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Theorem ruclem12 16186
Description: Lemma for ruc 16188. The supremum of the increasing sequence 1st ∘ 𝐺 is a real number that is not in the range of 𝐹. (Contributed by Mario Carneiro, 28-May-2014.)
Hypotheses
Ref Expression
ruc.1 (πœ‘ β†’ 𝐹:β„•βŸΆβ„)
ruc.2 (πœ‘ β†’ 𝐷 = (π‘₯ ∈ (ℝ Γ— ℝ), 𝑦 ∈ ℝ ↦ ⦋(((1st β€˜π‘₯) + (2nd β€˜π‘₯)) / 2) / π‘šβ¦Œif(π‘š < 𝑦, ⟨(1st β€˜π‘₯), π‘šβŸ©, ⟨((π‘š + (2nd β€˜π‘₯)) / 2), (2nd β€˜π‘₯)⟩)))
ruc.4 𝐢 = ({⟨0, ⟨0, 1⟩⟩} βˆͺ 𝐹)
ruc.5 𝐺 = seq0(𝐷, 𝐢)
ruc.6 𝑆 = sup(ran (1st ∘ 𝐺), ℝ, < )
Assertion
Ref Expression
ruclem12 (πœ‘ β†’ 𝑆 ∈ (ℝ βˆ– ran 𝐹))
Distinct variable groups:   π‘₯,π‘š,𝑦,𝐹   π‘š,𝐺,π‘₯,𝑦
Allowed substitution hints:   πœ‘(π‘₯,𝑦,π‘š)   𝐢(π‘₯,𝑦,π‘š)   𝐷(π‘₯,𝑦,π‘š)   𝑆(π‘₯,𝑦,π‘š)

Proof of Theorem ruclem12
Dummy variables 𝑧 𝑛 π‘˜ are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ruc.6 . . 3 𝑆 = sup(ran (1st ∘ 𝐺), ℝ, < )
2 ruc.1 . . . . . 6 (πœ‘ β†’ 𝐹:β„•βŸΆβ„)
3 ruc.2 . . . . . 6 (πœ‘ β†’ 𝐷 = (π‘₯ ∈ (ℝ Γ— ℝ), 𝑦 ∈ ℝ ↦ ⦋(((1st β€˜π‘₯) + (2nd β€˜π‘₯)) / 2) / π‘šβ¦Œif(π‘š < 𝑦, ⟨(1st β€˜π‘₯), π‘šβŸ©, ⟨((π‘š + (2nd β€˜π‘₯)) / 2), (2nd β€˜π‘₯)⟩)))
4 ruc.4 . . . . . 6 𝐢 = ({⟨0, ⟨0, 1⟩⟩} βˆͺ 𝐹)
5 ruc.5 . . . . . 6 𝐺 = seq0(𝐷, 𝐢)
62, 3, 4, 5ruclem11 16185 . . . . 5 (πœ‘ β†’ (ran (1st ∘ 𝐺) βŠ† ℝ ∧ ran (1st ∘ 𝐺) β‰  βˆ… ∧ βˆ€π‘§ ∈ ran (1st ∘ 𝐺)𝑧 ≀ 1))
76simp1d 1142 . . . 4 (πœ‘ β†’ ran (1st ∘ 𝐺) βŠ† ℝ)
86simp2d 1143 . . . 4 (πœ‘ β†’ ran (1st ∘ 𝐺) β‰  βˆ…)
9 1re 11216 . . . . 5 1 ∈ ℝ
106simp3d 1144 . . . . 5 (πœ‘ β†’ βˆ€π‘§ ∈ ran (1st ∘ 𝐺)𝑧 ≀ 1)
11 brralrspcev 5208 . . . . 5 ((1 ∈ ℝ ∧ βˆ€π‘§ ∈ ran (1st ∘ 𝐺)𝑧 ≀ 1) β†’ βˆƒπ‘› ∈ ℝ βˆ€π‘§ ∈ ran (1st ∘ 𝐺)𝑧 ≀ 𝑛)
129, 10, 11sylancr 587 . . . 4 (πœ‘ β†’ βˆƒπ‘› ∈ ℝ βˆ€π‘§ ∈ ran (1st ∘ 𝐺)𝑧 ≀ 𝑛)
137, 8, 12suprcld 12179 . . 3 (πœ‘ β†’ sup(ran (1st ∘ 𝐺), ℝ, < ) ∈ ℝ)
141, 13eqeltrid 2837 . 2 (πœ‘ β†’ 𝑆 ∈ ℝ)
