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Mirrors  >  Home  >  MPE Home  >  Th. List  >  volivth Structured version   Visualization version   GIF version
Theorem volivth 25124
Description: The Intermediate Value Theorem for the Lebesgue volume function. For any positive 𝐡 ≀ (volβ€˜π΄), there is a measurable subset of 𝐴 whose volume is 𝐡. (Contributed by Mario Carneiro, 30-Aug-2014.)
Assertion
Ref Expression
volivth ((𝐴 ∈ dom vol ∧ 𝐡 ∈ (0[,](volβ€˜π΄))) β†’ βˆƒπ‘₯ ∈ dom vol(π‘₯ βŠ† 𝐴 ∧ (volβ€˜π‘₯) = 𝐡))
Distinct variable groups:   π‘₯,𝐴   π‘₯,𝐡
Proof of Theorem volivth
Dummy variables 𝑒 𝑛 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpll 766 . . . 4 (((𝐴 ∈ dom vol ∧ 𝐡 ∈ (0[,](volβ€˜π΄))) ∧ 𝐡 < (volβ€˜π΄)) β†’ 𝐴 ∈ dom vol)
2 mnfxr 11271 . . . . . 6 -∞ ∈ ℝ*
32a1i 11 . . . . 5 (((𝐴 ∈ dom vol ∧ 𝐡 ∈ (0[,](volβ€˜π΄))) ∧ 𝐡 < (volβ€˜π΄)) β†’ -∞ ∈ ℝ*)
4 iccssxr 13407 . . . . . . 7 (0[,](volβ€˜π΄)) βŠ† ℝ*
5 simpr 486 . . . . . . 7 ((𝐴 ∈ dom vol ∧ 𝐡 ∈ (0[,](volβ€˜π΄))) β†’ 𝐡 ∈ (0[,](volβ€˜π΄)))
64, 5sselid 3981 . . . . . 6 ((𝐴 ∈ dom vol ∧ 𝐡 ∈ (0[,](volβ€˜π΄))) β†’ 𝐡 ∈ ℝ*)
76adantr 482 . . . . 5 (((𝐴 ∈ dom vol ∧ 𝐡 ∈ (0[,](volβ€˜π΄))) ∧ 𝐡 < (volβ€˜π΄)) β†’ 𝐡 ∈ ℝ*)
8 iccssxr 13407 . . . . . . . 8 (0[,]+∞) βŠ† ℝ*
9 volf 25046 . . . . . . . . 9 vol:dom vol⟢(0[,]+∞)
109ffvelcdmi 7086 . . . . . . . 8 (𝐴 ∈ dom vol β†’ (volβ€˜π΄) ∈ (0[,]+∞))
118, 10sselid 3981 . . . . . . 7 (𝐴 ∈ dom vol β†’ (volβ€˜π΄) ∈ ℝ*)
1211adantr 482 . . . . . 6 ((𝐴 ∈ dom vol ∧ 𝐡 ∈ (0[,](volβ€˜π΄))) β†’ (volβ€˜π΄) ∈ ℝ*)
1312adantr 482 . . . . 5 (((𝐴 ∈ dom vol ∧ 𝐡 ∈ (0[,](volβ€˜π΄))) ∧ 𝐡 < (volβ€˜π΄)) β†’ (volβ€˜π΄) ∈ ℝ*)
14 0xr 11261 . . . . . . . . . 10 0 ∈ ℝ*
15 elicc1 13368 . . . . . . . . . 10 ((0 ∈ ℝ* ∧ (volβ€˜π΄) ∈ ℝ*) β†’ (𝐡 ∈ (0[,](volβ€˜π΄)) ↔ (𝐡 ∈ ℝ* ∧ 0 ≀ 𝐡 ∧ 𝐡 ≀ (volβ€˜π΄))))
1614, 12, 15sylancr 588 . . . . . . . . 9 ((𝐴 ∈ dom vol ∧ 𝐡 ∈ (0[,](volβ€˜π΄))) β†’ (𝐡 ∈ (0[,](volβ€˜π΄)) ↔ (𝐡 ∈ ℝ* ∧ 0 ≀ 𝐡 ∧ 𝐡 ≀ (volβ€˜π΄))))
175, 16mpbid 231 . . . . . . . 8 ((𝐴 ∈ dom vol ∧ 𝐡 ∈ (0[,](volβ€˜π΄))) β†’ (𝐡 ∈ ℝ* ∧ 0 ≀ 𝐡 ∧ 𝐡 ≀ (volβ€˜π΄)))
1817simp2d 1144 . . . . . . 7 ((𝐴 ∈ dom vol ∧ 𝐡 ∈ (0[,](volβ€˜π΄))) β†’ 0 ≀ 𝐡)
1918adantr 482 . . . . . 6 (((𝐴 ∈ dom vol ∧ 𝐡 ∈ (0[,](volβ€˜π΄))) ∧ 𝐡 < (volβ€˜π΄)) β†’ 0 ≀ 𝐡)
20 mnflt0 13105 . . . . . . . 8 -∞ < 0
21 xrltletr 13136 . . . . . . . 8 ((-∞ ∈ ℝ* ∧ 0 ∈ ℝ* ∧ 𝐡 ∈ ℝ*) β†’ ((-∞ < 0 ∧ 0 ≀ 𝐡) β†’ -∞ < 𝐡))
2220, 21mpani 695 . . . . . . 7 ((-∞ ∈ ℝ* ∧ 0 ∈ ℝ* ∧ 𝐡 ∈ ℝ*) β†’ (0 ≀ 𝐡 β†’ -∞ < 𝐡))
232, 14, 22mp3an12 1452 . . . . . 6 (𝐡 ∈ ℝ* β†’ (0 ≀ 𝐡 β†’ -∞ < 𝐡))
247, 19, 23sylc 65 . . . . 5 (((𝐴 ∈ dom vol ∧ 𝐡 ∈ (0[,](volβ€˜π΄))) ∧ 𝐡 < (volβ€˜π΄)) β†’ -∞ < 𝐡)
