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Theorem iseqf1olemnab 10417
Description: Lemma for seq3f1o 10433. (Contributed by Jim Kingdon, 27-Aug-2022.)
Hypotheses
Ref Expression
iseqf1olemqcl.k (𝜑𝐾 ∈ (𝑀...𝑁))
iseqf1olemqcl.j (𝜑𝐽:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁))
iseqf1olemqcl.a (𝜑𝐴 ∈ (𝑀...𝑁))
iseqf1olemnab.b (𝜑𝐵 ∈ (𝑀...𝑁))
iseqf1olemnab.eq (𝜑 → (𝑄𝐴) = (𝑄𝐵))
iseqf1olemnab.q 𝑄 = (𝑢 ∈ (𝑀...𝑁) ↦ if(𝑢 ∈ (𝐾...(𝐽𝐾)), if(𝑢 = 𝐾, 𝐾, (𝐽‘(𝑢 − 1))), (𝐽𝑢)))
Assertion
Ref Expression
iseqf1olemnab (𝜑 → ¬ (𝐴 ∈ (𝐾...(𝐽𝐾)) ∧ ¬ 𝐵 ∈ (𝐾...(𝐽𝐾))))
Distinct variable groups:   𝑢,𝐴   𝑢,𝐵   𝑢,𝐽   𝑢,𝐾   𝑢,𝑀   𝑢,𝑁
Allowed substitution hints:   𝜑(𝑢)   𝑄(𝑢)

Proof of Theorem iseqf1olemnab
StepHypRef Expression
1 iseqf1olemnab.eq . . . 4 (𝜑 → (𝑄𝐴) = (𝑄𝐵))
21adantr 274 . . 3 ((𝜑 ∧ (𝐴 ∈ (𝐾...(𝐽𝐾)) ∧ ¬ 𝐵 ∈ (𝐾...(𝐽𝐾)))) → (𝑄𝐴) = (𝑄𝐵))
3 iseqf1olemqcl.k . . . . . . 7 (𝜑𝐾 ∈ (𝑀...𝑁))
4 iseqf1olemqcl.j . . . . . . 7 (𝜑𝐽:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁))
5 iseqf1olemqcl.a . . . . . . 7 (𝜑𝐴 ∈ (𝑀...𝑁))
6 iseqf1olemnab.q . . . . . . 7 𝑄 = (𝑢 ∈ (𝑀...𝑁) ↦ if(𝑢 ∈ (𝐾...(𝐽𝐾)), if(𝑢 = 𝐾, 𝐾, (𝐽‘(𝑢 − 1))), (𝐽𝑢)))
73, 4, 5, 6iseqf1olemqval 10416 . . . . . 6 (𝜑 → (𝑄𝐴) = if(𝐴 ∈ (𝐾...(𝐽𝐾)), if(𝐴 = 𝐾, 𝐾, (𝐽‘(𝐴 − 1))), (𝐽𝐴)))
87adantr 274 . . . . 5 ((𝜑 ∧ (𝐴 ∈ (𝐾...(𝐽𝐾)) ∧ ¬ 𝐵 ∈ (𝐾...(𝐽𝐾)))) → (𝑄𝐴) = if(𝐴 ∈ (𝐾...(𝐽𝐾)), if(𝐴 = 𝐾, 𝐾, (𝐽‘(𝐴 − 1))), (𝐽𝐴)))
9 simprl 521 . . . . . 6 ((𝜑 ∧ (𝐴 ∈ (𝐾...(𝐽𝐾)) ∧ ¬ 𝐵 ∈ (𝐾...(𝐽𝐾)))) → 𝐴 ∈ (𝐾...(𝐽𝐾)))