152adantr 481 . . . . . . . . 9 ((πœ‘ ∧ 𝑛 ∈ β„•) β†’ 𝐹:β„•βŸΆβ„)
163adantr 481 . . . . . . . . 9 ((πœ‘ ∧ 𝑛 ∈ β„•) β†’ 𝐷 = (π‘₯ ∈ (ℝ Γ— ℝ), 𝑦 ∈ ℝ ↦ ⦋(((1st β€˜π‘₯) + (2nd β€˜π‘₯)) / 2) / π‘šβ¦Œif(π‘š < 𝑦, ⟨(1st β€˜π‘₯), π‘šβŸ©, ⟨((π‘š + (2nd β€˜π‘₯)) / 2), (2nd β€˜π‘₯)⟩)))
172, 3, 4, 5ruclem6 16180 . . . . . . . . . . 11 (πœ‘ β†’ 𝐺:β„•0⟢(ℝ Γ— ℝ))
18 nnm1nn0 12515 . . . . . . . . . . 11 (𝑛 ∈ β„• β†’ (𝑛 βˆ’ 1) ∈ β„•0)
19 ffvelcdm 7083 . . . . . . . . . . 11 ((𝐺:β„•0⟢(ℝ Γ— ℝ) ∧ (𝑛 βˆ’ 1) ∈ β„•0) β†’ (πΊβ€˜(𝑛 βˆ’ 1)) ∈ (ℝ Γ— ℝ))
2017, 18, 19syl2an 596 . . . . . . . . . 10 ((πœ‘ ∧ 𝑛 ∈ β„•) β†’ (πΊβ€˜(𝑛 βˆ’ 1)) ∈ (ℝ Γ— ℝ))
21 xp1st 8009 . . . . . . . . . 10 ((πΊβ€˜(𝑛 βˆ’ 1)) ∈ (ℝ Γ— ℝ) β†’ (1st β€˜(πΊβ€˜(𝑛 βˆ’ 1))) ∈ ℝ)
2220, 21syl 17 . . . . . . . . 9 ((πœ‘ ∧ 𝑛 ∈ β„•) β†’ (1st β€˜(πΊβ€˜(𝑛 βˆ’ 1))) ∈ ℝ)
23 xp2nd 8010 . . . . . . . . . 10 ((πΊβ€˜(𝑛 βˆ’ 1)) ∈ (ℝ Γ— ℝ) β†’ (2nd β€˜(πΊβ€˜(𝑛 βˆ’ 1))) ∈ ℝ)
2420, 23syl 17 . . . . . . . . 9 ((πœ‘ ∧ 𝑛 ∈ β„•) β†’ (2nd β€˜(πΊβ€˜(𝑛 βˆ’ 1))) ∈ ℝ)
252ffvelcdmda 7086 . . . . . . . . 9 ((πœ‘ ∧ 𝑛 ∈ β„•) β†’ (πΉβ€˜π‘›) ∈ ℝ)
26 eqid 2732 . . . . . . . . 9 (1st β€˜(⟨(1st β€˜(πΊβ€˜(𝑛 βˆ’ 1))), (2nd β€˜(πΊβ€˜(𝑛 βˆ’ 1)))⟩𝐷(πΉβ€˜π‘›))) = (1st β€˜(⟨(1st β€˜(πΊβ€˜(𝑛 βˆ’ 1))), (2nd β€˜(πΊβ€˜(𝑛 βˆ’ 1)))⟩𝐷(πΉβ€˜π‘›)))
27 eqid 2732 . . . . . . . . 9 (2nd β€˜(⟨(1st β€˜(πΊβ€˜(𝑛 βˆ’ 1))), (2nd β€˜(πΊβ€˜(𝑛 βˆ’ 1)))⟩𝐷(πΉβ€˜π‘›))) = (2nd β€˜(⟨(1st β€˜(πΊβ€˜(𝑛 βˆ’ 1))), (2nd β€˜(πΊβ€˜(𝑛 βˆ’ 1)))⟩𝐷(πΉβ€˜π‘›)))
282, 3, 4, 5ruclem8 16182 . . . . . . . . . 10 ((πœ‘ ∧ (𝑛 βˆ’ 1) ∈ β„•0) β†’ (1st β€˜(πΊβ€˜(𝑛 βˆ’ 1))) < (2nd β€˜(πΊβ€˜(𝑛 βˆ’ 1))))
2918, 28sylan2 593 . . . . . . . . 9 ((πœ‘ ∧ 𝑛 ∈ β„•) β†’ (1st β€˜(πΊβ€˜(𝑛 βˆ’ 1))) < (2nd β€˜(πΊβ€˜(𝑛 βˆ’ 1))))
3015, 16, 22, 24, 25, 26, 27, 29ruclem3 16178 . . . . . . . 8 ((πœ‘ ∧ 𝑛 ∈ β„•) β†’ ((πΉβ€˜π‘›) < (1st β€˜(⟨(1st β€˜(πΊβ€˜(𝑛 βˆ’ 1))), (2nd β€˜(πΊβ€˜(𝑛 βˆ’ 1)))⟩𝐷(πΉβ€˜π‘›))) ∨ (2nd β€˜(⟨(1st β€˜(πΊβ€˜(𝑛 βˆ’ 1))), (2nd β€˜(πΊβ€˜(𝑛 βˆ’ 1)))⟩𝐷(πΉβ€˜π‘›))) < (πΉβ€˜π‘›)))