25 simpr 486 . . . . 5 (((𝐴 ∈ dom vol ∧ 𝐡 ∈ (0[,](volβ€˜π΄))) ∧ 𝐡 < (volβ€˜π΄)) β†’ 𝐡 < (volβ€˜π΄))
26 xrre2 13149 . . . . 5 (((-∞ ∈ ℝ* ∧ 𝐡 ∈ ℝ* ∧ (volβ€˜π΄) ∈ ℝ*) ∧ (-∞ < 𝐡 ∧ 𝐡 < (volβ€˜π΄))) β†’ 𝐡 ∈ ℝ)
273, 7, 13, 24, 25, 26syl32anc 1379 . . . 4 (((𝐴 ∈ dom vol ∧ 𝐡 ∈ (0[,](volβ€˜π΄))) ∧ 𝐡 < (volβ€˜π΄)) β†’ 𝐡 ∈ ℝ)
28 volsup2 25122 . . . 4 ((𝐴 ∈ dom vol ∧ 𝐡 ∈ ℝ ∧ 𝐡 < (volβ€˜π΄)) β†’ βˆƒπ‘› ∈ β„• 𝐡 < (volβ€˜(𝐴 ∩ (-𝑛[,]𝑛))))
291, 27, 25, 28syl3anc 1372 . . 3 (((𝐴 ∈ dom vol ∧ 𝐡 ∈ (0[,](volβ€˜π΄))) ∧ 𝐡 < (volβ€˜π΄)) β†’ βˆƒπ‘› ∈ β„• 𝐡 < (volβ€˜(𝐴 ∩ (-𝑛[,]𝑛))))
30 nnre 12219 . . . . . . 7 (𝑛 ∈ β„• β†’ 𝑛 ∈ ℝ)
3130ad2antrl 727 . . . . . 6 ((((𝐴 ∈ dom vol ∧ 𝐡 ∈ (0[,](volβ€˜π΄))) ∧ 𝐡 < (volβ€˜π΄)) ∧ (𝑛 ∈ β„• ∧ 𝐡 < (volβ€˜(𝐴 ∩ (-𝑛[,]𝑛))))) β†’ 𝑛 ∈ ℝ)
3231renegcld 11641 . . . . 5 ((((𝐴 ∈ dom vol ∧ 𝐡 ∈ (0[,](volβ€˜π΄))) ∧ 𝐡 < (volβ€˜π΄)) ∧ (𝑛 ∈ β„• ∧ 𝐡 < (volβ€˜(𝐴 ∩ (-𝑛[,]𝑛))))) β†’ -𝑛 ∈ ℝ)
3327adantr 482 . . . . 5 ((((𝐴 ∈ dom vol ∧ 𝐡 ∈ (0[,](volβ€˜π΄))) ∧ 𝐡 < (volβ€˜π΄)) ∧ (𝑛 ∈ β„• ∧ 𝐡 < (volβ€˜(𝐴 ∩ (-𝑛[,]𝑛))))) β†’ 𝐡 ∈ ℝ)
34 0red 11217 . . . . . 6 ((((𝐴 ∈ dom vol ∧ 𝐡 ∈ (0[,](volβ€˜π΄))) ∧ 𝐡 < (volβ€˜π΄)) ∧ (𝑛 ∈ β„• ∧ 𝐡 < (volβ€˜(𝐴 ∩ (-𝑛[,]𝑛))))) β†’ 0 ∈ ℝ)
35 nngt0 12243 . . . . . . . 8 (𝑛 ∈ β„• β†’ 0 < 𝑛)
3635ad2antrl 727 . . . . . . 7 ((((𝐴 ∈ dom vol ∧ 𝐡 ∈ (0[,](volβ€˜π΄))) ∧ 𝐡 < (volβ€˜π΄)) ∧ (𝑛 ∈ β„• ∧ 𝐡 < (volβ€˜(𝐴 ∩ (-𝑛[,]𝑛))))) β†’ 0 < 𝑛)
3731lt0neg2d 11784 . . . . . . 7 ((((𝐴 ∈ dom vol ∧ 𝐡 ∈ (0[,](volβ€˜π΄))) ∧ 𝐡 < (volβ€˜π΄)) ∧ (𝑛 ∈ β„• ∧ 𝐡 < (volβ€˜(𝐴 ∩ (-𝑛[,]𝑛))))) β†’ (0 < 𝑛 ↔ -𝑛 < 0))
3836, 37mpbid 231 . . . . . 6 ((((𝐴 ∈ dom vol ∧ 𝐡 ∈ (0[,](volβ€˜π΄))) ∧ 𝐡 < (volβ€˜π΄)) ∧ (𝑛 ∈ β„• ∧ 𝐡 < (volβ€˜(𝐴 ∩ (-𝑛[,]𝑛))))) β†’ -𝑛 < 0)
3932, 34, 31, 38, 36lttrd 11375 . . . . 5 ((((𝐴 ∈ dom vol ∧ 𝐡 ∈ (0[,](volβ€˜π΄))) ∧ 𝐡 < (volβ€˜π΄)) ∧ (𝑛 ∈ β„• ∧ 𝐡 < (volβ€˜(𝐴 ∩ (-𝑛[,]𝑛))))) β†’ -𝑛 < 𝑛)
40 iccssre 13406 . . . . . 6 ((-𝑛 ∈ ℝ ∧ 𝑛 ∈ ℝ) β†’ (-𝑛[,]𝑛) βŠ† ℝ)
4132, 31, 40syl2anc 585 . . . . 5 ((((𝐴 ∈ dom vol ∧ 𝐡 ∈ (0[,](volβ€˜π΄))) ∧ 𝐡 < (volβ€˜π΄)) ∧ (𝑛 ∈ β„• ∧ 𝐡 < (volβ€˜(𝐴 ∩ (-𝑛[,]𝑛))))) β†’ (-𝑛[,]𝑛) βŠ† ℝ)
42 ax-resscn 11167 . . . . . . 7 ℝ βŠ† β„‚
43 ssid 4005 . . . . . . 7 β„‚ βŠ† β„‚
44 cncfss 24415 . . . . . . 7 ((ℝ βŠ† β„‚ ∧ β„‚ βŠ† β„‚) β†’ (ℝ–cn→ℝ) βŠ† (ℝ–cnβ†’β„‚))
4542, 43, 44mp2an 691 . . . . . 6 (ℝ–cn→ℝ) βŠ† (ℝ–cnβ†’β„‚)
461adantr 482 . . . . . . 7 ((((𝐴 ∈ dom vol ∧ 𝐡 ∈ (0[,](volβ€˜π΄))) ∧ 𝐡 < (volβ€˜π΄)) ∧ (𝑛 ∈ β„• ∧ 𝐡 < (volβ€˜(𝐴 ∩ (-𝑛[,]𝑛))))) β†’ 𝐴 ∈ dom vol)
47 eqid 2733 . . . . . . . 8 (𝑦 ∈ ℝ ↦ (volβ€˜(𝐴 ∩ (-𝑛[,]𝑦)))) = (𝑦 ∈ ℝ ↦ (volβ€˜(𝐴 ∩ (-𝑛[,]𝑦))))
4847volcn 25123 . . . . . . 7 ((𝐴 ∈ dom vol ∧ -𝑛 ∈ ℝ) β†’ (𝑦 ∈ ℝ ↦ (volβ€˜(𝐴 ∩ (-𝑛[,]𝑦)))) ∈ (ℝ–cn→ℝ))
