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Theorem iseqf1olemnab 10474
Description: Lemma for seq3f1o 10490. (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 276 . . 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 10473 . . . . . 6 (𝜑 → (𝑄𝐴) = if(𝐴 ∈ (𝐾...(𝐽𝐾)), if(𝐴 = 𝐾, 𝐾, (𝐽‘(𝐴 − 1))), (𝐽𝐴)))
87adantr 276 . . . . 5 ((𝜑 ∧ (𝐴 ∈ (𝐾...(𝐽𝐾)) ∧ ¬ 𝐵 ∈ (𝐾...(𝐽𝐾)))) → (𝑄𝐴) = if(𝐴 ∈ (𝐾...(𝐽𝐾)), if(𝐴 = 𝐾, 𝐾, (𝐽‘(𝐴 − 1))), (𝐽𝐴)))
9 simprl 529 . . . . . 6 ((𝜑 ∧ (𝐴 ∈ (𝐾...(𝐽𝐾)) ∧ ¬ 𝐵 ∈ (𝐾...(𝐽𝐾)))) → 𝐴 ∈ (𝐾...(𝐽𝐾)))
109iftrued 3541 . . . . 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 2210 . . . 4 ((𝜑 ∧ (𝐴 ∈ (𝐾...(𝐽𝐾)) ∧ ¬ 𝐵 ∈ (𝐾...(𝐽𝐾)))) → (𝑄𝐴) = if(𝐴 = 𝐾, 𝐾, (𝐽‘(𝐴 − 1))))
12 f1ocnvfv2 5773 . . . . . . . 8 ((𝐽:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ 𝐾 ∈ (𝑀...𝑁)) → (𝐽‘(𝐽𝐾)) = 𝐾)
134, 3, 12syl2anc 411 . . . . . . 7 (𝜑 → (𝐽‘(𝐽𝐾)) = 𝐾)
1413ad2antrr 488 . . . . . 6 (((𝜑 ∧ (𝐴 ∈ (𝐾...(𝐽𝐾)) ∧ ¬ 𝐵 ∈ (𝐾...(𝐽𝐾)))) ∧ 𝐴 = 𝐾) → (𝐽‘(𝐽𝐾)) = 𝐾)
15 f1ofn 5458 . . . . . . . . 9 (𝐽:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) → 𝐽 Fn (𝑀...𝑁))
164, 15syl 14 . . . . . . . 8 (𝜑𝐽 Fn (𝑀...𝑁))
1716ad2antrr 488 . . . . . . 7 (((𝜑 ∧ (𝐴 ∈ (𝐾...(𝐽𝐾)) ∧ ¬ 𝐵 ∈ (𝐾...(𝐽𝐾)))) ∧ 𝐴 = 𝐾) → 𝐽 Fn (𝑀...𝑁))
18 elfzuz 10007 . . . . . . . . . 10 (𝐾 ∈ (𝑀...𝑁) → 𝐾 ∈ (ℤ𝑀))
19 fzss1 10049 . . . . . . . . . 10 (𝐾 ∈ (ℤ𝑀) → (𝐾...(𝐽𝐾)) ⊆ (𝑀...(𝐽𝐾)))
203, 18, 193syl 17 . . . . . . . . 9 (𝜑 → (𝐾...(𝐽𝐾)) ⊆ (𝑀...(𝐽𝐾)))
21 f1ocnv 5470 . . . . . . . . . . . 12 (𝐽:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) → 𝐽:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁))
