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Theorem ovolicc2lem1 24904
Description: Lemma for ovolicc2 24909. (Contributed by Mario Carneiro, 14-Jun-2014.)
Hypotheses
Ref Expression
ovolicc.1 (πœ‘ β†’ 𝐴 ∈ ℝ)
ovolicc.2 (πœ‘ β†’ 𝐡 ∈ ℝ)
ovolicc.3 (πœ‘ β†’ 𝐴 ≀ 𝐡)
ovolicc2.4 𝑆 = seq1( + , ((abs ∘ βˆ’ ) ∘ 𝐹))
ovolicc2.5 (πœ‘ β†’ 𝐹:β„•βŸΆ( ≀ ∩ (ℝ Γ— ℝ)))
ovolicc2.6 (πœ‘ β†’ π‘ˆ ∈ (𝒫 ran ((,) ∘ 𝐹) ∩ Fin))
ovolicc2.7 (πœ‘ β†’ (𝐴[,]𝐡) βŠ† βˆͺ π‘ˆ)
ovolicc2.8 (πœ‘ β†’ 𝐺:π‘ˆβŸΆβ„•)
ovolicc2.9 ((πœ‘ ∧ 𝑑 ∈ π‘ˆ) β†’ (((,) ∘ 𝐹)β€˜(πΊβ€˜π‘‘)) = 𝑑)
Assertion
Ref Expression
ovolicc2lem1 ((πœ‘ ∧ 𝑋 ∈ π‘ˆ) β†’ (𝑃 ∈ 𝑋 ↔ (𝑃 ∈ ℝ ∧ (1st β€˜(πΉβ€˜(πΊβ€˜π‘‹))) < 𝑃 ∧ 𝑃 < (2nd β€˜(πΉβ€˜(πΊβ€˜π‘‹))))))
Distinct variable groups:   𝑑,𝐴   𝑑,𝐡   𝑑,𝐹   𝑑,𝐺   πœ‘,𝑑   𝑑,π‘ˆ   𝑑,𝑋
Allowed substitution hints:   𝑃(𝑑)   𝑆(𝑑)

Proof of Theorem ovolicc2lem1
StepHypRef Expression
1 ovolicc2.5 . . . . . 6 (πœ‘ β†’ 𝐹:β„•βŸΆ( ≀ ∩ (ℝ Γ— ℝ)))
2 inss2 4193 . . . . . 6 ( ≀ ∩ (ℝ Γ— ℝ)) βŠ† (ℝ Γ— ℝ)
3 fss 6689 . . . . . 6 ((𝐹:β„•βŸΆ( ≀ ∩ (ℝ Γ— ℝ)) ∧ ( ≀ ∩ (ℝ Γ— ℝ)) βŠ† (ℝ Γ— ℝ)) β†’ 𝐹:β„•βŸΆ(ℝ Γ— ℝ))
41, 2, 3sylancl 587 . . . . 5 (πœ‘ β†’ 𝐹:β„•βŸΆ(ℝ Γ— ℝ))
5 ovolicc2.8 . . . . . 6 (πœ‘ β†’ 𝐺:π‘ˆβŸΆβ„•)
65ffvelcdmda 7039 . . . . 5 ((πœ‘ ∧ 𝑋 ∈ π‘ˆ) β†’ (πΊβ€˜π‘‹) ∈ β„•)
7 fvco3 6944 . . . . 5 ((𝐹:β„•βŸΆ(ℝ Γ— ℝ) ∧ (πΊβ€˜π‘‹) ∈ β„•) β†’ (((,) ∘ 𝐹)β€˜(πΊβ€˜π‘‹)) = ((,)β€˜(πΉβ€˜(πΊβ€˜π‘‹))))
84, 6, 7syl2an2r 684 . . . 4 ((πœ‘ ∧ 𝑋 ∈ π‘ˆ) β†’ (((,) ∘ 𝐹)β€˜(πΊβ€˜π‘‹)) = ((,)β€˜(πΉβ€˜(πΊβ€˜π‘‹))))
9 ovolicc2.9 . . . . . 6 ((πœ‘ ∧ 𝑑 ∈ π‘ˆ) β†’ (((,) ∘ 𝐹)β€˜(πΊβ€˜π‘‘)) = 𝑑)
109ralrimiva 3140 . . . . 5 (πœ‘ β†’ βˆ€π‘‘ ∈ π‘ˆ (((,) ∘ 𝐹)β€˜(πΊβ€˜π‘‘)) = 𝑑)
11 2fveq3 6851 . . . . . . 7 (𝑑 = 𝑋 β†’ (((,) ∘ 𝐹)β€˜(πΊβ€˜π‘‘)) = (((,) ∘ 𝐹)β€˜(πΊβ€˜π‘‹)))
12 id 22 . . . . . . 7 (𝑑 = 𝑋 β†’ 𝑑 = 𝑋)
1311, 12eqeq12d 2749 . . . . . 6 (𝑑 = 𝑋 β†’ ((((,) ∘ 𝐹)β€˜(πΊβ€˜π‘‘)) = 𝑑 ↔ (((,) ∘ 𝐹)β€˜(πΊβ€˜π‘‹)) = 𝑋))
1413rspccva 3582 . . . . 5 ((βˆ€π‘‘ ∈ π‘ˆ (((,) ∘ 𝐹)β€˜(πΊβ€˜π‘‘)) = 𝑑 ∧ 𝑋 ∈ π‘ˆ) β†’ (((,) ∘ 𝐹)β€˜(πΊβ€˜π‘‹)) = 𝑋)