4946, 32, 48syl2anc 585 . . . . . 6 ((((𝐴 ∈ dom vol ∧ 𝐡 ∈ (0[,](volβ€˜π΄))) ∧ 𝐡 < (volβ€˜π΄)) ∧ (𝑛 ∈ β„• ∧ 𝐡 < (volβ€˜(𝐴 ∩ (-𝑛[,]𝑛))))) β†’ (𝑦 ∈ ℝ ↦ (volβ€˜(𝐴 ∩ (-𝑛[,]𝑦)))) ∈ (ℝ–cn→ℝ))
5045, 49sselid 3981 . . . . 5 ((((𝐴 ∈ dom vol ∧ 𝐡 ∈ (0[,](volβ€˜π΄))) ∧ 𝐡 < (volβ€˜π΄)) ∧ (𝑛 ∈ β„• ∧ 𝐡 < (volβ€˜(𝐴 ∩ (-𝑛[,]𝑛))))) β†’ (𝑦 ∈ ℝ ↦ (volβ€˜(𝐴 ∩ (-𝑛[,]𝑦)))) ∈ (ℝ–cnβ†’β„‚))
5141sselda 3983 . . . . . 6 (((((𝐴 ∈ dom vol ∧ 𝐡 ∈ (0[,](volβ€˜π΄))) ∧ 𝐡 < (volβ€˜π΄)) ∧ (𝑛 ∈ β„• ∧ 𝐡 < (volβ€˜(𝐴 ∩ (-𝑛[,]𝑛))))) ∧ 𝑒 ∈ (-𝑛[,]𝑛)) β†’ 𝑒 ∈ ℝ)
52 cncff 24409 . . . . . . . 8 ((𝑦 ∈ ℝ ↦ (volβ€˜(𝐴 ∩ (-𝑛[,]𝑦)))) ∈ (ℝ–cn→ℝ) β†’ (𝑦 ∈ ℝ ↦ (volβ€˜(𝐴 ∩ (-𝑛[,]𝑦)))):β„βŸΆβ„)
5349, 52syl 17 . . . . . . 7 ((((𝐴 ∈ dom vol ∧ 𝐡 ∈ (0[,](volβ€˜π΄))) ∧ 𝐡 < (volβ€˜π΄)) ∧ (𝑛 ∈ β„• ∧ 𝐡 < (volβ€˜(𝐴 ∩ (-𝑛[,]𝑛))))) β†’ (𝑦 ∈ ℝ ↦ (volβ€˜(𝐴 ∩ (-𝑛[,]𝑦)))):β„βŸΆβ„)
5453ffvelcdmda 7087 . . . . . 6 (((((𝐴 ∈ dom vol ∧ 𝐡 ∈ (0[,](volβ€˜π΄))) ∧ 𝐡 < (volβ€˜π΄)) ∧ (𝑛 ∈ β„• ∧ 𝐡 < (volβ€˜(𝐴 ∩ (-𝑛[,]𝑛))))) ∧ 𝑒 ∈ ℝ) β†’ ((𝑦 ∈ ℝ ↦ (volβ€˜(𝐴 ∩ (-𝑛[,]𝑦))))β€˜π‘’) ∈ ℝ)
5551, 54syldan 592 . . . . 5 (((((𝐴 ∈ dom vol ∧ 𝐡 ∈ (0[,](volβ€˜π΄))) ∧ 𝐡 < (volβ€˜π΄)) ∧ (𝑛 ∈ β„• ∧ 𝐡 < (volβ€˜(𝐴 ∩ (-𝑛[,]𝑛))))) ∧ 𝑒 ∈ (-𝑛[,]𝑛)) β†’ ((𝑦 ∈ ℝ ↦ (volβ€˜(𝐴 ∩ (-𝑛[,]𝑦))))β€˜π‘’) ∈ ℝ)
56 oveq2 7417 . . . . . . . . . . . 12 (𝑦 = -𝑛 β†’ (-𝑛[,]𝑦) = (-𝑛[,]-𝑛))
5756ineq2d 4213 . . . . . . . . . . 11 (𝑦 = -𝑛 β†’ (𝐴 ∩ (-𝑛[,]𝑦)) = (𝐴 ∩ (-𝑛[,]-𝑛)))
5857fveq2d 6896 . . . . . . . . . 10 (𝑦 = -𝑛 β†’ (volβ€˜(𝐴 ∩ (-𝑛[,]𝑦))) = (volβ€˜(𝐴 ∩ (-𝑛[,]-𝑛))))
59 fvex 6905 . . . . . . . . . 10 (volβ€˜(𝐴 ∩ (-𝑛[,]-𝑛))) ∈ V
6058, 47, 59fvmpt 6999 . . . . . . . . 9 (-𝑛 ∈ ℝ β†’ ((𝑦 ∈ ℝ ↦ (volβ€˜(𝐴 ∩ (-𝑛[,]𝑦))))β€˜-𝑛) = (volβ€˜(𝐴 ∩ (-𝑛[,]-𝑛))))
6132, 60syl 17 . . . . . . . 8 ((((𝐴 ∈ dom vol ∧ 𝐡 ∈ (0[,](volβ€˜π΄))) ∧ 𝐡 < (volβ€˜π΄)) ∧ (𝑛 ∈ β„• ∧ 𝐡 < (volβ€˜(𝐴 ∩ (-𝑛[,]𝑛))))) β†’ ((𝑦 ∈ ℝ ↦ (volβ€˜(𝐴 ∩ (-𝑛[,]𝑦))))β€˜-𝑛) = (volβ€˜(𝐴 ∩ (-𝑛[,]-𝑛))))
62 inss2 4230 . . . . . . . . . . . 12 (𝐴 ∩ (-𝑛[,]-𝑛)) βŠ† (-𝑛[,]-𝑛)
6332rexrd 11264 . . . . . . . . . . . . 13 ((((𝐴 ∈ dom vol ∧ 𝐡 ∈ (0[,](volβ€˜π΄))) ∧ 𝐡 < (volβ€˜π΄)) ∧ (𝑛 ∈ β„• ∧ 𝐡 < (volβ€˜(𝐴 ∩ (-𝑛[,]𝑛))))) β†’ -𝑛 ∈ ℝ*)
64 iccid 13369 . . . . . . . . . . . . 13 (-𝑛 ∈ ℝ* β†’ (-𝑛[,]-𝑛) = {-𝑛})
6563, 64syl 17 . . . . . . . . . . . 12 ((((𝐴 ∈ dom vol ∧ 𝐡 ∈ (0[,](volβ€˜π΄))) ∧ 𝐡 < (volβ€˜π΄)) ∧ (𝑛 ∈ β„• ∧ 𝐡 < (volβ€˜(𝐴 ∩ (-𝑛[,]𝑛))))) β†’ (-𝑛[,]-𝑛) = {-𝑛})
6662, 65sseqtrid 4035 . . . . . . . . . . 11 ((((𝐴 ∈ dom vol ∧ 𝐡 ∈ (0[,](volβ€˜π΄))) ∧ 𝐡 < (volβ€˜π΄)) ∧ (𝑛 ∈ β„• ∧ 𝐡 < (volβ€˜(𝐴 ∩ (-𝑛[,]𝑛))))) β†’ (𝐴 ∩ (-𝑛[,]-𝑛)) βŠ† {-𝑛})
6732snssd 4813 . . . . . . . . . . 11 ((((𝐴 ∈ dom vol ∧ 𝐡 ∈ (0[,](volβ€˜π΄))) ∧ 𝐡 < (volβ€˜π΄)) ∧ (𝑛 ∈ β„• ∧ 𝐡 < (volβ€˜(𝐴 ∩ (-𝑛[,]𝑛))))) β†’ {-𝑛} βŠ† ℝ)