22 f1of 5457 . . . . . . . . . . . 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, 3ffvelcdmd 5648 . . . . . . . . . 10 (𝜑 → (𝐽𝐾) ∈ (𝑀...𝑁))
25 elfzuz3 10008 . . . . . . . . . 10 ((𝐽𝐾) ∈ (𝑀...𝑁) → 𝑁 ∈ (ℤ‘(𝐽𝐾)))
26 fzss2 10050 . . . . . . . . . 10 (𝑁 ∈ (ℤ‘(𝐽𝐾)) → (𝑀...(𝐽𝐾)) ⊆ (𝑀...𝑁))
2724, 25, 263syl 17 . . . . . . . . 9 (𝜑 → (𝑀...(𝐽𝐾)) ⊆ (𝑀...𝑁))
2820, 27sstrd 3165 . . . . . . . 8 (𝜑 → (𝐾...(𝐽𝐾)) ⊆ (𝑀...𝑁))
2928ad2antrr 488 . . . . . . 7 (((𝜑 ∧ (𝐴 ∈ (𝐾...(𝐽𝐾)) ∧ ¬ 𝐵 ∈ (𝐾...(𝐽𝐾)))) ∧ 𝐴 = 𝐾) → (𝐾...(𝐽𝐾)) ⊆ (𝑀...𝑁))
30 elfzubelfz 10022 . . . . . . . . 9 (𝐴 ∈ (𝐾...(𝐽𝐾)) → (𝐽𝐾) ∈ (𝐾...(𝐽𝐾)))
3130adantr 276 . . . . . . . 8 ((𝐴 ∈ (𝐾...(𝐽𝐾)) ∧ ¬ 𝐵 ∈ (𝐾...(𝐽𝐾))) → (𝐽𝐾) ∈ (𝐾...(𝐽𝐾)))
3231ad2antlr 489 . . . . . . 7 (((𝜑 ∧ (𝐴 ∈ (𝐾...(𝐽𝐾)) ∧ ¬ 𝐵 ∈ (𝐾...(𝐽𝐾)))) ∧ 𝐴 = 𝐾) → (𝐽𝐾) ∈ (𝐾...(𝐽𝐾)))
33 fnfvima 5746 . . . . . . 7 ((𝐽 Fn (𝑀...𝑁) ∧ (𝐾...(𝐽𝐾)) ⊆ (𝑀...𝑁) ∧ (𝐽𝐾) ∈ (𝐾...(𝐽𝐾))) → (𝐽‘(𝐽𝐾)) ∈ (𝐽 “ (𝐾...(𝐽𝐾))))
3417, 29, 32, 33syl3anc 1238 . . . . . 6 (((𝜑 ∧ (𝐴 ∈ (𝐾...(𝐽𝐾)) ∧ ¬ 𝐵 ∈ (𝐾...(𝐽𝐾)))) ∧ 𝐴 = 𝐾) → (𝐽‘(𝐽𝐾)) ∈ (𝐽 “ (𝐾...(𝐽𝐾))))
3514, 34eqeltrrd 2255 . . . . 5 (((𝜑 ∧ (𝐴 ∈ (𝐾...(𝐽𝐾)) ∧ ¬ 𝐵 ∈ (𝐾...(𝐽𝐾)))) ∧ 𝐴 = 𝐾) → 𝐾 ∈ (𝐽 “ (𝐾...(𝐽𝐾))))
3616ad2antrr 488 . . . . . 6 (((𝜑 ∧ (𝐴 ∈ (𝐾...(𝐽𝐾)) ∧ ¬ 𝐵 ∈ (𝐾...(𝐽𝐾)))) ∧ ¬ 𝐴 = 𝐾) → 𝐽 Fn (𝑀...𝑁))
3728ad2antrr 488 . . . . . 6 (((𝜑 ∧ (𝐴 ∈ (𝐾...(𝐽𝐾)) ∧ ¬ 𝐵 ∈ (𝐾...(𝐽𝐾)))) ∧ ¬ 𝐴 = 𝐾) → (𝐾...(𝐽𝐾)) ⊆ (𝑀...𝑁))
383adantr 276 . . . . . . . . . 10 ((𝜑 ∧ (𝐴 ∈ (𝐾...(𝐽𝐾)) ∧ ¬ 𝐵 ∈ (𝐾...(𝐽𝐾)))) → 𝐾 ∈ (𝑀...𝑁))
39 elfzelz 10011 . . . . . . . . . 10 (𝐾 ∈ (𝑀...𝑁) → 𝐾 ∈ ℤ)
4038, 39syl 14 . . . . . . . . 9 ((𝜑 ∧ (𝐴 ∈ (𝐾...(𝐽𝐾)) ∧ ¬ 𝐵 ∈ (𝐾...(𝐽𝐾)))) → 𝐾 ∈ ℤ)
4140adantr 276 . . . . . . . 8 (((𝜑 ∧ (𝐴 ∈ (𝐾...(𝐽𝐾)) ∧ ¬ 𝐵 ∈ (𝐾...(𝐽𝐾)))) ∧ ¬ 𝐴 = 𝐾) → 𝐾 ∈ ℤ)
