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Theorem ruclem9 16181
Description: Lemma for ruc 16186. The first components of the 𝐺 sequence are increasing, and the second components are decreasing. (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)
ruclem9.6 (πœ‘ β†’ 𝑀 ∈ β„•0)
ruclem9.7 (πœ‘ β†’ 𝑁 ∈ (β„€β‰₯β€˜π‘€))
Assertion
Ref Expression
ruclem9 (πœ‘ β†’ ((1st β€˜(πΊβ€˜π‘€)) ≀ (1st β€˜(πΊβ€˜π‘)) ∧ (2nd β€˜(πΊβ€˜π‘)) ≀ (2nd β€˜(πΊβ€˜π‘€))))
Distinct variable groups:   π‘₯,π‘š,𝑦,𝐹   π‘š,𝐺,π‘₯,𝑦   π‘š,𝑀,π‘₯,𝑦   π‘š,𝑁,π‘₯,𝑦
Allowed substitution hints:   πœ‘(π‘₯,𝑦,π‘š)   𝐢(π‘₯,𝑦,π‘š)   𝐷(π‘₯,𝑦,π‘š)

Proof of Theorem ruclem9
Dummy variables 𝑛 π‘˜ are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ruclem9.7 . 2 (πœ‘ β†’ 𝑁 ∈ (β„€β‰₯β€˜π‘€))
2 2fveq3 6897 . . . . . 6 (π‘˜ = 𝑀 β†’ (1st β€˜(πΊβ€˜π‘˜)) = (1st β€˜(πΊβ€˜π‘€)))
32breq2d 5161 . . . . 5 (π‘˜ = 𝑀 β†’ ((1st β€˜(πΊβ€˜π‘€)) ≀ (1st β€˜(πΊβ€˜π‘˜)) ↔ (1st β€˜(πΊβ€˜π‘€)) ≀ (1st β€˜(πΊβ€˜π‘€))))
4 2fveq3 6897 . . . . . 6 (π‘˜ = 𝑀 β†’ (2nd β€˜(πΊβ€˜π‘˜)) = (2nd β€˜(πΊβ€˜π‘€)))
54breq1d 5159 . . . . 5 (π‘˜ = 𝑀 β†’ ((2nd β€˜(πΊβ€˜π‘˜)) ≀ (2nd β€˜(πΊβ€˜π‘€)) ↔ (2nd β€˜(πΊβ€˜π‘€)) ≀ (2nd β€˜(πΊβ€˜π‘€))))
63, 5anbi12d 632 . . . 4 (π‘˜ = 𝑀 β†’ (((1st β€˜(πΊβ€˜π‘€)) ≀ (1st β€˜(πΊβ€˜π‘˜)) ∧ (2nd β€˜(πΊβ€˜π‘˜)) ≀ (2nd β€˜(πΊβ€˜π‘€))) ↔ ((1st β€˜(πΊβ€˜π‘€)) ≀ (1st β€˜(πΊβ€˜π‘€)) ∧ (2nd β€˜(πΊβ€˜π‘€)) ≀ (2nd β€˜(πΊβ€˜π‘€)))))
76imbi2d 341 . . 3 (π‘˜ = 𝑀 β†’ ((πœ‘ β†’ ((1st β€˜(πΊβ€˜π‘€)) ≀ (1st β€˜(πΊβ€˜π‘˜)) ∧ (2nd β€˜(πΊβ€˜π‘˜)) ≀ (2nd β€˜(πΊβ€˜π‘€)))) ↔ (πœ‘ β†’ ((1st β€˜(πΊβ€˜π‘€)) ≀ (1st β€˜(πΊβ€˜π‘€)) ∧ (2nd β€˜(πΊβ€˜π‘€)) ≀ (2nd β€˜(πΊβ€˜π‘€))))))
8 2fveq3 6897 . . . . . 6 (π‘˜ = 𝑛 β†’ (1st β€˜(πΊβ€˜π‘˜)) = (1st β€˜(πΊβ€˜π‘›)))