312, 3, 4, 5ruclem7 16181 . . . . . . . . . . . . 13 ((πœ‘ ∧ (𝑛 βˆ’ 1) ∈ β„•0) β†’ (πΊβ€˜((𝑛 βˆ’ 1) + 1)) = ((πΊβ€˜(𝑛 βˆ’ 1))𝐷(πΉβ€˜((𝑛 βˆ’ 1) + 1))))
3218, 31sylan2 593 . . . . . . . . . . . 12 ((πœ‘ ∧ 𝑛 ∈ β„•) β†’ (πΊβ€˜((𝑛 βˆ’ 1) + 1)) = ((πΊβ€˜(𝑛 βˆ’ 1))𝐷(πΉβ€˜((𝑛 βˆ’ 1) + 1))))
33 nncn 12222 . . . . . . . . . . . . . . 15 (𝑛 ∈ β„• β†’ 𝑛 ∈ β„‚)
3433adantl 482 . . . . . . . . . . . . . 14 ((πœ‘ ∧ 𝑛 ∈ β„•) β†’ 𝑛 ∈ β„‚)
35 ax-1cn 11170 . . . . . . . . . . . . . 14 1 ∈ β„‚
36 npcan 11471 . . . . . . . . . . . . . 14 ((𝑛 ∈ β„‚ ∧ 1 ∈ β„‚) β†’ ((𝑛 βˆ’ 1) + 1) = 𝑛)
3734, 35, 36sylancl 586 . . . . . . . . . . . . 13 ((πœ‘ ∧ 𝑛 ∈ β„•) β†’ ((𝑛 βˆ’ 1) + 1) = 𝑛)
3837fveq2d 6895 . . . . . . . . . . . 12 ((πœ‘ ∧ 𝑛 ∈ β„•) β†’ (πΊβ€˜((𝑛 βˆ’ 1) + 1)) = (πΊβ€˜π‘›))
39 1st2nd2 8016 . . . . . . . . . . . . . 14 ((πΊβ€˜(𝑛 βˆ’ 1)) ∈ (ℝ Γ— ℝ) β†’ (πΊβ€˜(𝑛 βˆ’ 1)) = ⟨(1st β€˜(πΊβ€˜(𝑛 βˆ’ 1))), (2nd β€˜(πΊβ€˜(𝑛 βˆ’ 1)))⟩)
4020, 39syl 17 . . . . . . . . . . . . 13 ((πœ‘ ∧ 𝑛 ∈ β„•) β†’ (πΊβ€˜(𝑛 βˆ’ 1)) = ⟨(1st β€˜(πΊβ€˜(𝑛 βˆ’ 1))), (2nd β€˜(πΊβ€˜(𝑛 βˆ’ 1)))⟩)
4137fveq2d 6895 . . . . . . . . . . . . 13 ((πœ‘ ∧ 𝑛 ∈ β„•) β†’ (πΉβ€˜((𝑛 βˆ’ 1) + 1)) = (πΉβ€˜π‘›))
4240, 41oveq12d 7429 . . . . . . . . . . . 12 ((πœ‘ ∧ 𝑛 ∈ β„•) β†’ ((πΊβ€˜(𝑛 βˆ’ 1))𝐷(πΉβ€˜((𝑛 βˆ’ 1) + 1))) = (⟨(1st β€˜(πΊβ€˜(𝑛 βˆ’ 1))), (2nd β€˜(πΊβ€˜(𝑛 βˆ’ 1)))⟩𝐷(πΉβ€˜π‘›)))
4332, 38, 423eqtr3d 2780 . . . . . . . . . . 11 ((πœ‘ ∧ 𝑛 ∈ β„•) β†’ (πΊβ€˜π‘›) = (⟨(1st β€˜(πΊβ€˜(𝑛 βˆ’ 1))), (2nd β€˜(πΊβ€˜(𝑛 βˆ’ 1)))⟩𝐷(πΉβ€˜π‘›)))
4443fveq2d 6895 . . . . . . . . . 10 ((πœ‘ ∧ 𝑛 ∈ β„•) β†’ (1st β€˜(πΊβ€˜π‘›)) = (1st β€˜(⟨(1st β€˜(πΊβ€˜(𝑛 βˆ’ 1))), (2nd β€˜(πΊβ€˜(𝑛 βˆ’ 1)))⟩𝐷(πΉβ€˜π‘›))))
4544breq2d 5160 . . . . . . . . 9 ((πœ‘ ∧ 𝑛 ∈ β„•) β†’ ((πΉβ€˜π‘›) < (1st β€˜(πΊβ€˜π‘›)) ↔ (πΉβ€˜π‘›) < (1st β€˜(⟨(1st β€˜(πΊβ€˜(𝑛 βˆ’ 1))), (2nd β€˜(πΊβ€˜(𝑛 βˆ’ 1)))⟩𝐷(πΉβ€˜π‘›)))))