4224ad2antrr 488 . . . . . . . . 9 (((𝜑 ∧ (𝐴 ∈ (𝐾...(𝐽𝐾)) ∧ ¬ 𝐵 ∈ (𝐾...(𝐽𝐾)))) ∧ ¬ 𝐴 = 𝐾) → (𝐽𝐾) ∈ (𝑀...𝑁))
43 elfzelz 10011 . . . . . . . . 9 ((𝐽𝐾) ∈ (𝑀...𝑁) → (𝐽𝐾) ∈ ℤ)
4442, 43syl 14 . . . . . . . 8 (((𝜑 ∧ (𝐴 ∈ (𝐾...(𝐽𝐾)) ∧ ¬ 𝐵 ∈ (𝐾...(𝐽𝐾)))) ∧ ¬ 𝐴 = 𝐾) → (𝐽𝐾) ∈ ℤ)
455adantr 276 . . . . . . . . . . 11 ((𝜑 ∧ (𝐴 ∈ (𝐾...(𝐽𝐾)) ∧ ¬ 𝐵 ∈ (𝐾...(𝐽𝐾)))) → 𝐴 ∈ (𝑀...𝑁))
46 elfzelz 10011 . . . . . . . . . . 11 (𝐴 ∈ (𝑀...𝑁) → 𝐴 ∈ ℤ)
4745, 46syl 14 . . . . . . . . . 10 ((𝜑 ∧ (𝐴 ∈ (𝐾...(𝐽𝐾)) ∧ ¬ 𝐵 ∈ (𝐾...(𝐽𝐾)))) → 𝐴 ∈ ℤ)
4847adantr 276 . . . . . . . . 9 (((𝜑 ∧ (𝐴 ∈ (𝐾...(𝐽𝐾)) ∧ ¬ 𝐵 ∈ (𝐾...(𝐽𝐾)))) ∧ ¬ 𝐴 = 𝐾) → 𝐴 ∈ ℤ)
49 peano2zm 9280 . . . . . . . . 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 1177 . . . . . . 7 (((𝜑 ∧ (𝐴 ∈ (𝐾...(𝐽𝐾)) ∧ ¬ 𝐵 ∈ (𝐾...(𝐽𝐾)))) ∧ ¬ 𝐴 = 𝐾) → (𝐾 ∈ ℤ ∧ (𝐽𝐾) ∈ ℤ ∧ (𝐴 − 1) ∈ ℤ))
52 simpr 110 . . . . . . . . . . 11 (((𝜑 ∧ (𝐴 ∈ (𝐾...(𝐽𝐾)) ∧ ¬ 𝐵 ∈ (𝐾...(𝐽𝐾)))) ∧ ¬ 𝐴 = 𝐾) → ¬ 𝐴 = 𝐾)
53 eqcom 2179 . . . . . . . . . . 11 (𝐴 = 𝐾𝐾 = 𝐴)
5452, 53sylnib 676 . . . . . . . . . 10 (((𝜑 ∧ (𝐴 ∈ (𝐾...(𝐽𝐾)) ∧ ¬ 𝐵 ∈ (𝐾...(𝐽𝐾)))) ∧ ¬ 𝐴 = 𝐾) → ¬ 𝐾 = 𝐴)
559adantr 276 . . . . . . . . . . . 12 (((𝜑 ∧ (𝐴 ∈ (𝐾...(𝐽𝐾)) ∧ ¬ 𝐵 ∈ (𝐾...(𝐽𝐾)))) ∧ ¬ 𝐴 = 𝐾) → 𝐴 ∈ (𝐾...(𝐽𝐾)))
56 elfzle1 10013 . . . . . . . . . . . 12 (𝐴 ∈ (𝐾...(𝐽𝐾)) → 𝐾𝐴)
5755, 56syl 14 . . . . . . . . . . 11 (((𝜑 ∧ (𝐴 ∈ (𝐾...(𝐽𝐾)) ∧ ¬ 𝐵 ∈ (𝐾...(𝐽𝐾)))) ∧ ¬ 𝐴 = 𝐾) → 𝐾𝐴)
58 zleloe 9289 . . . . . . . . . . . 12 ((𝐾 ∈ ℤ ∧ 𝐴 ∈ ℤ) → (𝐾𝐴 ↔ (𝐾 < 𝐴𝐾 = 𝐴)))
5941, 48, 58syl2anc 411 . . . . . . . . . . 11 (((𝜑 ∧ (𝐴 ∈ (𝐾...(𝐽𝐾)) ∧ ¬ 𝐵 ∈ (𝐾...(𝐽𝐾)))) ∧ ¬ 𝐴 = 𝐾) → (𝐾𝐴 ↔ (𝐾 < 𝐴𝐾 = 𝐴)))