1510, 14sylan 581 . . . 4 ((πœ‘ ∧ 𝑋 ∈ π‘ˆ) β†’ (((,) ∘ 𝐹)β€˜(πΊβ€˜π‘‹)) = 𝑋)
164adantr 482 . . . . . . . 8 ((πœ‘ ∧ 𝑋 ∈ π‘ˆ) β†’ 𝐹:β„•βŸΆ(ℝ Γ— ℝ))
1716, 6ffvelcdmd 7040 . . . . . . 7 ((πœ‘ ∧ 𝑋 ∈ π‘ˆ) β†’ (πΉβ€˜(πΊβ€˜π‘‹)) ∈ (ℝ Γ— ℝ))
18 1st2nd2 7964 . . . . . . 7 ((πΉβ€˜(πΊβ€˜π‘‹)) ∈ (ℝ Γ— ℝ) β†’ (πΉβ€˜(πΊβ€˜π‘‹)) = ⟨(1st β€˜(πΉβ€˜(πΊβ€˜π‘‹))), (2nd β€˜(πΉβ€˜(πΊβ€˜π‘‹)))⟩)
1917, 18syl 17 . . . . . 6 ((πœ‘ ∧ 𝑋 ∈ π‘ˆ) β†’ (πΉβ€˜(πΊβ€˜π‘‹)) = ⟨(1st β€˜(πΉβ€˜(πΊβ€˜π‘‹))), (2nd β€˜(πΉβ€˜(πΊβ€˜π‘‹)))⟩)
2019fveq2d 6850 . . . . 5 ((πœ‘ ∧ 𝑋 ∈ π‘ˆ) β†’ ((,)β€˜(πΉβ€˜(πΊβ€˜π‘‹))) = ((,)β€˜βŸ¨(1st β€˜(πΉβ€˜(πΊβ€˜π‘‹))), (2nd β€˜(πΉβ€˜(πΊβ€˜π‘‹)))⟩))
21 df-ov 7364 . . . . 5 ((1st β€˜(πΉβ€˜(πΊβ€˜π‘‹)))(,)(2nd β€˜(πΉβ€˜(πΊβ€˜π‘‹)))) = ((,)β€˜βŸ¨(1st β€˜(πΉβ€˜(πΊβ€˜π‘‹))), (2nd β€˜(πΉβ€˜(πΊβ€˜π‘‹)))⟩)
2220, 21eqtr4di 2791 . . . 4 ((πœ‘ ∧ 𝑋 ∈ π‘ˆ) β†’ ((,)β€˜(πΉβ€˜(πΊβ€˜π‘‹))) = ((1st β€˜(πΉβ€˜(πΊβ€˜π‘‹)))(,)(2nd β€˜(πΉβ€˜(πΊβ€˜π‘‹)))))
238, 15, 223eqtr3d 2781 . . 3 ((πœ‘ ∧ 𝑋 ∈ π‘ˆ) β†’ 𝑋 = ((1st β€˜(πΉβ€˜(πΊβ€˜π‘‹)))(,)(2nd β€˜(πΉβ€˜(πΊβ€˜π‘‹)))))
2423eleq2d 2820 . 2 ((πœ‘ ∧ 𝑋 ∈ π‘ˆ) β†’ (𝑃 ∈ 𝑋 ↔ 𝑃 ∈ ((1st β€˜(πΉβ€˜(πΊβ€˜π‘‹)))(,)(2nd β€˜(πΉβ€˜(πΊβ€˜π‘‹))))))
25 xp1st 7957 . . . 4 ((πΉβ€˜(πΊβ€˜π‘‹)) ∈ (ℝ Γ— ℝ) β†’ (1st β€˜(πΉβ€˜(πΊβ€˜π‘‹))) ∈ ℝ)
2617, 25syl 17 . . 3 ((πœ‘ ∧ 𝑋 ∈ π‘ˆ) β†’ (1st β€˜(πΉβ€˜(πΊβ€˜π‘‹))) ∈ ℝ)
27 xp2nd 7958 . . . 4 ((πΉβ€˜(πΊβ€˜π‘‹)) ∈ (ℝ Γ— ℝ) β†’ (2nd β€˜(πΉβ€˜(πΊβ€˜π‘‹))) ∈ ℝ)
2817, 27syl 17 . . 3 ((πœ‘ ∧ 𝑋 ∈ π‘ˆ) β†’ (2nd β€˜(πΉβ€˜(πΊβ€˜π‘‹))) ∈ ℝ)
29 rexr 11209 . . . 4 ((1st β€˜(πΉβ€˜(πΊβ€˜π‘‹))) ∈ ℝ β†’ (1st β€˜(πΉβ€˜(πΊβ€˜π‘‹))) ∈ ℝ*)
30 rexr 11209 . . . 4 ((2nd β€˜(πΉβ€˜(πΊβ€˜π‘‹))) ∈ ℝ β†’ (2nd β€˜(πΉβ€˜(πΊβ€˜π‘‹))) ∈ ℝ*)
31 elioo2 13314 . . . 4 (((1st β€˜(πΉβ€˜(πΊβ€˜π‘‹))) ∈ ℝ* ∧ (2nd β€˜(πΉβ€˜(πΊβ€˜π‘‹))) ∈ ℝ*) β†’ (𝑃 ∈ ((1st β€˜(πΉβ€˜(πΊβ€˜π‘‹)))(,)(2nd β€˜(πΉβ€˜(πΊβ€˜π‘‹)))) ↔ (𝑃 ∈ ℝ ∧ (1st β€˜(πΉβ€˜(πΊβ€˜π‘‹))) < 𝑃 ∧ 𝑃 < (2nd β€˜(πΉβ€˜(πΊβ€˜π‘‹))))))