109iftrued 3525 . . . . 5 ((𝜑 ∧ (𝐴 ∈ (𝐾...(𝐽𝐾)) ∧ ¬ 𝐵 ∈ (𝐾...(𝐽𝐾)))) → if(𝐴 ∈ (𝐾...(𝐽𝐾)), if(𝐴 = 𝐾, 𝐾, (𝐽‘(𝐴 − 1))), (𝐽𝐴)) = if(𝐴 = 𝐾, 𝐾, (𝐽‘(𝐴 − 1))))
118, 10eqtrd 2197 . . . 4 ((𝜑 ∧ (𝐴 ∈ (𝐾...(𝐽𝐾)) ∧ ¬ 𝐵 ∈ (𝐾...(𝐽𝐾)))) → (𝑄𝐴) = if(𝐴 = 𝐾, 𝐾, (𝐽‘(𝐴 − 1))))
12 f1ocnvfv2 5743 . . . . . . . 8 ((𝐽:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ 𝐾 ∈ (𝑀...𝑁)) → (𝐽‘(𝐽𝐾)) = 𝐾)
134, 3, 12syl2anc 409 . . . . . . 7 (𝜑 → (𝐽‘(𝐽𝐾)) = 𝐾)
1413ad2antrr 480 . . . . . 6 (((𝜑 ∧ (𝐴 ∈ (𝐾...(𝐽𝐾)) ∧ ¬ 𝐵 ∈ (𝐾...(𝐽𝐾)))) ∧ 𝐴 = 𝐾) → (𝐽‘(𝐽𝐾)) = 𝐾)
15 f1ofn 5430 . . . . . . . . 9 (𝐽:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) → 𝐽 Fn (𝑀...𝑁))
164, 15syl 14 . . . . . . . 8 (𝜑𝐽 Fn (𝑀...𝑁))
1716ad2antrr 480 . . . . . . 7 (((𝜑 ∧ (𝐴 ∈ (𝐾...(𝐽𝐾)) ∧ ¬ 𝐵 ∈ (𝐾...(𝐽𝐾)))) ∧ 𝐴 = 𝐾) → 𝐽 Fn (𝑀...𝑁))
18 elfzuz 9950 . . . . . . . . . 10 (𝐾 ∈ (𝑀...𝑁) → 𝐾 ∈ (ℤ𝑀))
19 fzss1 9992 . . . . . . . . . 10 (𝐾 ∈ (ℤ𝑀) → (𝐾...(𝐽𝐾)) ⊆ (𝑀...(𝐽𝐾)))
203, 18, 193syl 17 . . . . . . . . 9 (𝜑 → (𝐾...(𝐽𝐾)) ⊆ (𝑀...(𝐽𝐾)))
21 f1ocnv 5442 . . . . . . . . . . . 12 (𝐽:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) → 𝐽:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁))
22 f1of 5429 . . . . . . . . . . . 12 (𝐽:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) → 𝐽:(𝑀...𝑁)⟶(𝑀...𝑁))
234, 21, 223syl 17 . . . . . . . . . . 11 (𝜑𝐽:(𝑀...𝑁)⟶(𝑀...𝑁))
2423, 3ffvelrnd 5618 . . . . . . . . . 10 (𝜑 → (𝐽𝐾) ∈ (𝑀...𝑁))
25 elfzuz3 9951 . . . . . . . . . 10 ((𝐽𝐾) ∈ (𝑀...𝑁) → 𝑁 ∈ (ℤ‘(𝐽𝐾)))