6057, 59mpbid 147 . . . . . . . . . 10 (((𝜑 ∧ (𝐴 ∈ (𝐾...(𝐽𝐾)) ∧ ¬ 𝐵 ∈ (𝐾...(𝐽𝐾)))) ∧ ¬ 𝐴 = 𝐾) → (𝐾 < 𝐴𝐾 = 𝐴))
6154, 60ecased 1349 . . . . . . . . 9 (((𝜑 ∧ (𝐴 ∈ (𝐾...(𝐽𝐾)) ∧ ¬ 𝐵 ∈ (𝐾...(𝐽𝐾)))) ∧ ¬ 𝐴 = 𝐾) → 𝐾 < 𝐴)
62 zltlem1 9299 . . . . . . . . . 10 ((𝐾 ∈ ℤ ∧ 𝐴 ∈ ℤ) → (𝐾 < 𝐴𝐾 ≤ (𝐴 − 1)))
6341, 48, 62syl2anc 411 . . . . . . . . 9 (((𝜑 ∧ (𝐴 ∈ (𝐾...(𝐽𝐾)) ∧ ¬ 𝐵 ∈ (𝐾...(𝐽𝐾)))) ∧ ¬ 𝐴 = 𝐾) → (𝐾 < 𝐴𝐾 ≤ (𝐴 − 1)))
6461, 63mpbid 147 . . . . . . . 8 (((𝜑 ∧ (𝐴 ∈ (𝐾...(𝐽𝐾)) ∧ ¬ 𝐵 ∈ (𝐾...(𝐽𝐾)))) ∧ ¬ 𝐴 = 𝐾) → 𝐾 ≤ (𝐴 − 1))
6550zred 9364 . . . . . . . . 9 (((𝜑 ∧ (𝐴 ∈ (𝐾...(𝐽𝐾)) ∧ ¬ 𝐵 ∈ (𝐾...(𝐽𝐾)))) ∧ ¬ 𝐴 = 𝐾) → (𝐴 − 1) ∈ ℝ)
6648zred 9364 . . . . . . . . 9 (((𝜑 ∧ (𝐴 ∈ (𝐾...(𝐽𝐾)) ∧ ¬ 𝐵 ∈ (𝐾...(𝐽𝐾)))) ∧ ¬ 𝐴 = 𝐾) → 𝐴 ∈ ℝ)
6744zred 9364 . . . . . . . . 9 (((𝜑 ∧ (𝐴 ∈ (𝐾...(𝐽𝐾)) ∧ ¬ 𝐵 ∈ (𝐾...(𝐽𝐾)))) ∧ ¬ 𝐴 = 𝐾) → (𝐽𝐾) ∈ ℝ)
6866lem1d 8879 . . . . . . . . 9 (((𝜑 ∧ (𝐴 ∈ (𝐾...(𝐽𝐾)) ∧ ¬ 𝐵 ∈ (𝐾...(𝐽𝐾)))) ∧ ¬ 𝐴 = 𝐾) → (𝐴 − 1) ≤ 𝐴)
69 elfzle2 10014 . . . . . . . . . 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 8071 . . . . . . . 8 (((𝜑 ∧ (𝐴 ∈ (𝐾...(𝐽𝐾)) ∧ ¬ 𝐵 ∈ (𝐾...(𝐽𝐾)))) ∧ ¬ 𝐴 = 𝐾) → (𝐴 − 1) ≤ (𝐽𝐾))
7264, 71jca 306 . . . . . . 7 (((𝜑 ∧ (𝐴 ∈ (𝐾...(𝐽𝐾)) ∧ ¬ 𝐵 ∈ (𝐾...(𝐽𝐾)))) ∧ ¬ 𝐴 = 𝐾) → (𝐾 ≤ (𝐴 − 1) ∧ (𝐴 − 1) ≤ (𝐽𝐾)))
73 elfz2 10002 . . . . . . 7 ((𝐴 − 1) ∈ (𝐾...(𝐽𝐾)) ↔ ((𝐾 ∈ ℤ ∧ (𝐽𝐾) ∈ ℤ ∧ (𝐴 − 1) ∈ ℤ) ∧ (𝐾 ≤ (𝐴 − 1) ∧ (𝐴 − 1) ≤ (𝐽𝐾))))
7451, 72, 73sylanbrc 417 . . . . . 6 (((𝜑 ∧ (𝐴 ∈ (𝐾...(𝐽𝐾)) ∧ ¬ 𝐵 ∈ (𝐾...(𝐽𝐾)))) ∧ ¬ 𝐴 = 𝐾) → (𝐴 − 1) ∈ (𝐾...(𝐽𝐾)))
75 fnfvima 5746 . . . . . 6 ((𝐽 Fn (𝑀...𝑁) ∧ (𝐾...(𝐽𝐾)) ⊆ (𝑀...𝑁) ∧ (𝐴 − 1) ∈ (𝐾...(𝐽𝐾))) → (𝐽‘(𝐴 − 1)) ∈ (𝐽 “ (𝐾...(𝐽𝐾))))
7636, 37, 74, 75syl3anc 1238 . . . . 5 (((𝜑 ∧ (𝐴 ∈ (𝐾...(𝐽𝐾)) ∧ ¬ 𝐵 ∈ (𝐾...(𝐽𝐾)))) ∧ ¬ 𝐴 = 𝐾) → (𝐽‘(𝐴 − 1)) ∈ (𝐽 “ (𝐾...(𝐽𝐾))))