4643fveq2d 6895 . . . . . . . . . 10 ((πœ‘ ∧ 𝑛 ∈ β„•) β†’ (2nd β€˜(πΊβ€˜π‘›)) = (2nd β€˜(⟨(1st β€˜(πΊβ€˜(𝑛 βˆ’ 1))), (2nd β€˜(πΊβ€˜(𝑛 βˆ’ 1)))⟩𝐷(πΉβ€˜π‘›))))
4746breq1d 5158 . . . . . . . . 9 ((πœ‘ ∧ 𝑛 ∈ β„•) β†’ ((2nd β€˜(πΊβ€˜π‘›)) < (πΉβ€˜π‘›) ↔ (2nd β€˜(⟨(1st β€˜(πΊβ€˜(𝑛 βˆ’ 1))), (2nd β€˜(πΊβ€˜(𝑛 βˆ’ 1)))⟩𝐷(πΉβ€˜π‘›))) < (πΉβ€˜π‘›)))
4845, 47orbi12d 917 . . . . . . . 8 ((πœ‘ ∧ 𝑛 ∈ β„•) β†’ (((πΉβ€˜π‘›) < (1st β€˜(πΊβ€˜π‘›)) ∨ (2nd β€˜(πΊβ€˜π‘›)) < (πΉβ€˜π‘›)) ↔ ((πΉβ€˜π‘›) < (1st β€˜(⟨(1st β€˜(πΊβ€˜(𝑛 βˆ’ 1))), (2nd β€˜(πΊβ€˜(𝑛 βˆ’ 1)))⟩𝐷(πΉβ€˜π‘›))) ∨ (2nd β€˜(⟨(1st β€˜(πΊβ€˜(𝑛 βˆ’ 1))), (2nd β€˜(πΊβ€˜(𝑛 βˆ’ 1)))⟩𝐷(πΉβ€˜π‘›))) < (πΉβ€˜π‘›))))
4930, 48mpbird 256 . . . . . . 7 ((πœ‘ ∧ 𝑛 ∈ β„•) β†’ ((πΉβ€˜π‘›) < (1st β€˜(πΊβ€˜π‘›)) ∨ (2nd β€˜(πΊβ€˜π‘›)) < (πΉβ€˜π‘›)))
507adantr 481 . . . . . . . . . . 11 ((πœ‘ ∧ 𝑛 ∈ β„•) β†’ ran (1st ∘ 𝐺) βŠ† ℝ)
518adantr 481 . . . . . . . . . . 11 ((πœ‘ ∧ 𝑛 ∈ β„•) β†’ ran (1st ∘ 𝐺) β‰  βˆ…)
5212adantr 481 . . . . . . . . . . 11 ((πœ‘ ∧ 𝑛 ∈ β„•) β†’ βˆƒπ‘› ∈ ℝ βˆ€π‘§ ∈ ran (1st ∘ 𝐺)𝑧 ≀ 𝑛)
53 nnnn0 12481 . . . . . . . . . . . . 13 (𝑛 ∈ β„• β†’ 𝑛 ∈ β„•0)
54 fvco3 6990 . . . . . . . . . . . . 13 ((𝐺:β„•0⟢(ℝ Γ— ℝ) ∧ 𝑛 ∈ β„•0) β†’ ((1st ∘ 𝐺)β€˜π‘›) = (1st β€˜(πΊβ€˜π‘›)))
5517, 53, 54syl2an 596 . . . . . . . . . . . 12 ((πœ‘ ∧ 𝑛 ∈ β„•) β†’ ((1st ∘ 𝐺)β€˜π‘›) = (1st β€˜(πΊβ€˜π‘›)))
5617adantr 481 . . . . . . . . . . . . . 14 ((πœ‘ ∧ 𝑛 ∈ β„•) β†’ 𝐺:β„•0⟢(ℝ Γ— ℝ))
57 1stcof 8007 . . . . . . . . . . . . . 14 (𝐺:β„•0⟢(ℝ Γ— ℝ) β†’ (1st ∘ 𝐺):β„•0βŸΆβ„)
58 ffn 6717 . . . . . . . . . . . . . 14 ((1st ∘ 𝐺):β„•0βŸΆβ„ β†’ (1st ∘ 𝐺) Fn β„•0)
5956, 57, 583syl 18 . . . . . . . . . . . . 13 ((πœ‘ ∧ 𝑛 ∈ β„•) β†’ (1st ∘ 𝐺) Fn β„•0)
6053adantl 482 . . . . . . . . . . . . 13 ((πœ‘ ∧ 𝑛 ∈ β„•) β†’ 𝑛 ∈ β„•0)