26 fzss2 9993 . . . . . . . . . 10 (𝑁 ∈ (ℤ‘(𝐽𝐾)) → (𝑀...(𝐽𝐾)) ⊆ (𝑀...𝑁))
2724, 25, 263syl 17 . . . . . . . . 9 (𝜑 → (𝑀...(𝐽𝐾)) ⊆ (𝑀...𝑁))
2820, 27sstrd 3150 . . . . . . . 8 (𝜑 → (𝐾...(𝐽𝐾)) ⊆ (𝑀...𝑁))
2928ad2antrr 480 . . . . . . 7 (((𝜑 ∧ (𝐴 ∈ (𝐾...(𝐽𝐾)) ∧ ¬ 𝐵 ∈ (𝐾...(𝐽𝐾)))) ∧ 𝐴 = 𝐾) → (𝐾...(𝐽𝐾)) ⊆ (𝑀...𝑁))
30 elfzubelfz 9965 . . . . . . . . 9 (𝐴 ∈ (𝐾...(𝐽𝐾)) → (𝐽𝐾) ∈ (𝐾...(𝐽𝐾)))
3130adantr 274 . . . . . . . 8 ((𝐴 ∈ (𝐾...(𝐽𝐾)) ∧ ¬ 𝐵 ∈ (𝐾...(𝐽𝐾))) → (𝐽𝐾) ∈ (𝐾...(𝐽𝐾)))
3231ad2antlr 481 . . . . . . 7 (((𝜑 ∧ (𝐴 ∈ (𝐾...(𝐽𝐾)) ∧ ¬ 𝐵 ∈ (𝐾...(𝐽𝐾)))) ∧ 𝐴 = 𝐾) → (𝐽𝐾) ∈ (𝐾...(𝐽𝐾)))
33 fnfvima 5716 . . . . . . 7 ((𝐽 Fn (𝑀...𝑁) ∧ (𝐾...(𝐽𝐾)) ⊆ (𝑀...𝑁) ∧ (𝐽𝐾) ∈ (𝐾...(𝐽𝐾))) → (𝐽‘(𝐽𝐾)) ∈ (𝐽 “ (𝐾...(𝐽𝐾))))
3417, 29, 32, 33syl3anc 1227 . . . . . 6 (((𝜑 ∧ (𝐴 ∈ (𝐾...(𝐽𝐾)) ∧ ¬ 𝐵 ∈ (𝐾...(𝐽𝐾)))) ∧ 𝐴 = 𝐾) → (𝐽‘(𝐽𝐾)) ∈ (𝐽 “ (𝐾...(𝐽𝐾))))
3514, 34eqeltrrd 2242 . . . . 5 (((𝜑 ∧ (𝐴 ∈ (𝐾...(𝐽𝐾)) ∧ ¬ 𝐵 ∈ (𝐾...(𝐽𝐾)))) ∧ 𝐴 = 𝐾) → 𝐾 ∈ (𝐽 “ (𝐾...(𝐽𝐾))))
3616ad2antrr 480 . . . . . 6 (((𝜑 ∧ (𝐴 ∈ (𝐾...(𝐽𝐾)) ∧ ¬ 𝐵 ∈ (𝐾...(𝐽𝐾)))) ∧ ¬ 𝐴 = 𝐾) → 𝐽 Fn (𝑀...𝑁))
3728ad2antrr 480 . . . . . 6 (((𝜑 ∧ (𝐴 ∈ (𝐾...(𝐽𝐾)) ∧ ¬ 𝐵 ∈ (𝐾...(𝐽𝐾)))) ∧ ¬ 𝐴 = 𝐾) → (𝐾...(𝐽𝐾)) ⊆ (𝑀...𝑁))
383adantr 274 . . . . . . . . . 10 ((𝜑 ∧ (𝐴 ∈ (𝐾...(𝐽𝐾)) ∧ ¬ 𝐵 ∈ (𝐾...(𝐽𝐾)))) → 𝐾 ∈ (𝑀...𝑁))
39 elfzelz 9954 . . . . . . . . . 10 (𝐾 ∈ (𝑀...𝑁) → 𝐾 ∈ ℤ)
4038, 39syl 14 . . . . . . . . 9 ((𝜑 ∧ (𝐴 ∈ (𝐾...(𝐽𝐾)) ∧ ¬ 𝐵 ∈ (𝐾...(𝐽𝐾)))) → 𝐾 ∈ ℤ)
4140adantr 274 . . . . . . . 8 (((𝜑 ∧ (𝐴 ∈ (𝐾...(𝐽𝐾)) ∧ ¬ 𝐵 ∈ (𝐾...(𝐽𝐾)))) ∧ ¬ 𝐴 = 𝐾) → 𝐾 ∈ ℤ)