6866, 67sstrd 3993 . . . . . . . . . 10 ((((𝐴 ∈ dom vol ∧ 𝐡 ∈ (0[,](volβ€˜π΄))) ∧ 𝐡 < (volβ€˜π΄)) ∧ (𝑛 ∈ β„• ∧ 𝐡 < (volβ€˜(𝐴 ∩ (-𝑛[,]𝑛))))) β†’ (𝐴 ∩ (-𝑛[,]-𝑛)) βŠ† ℝ)
69 ovolsn 25012 . . . . . . . . . . . 12 (-𝑛 ∈ ℝ β†’ (vol*β€˜{-𝑛}) = 0)
7032, 69syl 17 . . . . . . . . . . 11 ((((𝐴 ∈ dom vol ∧ 𝐡 ∈ (0[,](volβ€˜π΄))) ∧ 𝐡 < (volβ€˜π΄)) ∧ (𝑛 ∈ β„• ∧ 𝐡 < (volβ€˜(𝐴 ∩ (-𝑛[,]𝑛))))) β†’ (vol*β€˜{-𝑛}) = 0)
71 ovolssnul 25004 . . . . . . . . . . 11 (((𝐴 ∩ (-𝑛[,]-𝑛)) βŠ† {-𝑛} ∧ {-𝑛} βŠ† ℝ ∧ (vol*β€˜{-𝑛}) = 0) β†’ (vol*β€˜(𝐴 ∩ (-𝑛[,]-𝑛))) = 0)
7266, 67, 70, 71syl3anc 1372 . . . . . . . . . 10 ((((𝐴 ∈ dom vol ∧ 𝐡 ∈ (0[,](volβ€˜π΄))) ∧ 𝐡 < (volβ€˜π΄)) ∧ (𝑛 ∈ β„• ∧ 𝐡 < (volβ€˜(𝐴 ∩ (-𝑛[,]𝑛))))) β†’ (vol*β€˜(𝐴 ∩ (-𝑛[,]-𝑛))) = 0)
73 nulmbl 25052 . . . . . . . . . 10 (((𝐴 ∩ (-𝑛[,]-𝑛)) βŠ† ℝ ∧ (vol*β€˜(𝐴 ∩ (-𝑛[,]-𝑛))) = 0) β†’ (𝐴 ∩ (-𝑛[,]-𝑛)) ∈ dom vol)
7468, 72, 73syl2anc 585 . . . . . . . . 9 ((((𝐴 ∈ dom vol ∧ 𝐡 ∈ (0[,](volβ€˜π΄))) ∧ 𝐡 < (volβ€˜π΄)) ∧ (𝑛 ∈ β„• ∧ 𝐡 < (volβ€˜(𝐴 ∩ (-𝑛[,]𝑛))))) β†’ (𝐴 ∩ (-𝑛[,]-𝑛)) ∈ dom vol)
75 mblvol 25047 . . . . . . . . 9 ((𝐴 ∩ (-𝑛[,]-𝑛)) ∈ dom vol β†’ (volβ€˜(𝐴 ∩ (-𝑛[,]-𝑛))) = (vol*β€˜(𝐴 ∩ (-𝑛[,]-𝑛))))
7674, 75syl 17 . . . . . . . 8 ((((𝐴 ∈ dom vol ∧ 𝐡 ∈ (0[,](volβ€˜π΄))) ∧ 𝐡 < (volβ€˜π΄)) ∧ (𝑛 ∈ β„• ∧ 𝐡 < (volβ€˜(𝐴 ∩ (-𝑛[,]𝑛))))) β†’ (volβ€˜(𝐴 ∩ (-𝑛[,]-𝑛))) = (vol*β€˜(𝐴 ∩ (-𝑛[,]-𝑛))))
7761, 76, 723eqtrd 2777 . . . . . . 7 ((((𝐴 ∈ dom vol ∧ 𝐡 ∈ (0[,](volβ€˜π΄))) ∧ 𝐡 < (volβ€˜π΄)) ∧ (𝑛 ∈ β„• ∧ 𝐡 < (volβ€˜(𝐴 ∩ (-𝑛[,]𝑛))))) β†’ ((𝑦 ∈ ℝ ↦ (volβ€˜(𝐴 ∩ (-𝑛[,]𝑦))))β€˜-𝑛) = 0)
7819adantr 482 . . . . . . 7 ((((𝐴 ∈ dom vol ∧ 𝐡 ∈ (0[,](volβ€˜π΄))) ∧ 𝐡 < (volβ€˜π΄)) ∧ (𝑛 ∈ β„• ∧ 𝐡 < (volβ€˜(𝐴 ∩ (-𝑛[,]𝑛))))) β†’ 0 ≀ 𝐡)
7977, 78eqbrtrd 5171 . . . . . 6 ((((𝐴 ∈ dom vol ∧ 𝐡 ∈ (0[,](volβ€˜π΄))) ∧ 𝐡 < (volβ€˜π΄)) ∧ (𝑛 ∈ β„• ∧ 𝐡 < (volβ€˜(𝐴 ∩ (-𝑛[,]𝑛))))) β†’ ((𝑦 ∈ ℝ ↦ (volβ€˜(𝐴 ∩ (-𝑛[,]𝑦))))β€˜-𝑛) ≀ 𝐡)
807adantr 482 . . . . . . . 8 ((((𝐴 ∈ dom vol ∧ 𝐡 ∈ (0[,](volβ€˜π΄))) ∧ 𝐡 < (volβ€˜π΄)) ∧ (𝑛 ∈ β„• ∧ 𝐡 < (volβ€˜(𝐴 ∩ (-𝑛[,]𝑛))))) β†’ 𝐡 ∈ ℝ*)
81 iccmbl 25083 . . . . . . . . . . 11 ((-𝑛 ∈ ℝ ∧ 𝑛 ∈ ℝ) β†’ (-𝑛[,]𝑛) ∈ dom vol)
8232, 31, 81syl2anc 585 . . . . . . . . . 10 ((((𝐴 ∈ dom vol ∧ 𝐡 ∈ (0[,](volβ€˜π΄))) ∧ 𝐡 < (volβ€˜π΄)) ∧ (𝑛 ∈ β„• ∧ 𝐡 < (volβ€˜(𝐴 ∩ (-𝑛[,]𝑛))))) β†’ (-𝑛[,]𝑛) ∈ dom vol)
83 inmbl 25059 . . . . . . . . . 10 ((𝐴 ∈ dom vol ∧ (-𝑛[,]𝑛) ∈ dom vol) β†’ (𝐴 ∩ (-𝑛[,]𝑛)) ∈ dom vol)
8446, 82, 83syl2anc 585 . . . . . . . . 9 ((((𝐴 ∈ dom vol ∧ 𝐡 ∈ (0[,](volβ€˜π΄))) ∧ 𝐡 < (volβ€˜π΄)) ∧ (𝑛 ∈ β„• ∧ 𝐡 < (volβ€˜(𝐴 ∩ (-𝑛[,]𝑛))))) β†’ (𝐴 ∩ (-𝑛[,]𝑛)) ∈ dom vol)