3229, 30, 31syl2an 597 . . 3 (((1st β€˜(πΉβ€˜(πΊβ€˜π‘‹))) ∈ ℝ ∧ (2nd β€˜(πΉβ€˜(πΊβ€˜π‘‹))) ∈ ℝ) β†’ (𝑃 ∈ ((1st β€˜(πΉβ€˜(πΊβ€˜π‘‹)))(,)(2nd β€˜(πΉβ€˜(πΊβ€˜π‘‹)))) ↔ (𝑃 ∈ ℝ ∧ (1st β€˜(πΉβ€˜(πΊβ€˜π‘‹))) < 𝑃 ∧ 𝑃 < (2nd β€˜(πΉβ€˜(πΊβ€˜π‘‹))))))
3326, 28, 32syl2anc 585 . 2 ((πœ‘ ∧ 𝑋 ∈ π‘ˆ) β†’ (𝑃 ∈ ((1st β€˜(πΉβ€˜(πΊβ€˜π‘‹)))(,)(2nd β€˜(πΉβ€˜(πΊβ€˜π‘‹)))) ↔ (𝑃 ∈ ℝ ∧ (1st β€˜(πΉβ€˜(πΊβ€˜π‘‹))) < 𝑃 ∧ 𝑃 < (2nd β€˜(πΉβ€˜(πΊβ€˜π‘‹))))))
3424, 33bitrd 279 1 ((πœ‘ ∧ 𝑋 ∈ π‘ˆ) β†’ (𝑃 ∈ 𝑋 ↔ (𝑃 ∈ ℝ ∧ (1st β€˜(πΉβ€˜(πΊβ€˜π‘‹))) < 𝑃 ∧ 𝑃 < (2nd β€˜(πΉβ€˜(πΊβ€˜π‘‹))))))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107  βˆ€wral 3061   ∩ cin 3913   βŠ† wss 3914  π’« cpw 4564  βŸ¨cop 4596  βˆͺ cuni 4869   class class class wbr 5109   Γ— cxp 5635  ran crn 5638   ∘ ccom 5641  βŸΆwf 6496  β€˜cfv 6500  (class class class)co 7361  1st c1st 7923  2nd c2nd 7924  Fincfn 8889  β„cr 11058  1c1 11060   + caddc 11062  β„*cxr 11196   < clt 11197   ≀ cle 11198   βˆ’ cmin 11393  β„•cn 12161  (,)cioo 13273  [,]cicc 13276  seqcseq 13915  abscabs 15128
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-sep 5260  ax-nul 5267  ax-pow 5324  ax-pr 5388  ax-un 7676  ax-cnex 11115  ax-resscn 11116  ax-pre-lttri 11133  ax-pre-lttrn 11134
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 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3449  df-sbc 3744  df-csb 3860  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-iun 4960  df-br 5110  df-opab 5172  df-mpt 5193  df-id 5535  df-po 5549  df-so 5550  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650  df-iota 6452  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-ov 7364  df-oprab 7365  df-mpo 7366  df-1st 7925  df-2nd 7926  df-er 8654  df-en 8890  df-dom 8891  df-sdom 8892  df-pnf 11199  df-mnf 11200  df-xr 11201  df-ltxr 11202  df-le 11203  df-ioo 13277
This theorem is referenced by:  ovolicc2lem2  24905  ovolicc2lem3  24906  ovolicc2lem4  24907
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