4224ad2antrr 480 . . . . . . . . 9 (((𝜑 ∧ (𝐴 ∈ (𝐾...(𝐽𝐾)) ∧ ¬ 𝐵 ∈ (𝐾...(𝐽𝐾)))) ∧ ¬ 𝐴 = 𝐾) → (𝐽𝐾) ∈ (𝑀...𝑁))
43 elfzelz 9954 . . . . . . . . 9 ((𝐽𝐾) ∈ (𝑀...𝑁) → (𝐽𝐾) ∈ ℤ)
4442, 43syl 14 . . . . . . . 8 (((𝜑 ∧ (𝐴 ∈ (𝐾...(𝐽𝐾)) ∧ ¬ 𝐵 ∈ (𝐾...(𝐽𝐾)))) ∧ ¬ 𝐴 = 𝐾) → (𝐽𝐾) ∈ ℤ)
455adantr 274 . . . . . . . . . . 11 ((𝜑 ∧ (𝐴 ∈ (𝐾...(𝐽𝐾)) ∧ ¬ 𝐵 ∈ (𝐾...(𝐽𝐾)))) → 𝐴 ∈ (𝑀...𝑁))
46 elfzelz 9954 . . . . . . . . . . 11 (𝐴 ∈ (𝑀...𝑁) → 𝐴 ∈ ℤ)
4745, 46syl 14 . . . . . . . . . 10 ((𝜑 ∧ (𝐴 ∈ (𝐾...(𝐽𝐾)) ∧ ¬ 𝐵 ∈ (𝐾...(𝐽𝐾)))) → 𝐴 ∈ ℤ)
4847adantr 274 . . . . . . . . 9 (((𝜑 ∧ (𝐴 ∈ (𝐾...(𝐽𝐾)) ∧ ¬ 𝐵 ∈ (𝐾...(𝐽𝐾)))) ∧ ¬ 𝐴 = 𝐾) → 𝐴 ∈ ℤ)
49 peano2zm 9223 . . . . . . . . 9 (𝐴 ∈ ℤ → (𝐴 − 1) ∈ ℤ)
5048, 49syl 14 . . . . . . . 8 (((𝜑 ∧ (𝐴 ∈ (𝐾...(𝐽𝐾)) ∧ ¬ 𝐵 ∈ (𝐾...(𝐽𝐾)))) ∧ ¬ 𝐴 = 𝐾) → (𝐴 − 1) ∈ ℤ)
5141, 44, 503jca 1166 . . . . . . 7 (((𝜑 ∧ (𝐴 ∈ (𝐾...(𝐽𝐾)) ∧ ¬ 𝐵 ∈ (𝐾...(𝐽𝐾)))) ∧ ¬ 𝐴 = 𝐾) → (𝐾 ∈ ℤ ∧ (𝐽𝐾) ∈ ℤ ∧ (𝐴 − 1) ∈ ℤ))
52 simpr 109 . . . . . . . . . . 11 (((𝜑 ∧ (𝐴 ∈ (𝐾...(𝐽𝐾)) ∧ ¬ 𝐵 ∈ (𝐾...(𝐽𝐾)))) ∧ ¬ 𝐴 = 𝐾) → ¬ 𝐴 = 𝐾)
53 eqcom 2166 . . . . . . . . . . 11 (𝐴 = 𝐾𝐾 = 𝐴)
5452, 53sylnib 666 . . . . . . . . . 10 (((𝜑 ∧ (𝐴 ∈ (𝐾...(𝐽𝐾)) ∧ ¬ 𝐵 ∈ (𝐾...(𝐽𝐾)))) ∧ ¬ 𝐴 = 𝐾) → ¬ 𝐾 = 𝐴)
559adantr 274 . . . . . . . . . . . 12 (((𝜑 ∧ (𝐴 ∈ (𝐾...(𝐽𝐾)) ∧ ¬ 𝐵 ∈ (𝐾...(𝐽𝐾)))) ∧ ¬ 𝐴 = 𝐾) → 𝐴 ∈ (𝐾...(𝐽𝐾)))
56 elfzle1 9956 . . . . . . . . . . . 12 (𝐴 ∈ (𝐾...(𝐽𝐾)) → 𝐾𝐴)
5755, 56syl 14 . . . . . . . . . . 11 (((𝜑 ∧ (𝐴 ∈ (𝐾...(𝐽𝐾)) ∧ ¬ 𝐵 ∈ (𝐾...(𝐽𝐾)))) ∧ ¬ 𝐴 = 𝐾) → 𝐾𝐴)
