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Theorem iseqf1olemnab 10212
 Description: Lemma for seq3f1o 10228. (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 272 . . 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 10211 . . . . . 6 (𝜑 → (𝑄𝐴) = if(𝐴 ∈ (𝐾...(𝐽𝐾)), if(𝐴 = 𝐾, 𝐾, (𝐽‘(𝐴 − 1))), (𝐽𝐴)))
87adantr 272 . . . . 5 ((𝜑 ∧ (𝐴 ∈ (𝐾...(𝐽𝐾)) ∧ ¬ 𝐵 ∈ (𝐾...(𝐽𝐾)))) → (𝑄𝐴) = if(𝐴 ∈ (𝐾...(𝐽𝐾)), if(𝐴 = 𝐾, 𝐾, (𝐽‘(𝐴 − 1))), (𝐽𝐴)))
9 simprl 503 . . . . . 6 ((𝜑 ∧ (𝐴 ∈ (𝐾...(𝐽𝐾)) ∧ ¬ 𝐵 ∈ (𝐾...(𝐽𝐾)))) → 𝐴 ∈ (𝐾...(𝐽𝐾)))
109iftrued 3449 . . . . 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 2148 . . . 4 ((𝜑 ∧ (𝐴 ∈ (𝐾...(𝐽𝐾)) ∧ ¬ 𝐵 ∈ (𝐾...(𝐽𝐾)))) → (𝑄𝐴) = if(𝐴 = 𝐾, 𝐾, (𝐽‘(𝐴 − 1))))
12 f1ocnvfv2 5645 . . . . . . . 8 ((𝐽:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ 𝐾 ∈ (𝑀...𝑁)) → (𝐽‘(𝐽𝐾)) = 𝐾)
134, 3, 12syl2anc 406 . . . . . . 7 (𝜑 → (𝐽‘(𝐽𝐾)) = 𝐾)
1413ad2antrr 477 . . . . . 6 (((𝜑 ∧ (𝐴 ∈ (𝐾...(𝐽𝐾)) ∧ ¬ 𝐵 ∈ (𝐾...(𝐽𝐾)))) ∧ 𝐴 = 𝐾) → (𝐽‘(𝐽𝐾)) = 𝐾)
15 f1ofn 5334 . . . . . . . . 9 (𝐽:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) → 𝐽 Fn (𝑀...𝑁))
164, 15syl 14 . . . . . . . 8 (𝜑𝐽 Fn (𝑀...𝑁))
1716ad2antrr 477 . . . . . . 7 (((𝜑 ∧ (𝐴 ∈ (𝐾...(𝐽𝐾)) ∧ ¬ 𝐵 ∈ (𝐾...(𝐽𝐾)))) ∧ 𝐴 = 𝐾) → 𝐽 Fn (𝑀...𝑁))
18 elfzuz 9753 . . . . . . . . . 10 (𝐾 ∈ (𝑀...𝑁) → 𝐾 ∈ (ℤ𝑀))
19 fzss1 9794 . . . . . . . . . 10 (𝐾 ∈ (ℤ𝑀) → (𝐾...(𝐽𝐾)) ⊆ (𝑀...(𝐽𝐾)))
203, 18, 193syl 17 . . . . . . . . 9 (𝜑 → (𝐾...(𝐽𝐾)) ⊆ (𝑀...(𝐽𝐾)))
21 f1ocnv 5346 . . . . . . . . . . . 12 (𝐽:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) → 𝐽:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁))