61 fnfvelrn 7082 . . . . . . . . . . . . 13 (((1st ∘ 𝐺) Fn β„•0 ∧ 𝑛 ∈ β„•0) β†’ ((1st ∘ 𝐺)β€˜π‘›) ∈ ran (1st ∘ 𝐺))
6259, 60, 61syl2anc 584 . . . . . . . . . . . 12 ((πœ‘ ∧ 𝑛 ∈ β„•) β†’ ((1st ∘ 𝐺)β€˜π‘›) ∈ ran (1st ∘ 𝐺))
6355, 62eqeltrrd 2834 . . . . . . . . . . 11 ((πœ‘ ∧ 𝑛 ∈ β„•) β†’ (1st β€˜(πΊβ€˜π‘›)) ∈ ran (1st ∘ 𝐺))
6450, 51, 52, 63suprubd 12178 . . . . . . . . . 10 ((πœ‘ ∧ 𝑛 ∈ β„•) β†’ (1st β€˜(πΊβ€˜π‘›)) ≀ sup(ran (1st ∘ 𝐺), ℝ, < ))
6564, 1breqtrrdi 5190 . . . . . . . . 9 ((πœ‘ ∧ 𝑛 ∈ β„•) β†’ (1st β€˜(πΊβ€˜π‘›)) ≀ 𝑆)
66 ffvelcdm 7083 . . . . . . . . . . . 12 ((𝐺:β„•0⟢(ℝ Γ— ℝ) ∧ 𝑛 ∈ β„•0) β†’ (πΊβ€˜π‘›) ∈ (ℝ Γ— ℝ))
6717, 53, 66syl2an 596 . . . . . . . . . . 11 ((πœ‘ ∧ 𝑛 ∈ β„•) β†’ (πΊβ€˜π‘›) ∈ (ℝ Γ— ℝ))
68 xp1st 8009 . . . . . . . . . . 11 ((πΊβ€˜π‘›) ∈ (ℝ Γ— ℝ) β†’ (1st β€˜(πΊβ€˜π‘›)) ∈ ℝ)
6967, 68syl 17 . . . . . . . . . 10 ((πœ‘ ∧ 𝑛 ∈ β„•) β†’ (1st β€˜(πΊβ€˜π‘›)) ∈ ℝ)
7014adantr 481 . . . . . . . . . 10 ((πœ‘ ∧ 𝑛 ∈ β„•) β†’ 𝑆 ∈ ℝ)
71 ltletr 11308 . . . . . . . . . 10 (((πΉβ€˜π‘›) ∈ ℝ ∧ (1st β€˜(πΊβ€˜π‘›)) ∈ ℝ ∧ 𝑆 ∈ ℝ) β†’ (((πΉβ€˜π‘›) < (1st β€˜(πΊβ€˜π‘›)) ∧ (1st β€˜(πΊβ€˜π‘›)) ≀ 𝑆) β†’ (πΉβ€˜π‘›) < 𝑆))
7225, 69, 70, 71syl3anc 1371 . . . . . . . . 9 ((πœ‘ ∧ 𝑛 ∈ β„•) β†’ (((πΉβ€˜π‘›) < (1st β€˜(πΊβ€˜π‘›)) ∧ (1st β€˜(πΊβ€˜π‘›)) ≀ 𝑆) β†’ (πΉβ€˜π‘›) < 𝑆))
7365, 72mpan2d 692 . . . . . . . 8 ((πœ‘ ∧ 𝑛 ∈ β„•) β†’ ((πΉβ€˜π‘›) < (1st β€˜(πΊβ€˜π‘›)) β†’ (πΉβ€˜π‘›) < 𝑆))
74 fvco3 6990 . . . . . . . . . . . . . . 15 ((𝐺:β„•0⟢(ℝ Γ— ℝ) ∧ π‘˜ ∈ β„•0) β†’ ((1st ∘ 𝐺)β€˜π‘˜) = (1st β€˜(πΊβ€˜π‘˜)))
7556, 74sylan 580 . . . . . . . . . . . . . 14 (((πœ‘ ∧ 𝑛 ∈ β„•) ∧ π‘˜ ∈ β„•0) β†’ ((1st ∘ 𝐺)β€˜π‘˜) = (1st β€˜(πΊβ€˜π‘˜)))
7656ffvelcdmda 7086 . . . . . . . . . . . . . . . 16 (((πœ‘ ∧ 𝑛 ∈ β„•) ∧ π‘˜ ∈ β„•0) β†’ (πΊβ€˜π‘˜) ∈ (ℝ Γ— ℝ))
77 xp1st 8009 . . . . . . . . . . . . . . . 16 ((πΊβ€˜π‘˜) ∈ (ℝ Γ— ℝ) β†’ (1st β€˜(πΊβ€˜π‘˜)) ∈ ℝ)