859ffvelcdmi 7086 . . . . . . . . . 10 ((𝐴 ∩ (-𝑛[,]𝑛)) ∈ dom vol β†’ (volβ€˜(𝐴 ∩ (-𝑛[,]𝑛))) ∈ (0[,]+∞))
868, 85sselid 3981 . . . . . . . . 9 ((𝐴 ∩ (-𝑛[,]𝑛)) ∈ dom vol β†’ (volβ€˜(𝐴 ∩ (-𝑛[,]𝑛))) ∈ ℝ*)
8784, 86syl 17 . . . . . . . 8 ((((𝐴 ∈ dom vol ∧ 𝐡 ∈ (0[,](volβ€˜π΄))) ∧ 𝐡 < (volβ€˜π΄)) ∧ (𝑛 ∈ β„• ∧ 𝐡 < (volβ€˜(𝐴 ∩ (-𝑛[,]𝑛))))) β†’ (volβ€˜(𝐴 ∩ (-𝑛[,]𝑛))) ∈ ℝ*)
88 simprr 772 . . . . . . . 8 ((((𝐴 ∈ dom vol ∧ 𝐡 ∈ (0[,](volβ€˜π΄))) ∧ 𝐡 < (volβ€˜π΄)) ∧ (𝑛 ∈ β„• ∧ 𝐡 < (volβ€˜(𝐴 ∩ (-𝑛[,]𝑛))))) β†’ 𝐡 < (volβ€˜(𝐴 ∩ (-𝑛[,]𝑛))))
8980, 87, 88xrltled 13129 . . . . . . 7 ((((𝐴 ∈ dom vol ∧ 𝐡 ∈ (0[,](volβ€˜π΄))) ∧ 𝐡 < (volβ€˜π΄)) ∧ (𝑛 ∈ β„• ∧ 𝐡 < (volβ€˜(𝐴 ∩ (-𝑛[,]𝑛))))) β†’ 𝐡 ≀ (volβ€˜(𝐴 ∩ (-𝑛[,]𝑛))))
90 oveq2 7417 . . . . . . . . . . 11 (𝑦 = 𝑛 β†’ (-𝑛[,]𝑦) = (-𝑛[,]𝑛))
9190ineq2d 4213 . . . . . . . . . 10 (𝑦 = 𝑛 β†’ (𝐴 ∩ (-𝑛[,]𝑦)) = (𝐴 ∩ (-𝑛[,]𝑛)))
9291fveq2d 6896 . . . . . . . . 9 (𝑦 = 𝑛 β†’ (volβ€˜(𝐴 ∩ (-𝑛[,]𝑦))) = (volβ€˜(𝐴 ∩ (-𝑛[,]𝑛))))
93 fvex 6905 . . . . . . . . 9 (volβ€˜(𝐴 ∩ (-𝑛[,]𝑛))) ∈ V
9492, 47, 93fvmpt 6999 . . . . . . . 8 (𝑛 ∈ ℝ β†’ ((𝑦 ∈ ℝ ↦ (volβ€˜(𝐴 ∩ (-𝑛[,]𝑦))))β€˜π‘›) = (volβ€˜(𝐴 ∩ (-𝑛[,]𝑛))))
9531, 94syl 17 . . . . . . 7 ((((𝐴 ∈ dom vol ∧ 𝐡 ∈ (0[,](volβ€˜π΄))) ∧ 𝐡 < (volβ€˜π΄)) ∧ (𝑛 ∈ β„• ∧ 𝐡 < (volβ€˜(𝐴 ∩ (-𝑛[,]𝑛))))) β†’ ((𝑦 ∈ ℝ ↦ (volβ€˜(𝐴 ∩ (-𝑛[,]𝑦))))β€˜π‘›) = (volβ€˜(𝐴 ∩ (-𝑛[,]𝑛))))
9689, 95breqtrrd 5177 . . . . . 6 ((((𝐴 ∈ dom vol ∧ 𝐡 ∈ (0[,](volβ€˜π΄))) ∧ 𝐡 < (volβ€˜π΄)) ∧ (𝑛 ∈ β„• ∧ 𝐡 < (volβ€˜(𝐴 ∩ (-𝑛[,]𝑛))))) β†’ 𝐡 ≀ ((𝑦 ∈ ℝ ↦ (volβ€˜(𝐴 ∩ (-𝑛[,]𝑦))))β€˜π‘›))
9779, 96jca 513 . . . . 5 ((((𝐴 ∈ dom vol ∧ 𝐡 ∈ (0[,](volβ€˜π΄))) ∧ 𝐡 < (volβ€˜π΄)) ∧ (𝑛 ∈ β„• ∧ 𝐡 < (volβ€˜(𝐴 ∩ (-𝑛[,]𝑛))))) β†’ (((𝑦 ∈ ℝ ↦ (volβ€˜(𝐴 ∩ (-𝑛[,]𝑦))))β€˜-𝑛) ≀ 𝐡 ∧ 𝐡 ≀ ((𝑦 ∈ ℝ ↦ (volβ€˜(𝐴 ∩ (-𝑛[,]𝑦))))β€˜π‘›)))
9832, 31, 33, 39, 41, 50, 55, 97ivthle 24973 . . . 4 ((((𝐴 ∈ dom vol ∧ 𝐡 ∈ (0[,](volβ€˜π΄))) ∧ 𝐡 < (volβ€˜π΄)) ∧ (𝑛 ∈ β„• ∧ 𝐡 < (volβ€˜(𝐴 ∩ (-𝑛[,]𝑛))))) β†’ βˆƒπ‘§ ∈ (-𝑛[,]𝑛)((𝑦 ∈ ℝ ↦ (volβ€˜(𝐴 ∩ (-𝑛[,]𝑦))))β€˜π‘§) = 𝐡)
9941sselda 3983 . . . . . . . 8 (((((𝐴 ∈ dom vol ∧ 𝐡 ∈ (0[,](volβ€˜π΄))) ∧ 𝐡 < (volβ€˜π΄)) ∧ (𝑛 ∈ β„• ∧ 𝐡 < (volβ€˜(𝐴 ∩ (-𝑛[,]𝑛))))) ∧ 𝑧 ∈ (-𝑛[,]𝑛)) β†’ 𝑧 ∈ ℝ)
100 oveq2 7417 . . . . . . . . . . 11 (𝑦 = 𝑧 β†’ (-𝑛[,]𝑦) = (-𝑛[,]𝑧))
101100ineq2d 4213 . . . . . . . . . 10 (𝑦 = 𝑧 β†’ (𝐴 ∩ (-𝑛[,]𝑦)) = (𝐴 ∩ (-𝑛[,]𝑧)))
102101fveq2d 6896 . . . . . . . . 9 (𝑦 = 𝑧 β†’ (volβ€˜(𝐴 ∩ (-𝑛[,]𝑦))) = (volβ€˜(𝐴 ∩ (-𝑛[,]𝑧))))
103 fvex 6905 . . . . . . . . 9 (volβ€˜(𝐴 ∩ (-𝑛[,]𝑧))) ∈ V
104102, 47, 103fvmpt 6999 . . . . . . . 8 (𝑧 ∈ ℝ β†’ ((𝑦 ∈ ℝ ↦ (volβ€˜(𝐴 ∩ (-𝑛[,]𝑦))))β€˜π‘§) = (volβ€˜(𝐴 ∩ (-𝑛[,]𝑧))))