22 f1of 5333 . . . . . . . . . . . 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 5522 . . . . . . . . . 10 (𝜑 → (𝐽𝐾) ∈ (𝑀...𝑁))
25 elfzuz3 9754 . . . . . . . . . 10 ((𝐽𝐾) ∈ (𝑀...𝑁) → 𝑁 ∈ (ℤ‘(𝐽𝐾)))
26 fzss2 9795 . . . . . . . . . 10 (𝑁 ∈ (ℤ‘(𝐽𝐾)) → (𝑀...(𝐽𝐾)) ⊆ (𝑀...𝑁))
2724, 25, 263syl 17 . . . . . . . . 9 (𝜑 → (𝑀...(𝐽𝐾)) ⊆ (𝑀...𝑁))
2820, 27sstrd 3075 . . . . . . . 8 (𝜑 → (𝐾...(𝐽𝐾)) ⊆ (𝑀...𝑁))
2928ad2antrr 477 . . . . . . 7 (((𝜑 ∧ (𝐴 ∈ (𝐾...(𝐽𝐾)) ∧ ¬ 𝐵 ∈ (𝐾...(𝐽𝐾)))) ∧ 𝐴 = 𝐾) → (𝐾...(𝐽𝐾)) ⊆ (𝑀...𝑁))
30 elfzubelfz 9767 . . . . . . . . 9 (𝐴 ∈ (𝐾...(𝐽𝐾)) → (𝐽𝐾) ∈ (𝐾...(𝐽𝐾)))
3130adantr 272 . . . . . . . 8 ((𝐴 ∈ (𝐾...(𝐽𝐾)) ∧ ¬ 𝐵 ∈ (𝐾...(𝐽𝐾))) → (𝐽𝐾) ∈ (𝐾...(𝐽𝐾)))
3231ad2antlr 478 . . . . . . 7 (((𝜑 ∧ (𝐴 ∈ (𝐾...(𝐽𝐾)) ∧ ¬ 𝐵 ∈ (𝐾...(𝐽𝐾)))) ∧ 𝐴 = 𝐾) → (𝐽𝐾) ∈ (𝐾...(𝐽𝐾)))
33 fnfvima 5618 . . . . . . 7 ((𝐽 Fn (𝑀...𝑁) ∧ (𝐾...(𝐽𝐾)) ⊆ (𝑀...𝑁) ∧ (𝐽𝐾) ∈ (𝐾...(𝐽𝐾))) → (𝐽‘(𝐽𝐾)) ∈ (𝐽 “ (𝐾...(𝐽𝐾))))
3417, 29, 32, 33syl3anc 1199 . . . . . 6 (((𝜑 ∧ (𝐴 ∈ (𝐾...(𝐽𝐾)) ∧ ¬ 𝐵 ∈ (𝐾...(𝐽𝐾)))) ∧ 𝐴 = 𝐾) → (𝐽‘(𝐽𝐾)) ∈ (𝐽 “ (𝐾...(𝐽𝐾))))
3514, 34eqeltrrd 2193 . . . . 5 (((𝜑 ∧ (𝐴 ∈ (𝐾...(𝐽𝐾)) ∧ ¬ 𝐵 ∈ (𝐾...(𝐽𝐾)))) ∧ 𝐴 = 𝐾) → 𝐾 ∈ (𝐽 “ (𝐾...(𝐽𝐾))))
3616ad2antrr 477 . . . . . 6 (((𝜑 ∧ (𝐴 ∈ (𝐾...(𝐽𝐾)) ∧ ¬ 𝐵 ∈ (𝐾...(𝐽𝐾)))) ∧ ¬ 𝐴 = 𝐾) → 𝐽 Fn (𝑀...𝑁))
3728ad2antrr 477 . . . . . 6 (((𝜑 ∧ (𝐴 ∈ (𝐾...(𝐽𝐾)) ∧ ¬ 𝐵 ∈ (𝐾...(𝐽𝐾)))) ∧ ¬ 𝐴 = 𝐾) → (𝐾...(𝐽𝐾)) ⊆ (𝑀...𝑁))
383adantr 272 . . . . . . . . . 10 ((𝜑 ∧ (𝐴 ∈ (𝐾...(𝐽𝐾)) ∧ ¬ 𝐵 ∈ (𝐾...(𝐽𝐾)))) → 𝐾 ∈ (𝑀...𝑁))
39 elfzelz 9757 . . . . . . . . . 10 (𝐾 ∈ (𝑀...𝑁) → 𝐾 ∈ ℤ)
4038, 39syl 14 . . . . . . . . 9 ((𝜑 ∧ (𝐴 ∈ (𝐾...(𝐽𝐾)) ∧ ¬ 𝐵 ∈ (𝐾...(𝐽𝐾)))) → 𝐾 ∈ ℤ)