98breq2d 5161 . . . . 5 (π‘˜ = 𝑛 β†’ ((1st β€˜(πΊβ€˜π‘€)) ≀ (1st β€˜(πΊβ€˜π‘˜)) ↔ (1st β€˜(πΊβ€˜π‘€)) ≀ (1st β€˜(πΊβ€˜π‘›))))
10 2fveq3 6897 . . . . . 6 (π‘˜ = 𝑛 β†’ (2nd β€˜(πΊβ€˜π‘˜)) = (2nd β€˜(πΊβ€˜π‘›)))
1110breq1d 5159 . . . . 5 (π‘˜ = 𝑛 β†’ ((2nd β€˜(πΊβ€˜π‘˜)) ≀ (2nd β€˜(πΊβ€˜π‘€)) ↔ (2nd β€˜(πΊβ€˜π‘›)) ≀ (2nd β€˜(πΊβ€˜π‘€))))
129, 11anbi12d 632 . . . 4 (π‘˜ = 𝑛 β†’ (((1st β€˜(πΊβ€˜π‘€)) ≀ (1st β€˜(πΊβ€˜π‘˜)) ∧ (2nd β€˜(πΊβ€˜π‘˜)) ≀ (2nd β€˜(πΊβ€˜π‘€))) ↔ ((1st β€˜(πΊβ€˜π‘€)) ≀ (1st β€˜(πΊβ€˜π‘›)) ∧ (2nd β€˜(πΊβ€˜π‘›)) ≀ (2nd β€˜(πΊβ€˜π‘€)))))
1312imbi2d 341 . . 3 (π‘˜ = 𝑛 β†’ ((πœ‘ β†’ ((1st β€˜(πΊβ€˜π‘€)) ≀ (1st β€˜(πΊβ€˜π‘˜)) ∧ (2nd β€˜(πΊβ€˜π‘˜)) ≀ (2nd β€˜(πΊβ€˜π‘€)))) ↔ (πœ‘ β†’ ((1st β€˜(πΊβ€˜π‘€)) ≀ (1st β€˜(πΊβ€˜π‘›)) ∧ (2nd β€˜(πΊβ€˜π‘›)) ≀ (2nd β€˜(πΊβ€˜π‘€))))))
14 2fveq3 6897 . . . . . 6 (π‘˜ = (𝑛 + 1) β†’ (1st β€˜(πΊβ€˜π‘˜)) = (1st β€˜(πΊβ€˜(𝑛 + 1))))
1514breq2d 5161 . . . . 5 (π‘˜ = (𝑛 + 1) β†’ ((1st β€˜(πΊβ€˜π‘€)) ≀ (1st β€˜(πΊβ€˜π‘˜)) ↔ (1st β€˜(πΊβ€˜π‘€)) ≀ (1st β€˜(πΊβ€˜(𝑛 + 1)))))
16 2fveq3 6897 . . . . . 6 (π‘˜ = (𝑛 + 1) β†’ (2nd β€˜(πΊβ€˜π‘˜)) = (2nd β€˜(πΊβ€˜(𝑛 + 1))))
1716breq1d 5159 . . . . 5 (π‘˜ = (𝑛 + 1) β†’ ((2nd β€˜(πΊβ€˜π‘˜)) ≀ (2nd β€˜(πΊβ€˜π‘€)) ↔ (2nd β€˜(πΊβ€˜(𝑛 + 1))) ≀ (2nd β€˜(πΊβ€˜π‘€))))
1815, 17anbi12d 632 . . . 4 (π‘˜ = (𝑛 + 1) β†’ (((1st β€˜(πΊβ€˜π‘€)) ≀ (1st β€˜(πΊβ€˜π‘˜)) ∧ (2nd β€˜(πΊβ€˜π‘˜)) ≀ (2nd β€˜(πΊβ€˜π‘€))) ↔ ((1st β€˜(πΊβ€˜π‘€)) ≀ (1st β€˜(πΊβ€˜(𝑛 + 1))) ∧ (2nd β€˜(πΊβ€˜(𝑛 + 1))) ≀ (2nd β€˜(πΊβ€˜π‘€)))))
1918imbi2d 341 . . 3 (π‘˜ = (𝑛 + 1) β†’ ((πœ‘ β†’ ((1st β€˜(πΊβ€˜π‘€)) ≀ (1st β€˜(πΊβ€˜π‘˜)) ∧ (2nd β€˜(πΊβ€˜π‘˜)) ≀ (2nd β€˜(πΊβ€˜π‘€)))) ↔ (πœ‘ β†’ ((1st β€˜(πΊβ€˜π‘€)) ≀ (1st β€˜(πΊβ€˜(𝑛 + 1))) ∧ (2nd β€˜(πΊβ€˜(𝑛 + 1))) ≀ (2nd β€˜(πΊβ€˜π‘€))))))