10599, 104syl 17 . . . . . . 7 (((((𝐴 ∈ dom vol ∧ 𝐡 ∈ (0[,](volβ€˜π΄))) ∧ 𝐡 < (volβ€˜π΄)) ∧ (𝑛 ∈ β„• ∧ 𝐡 < (volβ€˜(𝐴 ∩ (-𝑛[,]𝑛))))) ∧ 𝑧 ∈ (-𝑛[,]𝑛)) β†’ ((𝑦 ∈ ℝ ↦ (volβ€˜(𝐴 ∩ (-𝑛[,]𝑦))))β€˜π‘§) = (volβ€˜(𝐴 ∩ (-𝑛[,]𝑧))))
106105eqeq1d 2735 . . . . . 6 (((((𝐴 ∈ dom vol ∧ 𝐡 ∈ (0[,](volβ€˜π΄))) ∧ 𝐡 < (volβ€˜π΄)) ∧ (𝑛 ∈ β„• ∧ 𝐡 < (volβ€˜(𝐴 ∩ (-𝑛[,]𝑛))))) ∧ 𝑧 ∈ (-𝑛[,]𝑛)) β†’ (((𝑦 ∈ ℝ ↦ (volβ€˜(𝐴 ∩ (-𝑛[,]𝑦))))β€˜π‘§) = 𝐡 ↔ (volβ€˜(𝐴 ∩ (-𝑛[,]𝑧))) = 𝐡))
10746adantr 482 . . . . . . . . 9 (((((𝐴 ∈ dom vol ∧ 𝐡 ∈ (0[,](volβ€˜π΄))) ∧ 𝐡 < (volβ€˜π΄)) ∧ (𝑛 ∈ β„• ∧ 𝐡 < (volβ€˜(𝐴 ∩ (-𝑛[,]𝑛))))) ∧ (𝑧 ∈ (-𝑛[,]𝑛) ∧ (volβ€˜(𝐴 ∩ (-𝑛[,]𝑧))) = 𝐡)) β†’ 𝐴 ∈ dom vol)
10832adantr 482 . . . . . . . . . 10 (((((𝐴 ∈ dom vol ∧ 𝐡 ∈ (0[,](volβ€˜π΄))) ∧ 𝐡 < (volβ€˜π΄)) ∧ (𝑛 ∈ β„• ∧ 𝐡 < (volβ€˜(𝐴 ∩ (-𝑛[,]𝑛))))) ∧ (𝑧 ∈ (-𝑛[,]𝑛) ∧ (volβ€˜(𝐴 ∩ (-𝑛[,]𝑧))) = 𝐡)) β†’ -𝑛 ∈ ℝ)
10999adantrr 716 . . . . . . . . . 10 (((((𝐴 ∈ dom vol ∧ 𝐡 ∈ (0[,](volβ€˜π΄))) ∧ 𝐡 < (volβ€˜π΄)) ∧ (𝑛 ∈ β„• ∧ 𝐡 < (volβ€˜(𝐴 ∩ (-𝑛[,]𝑛))))) ∧ (𝑧 ∈ (-𝑛[,]𝑛) ∧ (volβ€˜(𝐴 ∩ (-𝑛[,]𝑧))) = 𝐡)) β†’ 𝑧 ∈ ℝ)
110 iccmbl 25083 . . . . . . . . . 10 ((-𝑛 ∈ ℝ ∧ 𝑧 ∈ ℝ) β†’ (-𝑛[,]𝑧) ∈ dom vol)
111108, 109, 110syl2anc 585 . . . . . . . . 9 (((((𝐴 ∈ dom vol ∧ 𝐡 ∈ (0[,](volβ€˜π΄))) ∧ 𝐡 < (volβ€˜π΄)) ∧ (𝑛 ∈ β„• ∧ 𝐡 < (volβ€˜(𝐴 ∩ (-𝑛[,]𝑛))))) ∧ (𝑧 ∈ (-𝑛[,]𝑛) ∧ (volβ€˜(𝐴 ∩ (-𝑛[,]𝑧))) = 𝐡)) β†’ (-𝑛[,]𝑧) ∈ dom vol)
112 inmbl 25059 . . . . . . . . 9 ((𝐴 ∈ dom vol ∧ (-𝑛[,]𝑧) ∈ dom vol) β†’ (𝐴 ∩ (-𝑛[,]𝑧)) ∈ dom vol)
113107, 111, 112syl2anc 585 . . . . . . . 8 (((((𝐴 ∈ dom vol ∧ 𝐡 ∈ (0[,](volβ€˜π΄))) ∧ 𝐡 < (volβ€˜π΄)) ∧ (𝑛 ∈ β„• ∧ 𝐡 < (volβ€˜(𝐴 ∩ (-𝑛[,]𝑛))))) ∧ (𝑧 ∈ (-𝑛[,]𝑛) ∧ (volβ€˜(𝐴 ∩ (-𝑛[,]𝑧))) = 𝐡)) β†’ (𝐴 ∩ (-𝑛[,]𝑧)) ∈ dom vol)
114 inss1 4229 . . . . . . . . 9 (𝐴 ∩ (-𝑛[,]𝑧)) βŠ† 𝐴
115114a1i 11 . . . . . . . 8 (((((𝐴 ∈ dom vol ∧ 𝐡 ∈ (0[,](volβ€˜π΄))) ∧ 𝐡 < (volβ€˜π΄)) ∧ (𝑛 ∈ β„• ∧ 𝐡 < (volβ€˜(𝐴 ∩ (-𝑛[,]𝑛))))) ∧ (𝑧 ∈ (-𝑛[,]𝑛) ∧ (volβ€˜(𝐴 ∩ (-𝑛[,]𝑧))) = 𝐡)) β†’ (𝐴 ∩ (-𝑛[,]𝑧)) βŠ† 𝐴)
116 simprr 772 . . . . . . . 8 (((((𝐴 ∈ dom vol ∧ 𝐡 ∈ (0[,](volβ€˜π΄))) ∧ 𝐡 < (volβ€˜π΄)) ∧ (𝑛 ∈ β„• ∧ 𝐡 < (volβ€˜(𝐴 ∩ (-𝑛[,]𝑛))))) ∧ (𝑧 ∈ (-𝑛[,]𝑛) ∧ (volβ€˜(𝐴 ∩ (-𝑛[,]𝑧))) = 𝐡)) β†’ (volβ€˜(𝐴 ∩ (-𝑛[,]𝑧))) = 𝐡)
117 sseq1 4008 . . . . . . . . . 10 (π‘₯ = (𝐴 ∩ (-𝑛[,]𝑧)) β†’ (π‘₯ βŠ† 𝐴 ↔ (𝐴 ∩ (-𝑛[,]𝑧)) βŠ† 𝐴))
118 fveqeq2 6901 . . . . . . . . . 10 (π‘₯ = (𝐴 ∩ (-𝑛[,]𝑧)) β†’ ((volβ€˜π‘₯) = 𝐡 ↔ (volβ€˜(𝐴 ∩ (-𝑛[,]𝑧))) = 𝐡))
119117, 118anbi12d 632 . . . . . . . . 9 (π‘₯ = (𝐴 ∩ (-𝑛[,]𝑧)) β†’ ((π‘₯ βŠ† 𝐴 ∧ (volβ€˜π‘₯) = 𝐡) ↔ ((𝐴 ∩ (-𝑛[,]𝑧)) βŠ† 𝐴 ∧ (volβ€˜(𝐴 ∩ (-𝑛[,]𝑧))) = 𝐡)))
120119rspcev 3613 . . . . . . . 8 (((𝐴 ∩ (-𝑛[,]𝑧)) ∈ dom vol ∧ ((𝐴 ∩ (-𝑛[,]𝑧)) βŠ† 𝐴 ∧ (volβ€˜(𝐴 ∩ (-𝑛[,]𝑧))) = 𝐡)) β†’ βˆƒπ‘₯ ∈ dom vol(π‘₯ βŠ† 𝐴 ∧ (volβ€˜π‘₯) = 𝐡))