77 zdceq 9317 . . . . . 6 ((𝐴 ∈ ℤ ∧ 𝐾 ∈ ℤ) → DECID 𝐴 = 𝐾)
7847, 40, 77syl2anc 411 . . . . 5 ((𝜑 ∧ (𝐴 ∈ (𝐾...(𝐽𝐾)) ∧ ¬ 𝐵 ∈ (𝐾...(𝐽𝐾)))) → DECID 𝐴 = 𝐾)
7935, 76, 78ifcldadc 3563 . . . 4 ((𝜑 ∧ (𝐴 ∈ (𝐾...(𝐽𝐾)) ∧ ¬ 𝐵 ∈ (𝐾...(𝐽𝐾)))) → if(𝐴 = 𝐾, 𝐾, (𝐽‘(𝐴 − 1))) ∈ (𝐽 “ (𝐾...(𝐽𝐾))))
8011, 79eqeltrd 2254 . . 3 ((𝜑 ∧ (𝐴 ∈ (𝐾...(𝐽𝐾)) ∧ ¬ 𝐵 ∈ (𝐾...(𝐽𝐾)))) → (𝑄𝐴) ∈ (𝐽 “ (𝐾...(𝐽𝐾))))
812, 80eqeltrrd 2255 . 2 ((𝜑 ∧ (𝐴 ∈ (𝐾...(𝐽𝐾)) ∧ ¬ 𝐵 ∈ (𝐾...(𝐽𝐾)))) → (𝑄𝐵) ∈ (𝐽 “ (𝐾...(𝐽𝐾))))
82 iseqf1olemnab.b . . . . . 6 (𝜑𝐵 ∈ (𝑀...𝑁))
833, 4, 82, 6iseqf1olemqval 10473 . . . . 5 (𝜑 → (𝑄𝐵) = if(𝐵 ∈ (𝐾...(𝐽𝐾)), if(𝐵 = 𝐾, 𝐾, (𝐽‘(𝐵 − 1))), (𝐽𝐵)))
8483adantr 276 . . . 4 ((𝜑 ∧ (𝐴 ∈ (𝐾...(𝐽𝐾)) ∧ ¬ 𝐵 ∈ (𝐾...(𝐽𝐾)))) → (𝑄𝐵) = if(𝐵 ∈ (𝐾...(𝐽𝐾)), if(𝐵 = 𝐾, 𝐾, (𝐽‘(𝐵 − 1))), (𝐽𝐵)))
85 simprr 531 . . . . 5 ((𝜑 ∧ (𝐴 ∈ (𝐾...(𝐽𝐾)) ∧ ¬ 𝐵 ∈ (𝐾...(𝐽𝐾)))) → ¬ 𝐵 ∈ (𝐾...(𝐽𝐾)))
8685iffalsed 3544 . . . 4 ((𝜑 ∧ (𝐴 ∈ (𝐾...(𝐽𝐾)) ∧ ¬ 𝐵 ∈ (𝐾...(𝐽𝐾)))) → if(𝐵 ∈ (𝐾...(𝐽𝐾)), if(𝐵 = 𝐾, 𝐾, (𝐽‘(𝐵 − 1))), (𝐽𝐵)) = (𝐽𝐵))
8784, 86eqtrd 2210 . . 3 ((𝜑 ∧ (𝐴 ∈ (𝐾...(𝐽𝐾)) ∧ ¬ 𝐵 ∈ (𝐾...(𝐽𝐾)))) → (𝑄𝐵) = (𝐽𝐵))
88 f1of1 5456 . . . . . . 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 5768 . . . . . 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 1238 . . . . 5 (𝜑 → ((𝐽𝐵) ∈ (𝐽 “ (𝐾...(𝐽𝐾))) ↔ 𝐵 ∈ (𝐾...(𝐽𝐾))))
9291adantr 276 . . . 4 ((𝜑 ∧ (𝐴 ∈ (𝐾...(𝐽𝐾)) ∧ ¬ 𝐵 ∈ (𝐾...(𝐽𝐾)))) → ((𝐽𝐵) ∈ (𝐽 “ (𝐾...(𝐽𝐾))) ↔ 𝐵 ∈ (𝐾...(𝐽𝐾))))
9385, 92mtbird 673 . . 3 ((𝜑 ∧ (𝐴 ∈ (𝐾...(𝐽𝐾)) ∧ ¬ 𝐵 ∈ (𝐾...(𝐽𝐾)))) → ¬ (𝐽𝐵) ∈ (𝐽 “ (𝐾...(𝐽𝐾))))
9487, 93eqneltrd 2273 . 2 ((𝜑 ∧ (𝐴 ∈ (𝐾...(𝐽𝐾)) ∧ ¬ 𝐵 ∈ (𝐾...(𝐽𝐾)))) → ¬ (𝑄𝐵) ∈ (𝐽 “ (𝐾...(𝐽𝐾))))