7876, 77syl 17 . . . . . . . . . . . . . . 15 (((πœ‘ ∧ 𝑛 ∈ β„•) ∧ π‘˜ ∈ β„•0) β†’ (1st β€˜(πΊβ€˜π‘˜)) ∈ ℝ)
79 xp2nd 8010 . . . . . . . . . . . . . . . . 17 ((πΊβ€˜π‘›) ∈ (ℝ Γ— ℝ) β†’ (2nd β€˜(πΊβ€˜π‘›)) ∈ ℝ)
8067, 79syl 17 . . . . . . . . . . . . . . . 16 ((πœ‘ ∧ 𝑛 ∈ β„•) β†’ (2nd β€˜(πΊβ€˜π‘›)) ∈ ℝ)
8180adantr 481 . . . . . . . . . . . . . . 15 (((πœ‘ ∧ 𝑛 ∈ β„•) ∧ π‘˜ ∈ β„•0) β†’ (2nd β€˜(πΊβ€˜π‘›)) ∈ ℝ)
8215adantr 481 . . . . . . . . . . . . . . . 16 (((πœ‘ ∧ 𝑛 ∈ β„•) ∧ π‘˜ ∈ β„•0) β†’ 𝐹:β„•βŸΆβ„)
8316adantr 481 . . . . . . . . . . . . . . . 16 (((πœ‘ ∧ 𝑛 ∈ β„•) ∧ π‘˜ ∈ β„•0) β†’ 𝐷 = (π‘₯ ∈ (ℝ Γ— ℝ), 𝑦 ∈ ℝ ↦ ⦋(((1st β€˜π‘₯) + (2nd β€˜π‘₯)) / 2) / π‘šβ¦Œif(π‘š < 𝑦, ⟨(1st β€˜π‘₯), π‘šβŸ©, ⟨((π‘š + (2nd β€˜π‘₯)) / 2), (2nd β€˜π‘₯)⟩)))
84 simpr 485 . . . . . . . . . . . . . . . 16 (((πœ‘ ∧ 𝑛 ∈ β„•) ∧ π‘˜ ∈ β„•0) β†’ π‘˜ ∈ β„•0)
8560adantr 481 . . . . . . . . . . . . . . . 16 (((πœ‘ ∧ 𝑛 ∈ β„•) ∧ π‘˜ ∈ β„•0) β†’ 𝑛 ∈ β„•0)
8682, 83, 4, 5, 84, 85ruclem10 16184 . . . . . . . . . . . . . . 15 (((πœ‘ ∧ 𝑛 ∈ β„•) ∧ π‘˜ ∈ β„•0) β†’ (1st β€˜(πΊβ€˜π‘˜)) < (2nd β€˜(πΊβ€˜π‘›)))
8778, 81, 86ltled 11364 . . . . . . . . . . . . . 14 (((πœ‘ ∧ 𝑛 ∈ β„•) ∧ π‘˜ ∈ β„•0) β†’ (1st β€˜(πΊβ€˜π‘˜)) ≀ (2nd β€˜(πΊβ€˜π‘›)))
8875, 87eqbrtrd 5170 . . . . . . . . . . . . 13 (((πœ‘ ∧ 𝑛 ∈ β„•) ∧ π‘˜ ∈ β„•0) β†’ ((1st ∘ 𝐺)β€˜π‘˜) ≀ (2nd β€˜(πΊβ€˜π‘›)))
8988ralrimiva 3146 . . . . . . . . . . . 12 ((πœ‘ ∧ 𝑛 ∈ β„•) β†’ βˆ€π‘˜ ∈ β„•0 ((1st ∘ 𝐺)β€˜π‘˜) ≀ (2nd β€˜(πΊβ€˜π‘›)))
90 breq1 5151 . . . . . . . . . . . . . 14 (𝑧 = ((1st ∘ 𝐺)β€˜π‘˜) β†’ (𝑧 ≀ (2nd β€˜(πΊβ€˜π‘›)) ↔ ((1st ∘ 𝐺)β€˜π‘˜) ≀ (2nd β€˜(πΊβ€˜π‘›))))
9190ralrn 7089 . . . . . . . . . . . . 13 ((1st ∘ 𝐺) Fn β„•0 β†’ (βˆ€π‘§ ∈ ran (1st ∘ 𝐺)𝑧 ≀ (2nd β€˜(πΊβ€˜π‘›)) ↔ βˆ€π‘˜ ∈ β„•0 ((1st ∘ 𝐺)β€˜π‘˜) ≀ (2nd β€˜(πΊβ€˜π‘›))))
9259, 91syl 17 . . . . . . . . . . . 12 ((πœ‘ ∧ 𝑛 ∈ β„•) β†’ (βˆ€π‘§ ∈ ran (1st ∘ 𝐺)𝑧 ≀ (2nd β€˜(πΊβ€˜π‘›)) ↔ βˆ€π‘˜ ∈ β„•0 ((1st ∘ 𝐺)β€˜π‘˜) ≀ (2nd β€˜(πΊβ€˜π‘›))))