121113, 115, 116, 120syl12anc 836 . . . . . . 7 (((((𝐴 ∈ dom vol ∧ 𝐡 ∈ (0[,](volβ€˜π΄))) ∧ 𝐡 < (volβ€˜π΄)) ∧ (𝑛 ∈ β„• ∧ 𝐡 < (volβ€˜(𝐴 ∩ (-𝑛[,]𝑛))))) ∧ (𝑧 ∈ (-𝑛[,]𝑛) ∧ (volβ€˜(𝐴 ∩ (-𝑛[,]𝑧))) = 𝐡)) β†’ βˆƒπ‘₯ ∈ dom vol(π‘₯ βŠ† 𝐴 ∧ (volβ€˜π‘₯) = 𝐡))
122121expr 458 . . . . . 6 (((((𝐴 ∈ dom vol ∧ 𝐡 ∈ (0[,](volβ€˜π΄))) ∧ 𝐡 < (volβ€˜π΄)) ∧ (𝑛 ∈ β„• ∧ 𝐡 < (volβ€˜(𝐴 ∩ (-𝑛[,]𝑛))))) ∧ 𝑧 ∈ (-𝑛[,]𝑛)) β†’ ((volβ€˜(𝐴 ∩ (-𝑛[,]𝑧))) = 𝐡 β†’ βˆƒπ‘₯ ∈ dom vol(π‘₯ βŠ† 𝐴 ∧ (volβ€˜π‘₯) = 𝐡)))
123106, 122sylbid 239 . . . . 5 (((((𝐴 ∈ dom vol ∧ 𝐡 ∈ (0[,](volβ€˜π΄))) ∧ 𝐡 < (volβ€˜π΄)) ∧ (𝑛 ∈ β„• ∧ 𝐡 < (volβ€˜(𝐴 ∩ (-𝑛[,]𝑛))))) ∧ 𝑧 ∈ (-𝑛[,]𝑛)) β†’ (((𝑦 ∈ ℝ ↦ (volβ€˜(𝐴 ∩ (-𝑛[,]𝑦))))β€˜π‘§) = 𝐡 β†’ βˆƒπ‘₯ ∈ dom vol(π‘₯ βŠ† 𝐴 ∧ (volβ€˜π‘₯) = 𝐡)))
124123rexlimdva 3156 . . . 4 ((((𝐴 ∈ dom vol ∧ 𝐡 ∈ (0[,](volβ€˜π΄))) ∧ 𝐡 < (volβ€˜π΄)) ∧ (𝑛 ∈ β„• ∧ 𝐡 < (volβ€˜(𝐴 ∩ (-𝑛[,]𝑛))))) β†’ (βˆƒπ‘§ ∈ (-𝑛[,]𝑛)((𝑦 ∈ ℝ ↦ (volβ€˜(𝐴 ∩ (-𝑛[,]𝑦))))β€˜π‘§) = 𝐡 β†’ βˆƒπ‘₯ ∈ dom vol(π‘₯ βŠ† 𝐴 ∧ (volβ€˜π‘₯) = 𝐡)))
12598, 124mpd 15 . . 3 ((((𝐴 ∈ dom vol ∧ 𝐡 ∈ (0[,](volβ€˜π΄))) ∧ 𝐡 < (volβ€˜π΄)) ∧ (𝑛 ∈ β„• ∧ 𝐡 < (volβ€˜(𝐴 ∩ (-𝑛[,]𝑛))))) β†’ βˆƒπ‘₯ ∈ dom vol(π‘₯ βŠ† 𝐴 ∧ (volβ€˜π‘₯) = 𝐡))
12629, 125rexlimddv 3162 . 2 (((𝐴 ∈ dom vol ∧ 𝐡 ∈ (0[,](volβ€˜π΄))) ∧ 𝐡 < (volβ€˜π΄)) β†’ βˆƒπ‘₯ ∈ dom vol(π‘₯ βŠ† 𝐴 ∧ (volβ€˜π‘₯) = 𝐡))
127 simpll 766 . . 3 (((𝐴 ∈ dom vol ∧ 𝐡 ∈ (0[,](volβ€˜π΄))) ∧ 𝐡 = (volβ€˜π΄)) β†’ 𝐴 ∈ dom vol)
128 ssid 4005 . . . 4 𝐴 βŠ† 𝐴
129128a1i 11 . . 3 (((𝐴 ∈ dom vol ∧ 𝐡 ∈ (0[,](volβ€˜π΄))) ∧ 𝐡 = (volβ€˜π΄)) β†’ 𝐴 βŠ† 𝐴)
130 simpr 486 . . . 4 (((𝐴 ∈ dom vol ∧ 𝐡 ∈ (0[,](volβ€˜π΄))) ∧ 𝐡 = (volβ€˜π΄)) β†’ 𝐡 = (volβ€˜π΄))
131130eqcomd 2739 . . 3 (((𝐴 ∈ dom vol ∧ 𝐡 ∈ (0[,](volβ€˜π΄))) ∧ 𝐡 = (volβ€˜π΄)) β†’ (volβ€˜π΄) = 𝐡)
132 sseq1 4008 . . . . 5 (π‘₯ = 𝐴 β†’ (π‘₯ βŠ† 𝐴 ↔ 𝐴 βŠ† 𝐴))
133 fveqeq2 6901 . . . . 5 (π‘₯ = 𝐴 β†’ ((volβ€˜π‘₯) = 𝐡 ↔ (volβ€˜π΄) = 𝐡))
134132, 133anbi12d 632 . . . 4 (π‘₯ = 𝐴 β†’ ((π‘₯ βŠ† 𝐴 ∧ (volβ€˜π‘₯) = 𝐡) ↔ (𝐴 βŠ† 𝐴 ∧ (volβ€˜π΄) = 𝐡)))
135134rspcev 3613 . . 3 ((𝐴 ∈ dom vol ∧ (𝐴 βŠ† 𝐴 ∧ (volβ€˜π΄) = 𝐡)) β†’ βˆƒπ‘₯ ∈ dom vol(π‘₯ βŠ† 𝐴 ∧ (volβ€˜π‘₯) = 𝐡))
136127, 129, 131, 135syl12anc 836 . 2 (((𝐴 ∈ dom vol ∧ 𝐡 ∈ (0[,](volβ€˜π΄))) ∧ 𝐡 = (volβ€˜π΄)) β†’ βˆƒπ‘₯ ∈ dom vol(π‘₯ βŠ† 𝐴 ∧ (volβ€˜π‘₯) = 𝐡))
13717simp3d 1145 . . 3 ((𝐴 ∈ dom vol ∧ 𝐡 ∈ (0[,](volβ€˜π΄))) β†’ 𝐡 ≀ (volβ€˜π΄))
138 xrleloe 13123 . . . 4 ((𝐡 ∈ ℝ* ∧ (volβ€˜π΄) ∈ ℝ*) β†’ (𝐡 ≀ (volβ€˜π΄) ↔ (𝐡 < (volβ€˜π΄) ∨ 𝐡 = (volβ€˜π΄))))
1396, 12, 138syl2anc 585 . . 3 ((𝐴 ∈ dom vol ∧ 𝐡 ∈ (0[,](volβ€˜π΄))) β†’ (𝐡 ≀ (volβ€˜π΄) ↔ (𝐡 < (volβ€˜π΄) ∨ 𝐡 = (volβ€˜π΄))))
140137, 139mpbid 231 . 2 ((𝐴 ∈ dom vol ∧ 𝐡 ∈ (0[,](volβ€˜π΄))) β†’ (𝐡 < (volβ€˜π΄) ∨ 𝐡 = (volβ€˜π΄)))