20 2fveq3 6897 . . . . . 6 (π‘˜ = 𝑁 β†’ (1st β€˜(πΊβ€˜π‘˜)) = (1st β€˜(πΊβ€˜π‘)))
2120breq2d 5161 . . . . 5 (π‘˜ = 𝑁 β†’ ((1st β€˜(πΊβ€˜π‘€)) ≀ (1st β€˜(πΊβ€˜π‘˜)) ↔ (1st β€˜(πΊβ€˜π‘€)) ≀ (1st β€˜(πΊβ€˜π‘))))
22 2fveq3 6897 . . . . . 6 (π‘˜ = 𝑁 β†’ (2nd β€˜(πΊβ€˜π‘˜)) = (2nd β€˜(πΊβ€˜π‘)))
2322breq1d 5159 . . . . 5 (π‘˜ = 𝑁 β†’ ((2nd β€˜(πΊβ€˜π‘˜)) ≀ (2nd β€˜(πΊβ€˜π‘€)) ↔ (2nd β€˜(πΊβ€˜π‘)) ≀ (2nd β€˜(πΊβ€˜π‘€))))
2421, 23anbi12d 632 . . . 4 (π‘˜ = 𝑁 β†’ (((1st β€˜(πΊβ€˜π‘€)) ≀ (1st β€˜(πΊβ€˜π‘˜)) ∧ (2nd β€˜(πΊβ€˜π‘˜)) ≀ (2nd β€˜(πΊβ€˜π‘€))) ↔ ((1st β€˜(πΊβ€˜π‘€)) ≀ (1st β€˜(πΊβ€˜π‘)) ∧ (2nd β€˜(πΊβ€˜π‘)) ≀ (2nd β€˜(πΊβ€˜π‘€)))))
2524imbi2d 341 . . 3 (π‘˜ = 𝑁 β†’ ((πœ‘ β†’ ((1st β€˜(πΊβ€˜π‘€)) ≀ (1st β€˜(πΊβ€˜π‘˜)) ∧ (2nd β€˜(πΊβ€˜π‘˜)) ≀ (2nd β€˜(πΊβ€˜π‘€)))) ↔ (πœ‘ β†’ ((1st β€˜(πΊβ€˜π‘€)) ≀ (1st β€˜(πΊβ€˜π‘)) ∧ (2nd β€˜(πΊβ€˜π‘)) ≀ (2nd β€˜(πΊβ€˜π‘€))))))
26 ruc.1 . . . . . . . 8 (πœ‘ β†’ 𝐹:β„•βŸΆβ„)
27 ruc.2 . . . . . . . 8 (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‘₯)⟩)))
28 ruc.4 . . . . . . . 8 𝐢 = ({⟨0, ⟨0, 1⟩⟩} βˆͺ 𝐹)
29 ruc.5 . . . . . . . 8 𝐺 = seq0(𝐷, 𝐢)
3026, 27, 28, 29ruclem6 16178 . . . . . . 7 (πœ‘ β†’ 𝐺:β„•0⟢(ℝ Γ— ℝ))
31 ruclem9.6 . . . . . . 7 (πœ‘ β†’ 𝑀 ∈ β„•0)
3230, 31ffvelcdmd 7088 . . . . . 6 (πœ‘ β†’ (πΊβ€˜π‘€) ∈ (ℝ Γ— ℝ))
33 xp1st 8007 . . . . . 6 ((πΊβ€˜π‘€) ∈ (ℝ Γ— ℝ) β†’ (1st β€˜(πΊβ€˜π‘€)) ∈ ℝ)
3432, 33syl 17 . . . . 5 (πœ‘ β†’ (1st β€˜(πΊβ€˜π‘€)) ∈ ℝ)
3534leidd 11780 . . . 4 (πœ‘ β†’ (1st β€˜(πΊβ€˜π‘€)) ≀ (1st β€˜(πΊβ€˜π‘€)))
36 xp2nd 8008 . . . . . 6 ((πΊβ€˜π‘€) ∈ (ℝ Γ— ℝ) β†’ (2nd β€˜(πΊβ€˜π‘€)) ∈ ℝ)