4140adantr 272 . . . . . . . 8 (((𝜑 ∧ (𝐴 ∈ (𝐾...(𝐽𝐾)) ∧ ¬ 𝐵 ∈ (𝐾...(𝐽𝐾)))) ∧ ¬ 𝐴 = 𝐾) → 𝐾 ∈ ℤ)
4224ad2antrr 477 . . . . . . . . 9 (((𝜑 ∧ (𝐴 ∈ (𝐾...(𝐽𝐾)) ∧ ¬ 𝐵 ∈ (𝐾...(𝐽𝐾)))) ∧ ¬ 𝐴 = 𝐾) → (𝐽𝐾) ∈ (𝑀...𝑁))
43 elfzelz 9757 . . . . . . . . 9 ((𝐽𝐾) ∈ (𝑀...𝑁) → (𝐽𝐾) ∈ ℤ)
4442, 43syl 14 . . . . . . . 8 (((𝜑 ∧ (𝐴 ∈ (𝐾...(𝐽𝐾)) ∧ ¬ 𝐵 ∈ (𝐾...(𝐽𝐾)))) ∧ ¬ 𝐴 = 𝐾) → (𝐽𝐾) ∈ ℤ)
455adantr 272 . . . . . . . . . . 11 ((𝜑 ∧ (𝐴 ∈ (𝐾...(𝐽𝐾)) ∧ ¬ 𝐵 ∈ (𝐾...(𝐽𝐾)))) → 𝐴 ∈ (𝑀...𝑁))
46 elfzelz 9757 . . . . . . . . . . 11 (𝐴 ∈ (𝑀...𝑁) → 𝐴 ∈ ℤ)
4745, 46syl 14 . . . . . . . . . 10 ((𝜑 ∧ (𝐴 ∈ (𝐾...(𝐽𝐾)) ∧ ¬ 𝐵 ∈ (𝐾...(𝐽𝐾)))) → 𝐴 ∈ ℤ)
4847adantr 272 . . . . . . . . 9 (((𝜑 ∧ (𝐴 ∈ (𝐾...(𝐽𝐾)) ∧ ¬ 𝐵 ∈ (𝐾...(𝐽𝐾)))) ∧ ¬ 𝐴 = 𝐾) → 𝐴 ∈ ℤ)
49 peano2zm 9046 . . . . . . . . 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 1144 . . . . . . 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 2117 . . . . . . . . . . 11 (𝐴 = 𝐾𝐾 = 𝐴)
5452, 53sylnib 648 . . . . . . . . . 10 (((𝜑 ∧ (𝐴 ∈ (𝐾...(𝐽𝐾)) ∧ ¬ 𝐵 ∈ (𝐾...(𝐽𝐾)))) ∧ ¬ 𝐴 = 𝐾) → ¬ 𝐾 = 𝐴)
559adantr 272 . . . . . . . . . . . 12 (((𝜑 ∧ (𝐴 ∈ (𝐾...(𝐽𝐾)) ∧ ¬ 𝐵 ∈ (𝐾...(𝐽𝐾)))) ∧ ¬ 𝐴 = 𝐾) → 𝐴 ∈ (𝐾...(𝐽𝐾)))
56 elfzle1 9758 . . . . . . . . . . . 12 (𝐴 ∈ (𝐾...(𝐽𝐾)) → 𝐾𝐴)
5755, 56syl 14 . . . . . . . . . . 11 (((𝜑 ∧ (𝐴 ∈ (𝐾...(𝐽𝐾)) ∧ ¬ 𝐵 ∈ (𝐾...(𝐽𝐾)))) ∧ ¬ 𝐴 = 𝐾) → 𝐾𝐴)
58 zleloe 9055 . . . . . . . . . . . 12 ((𝐾 ∈ ℤ ∧ 𝐴 ∈ ℤ) → (𝐾𝐴 ↔ (𝐾 < 𝐴𝐾 = 𝐴)))
5941, 48, 58syl2anc 406 . . . . . . . . . . 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 1310 . . . . . . . . 9 (((𝜑 ∧ (𝐴 ∈ (𝐾...(𝐽𝐾)) ∧ ¬ 𝐵 ∈ (𝐾...(𝐽𝐾)))) ∧ ¬ 𝐴 = 𝐾) → 𝐾 < 𝐴)