9581, 94pm2.65da 661 1 (𝜑 → ¬ (𝐴 ∈ (𝐾...(𝐽𝐾)) ∧ ¬ 𝐵 ∈ (𝐾...(𝐽𝐾))))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 104  wb 105  wo 708  DECID wdc 834  w3a 978   = wceq 1353  wcel 2148  wss 3129  ifcif 3534   class class class wbr 4000  cmpt 4061  ccnv 4622  cima 4626   Fn wfn 5207  wf 5208  1-1wf1 5209  1-1-ontowf1o 5211  cfv 5212  (class class class)co 5869  1c1 7803   < clt 7982  cle 7983  cmin 8118  cz 9242  cuz 9517  ...cfz 9995
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-pow 4171  ax-pr 4206  ax-un 4430  ax-setind 4533  ax-cnex 7893  ax-resscn 7894  ax-1cn 7895  ax-1re 7896  ax-icn 7897  ax-addcl 7898  ax-addrcl 7899  ax-mulcl 7900  ax-addcom 7902  ax-addass 7904  ax-distr 7906  ax-i2m1 7907  ax-0lt1 7908  ax-0id 7910  ax-rnegex 7911  ax-cnre 7913  ax-pre-ltirr 7914  ax-pre-ltwlin 7915  ax-pre-lttrn 7916  ax-pre-ltadd 7918
This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2739  df-sbc 2963  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-if 3535  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-int 3843  df-br 4001  df-opab 4062  df-mpt 4063  df-id 4290  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-rn 4634  df-res 4635  df-ima 4636  df-iota 5174  df-fun 5214  df-fn 5215  df-f 5216  df-f1 5217  df-fo 5218  df-f1o 5219  df-fv 5220  df-riota 5825  df-ov 5872  df-oprab 5873  df-mpo 5874  df-pnf 7984  df-mnf 7985  df-xr 7986  df-ltxr 7987  df-le 7988  df-sub 8120  df-neg 8121  df-inn 8909  df-n0 9166  df-z 9243  df-uz 9518  df-fz 9996
This theorem is referenced by:  iseqf1olemmo  10478
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