3732, 36syl 17 . . . . 5 (πœ‘ β†’ (2nd β€˜(πΊβ€˜π‘€)) ∈ ℝ)
3837leidd 11780 . . . 4 (πœ‘ β†’ (2nd β€˜(πΊβ€˜π‘€)) ≀ (2nd β€˜(πΊβ€˜π‘€)))
3935, 38jca 513 . . 3 (πœ‘ β†’ ((1st β€˜(πΊβ€˜π‘€)) ≀ (1st β€˜(πΊβ€˜π‘€)) ∧ (2nd β€˜(πΊβ€˜π‘€)) ≀ (2nd β€˜(πΊβ€˜π‘€))))
4026adantr 482 . . . . . . . . . 10 ((πœ‘ ∧ 𝑛 ∈ (β„€β‰₯β€˜π‘€)) β†’ 𝐹:β„•βŸΆβ„)
4127adantr 482 . . . . . . . . . 10 ((πœ‘ ∧ 𝑛 ∈ (β„€β‰₯β€˜π‘€)) 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‘₯)⟩)))
4230adantr 482 . . . . . . . . . . . 12 ((πœ‘ ∧ 𝑛 ∈ (β„€β‰₯β€˜π‘€)) β†’ 𝐺:β„•0⟢(ℝ Γ— ℝ))
43 eluznn0 12901 . . . . . . . . . . . . 13 ((𝑀 ∈ β„•0 ∧ 𝑛 ∈ (β„€β‰₯β€˜π‘€)) β†’ 𝑛 ∈ β„•0)
4431, 43sylan 581 . . . . . . . . . . . 12 ((πœ‘ ∧ 𝑛 ∈ (β„€β‰₯β€˜π‘€)) β†’ 𝑛 ∈ β„•0)
4542, 44ffvelcdmd 7088 . . . . . . . . . . 11 ((πœ‘ ∧ 𝑛 ∈ (β„€β‰₯β€˜π‘€)) β†’ (πΊβ€˜π‘›) ∈ (ℝ Γ— ℝ))
46 xp1st 8007 . . . . . . . . . . 11 ((πΊβ€˜π‘›) ∈ (ℝ Γ— ℝ) β†’ (1st β€˜(πΊβ€˜π‘›)) ∈ ℝ)
4745, 46syl 17 . . . . . . . . . 10 ((πœ‘ ∧ 𝑛 ∈ (β„€β‰₯β€˜π‘€)) β†’ (1st β€˜(πΊβ€˜π‘›)) ∈ ℝ)
48 xp2nd 8008 . . . . . . . . . . 11 ((πΊβ€˜π‘›) ∈ (ℝ Γ— ℝ) β†’ (2nd β€˜(πΊβ€˜π‘›)) ∈ ℝ)
4945, 48syl 17 . . . . . . . . . 10 ((πœ‘ ∧ 𝑛 ∈ (β„€β‰₯β€˜π‘€)) β†’ (2nd β€˜(πΊβ€˜π‘›)) ∈ ℝ)
50 nn0p1nn 12511 . . . . . . . . . . . 12 (𝑛 ∈ β„•0 β†’ (𝑛 + 1) ∈ β„•)
5144, 50syl 17 . . . . . . . . . . 11 ((πœ‘ ∧ 𝑛 ∈ (β„€β‰₯β€˜π‘€)) β†’ (𝑛 + 1) ∈ β„•)
5240, 51ffvelcdmd 7088 . . . . . . . . . 10 ((πœ‘ ∧ 𝑛 ∈ (β„€β‰₯β€˜π‘€)) β†’ (πΉβ€˜(𝑛 + 1)) ∈ ℝ)
53 eqid 2733 . . . . . . . . . 10 (1st β€˜(⟨(1st β€˜(πΊβ€˜π‘›)), (2nd β€˜(πΊβ€˜π‘›))⟩𝐷(πΉβ€˜(𝑛 + 1)))) = (1st β€˜(⟨(1st β€˜(πΊβ€˜π‘›)), (2nd β€˜(πΊβ€˜π‘›))⟩𝐷(πΉβ€˜(𝑛 + 1))))