58 zleloe 9232 . . . . . . . . . . . 12 ((𝐾 ∈ ℤ ∧ 𝐴 ∈ ℤ) → (𝐾𝐴 ↔ (𝐾 < 𝐴𝐾 = 𝐴)))
5941, 48, 58syl2anc 409 . . . . . . . . . . 11 (((𝜑 ∧ (𝐴 ∈ (𝐾...(𝐽𝐾)) ∧ ¬ 𝐵 ∈ (𝐾...(𝐽𝐾)))) ∧ ¬ 𝐴 = 𝐾) → (𝐾𝐴 ↔ (𝐾 < 𝐴𝐾 = 𝐴)))
6057, 59mpbid 146 . . . . . . . . . 10 (((𝜑 ∧ (𝐴 ∈ (𝐾...(𝐽𝐾)) ∧ ¬ 𝐵 ∈ (𝐾...(𝐽𝐾)))) ∧ ¬ 𝐴 = 𝐾) → (𝐾 < 𝐴𝐾 = 𝐴))
6154, 60ecased 1338 . . . . . . . . 9 (((𝜑 ∧ (𝐴 ∈ (𝐾...(𝐽𝐾)) ∧ ¬ 𝐵 ∈ (𝐾...(𝐽𝐾)))) ∧ ¬ 𝐴 = 𝐾) → 𝐾 < 𝐴)
62 zltlem1 9242 . . . . . . . . . 10 ((𝐾 ∈ ℤ ∧ 𝐴 ∈ ℤ) → (𝐾 < 𝐴𝐾 ≤ (𝐴 − 1)))
6341, 48, 62syl2anc 409 . . . . . . . . 9 (((𝜑 ∧ (𝐴 ∈ (𝐾...(𝐽𝐾)) ∧ ¬ 𝐵 ∈ (𝐾...(𝐽𝐾)))) ∧ ¬ 𝐴 = 𝐾) → (𝐾 < 𝐴𝐾 ≤ (𝐴 − 1)))
6461, 63mpbid 146 . . . . . . . 8 (((𝜑 ∧ (𝐴 ∈ (𝐾...(𝐽𝐾)) ∧ ¬ 𝐵 ∈ (𝐾...(𝐽𝐾)))) ∧ ¬ 𝐴 = 𝐾) → 𝐾 ≤ (𝐴 − 1))
6550zred 9307 . . . . . . . . 9 (((𝜑 ∧ (𝐴 ∈ (𝐾...(𝐽𝐾)) ∧ ¬ 𝐵 ∈ (𝐾...(𝐽𝐾)))) ∧ ¬ 𝐴 = 𝐾) → (𝐴 − 1) ∈ ℝ)
6648zred 9307 . . . . . . . . 9 (((𝜑 ∧ (𝐴 ∈ (𝐾...(𝐽𝐾)) ∧ ¬ 𝐵 ∈ (𝐾...(𝐽𝐾)))) ∧ ¬ 𝐴 = 𝐾) → 𝐴 ∈ ℝ)
6744zred 9307 . . . . . . . . 9 (((𝜑 ∧ (𝐴 ∈ (𝐾...(𝐽𝐾)) ∧ ¬ 𝐵 ∈ (𝐾...(𝐽𝐾)))) ∧ ¬ 𝐴 = 𝐾) → (𝐽𝐾) ∈ ℝ)
6866lem1d 8822 . . . . . . . . 9 (((𝜑 ∧ (𝐴 ∈ (𝐾...(𝐽𝐾)) ∧ ¬ 𝐵 ∈ (𝐾...(𝐽𝐾)))) ∧ ¬ 𝐴 = 𝐾) → (𝐴 − 1) ≤ 𝐴)
69 elfzle2 9957 . . . . . . . . . 10 (𝐴 ∈ (𝐾...(𝐽𝐾)) → 𝐴 ≤ (𝐽𝐾))
7055, 69syl 14 . . . . . . . . 9 (((𝜑 ∧ (𝐴 ∈ (𝐾...(𝐽𝐾)) ∧ ¬ 𝐵 ∈ (𝐾...(𝐽𝐾)))) ∧ ¬ 𝐴 = 𝐾) → 𝐴 ≤ (𝐽𝐾))
7165, 66, 67, 68, 70letrd 8016 . . . . . . . 8 (((𝜑 ∧ (𝐴 ∈ (𝐾...(𝐽𝐾)) ∧ ¬ 𝐵 ∈ (𝐾...(𝐽𝐾)))) ∧ ¬ 𝐴 = 𝐾) → (𝐴 − 1) ≤ (𝐽𝐾))
