ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  iseqf1olemnab GIF version

Theorem iseqf1olemnab 9882
Description: Lemma for seq3f1o 9898. (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 270 . . 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 9881 . . . . . 6 (𝜑 → (𝑄𝐴) = if(𝐴 ∈ (𝐾...(𝐽𝐾)), if(𝐴 = 𝐾, 𝐾, (𝐽‘(𝐴 − 1))), (𝐽𝐴)))
87adantr 270 . . . . 5 ((𝜑 ∧ (𝐴 ∈ (𝐾...(𝐽𝐾)) ∧ ¬ 𝐵 ∈ (𝐾...(𝐽𝐾)))) → (𝑄𝐴) = if(𝐴 ∈ (𝐾...(𝐽𝐾)), if(𝐴 = 𝐾, 𝐾, (𝐽‘(𝐴 − 1))), (𝐽𝐴)))
9 simprl 498 . . . . . 6 ((𝜑 ∧ (𝐴 ∈ (𝐾...(𝐽𝐾)) ∧ ¬ 𝐵 ∈ (𝐾...(𝐽𝐾)))) → 𝐴 ∈ (𝐾...(𝐽𝐾)))
109iftrued 3396 . . . . 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 2120 . . . 4 ((𝜑 ∧ (𝐴 ∈ (𝐾...(𝐽𝐾)) ∧ ¬ 𝐵 ∈ (𝐾...(𝐽𝐾)))) → (𝑄𝐴) = if(𝐴 = 𝐾, 𝐾, (𝐽‘(𝐴 − 1))))
12 f1ocnvfv2 5539 . . . . . . . 8 ((𝐽:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ 𝐾 ∈ (𝑀...𝑁)) → (𝐽‘(𝐽𝐾)) = 𝐾)
134, 3, 12syl2anc 403 . . . . . . 7 (𝜑 → (𝐽‘(𝐽𝐾)) = 𝐾)
1413ad2antrr 472 . . . . . 6 (((𝜑 ∧ (𝐴 ∈ (𝐾...(𝐽𝐾)) ∧ ¬ 𝐵 ∈ (𝐾...(𝐽𝐾)))) ∧ 𝐴 = 𝐾) → (𝐽‘(𝐽𝐾)) = 𝐾)
15 f1ofn 5238 . . . . . . . . 9 (𝐽:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) → 𝐽 Fn (𝑀...𝑁))
164, 15syl 14 . . . . . . . 8 (𝜑𝐽 Fn (𝑀...𝑁))
1716ad2antrr 472 . . . . . . 7 (((𝜑 ∧ (𝐴 ∈ (𝐾...(𝐽𝐾)) ∧ ¬ 𝐵 ∈ (𝐾...(𝐽𝐾)))) ∧ 𝐴 = 𝐾) → 𝐽 Fn (𝑀...𝑁))
18 elfzuz 9405 . . . . . . . . . 10 (𝐾 ∈ (𝑀...𝑁) → 𝐾 ∈ (ℤ𝑀))
19 fzss1 9445 . . . . . . . . . 10 (𝐾 ∈ (ℤ𝑀) → (𝐾...(𝐽𝐾)) ⊆ (𝑀...(𝐽𝐾)))
203, 18, 193syl 17 . . . . . . . . 9 (𝜑 → (𝐾...(𝐽𝐾)) ⊆ (𝑀...(𝐽𝐾)))
21 f1ocnv 5250 . . . . . . . . . . . 12 (𝐽:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) → 𝐽:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁))
22 f1of 5237 . . . . . . . . . . . 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 5419 . . . . . . . . . 10 (𝜑 → (𝐽𝐾) ∈ (𝑀...𝑁))