54 eqid 2733 . . . . . . . . . 10 (2nd β€˜(⟨(1st β€˜(πΊβ€˜π‘›)), (2nd β€˜(πΊβ€˜π‘›))⟩𝐷(πΉβ€˜(𝑛 + 1)))) = (2nd β€˜(⟨(1st β€˜(πΊβ€˜π‘›)), (2nd β€˜(πΊβ€˜π‘›))⟩𝐷(πΉβ€˜(𝑛 + 1))))
5526, 27, 28, 29ruclem8 16180 . . . . . . . . . . 11 ((πœ‘ ∧ 𝑛 ∈ β„•0) β†’ (1st β€˜(πΊβ€˜π‘›)) < (2nd β€˜(πΊβ€˜π‘›)))
5644, 55syldan 592 . . . . . . . . . 10 ((πœ‘ ∧ 𝑛 ∈ (β„€β‰₯β€˜π‘€)) β†’ (1st β€˜(πΊβ€˜π‘›)) < (2nd β€˜(πΊβ€˜π‘›)))
5740, 41, 47, 49, 52, 53, 54, 56ruclem2 16175 . . . . . . . . 9 ((πœ‘ ∧ 𝑛 ∈ (β„€β‰₯β€˜π‘€)) β†’ ((1st β€˜(πΊβ€˜π‘›)) ≀ (1st β€˜(⟨(1st β€˜(πΊβ€˜π‘›)), (2nd β€˜(πΊβ€˜π‘›))⟩𝐷(πΉβ€˜(𝑛 + 1)))) ∧ (1st β€˜(⟨(1st β€˜(πΊβ€˜π‘›)), (2nd β€˜(πΊβ€˜π‘›))⟩𝐷(πΉβ€˜(𝑛 + 1)))) < (2nd β€˜(⟨(1st β€˜(πΊβ€˜π‘›)), (2nd β€˜(πΊβ€˜π‘›))⟩𝐷(πΉβ€˜(𝑛 + 1)))) ∧ (2nd β€˜(⟨(1st β€˜(πΊβ€˜π‘›)), (2nd β€˜(πΊβ€˜π‘›))⟩𝐷(πΉβ€˜(𝑛 + 1)))) ≀ (2nd β€˜(πΊβ€˜π‘›))))
5857simp1d 1143 . . . . . . . 8 ((πœ‘ ∧ 𝑛 ∈ (β„€β‰₯β€˜π‘€)) β†’ (1st β€˜(πΊβ€˜π‘›)) ≀ (1st β€˜(⟨(1st β€˜(πΊβ€˜π‘›)), (2nd β€˜(πΊβ€˜π‘›))⟩𝐷(πΉβ€˜(𝑛 + 1)))))
5926, 27, 28, 29ruclem7 16179 . . . . . . . . . . 11 ((πœ‘ ∧ 𝑛 ∈ β„•0) β†’ (πΊβ€˜(𝑛 + 1)) = ((πΊβ€˜π‘›)𝐷(πΉβ€˜(𝑛 + 1))))
6044, 59syldan 592 . . . . . . . . . 10 ((πœ‘ ∧ 𝑛 ∈ (β„€β‰₯β€˜π‘€)) β†’ (πΊβ€˜(𝑛 + 1)) = ((πΊβ€˜π‘›)𝐷(πΉβ€˜(𝑛 + 1))))
61 1st2nd2 8014 . . . . . . . . . . . 12 ((πΊβ€˜π‘›) ∈ (ℝ Γ— ℝ) β†’ (πΊβ€˜π‘›) = ⟨(1st β€˜(πΊβ€˜π‘›)), (2nd β€˜(πΊβ€˜π‘›))⟩)
6245, 61syl 17 . . . . . . . . . . 11 ((πœ‘ ∧ 𝑛 ∈ (β„€β‰₯β€˜π‘€)) β†’ (πΊβ€˜π‘›) = ⟨(1st β€˜(πΊβ€˜π‘›)), (2nd β€˜(πΊβ€˜π‘›))⟩)
6362oveq1d 7424 . . . . . . . . . 10 ((πœ‘ ∧ 𝑛 ∈ (β„€β‰₯β€˜π‘€)) β†’ ((πΊβ€˜π‘›)𝐷(πΉβ€˜(𝑛 + 1))) = (⟨(1st β€˜(πΊβ€˜π‘›)), (2nd β€˜(πΊβ€˜π‘›))⟩𝐷(πΉβ€˜(𝑛 + 1))))
6460, 63eqtrd 2773 . . . . . . . . 9 ((πœ‘ ∧ 𝑛 ∈ (β„€β‰₯β€˜π‘€)) β†’ (πΊβ€˜(𝑛 + 1)) = (⟨(1st β€˜(πΊβ€˜π‘›)), (2nd β€˜(πΊβ€˜π‘›))⟩𝐷(πΉβ€˜(𝑛 + 1))))