7264, 71jca 304 . . . . . . 7 (((𝜑 ∧ (𝐴 ∈ (𝐾...(𝐽𝐾)) ∧ ¬ 𝐵 ∈ (𝐾...(𝐽𝐾)))) ∧ ¬ 𝐴 = 𝐾) → (𝐾 ≤ (𝐴 − 1) ∧ (𝐴 − 1) ≤ (𝐽𝐾)))
73 elfz2 9945 . . . . . . 7 ((𝐴 − 1) ∈ (𝐾...(𝐽𝐾)) ↔ ((𝐾 ∈ ℤ ∧ (𝐽𝐾) ∈ ℤ ∧ (𝐴 − 1) ∈ ℤ) ∧ (𝐾 ≤ (𝐴 − 1) ∧ (𝐴 − 1) ≤ (𝐽𝐾))))
7451, 72, 73sylanbrc 414 . . . . . 6 (((𝜑 ∧ (𝐴 ∈ (𝐾...(𝐽𝐾)) ∧ ¬ 𝐵 ∈ (𝐾...(𝐽𝐾)))) ∧ ¬ 𝐴 = 𝐾) → (𝐴 − 1) ∈ (𝐾...(𝐽𝐾)))
75 fnfvima 5716 . . . . . 6 ((𝐽 Fn (𝑀...𝑁) ∧ (𝐾...(𝐽𝐾)) ⊆ (𝑀...𝑁) ∧ (𝐴 − 1) ∈ (𝐾...(𝐽𝐾))) → (𝐽‘(𝐴 − 1)) ∈ (𝐽 “ (𝐾...(𝐽𝐾))))
7636, 37, 74, 75syl3anc 1227 . . . . 5 (((𝜑 ∧ (𝐴 ∈ (𝐾...(𝐽𝐾)) ∧ ¬ 𝐵 ∈ (𝐾...(𝐽𝐾)))) ∧ ¬ 𝐴 = 𝐾) → (𝐽‘(𝐴 − 1)) ∈ (𝐽 “ (𝐾...(𝐽𝐾))))
77 zdceq 9260 . . . . . 6 ((𝐴 ∈ ℤ ∧ 𝐾 ∈ ℤ) → DECID 𝐴 = 𝐾)
7847, 40, 77syl2anc 409 . . . . 5 ((𝜑 ∧ (𝐴 ∈ (𝐾...(𝐽𝐾)) ∧ ¬ 𝐵 ∈ (𝐾...(𝐽𝐾)))) → DECID 𝐴 = 𝐾)
7935, 76, 78ifcldadc 3547 . . . 4 ((𝜑 ∧ (𝐴 ∈ (𝐾...(𝐽𝐾)) ∧ ¬ 𝐵 ∈ (𝐾...(𝐽𝐾)))) → if(𝐴 = 𝐾, 𝐾, (𝐽‘(𝐴 − 1))) ∈ (𝐽 “ (𝐾...(𝐽𝐾))))
8011, 79eqeltrd 2241 . . 3 ((𝜑 ∧ (𝐴 ∈ (𝐾...(𝐽𝐾)) ∧ ¬ 𝐵 ∈ (𝐾...(𝐽𝐾)))) → (𝑄𝐴) ∈ (𝐽 “ (𝐾...(𝐽𝐾))))
812, 80eqeltrrd 2242 . 2 ((𝜑 ∧ (𝐴 ∈ (𝐾...(𝐽𝐾)) ∧ ¬ 𝐵 ∈ (𝐾...(𝐽𝐾)))) → (𝑄𝐵) ∈ (𝐽 “ (𝐾...(𝐽𝐾))))
82 iseqf1olemnab.b . . . . . 6 (𝜑𝐵 ∈ (𝑀...𝑁))
833, 4, 82, 6iseqf1olemqval 10416 . . . . 5 (𝜑 → (𝑄𝐵) = if(𝐵 ∈ (𝐾...(𝐽𝐾)), if(𝐵 = 𝐾, 𝐾, (𝐽‘(𝐵 − 1))), (𝐽𝐵)))
8483adantr 274 . . . 4 ((𝜑 ∧ (𝐴 ∈ (𝐾...(𝐽𝐾)) ∧ ¬ 𝐵 ∈ (𝐾...(𝐽𝐾)))) → (𝑄𝐵) = if(𝐵 ∈ (𝐾...(𝐽𝐾)), if(𝐵 = 𝐾, 𝐾, (𝐽‘(𝐵 − 1))), (𝐽𝐵)))
85 simprr 522 . . . . 5 ((𝜑 ∧ (𝐴 ∈ (𝐾...(𝐽𝐾)) ∧ ¬ 𝐵 ∈ (𝐾...(𝐽𝐾)))) → ¬ 𝐵 ∈ (𝐾...(𝐽𝐾)))