9389, 92mpbird 256 . . . . . . . . . . 11 ((πœ‘ ∧ 𝑛 ∈ β„•) β†’ βˆ€π‘§ ∈ ran (1st ∘ 𝐺)𝑧 ≀ (2nd β€˜(πΊβ€˜π‘›)))
94 suprleub 12182 . . . . . . . . . . . 12 (((ran (1st ∘ 𝐺) βŠ† ℝ ∧ ran (1st ∘ 𝐺) β‰  βˆ… ∧ βˆƒπ‘› ∈ ℝ βˆ€π‘§ ∈ ran (1st ∘ 𝐺)𝑧 ≀ 𝑛) ∧ (2nd β€˜(πΊβ€˜π‘›)) ∈ ℝ) β†’ (sup(ran (1st ∘ 𝐺), ℝ, < ) ≀ (2nd β€˜(πΊβ€˜π‘›)) ↔ βˆ€π‘§ ∈ ran (1st ∘ 𝐺)𝑧 ≀ (2nd β€˜(πΊβ€˜π‘›))))
9550, 51, 52, 80, 94syl31anc 1373 . . . . . . . . . . 11 ((πœ‘ ∧ 𝑛 ∈ β„•) β†’ (sup(ran (1st ∘ 𝐺), ℝ, < ) ≀ (2nd β€˜(πΊβ€˜π‘›)) ↔ βˆ€π‘§ ∈ ran (1st ∘ 𝐺)𝑧 ≀ (2nd β€˜(πΊβ€˜π‘›))))
9693, 95mpbird 256 . . . . . . . . . 10 ((πœ‘ ∧ 𝑛 ∈ β„•) β†’ sup(ran (1st ∘ 𝐺), ℝ, < ) ≀ (2nd β€˜(πΊβ€˜π‘›)))
971, 96eqbrtrid 5183 . . . . . . . . 9 ((πœ‘ ∧ 𝑛 ∈ β„•) β†’ 𝑆 ≀ (2nd β€˜(πΊβ€˜π‘›)))
98 lelttr 11306 . . . . . . . . . 10 ((𝑆 ∈ ℝ ∧ (2nd β€˜(πΊβ€˜π‘›)) ∈ ℝ ∧ (πΉβ€˜π‘›) ∈ ℝ) β†’ ((𝑆 ≀ (2nd β€˜(πΊβ€˜π‘›)) ∧ (2nd β€˜(πΊβ€˜π‘›)) < (πΉβ€˜π‘›)) β†’ 𝑆 < (πΉβ€˜π‘›)))
9970, 80, 25, 98syl3anc 1371 . . . . . . . . 9 ((πœ‘ ∧ 𝑛 ∈ β„•) β†’ ((𝑆 ≀ (2nd β€˜(πΊβ€˜π‘›)) ∧ (2nd β€˜(πΊβ€˜π‘›)) < (πΉβ€˜π‘›)) β†’ 𝑆 < (πΉβ€˜π‘›)))
10097, 99mpand 693 . . . . . . . 8 ((πœ‘ ∧ 𝑛 ∈ β„•) β†’ ((2nd β€˜(πΊβ€˜π‘›)) < (πΉβ€˜π‘›) β†’ 𝑆 < (πΉβ€˜π‘›)))
10173, 100orim12d 963 . . . . . . 7 ((πœ‘ ∧ 𝑛 ∈ β„•) β†’ (((πΉβ€˜π‘›) < (1st β€˜(πΊβ€˜π‘›)) ∨ (2nd β€˜(πΊβ€˜π‘›)) < (πΉβ€˜π‘›)) β†’ ((πΉβ€˜π‘›) < 𝑆 ∨ 𝑆 < (πΉβ€˜π‘›))))
10249, 101mpd 15 . . . . . 6 ((πœ‘ ∧ 𝑛 ∈ β„•) β†’ ((πΉβ€˜π‘›) < 𝑆 ∨ 𝑆 < (πΉβ€˜π‘›)))
10325, 70lttri2d 11355 . . . . . 6 ((πœ‘ ∧ 𝑛 ∈ β„•) β†’ ((πΉβ€˜π‘›) β‰  𝑆 ↔ ((πΉβ€˜π‘›) < 𝑆 ∨ 𝑆 < (πΉβ€˜π‘›))))
104102, 103mpbird 256 . . . . 5 ((πœ‘ ∧ 𝑛 ∈ β„•) β†’ (πΉβ€˜π‘›) β‰  𝑆)
105104neneqd 2945 . . . 4 ((πœ‘ ∧ 𝑛 ∈ β„•) β†’ Β¬ (πΉβ€˜π‘›) = 𝑆)
106105nrexdv 3149 . . 3 (πœ‘ β†’ Β¬ βˆƒπ‘› ∈ β„• (πΉβ€˜π‘›) = 𝑆)
107 risset 3230 . . . 4 (𝑆 ∈ ran 𝐹 ↔ βˆƒπ‘§ ∈ ran 𝐹 𝑧 = 𝑆)