62 zltlem1 9065 . . . . . . . . . 10 ((𝐾 ∈ ℤ ∧ 𝐴 ∈ ℤ) → (𝐾 < 𝐴𝐾 ≤ (𝐴 − 1)))
6341, 48, 62syl2anc 406 . . . . . . . . 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 9127 . . . . . . . . 9 (((𝜑 ∧ (𝐴 ∈ (𝐾...(𝐽𝐾)) ∧ ¬ 𝐵 ∈ (𝐾...(𝐽𝐾)))) ∧ ¬ 𝐴 = 𝐾) → (𝐴 − 1) ∈ ℝ)
6648zred 9127 . . . . . . . . 9 (((𝜑 ∧ (𝐴 ∈ (𝐾...(𝐽𝐾)) ∧ ¬ 𝐵 ∈ (𝐾...(𝐽𝐾)))) ∧ ¬ 𝐴 = 𝐾) → 𝐴 ∈ ℝ)
6744zred 9127 . . . . . . . . 9 (((𝜑 ∧ (𝐴 ∈ (𝐾...(𝐽𝐾)) ∧ ¬ 𝐵 ∈ (𝐾...(𝐽𝐾)))) ∧ ¬ 𝐴 = 𝐾) → (𝐽𝐾) ∈ ℝ)
6866lem1d 8651 . . . . . . . . 9 (((𝜑 ∧ (𝐴 ∈ (𝐾...(𝐽𝐾)) ∧ ¬ 𝐵 ∈ (𝐾...(𝐽𝐾)))) ∧ ¬ 𝐴 = 𝐾) → (𝐴 − 1) ≤ 𝐴)
69 elfzle2 9759 . . . . . . . . . 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 7850 . . . . . . . 8 (((𝜑 ∧ (𝐴 ∈ (𝐾...(𝐽𝐾)) ∧ ¬ 𝐵 ∈ (𝐾...(𝐽𝐾)))) ∧ ¬ 𝐴 = 𝐾) → (𝐴 − 1) ≤ (𝐽𝐾))
7264, 71jca 302 . . . . . . 7 (((𝜑 ∧ (𝐴 ∈ (𝐾...(𝐽𝐾)) ∧ ¬ 𝐵 ∈ (𝐾...(𝐽𝐾)))) ∧ ¬ 𝐴 = 𝐾) → (𝐾 ≤ (𝐴 − 1) ∧ (𝐴 − 1) ≤ (𝐽𝐾)))
73 elfz2 9748 . . . . . . 7 ((𝐴 − 1) ∈ (𝐾...(𝐽𝐾)) ↔ ((𝐾 ∈ ℤ ∧ (𝐽𝐾) ∈ ℤ ∧ (𝐴 − 1) ∈ ℤ) ∧ (𝐾 ≤ (𝐴 − 1) ∧ (𝐴 − 1) ≤ (𝐽𝐾))))
7451, 72, 73sylanbrc 411 . . . . . 6 (((𝜑 ∧ (𝐴 ∈ (𝐾...(𝐽𝐾)) ∧ ¬ 𝐵 ∈ (𝐾...(𝐽𝐾)))) ∧ ¬ 𝐴 = 𝐾) → (𝐴 − 1) ∈ (𝐾...(𝐽𝐾)))
75 fnfvima 5618 . . . . . 6 ((𝐽 Fn (𝑀...𝑁) ∧ (𝐾...(𝐽𝐾)) ⊆ (𝑀...𝑁) ∧ (𝐴 − 1) ∈ (𝐾...(𝐽𝐾))) → (𝐽‘(𝐴 − 1)) ∈ (𝐽 “ (𝐾...(𝐽𝐾))))
7636, 37, 74, 75syl3anc 1199 . . . . 5 (((𝜑 ∧ (𝐴 ∈ (𝐾...(𝐽𝐾)) ∧ ¬ 𝐵 ∈ (𝐾...(𝐽𝐾)))) ∧ ¬ 𝐴 = 𝐾) → (𝐽‘(𝐴 − 1)) ∈ (𝐽 “ (𝐾...(𝐽𝐾))))
77 zdceq 9080 . . . . . 6 ((𝐴 ∈ ℤ ∧ 𝐾 ∈ ℤ) → DECID 𝐴 = 𝐾)
7847, 40, 77syl2anc 406 . . . . 5 ((𝜑 ∧ (𝐴 ∈ (𝐾...(𝐽𝐾)) ∧ ¬ 𝐵 ∈ (𝐾...(𝐽𝐾)))) → DECID 𝐴 = 𝐾)
7935, 76, 78ifcldadc 3469 . . . 4 ((𝜑 ∧ (𝐴 ∈ (𝐾...(𝐽𝐾)) ∧ ¬ 𝐵 ∈ (𝐾...(𝐽𝐾)))) → if(𝐴 = 𝐾, 𝐾, (𝐽‘(𝐴 − 1))) ∈ (𝐽 “ (𝐾...(𝐽𝐾))))
