) ≀ (vol*β€˜βˆͺ ran 𝐹)))
146145ralrn 7089 . . . . . 6 (𝑆 Fn β„• β†’ (βˆ€π‘§ ∈ ran 𝑆 𝑧 ≀ (vol*β€˜βˆͺ ran 𝐹) ↔ βˆ€π‘˜ ∈ β„• (π‘†β€˜π‘˜) ≀ (vol*β€˜βˆͺ ran 𝐹)))
147144, 146syl 17 . . . . 5 (πœ‘ β†’ (βˆ€π‘§ ∈ ran 𝑆 𝑧 ≀ (vol*β€˜βˆͺ ran 𝐹) ↔ βˆ€π‘˜ ∈ β„• (π‘†β€˜π‘˜) ≀ (vol*β€˜βˆͺ ran 𝐹)))
148143, 147mpbird 256 . . . 4 (πœ‘ β†’ βˆ€π‘§ ∈ ran 𝑆 𝑧 ≀ (vol*β€˜βˆͺ ran 𝐹))
149 supxrleub 13307 . . . . 5 ((ran 𝑆 βŠ† ℝ* ∧ (vol*β€˜βˆͺ ran 𝐹) ∈ ℝ*) β†’ (sup(ran 𝑆, ℝ*, < ) ≀ (vol*β€˜βˆͺ ran 𝐹) ↔ βˆ€π‘§ ∈ ran 𝑆 𝑧 ≀ (vol*β€˜βˆͺ ran 𝐹)))
15037, 17, 149syl2anc 584 . . . 4 (πœ‘ β†’ (sup(ran 𝑆, ℝ*, < ) ≀ (vol*β€˜βˆͺ ran 𝐹) ↔ βˆ€π‘§ ∈ ran 𝑆 𝑧 ≀ (vol*β€˜βˆͺ ran 𝐹)))
151148, 150mpbird 256 . . 3 (πœ‘ β†’ sup(ran 𝑆, ℝ*, < ) ≀ (vol*β€˜βˆͺ ran 𝐹))
15217, 39, 63, 151xrletrid 13136 . 2 (πœ‘ β†’ (vol*β€˜βˆͺ ran 𝐹) = sup(ran 𝑆, ℝ*, < ))
1536, 152eqtrd 2772 1 (πœ‘ β†’ (volβ€˜βˆͺ ran 𝐹) = sup(ran 𝑆, ℝ*, < ))
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106  βˆ€wral 3061   βˆ– cdif 3945   ∩ cin 3947   βŠ† wss 3948  βˆ…c0 4322  π’« cpw 4602  βˆͺ cuni 4908  βˆͺ ciun 4997  Disj wdisj 5113   class class class wbr 5148   ↦ cmpt 5231  dom cdm 5676  ran crn 5677   Fn wfn 6538  βŸΆwf 6539  β€˜cfv 6543  (class class class)co 7411  supcsup 9437  β„cr 11111  0cc0 11112  1c1 11113   + caddc 11115  +∞cpnf 11247  -∞cmnf 11248  β„*cxr 11249   < clt 11250   ≀ cle 11251  β„•cn 12214  seqcseq 13968  vol*covol 24986  volcvol 24987
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7727  ax-inf2 9638  ax-cc 10432  ax-cnex 11168  ax-resscn 11169  ax-1cn 11170  ax-icn 11171  ax-addcl 11172  ax-addrcl 11173  ax-mulcl 11174  ax-mulrcl 11175  ax-mulcom 11176  ax-addass 11177  ax-mulass 11178  ax-distr 11179  ax-i2m1 11180  ax-1ne0 11181  ax-1rid 11182  ax-rnegex 11183  ax-rrecex 11184  ax-cnre 11185  ax-pre-lttri 11186  ax-pre-lttrn 11187  ax-pre-ltadd 11188  ax-pre-mulgt0 11189  ax-pre-sup 11190
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-int 4951  df-iun 4999  df-disj 5114  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-se 5632  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-isom 6552  df-riota 7367  df-ov 7414  df-oprab 7415  df-mpo 7416  df-of 7672  df-om 7858  df-1st 7977  df-2nd 7978  df-frecs 8268  df-wrecs 8299  df-recs 8373  df-rdg 8412  df-1o 8468  df-2o 8469  df-er 8705  df-map 8824  df-pm 8825  df-en 8942  df-dom 8943  df-sdom 8944  df-fin 8945  df-sup 9439  df-inf 9440  df-oi 9507  df-dju 9898  df-card 9936  df-pnf 11252  df-mnf 11253  df-xr 11254  df-ltxr 11255  df-le 11256  df-sub 11448  df-neg 11449  df-div 11874  df-nn 12215  df-2 12277  df-3 12278  df-n0 12475  df-z 12561  df-uz 12825  df-q 12935  df-rp 12977  df-xadd 13095  df-ioo 13330  df-ico 13332  df-icc 13333  df-fz 13487  df-fzo 13630  df-fl 13759  df-seq 13969  df-exp 14030  df-hash 14293  df-cj 15048  df-re 15049  df-im 15050  df-sqrt 15184  df-abs 15185  df-clim 15434  df-rlim 15435  df-sum 15635  df-xmet 20943  df-met 20944  df-ovol 24988  df-vol 24989
This theorem is referenced by:  voliun  25078
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