108 ffn 6717 . . . . 5 (𝐹:β„•βŸΆβ„ β†’ 𝐹 Fn β„•)
109 eqeq1 2736 . . . . . 6 (𝑧 = (πΉβ€˜π‘›) β†’ (𝑧 = 𝑆 ↔ (πΉβ€˜π‘›) = 𝑆))
110109rexrn 7088 . . . . 5 (𝐹 Fn β„• β†’ (βˆƒπ‘§ ∈ ran 𝐹 𝑧 = 𝑆 ↔ βˆƒπ‘› ∈ β„• (πΉβ€˜π‘›) = 𝑆))
1112, 108, 1103syl 18 . . . 4 (πœ‘ β†’ (βˆƒπ‘§ ∈ ran 𝐹 𝑧 = 𝑆 ↔ βˆƒπ‘› ∈ β„• (πΉβ€˜π‘›) = 𝑆))
112107, 111bitrid 282 . . 3 (πœ‘ β†’ (𝑆 ∈ ran 𝐹 ↔ βˆƒπ‘› ∈ β„• (πΉβ€˜π‘›) = 𝑆))
113106, 112mtbird 324 . 2 (πœ‘ β†’ Β¬ 𝑆 ∈ ran 𝐹)
11414, 113eldifd 3959 1 (πœ‘ β†’ 𝑆 ∈ (ℝ βˆ– ran 𝐹))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 396   ∨ wo 845   = wceq 1541   ∈ wcel 2106   β‰  wne 2940  βˆ€wral 3061  βˆƒwrex 3070  β¦‹csb 3893   βˆ– cdif 3945   βˆͺ cun 3946   βŠ† wss 3948  βˆ…c0 4322  ifcif 4528  {csn 4628  βŸ¨cop 4634   class class class wbr 5148   Γ— cxp 5674  ran crn 5677   ∘ ccom 5680   Fn wfn 6538  βŸΆwf 6539  β€˜cfv 6543  (class class class)co 7411   ∈ cmpo 7413  1st c1st 7975  2nd c2nd 7976  supcsup 9437  β„‚cc 11110  β„cr 11111  0cc0 11112  1c1 11113   + caddc 11115   < clt 11250   ≀ cle 11251   βˆ’ cmin 11446   / cdiv 11873  β„•cn 12214  2c2 12269  β„•0cn0 12474  seqcseq 13968
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-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7727  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-iun 4999  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-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-riota 7367  df-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7858  df-1st 7977  df-2nd 7978  df-frecs 8268  df-wrecs 8299  df-recs 8373  df-rdg 8412  df-er 8705  df-en 8942  df-dom 8943  df-sdom 8944  df-sup 9439  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-n0 12475  df-z 12561  df-uz 12825  df-fz 13487  df-seq 13969
This theorem is referenced by:  ruclem13  16187
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