6564fveq2d 6896 . . . . . . . 8 ((πœ‘ ∧ 𝑛 ∈ (β„€β‰₯β€˜π‘€)) β†’ (1st β€˜(πΊβ€˜(𝑛 + 1))) = (1st β€˜(⟨(1st β€˜(πΊβ€˜π‘›)), (2nd β€˜(πΊβ€˜π‘›))⟩𝐷(πΉβ€˜(𝑛 + 1)))))
6658, 65breqtrrd 5177 . . . . . . 7 ((πœ‘ ∧ 𝑛 ∈ (β„€β‰₯β€˜π‘€)) β†’ (1st β€˜(πΊβ€˜π‘›)) ≀ (1st β€˜(πΊβ€˜(𝑛 + 1))))
6734adantr 482 . . . . . . . 8 ((πœ‘ ∧ 𝑛 ∈ (β„€β‰₯β€˜π‘€)) β†’ (1st β€˜(πΊβ€˜π‘€)) ∈ ℝ)
68 peano2nn0 12512 . . . . . . . . . . 11 (𝑛 ∈ β„•0 β†’ (𝑛 + 1) ∈ β„•0)
6944, 68syl 17 . . . . . . . . . 10 ((πœ‘ ∧ 𝑛 ∈ (β„€β‰₯β€˜π‘€)) β†’ (𝑛 + 1) ∈ β„•0)
7042, 69ffvelcdmd 7088 . . . . . . . . 9 ((πœ‘ ∧ 𝑛 ∈ (β„€β‰₯β€˜π‘€)) β†’ (πΊβ€˜(𝑛 + 1)) ∈ (ℝ Γ— ℝ))
71 xp1st 8007 . . . . . . . . 9 ((πΊβ€˜(𝑛 + 1)) ∈ (ℝ Γ— ℝ) β†’ (1st β€˜(πΊβ€˜(𝑛 + 1))) ∈ ℝ)
7270, 71syl 17 . . . . . . . 8 ((πœ‘ ∧ 𝑛 ∈ (β„€β‰₯β€˜π‘€)) β†’ (1st β€˜(πΊβ€˜(𝑛 + 1))) ∈ ℝ)
73 letr 11308 . . . . . . . 8 (((1st β€˜(πΊβ€˜π‘€)) ∈ ℝ ∧ (1st β€˜(πΊβ€˜π‘›)) ∈ ℝ ∧ (1st β€˜(πΊβ€˜(𝑛 + 1))) ∈ ℝ) β†’ (((1st β€˜(πΊβ€˜π‘€)) ≀ (1st β€˜(πΊβ€˜π‘›)) ∧ (1st β€˜(πΊβ€˜π‘›)) ≀ (1st β€˜(πΊβ€˜(𝑛 + 1)))) β†’ (1st β€˜(πΊβ€˜π‘€)) ≀ (1st β€˜(πΊβ€˜(𝑛 + 1)))))
7467, 47, 72, 73syl3anc 1372 . . . . . . 7 ((πœ‘ ∧ 𝑛 ∈ (β„€β‰₯β€˜π‘€)) β†’ (((1st β€˜(πΊβ€˜π‘€)) ≀ (1st β€˜(πΊβ€˜π‘›)) ∧ (1st β€˜(πΊβ€˜π‘›)) ≀ (1st β€˜(πΊβ€˜(𝑛 + 1)))) β†’ (1st β€˜(πΊβ€˜π‘€)) ≀ (1st β€˜(πΊβ€˜(𝑛 + 1)))))
7566, 74mpan2d 693 . . . . . 6 ((πœ‘ ∧ 𝑛 ∈ (β„€β‰₯β€˜π‘€)) β†’ ((1st β€˜(πΊβ€˜π‘€)) ≀ (1st β€˜(πΊβ€˜π‘›)) β†’ (1st β€˜(πΊβ€˜π‘€)) ≀ (1st β€˜(πΊβ€˜(𝑛 + 1)))))
7664fveq2d 6896 . . . . . . . 8 ((πœ‘ ∧ 𝑛 ∈ (β„€β‰₯β€˜π‘€)) β†’ (2nd β€˜(πΊβ€˜(𝑛 + 1))) = (2nd β€˜(⟨(1st β€˜(πΊβ€˜π‘›)), (2nd β€˜(πΊβ€˜π‘›))⟩𝐷(πΉβ€˜(𝑛 + 1)))))