8685iffalsed 3528 . . . 4 ((𝜑 ∧ (𝐴 ∈ (𝐾...(𝐽𝐾)) ∧ ¬ 𝐵 ∈ (𝐾...(𝐽𝐾)))) → if(𝐵 ∈ (𝐾...(𝐽𝐾)), if(𝐵 = 𝐾, 𝐾, (𝐽‘(𝐵 − 1))), (𝐽𝐵)) = (𝐽𝐵))
8784, 86eqtrd 2197 . . 3 ((𝜑 ∧ (𝐴 ∈ (𝐾...(𝐽𝐾)) ∧ ¬ 𝐵 ∈ (𝐾...(𝐽𝐾)))) → (𝑄𝐵) = (𝐽𝐵))
88 f1of1 5428 . . . . . . 7 (𝐽:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) → 𝐽:(𝑀...𝑁)–1-1→(𝑀...𝑁))
894, 88syl 14 . . . . . 6 (𝜑𝐽:(𝑀...𝑁)–1-1→(𝑀...𝑁))
90 f1elima 5738 . . . . . 6 ((𝐽:(𝑀...𝑁)–1-1→(𝑀...𝑁) ∧ 𝐵 ∈ (𝑀...𝑁) ∧ (𝐾...(𝐽𝐾)) ⊆ (𝑀...𝑁)) → ((𝐽𝐵) ∈ (𝐽 “ (𝐾...(𝐽𝐾))) ↔ 𝐵 ∈ (𝐾...(𝐽𝐾))))
9189, 82, 28, 90syl3anc 1227 . . . . 5 (𝜑 → ((𝐽𝐵) ∈ (𝐽 “ (𝐾...(𝐽𝐾))) ↔ 𝐵 ∈ (𝐾...(𝐽𝐾))))
9291adantr 274 . . . 4 ((𝜑 ∧ (𝐴 ∈ (𝐾...(𝐽𝐾)) ∧ ¬ 𝐵 ∈ (𝐾...(𝐽𝐾)))) → ((𝐽𝐵) ∈ (𝐽 “ (𝐾...(𝐽𝐾))) ↔ 𝐵 ∈ (𝐾...(𝐽𝐾))))
9385, 92mtbird 663 . . 3 ((𝜑 ∧ (𝐴 ∈ (𝐾...(𝐽𝐾)) ∧ ¬ 𝐵 ∈ (𝐾...(𝐽𝐾)))) → ¬ (𝐽𝐵) ∈ (𝐽 “ (𝐾...(𝐽𝐾))))
9487, 93eqneltrd 2260 . 2 ((𝜑 ∧ (𝐴 ∈ (𝐾...(𝐽𝐾)) ∧ ¬ 𝐵 ∈ (𝐾...(𝐽𝐾)))) → ¬ (𝑄𝐵) ∈ (𝐽 “ (𝐾...(𝐽𝐾))))
9581, 94pm2.65da 651 1 (𝜑 → ¬ (𝐴 ∈ (𝐾...(𝐽𝐾)) ∧ ¬ 𝐵 ∈ (𝐾...(𝐽𝐾))))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 103  wb 104  wo 698  DECID wdc 824  w3a 967   = wceq 1342  wcel 2135  wss 3114  ifcif 3518   class class class wbr 3979  cmpt 4040  ccnv 4600  cima 4604   Fn wfn 5180  wf 5181  1-1wf1 5182  1-1-ontowf1o 5184  cfv 5185  (class class class)co 5839  1c1 7748   < clt 7927  cle 7928  cmin 8063  cz 9185  cuz 9460  ...cfz 9938