141126, 136, 140mpjaodan 958 1 ((𝐴 ∈ dom vol ∧ 𝐡 ∈ (0[,](volβ€˜π΄))) β†’ βˆƒπ‘₯ ∈ dom vol(π‘₯ βŠ† 𝐴 ∧ (volβ€˜π‘₯) = 𝐡))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 397   ∨ wo 846   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107  βˆƒwrex 3071   ∩ cin 3948   βŠ† wss 3949  {csn 4629   class class class wbr 5149   ↦ cmpt 5232  dom cdm 5677  βŸΆwf 6540  β€˜cfv 6544  (class class class)co 7409  β„‚cc 11108  β„cr 11109  0cc0 11110  +∞cpnf 11245  -∞cmnf 11246  β„*cxr 11247   < clt 11248   ≀ cle 11249  -cneg 11445  β„•cn 12212  [,]cicc 13327  β€“cnβ†’ccncf 24392  vol*covol 24979  volcvol 24980
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 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725  ax-inf2 9636  ax-cc 10430  ax-cnex 11166  ax-resscn 11167  ax-1cn 11168  ax-icn 11169  ax-addcl 11170  ax-addrcl 11171  ax-mulcl 11172  ax-mulrcl 11173  ax-mulcom 11174  ax-addass 11175  ax-mulass 11176  ax-distr 11177  ax-i2m1 11178  ax-1ne0 11179  ax-1rid 11180  ax-rnegex 11181  ax-rrecex 11182  ax-cnre 11183  ax-pre-lttri 11184  ax-pre-lttrn 11185  ax-pre-ltadd 11186  ax-pre-mulgt0 11187  ax-pre-sup 11188
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 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-int 4952  df-iun 5000  df-disj 5115  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-se 5633  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-isom 6553  df-riota 7365  df-ov 7412  df-oprab 7413  df-mpo 7414  df-of 7670  df-om 7856  df-1st 7975  df-2nd 7976  df-frecs 8266  df-wrecs 8297  df-recs 8371  df-rdg 8410  df-1o 8466  df-2o 8467  df-er 8703  df-map 8822  df-pm 8823  df-en 8940  df-dom 8941  df-sdom 8942  df-fin 8943  df-fi 9406  df-sup 9437  df-inf 9438  df-oi 9505  df-dju 9896  df-card 9934  df-pnf 11250  df-mnf 11251  df-xr 11252  df-ltxr 11253  df-le 11254  df-sub 11446  df-neg 11447  df-div 11872  df-nn 12213  df-2 12275  df-3 12276  df-n0 12473  df-z 12559  df-uz 12823  df-q 12933  df-rp 12975  df-xneg 13092  df-xadd 13093  df-xmul 13094  df-ioo 13328  df-ico 13330  df-icc 13331  df-fz 13485  df-fzo 13628  df-fl 13757  df-seq 13967  df-exp 14028  df-hash 14291  df-cj 15046  df-re 15047  df-im 15048  df-sqrt 15182  df-abs 15183  df-clim 15432  df-rlim 15433  df-sum 15633  df-rest 17368  df-topgen 17389  df-psmet 20936  df-xmet 20937  df-met 20938  df-bl 20939  df-mopn 20940  df-top 22396  df-topon 22413  df-bases 22449  df-cmp 22891  df-cncf 24394  df-ovol 24981  df-vol 24982
This theorem is referenced by:  itg2const2  25259
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