7757simp3d 1145 . . . . . . . 8 ((πœ‘ ∧ 𝑛 ∈ (β„€β‰₯β€˜π‘€)) β†’ (2nd β€˜(⟨(1st β€˜(πΊβ€˜π‘›)), (2nd β€˜(πΊβ€˜π‘›))⟩𝐷(πΉβ€˜(𝑛 + 1)))) ≀ (2nd β€˜(πΊβ€˜π‘›)))
7876, 77eqbrtrd 5171 . . . . . . 7 ((πœ‘ ∧ 𝑛 ∈ (β„€β‰₯β€˜π‘€)) β†’ (2nd β€˜(πΊβ€˜(𝑛 + 1))) ≀ (2nd β€˜(πΊβ€˜π‘›)))
79 xp2nd 8008 . . . . . . . . 9 ((πΊβ€˜(𝑛 + 1)) ∈ (ℝ Γ— ℝ) β†’ (2nd β€˜(πΊβ€˜(𝑛 + 1))) ∈ ℝ)
8070, 79syl 17 . . . . . . . 8 ((πœ‘ ∧ 𝑛 ∈ (β„€β‰₯β€˜π‘€)) β†’ (2nd β€˜(πΊβ€˜(𝑛 + 1))) ∈ ℝ)
8137adantr 482 . . . . . . . 8 ((πœ‘ ∧ 𝑛 ∈ (β„€β‰₯β€˜π‘€)) β†’ (2nd β€˜(πΊβ€˜π‘€)) ∈ ℝ)
82 letr 11308 . . . . . . . 8 (((2nd β€˜(πΊβ€˜(𝑛 + 1))) ∈ ℝ ∧ (2nd β€˜(πΊβ€˜π‘›)) ∈ ℝ ∧ (2nd β€˜(πΊβ€˜π‘€)) ∈ ℝ) β†’ (((2nd β€˜(πΊβ€˜(𝑛 + 1))) ≀ (2nd β€˜(πΊβ€˜π‘›)) ∧ (2nd β€˜(πΊβ€˜π‘›)) ≀ (2nd β€˜(πΊβ€˜π‘€))) β†’ (2nd β€˜(πΊβ€˜(𝑛 + 1))) ≀ (2nd β€˜(πΊβ€˜π‘€))))
8380, 49, 81, 82syl3anc 1372 . . . . . . 7 ((πœ‘ ∧ 𝑛 ∈ (β„€β‰₯β€˜π‘€)) β†’ (((2nd β€˜(πΊβ€˜(𝑛 + 1))) ≀ (2nd β€˜(πΊβ€˜π‘›)) ∧ (2nd β€˜(πΊβ€˜π‘›)) ≀ (2nd β€˜(πΊβ€˜π‘€))) β†’ (2nd β€˜(πΊβ€˜(𝑛 + 1))) ≀ (2nd β€˜(πΊβ€˜π‘€))))
8478, 83mpand 694 . . . . . 6 ((πœ‘ ∧ 𝑛 ∈ (β„€β‰₯β€˜π‘€)) β†’ ((2nd β€˜(πΊβ€˜π‘›)) ≀ (2nd β€˜(πΊβ€˜π‘€)) β†’ (2nd β€˜(πΊβ€˜(𝑛 + 1))) ≀ (2nd β€˜(πΊβ€˜π‘€))))
8575, 84anim12d 610 . . . . 5 ((πœ‘ ∧ 𝑛 ∈ (β„€β‰₯β€˜π‘€)) β†’ (((1st β€˜(πΊβ€˜π‘€)) ≀ (1st β€˜(πΊβ€˜π‘›)) ∧ (2nd β€˜(πΊβ€˜π‘›)) ≀ (2nd β€˜(πΊβ€˜π‘€))) β†’ ((1st β€˜(πΊβ€˜π‘€)) ≀ (1st β€˜(πΊβ€˜(𝑛 + 1))) ∧ (2nd β€˜(πΊβ€˜(𝑛 + 1))) ≀ (2nd β€˜(πΊβ€˜π‘€)))))
8685expcom 415 . . . 4 (𝑛 ∈ (β„€β‰₯β€˜π‘€) β†’ (πœ‘ β†’ (((1st β€˜(πΊβ€˜π‘€)) ≀ (1st β€˜(πΊβ€˜π‘›)) ∧ (2nd β€˜(πΊβ€˜π‘›)) ≀ (2nd β€˜(πΊβ€˜π‘€))) β†’ ((1st β€˜(πΊβ€˜π‘€)) ≀ (1st β€˜(πΊβ€˜(𝑛 + 1))) ∧ (2nd β€˜(πΊβ€˜(𝑛 + 1))) ≀ (2nd β€˜(πΊβ€˜π‘€))))))