8011, 79eqeltrd 2192 . . 3 ((𝜑 ∧ (𝐴 ∈ (𝐾...(𝐽𝐾)) ∧ ¬ 𝐵 ∈ (𝐾...(𝐽𝐾)))) → (𝑄𝐴) ∈ (𝐽 “ (𝐾...(𝐽𝐾))))
812, 80eqeltrrd 2193 . 2 ((𝜑 ∧ (𝐴 ∈ (𝐾...(𝐽𝐾)) ∧ ¬ 𝐵 ∈ (𝐾...(𝐽𝐾)))) → (𝑄𝐵) ∈ (𝐽 “ (𝐾...(𝐽𝐾))))
82 iseqf1olemnab.b . . . . . 6 (𝜑𝐵 ∈ (𝑀...𝑁))
833, 4, 82, 6iseqf1olemqval 10211 . . . . 5 (𝜑 → (𝑄𝐵) = if(𝐵 ∈ (𝐾...(𝐽𝐾)), if(𝐵 = 𝐾, 𝐾, (𝐽‘(𝐵 − 1))), (𝐽𝐵)))
8483adantr 272 . . . 4 ((𝜑 ∧ (𝐴 ∈ (𝐾...(𝐽𝐾)) ∧ ¬ 𝐵 ∈ (𝐾...(𝐽𝐾)))) → (𝑄𝐵) = if(𝐵 ∈ (𝐾...(𝐽𝐾)), if(𝐵 = 𝐾, 𝐾, (𝐽‘(𝐵 − 1))), (𝐽𝐵)))
85 simprr 504 . . . . 5 ((𝜑 ∧ (𝐴 ∈ (𝐾...(𝐽𝐾)) ∧ ¬ 𝐵 ∈ (𝐾...(𝐽𝐾)))) → ¬ 𝐵 ∈ (𝐾...(𝐽𝐾)))
8685iffalsed 3452 . . . 4 ((𝜑 ∧ (𝐴 ∈ (𝐾...(𝐽𝐾)) ∧ ¬ 𝐵 ∈ (𝐾...(𝐽𝐾)))) → if(𝐵 ∈ (𝐾...(𝐽𝐾)), if(𝐵 = 𝐾, 𝐾, (𝐽‘(𝐵 − 1))), (𝐽𝐵)) = (𝐽𝐵))
8784, 86eqtrd 2148 . . 3 ((𝜑 ∧ (𝐴 ∈ (𝐾...(𝐽𝐾)) ∧ ¬ 𝐵 ∈ (𝐾...(𝐽𝐾)))) → (𝑄𝐵) = (𝐽𝐵))
88 f1of1 5332 . . . . . . 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 5640 . . . . . 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 1199 . . . . 5 (𝜑 → ((𝐽𝐵) ∈ (𝐽 “ (𝐾...(𝐽𝐾))) ↔ 𝐵 ∈ (𝐾...(𝐽𝐾))))
9291adantr 272 . . . 4 ((𝜑 ∧ (𝐴 ∈ (𝐾...(𝐽𝐾)) ∧ ¬ 𝐵 ∈ (𝐾...(𝐽𝐾)))) → ((𝐽𝐵) ∈ (𝐽 “ (𝐾...(𝐽𝐾))) ↔ 𝐵 ∈ (𝐾...(𝐽𝐾))))
9385, 92mtbird 645 . . 3 ((𝜑 ∧ (𝐴 ∈ (𝐾...(𝐽𝐾)) ∧ ¬ 𝐵 ∈ (𝐾...(𝐽𝐾)))) → ¬ (𝐽𝐵) ∈ (𝐽 “ (𝐾...(𝐽𝐾))))
9487, 93eqneltrd 2211 . 2 ((𝜑 ∧ (𝐴 ∈ (𝐾...(𝐽𝐾)) ∧ ¬ 𝐵 ∈ (𝐾...(𝐽𝐾)))) → ¬ (𝑄𝐵) ∈ (𝐽 “ (𝐾...(𝐽𝐾))))
9581, 94pm2.65da 633 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 680  DECID wdc 802   ∧ w3a 945   = wceq 1314   ∈ wcel 1463   ⊆ wss 3039  ifcif 3442   class class class wbr 3897   ↦ cmpt 3957  ◡ccnv 4506   “ cima 4510   Fn wfn 5086  ⟶wf 5087  –1-1→wf1 