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-10 1492  ax-11 1493  ax-i12 1494  ax-bndl 1496  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-13 2137  ax-14 2138  ax-ext 2146  ax-sep 4097  ax-pow 4150  ax-pr 4184  ax-un 4408  ax-setind 4511  ax-cnex 7838  ax-resscn 7839  ax-1cn 7840  ax-1re 7841  ax-icn 7842  ax-addcl 7843  ax-addrcl 7844  ax-mulcl 7845  ax-addcom 7847  ax-addass 7849  ax-distr 7851  ax-i2m1 7852  ax-0lt1 7853  ax-0id 7855  ax-rnegex 7856  ax-cnre 7858  ax-pre-ltirr 7859  ax-pre-ltwlin 7860  ax-pre-lttrn 7861  ax-pre-ltadd 7863
This theorem depends on definitions:  df-bi 116  df-dc 825  df-3or 968  df-3an 969  df-tru 1345  df-fal 1348  df-nf 1448  df-sb 1750  df-eu 2016  df-mo 2017  df-clab 2151  df-cleq 2157  df-clel 2160  df-nfc 2295  df-ne 2335  df-nel 2430  df-ral 2447  df-rex 2448  df-reu 2449  df-rab 2451  df-v 2726  df-sbc 2950  df-dif 3116  df-un 3118  df-in 3120  df-ss 3127  df-if 3519  df-pw 3558  df-sn 3579  df-pr 3580  df-op 3582  df-uni 3787  df-int 3822  df-br 3980  df-opab 4041  df-mpt 4042  df-id 4268  df-xp 4607  df-rel 4608  df-cnv 4609  df-co 4610  df-dm 4611  df-rn 4612  df-res 4613  df-ima 4614  df-iota 5150  df-fun 5187  df-fn 5188  df-f 5189  df-f1 5190  df-fo 5191  df-f1o 5192  df-fv 5193  df-riota 5795  df-ov 5842  df-oprab 5843  df-mpo 5844  df-pnf 7929  df-mnf 7930  df-xr 7931  df-ltxr 7932  df-le 7933  df-sub 8065  df-neg 8066  df-inn 8852  df-n0 9109  df-z 9186  df-uz 9461  df-fz 9939
This theorem is referenced by:  iseqf1olemmo  10421
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