8786a2d 29 . . 3 (𝑛 ∈ (β„€β‰₯β€˜π‘€) β†’ ((πœ‘ β†’ ((1st β€˜(πΊβ€˜π‘€)) ≀ (1st β€˜(πΊβ€˜π‘›)) ∧ (2nd β€˜(πΊβ€˜π‘›)) ≀ (2nd β€˜(πΊβ€˜π‘€)))) β†’ (πœ‘ β†’ ((1st β€˜(πΊβ€˜π‘€)) ≀ (1st β€˜(πΊβ€˜(𝑛 + 1))) ∧ (2nd β€˜(πΊβ€˜(𝑛 + 1))) ≀ (2nd β€˜(πΊβ€˜π‘€))))))
887, 13, 19, 25, 39, 87uzind4i 12894 . 2 (𝑁 ∈ (β„€β‰₯β€˜π‘€) β†’ (πœ‘ β†’ ((1st β€˜(πΊβ€˜π‘€)) ≀ (1st β€˜(πΊβ€˜π‘)) ∧ (2nd β€˜(πΊβ€˜π‘)) ≀ (2nd β€˜(πΊβ€˜π‘€)))))
891, 88mpcom 38 1 (πœ‘ β†’ ((1st β€˜(πΊβ€˜π‘€)) ≀ (1st β€˜(πΊβ€˜π‘)) ∧ (2nd β€˜(πΊβ€˜π‘)) ≀ (2nd β€˜(πΊβ€˜π‘€))))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 397   = wceq 1542   ∈ wcel 2107  β¦‹csb 3894   βˆͺ cun 3947  ifcif 4529  {csn 4629  βŸ¨cop 4635   class class class wbr 5149   Γ— cxp 5675  βŸΆwf 6540  β€˜cfv 6544  (class class class)co 7409   ∈ cmpo 7411  1st c1st 7973  2nd c2nd 7974  β„cr 11109  0cc0 11110  1c1 11111   + caddc 11113   < clt 11248   ≀ cle 11249   / cdiv 11871  β„•cn 12212  2c2 12267  β„•0cn0 12472  β„€β‰₯cuz 12822  seqcseq 13966
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 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725  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
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-iun 5000  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-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-riota 7365  df-ov 7412  df-oprab 7413  df-mpo 7414  df-om 7856  df-1st 7975  df-2nd 7976  df-frecs 8266  df-wrecs 8297  df-recs 8371  df-rdg 8410  df-er 8703  df-en 8940  df-dom 8941  df-sdom 8942  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-n0 12473  df-z 12559  df-uz 12823  df-fz 13485  df-seq 13967
This theorem is referenced by:  ruclem10  16182
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