5088  –1-1-onto→wf1o 5090  ‘cfv 5091  (class class class)co 5740  1c1 7585   < clt 7764   ≤ cle 7765   − cmin 7897  ℤcz 9008  ℤ≥cuz 9278  ...cfz 9741 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 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4014  ax-pow 4066  ax-pr 4099  ax-un 4323  ax-setind 4420  ax-cnex 7675  ax-resscn 7676  ax-1cn 7677  ax-1re 7678  ax-icn 7679  ax-addcl 7680  ax-addrcl 7681  ax-mulcl 7682  ax-addcom 7684  ax-addass 7686  ax-distr 7688  ax-i2m1 7689  ax-0lt1 7690  ax-0id 7692  ax-rnegex 7693  ax-cnre 7695  ax-pre-ltirr 7696  ax-pre-ltwlin 7697  ax-pre-lttrn 7698  ax-pre-ltadd 7700 This theorem depends on definitions:  df-bi 116  df-dc 803  df-3or 946  df-3an 947  df-tru 1317  df-fal 1320  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ne 2284  df-nel 2379  df-ral 2396  df-rex 2397  df-reu 2398  df-rab 2400  df-v 2660  df-sbc 2881  df-dif 3041  df-un 3043  df-in 3045  df-ss 3052  df-if 3443  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-uni 3705  df-int 3740  df-br 3898  df-opab 3958  df-mpt 3959  df-id 4183  df-xp 4513  df-rel 4514  df-cnv 4515  df-co 4516  df-dm 4517  df-rn 4518  df-res 4519  df-ima 4520  df-iota 5056  df-fun 5093  df-fn 5094  df-f 5095  df-f1 5096  df-fo 5097  df-f1o 5098  df-fv 5099  df-riota 5696  df-ov 5743  df-oprab 5744  df-mpo 5745  df-pnf 7766  df-mnf 7767  df-xr 7768  df-ltxr 7769  df-le 7770  df-sub 7899  df-neg 7900  df-inn 8681  df-n0 8932  df-z 9009  df-uz 9279  df-fz 9742 This theorem is referenced by:  iseqf1olemmo  10216
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