25 elfzuz3 9406 . . . . . . . . . 10 ((𝐽𝐾) ∈ (𝑀...𝑁) → 𝑁 ∈ (ℤ‘(𝐽𝐾)))
26 fzss2 9446 . . . . . . . . . 10 (𝑁 ∈ (ℤ‘(𝐽𝐾)) → (𝑀...(𝐽𝐾)) ⊆ (𝑀...𝑁))
2724, 25, 263syl 17 . . . . . . . . 9 (𝜑 → (𝑀...(𝐽𝐾)) ⊆ (𝑀...𝑁))
2820, 27sstrd 3033 . . . . . . . 8 (𝜑 → (𝐾...(𝐽𝐾)) ⊆ (𝑀...𝑁))
2928ad2antrr 472 . . . . . . 7 (((𝜑 ∧ (𝐴 ∈ (𝐾...(𝐽𝐾)) ∧ ¬ 𝐵 ∈ (𝐾...(𝐽𝐾)))) ∧ 𝐴 = 𝐾) → (𝐾...(𝐽𝐾)) ⊆ (𝑀...𝑁))
30 elfzubelfz 9419 . . . . . . . . 9 (𝐴 ∈ (𝐾...(𝐽𝐾)) → (𝐽𝐾) ∈ (𝐾...(𝐽𝐾)))
3130adantr 270 . . . . . . . 8 ((𝐴 ∈ (𝐾...(𝐽𝐾)) ∧ ¬ 𝐵 ∈ (𝐾...(𝐽𝐾))) → (𝐽𝐾) ∈ (𝐾...(𝐽𝐾)))
3231ad2antlr 473 . . . . . . 7 (((𝜑 ∧ (𝐴 ∈ (𝐾...(𝐽𝐾)) ∧ ¬ 𝐵 ∈ (𝐾...(𝐽𝐾)))) ∧ 𝐴 = 𝐾) → (𝐽𝐾) ∈ (𝐾...(𝐽𝐾)))
33 fnfvima 5511 . . . . . . 7 ((𝐽 Fn (𝑀...𝑁) ∧ (𝐾...(𝐽𝐾)) ⊆ (𝑀...𝑁) ∧ (𝐽𝐾) ∈ (𝐾...(𝐽𝐾))) → (𝐽‘(𝐽𝐾)) ∈ (𝐽 “ (𝐾...(𝐽𝐾))))
3417, 29, 32, 33syl3anc 1174 . . . . . 6 (((𝜑 ∧ (𝐴 ∈ (𝐾...(𝐽𝐾)) ∧ ¬ 𝐵 ∈ (𝐾...(𝐽𝐾)))) ∧ 𝐴 = 𝐾) → (𝐽‘(𝐽𝐾)) ∈ (𝐽 “ (𝐾...(𝐽𝐾))))
3514, 34eqeltrrd 2165 . . . . 5 (((𝜑 ∧ (𝐴 ∈ (𝐾...(𝐽𝐾)) ∧ ¬ 𝐵 ∈ (𝐾...(𝐽𝐾)))) ∧ 𝐴 = 𝐾) → 𝐾 ∈ (𝐽 “ (𝐾...(𝐽𝐾))))
3616ad2antrr 472 . . . . . 6 (((𝜑 ∧ (𝐴 ∈ (𝐾...(𝐽𝐾)) ∧ ¬ 𝐵 ∈ (𝐾...(𝐽𝐾)))) ∧ ¬ 𝐴 = 𝐾) → 𝐽 Fn (𝑀...𝑁))
3728ad2antrr 472 . . . . . 6 (((𝜑 ∧ (𝐴 ∈ (𝐾...(𝐽𝐾)) ∧ ¬ 𝐵 ∈ (𝐾...(𝐽𝐾)))) ∧ ¬ 𝐴 = 𝐾) → (𝐾...(𝐽𝐾)) ⊆ (𝑀...𝑁))
383adantr 270 . . . . . . . . . 10 ((𝜑 ∧ (𝐴 ∈ (𝐾...(𝐽𝐾)) ∧ ¬ 𝐵 ∈ (𝐾...(𝐽𝐾)))) → 𝐾 ∈ (𝑀...𝑁))
39 elfzelz 9409 . . . . . . . . . 10 (𝐾 ∈ (𝑀...𝑁) → 𝐾 ∈ ℤ)
4038, 39syl 14 . . . . . . . . 9 ((𝜑 ∧ (𝐴 ∈ (𝐾...(𝐽𝐾)) ∧ ¬ 𝐵 ∈ (𝐾...(𝐽𝐾)))) → 𝐾 ∈ ℤ)
4140adantr 270 . . . . . . . 8 (((𝜑 ∧ (𝐴 ∈ (𝐾...(𝐽𝐾)) ∧ ¬ 𝐵 ∈ (𝐾...(𝐽𝐾)))) ∧ ¬ 𝐴 = 𝐾) → 𝐾 ∈ ℤ)
4224ad2antrr 472 . . . . . . . . 9 (((𝜑 ∧ (𝐴 ∈ (𝐾...(𝐽𝐾)) ∧ ¬ 𝐵 ∈ (𝐾...(𝐽𝐾)))) ∧ ¬ 𝐴 = 𝐾) → (𝐽𝐾) ∈ (𝑀...𝑁))
43 elfzelz 9409 . . . . . . . . 9 ((𝐽𝐾) ∈ (𝑀...𝑁) → (𝐽𝐾) ∈ ℤ)
4442, 43syl 14 . . . . . . . 8 (((𝜑 ∧ (𝐴 ∈ (𝐾...(𝐽𝐾)) ∧ ¬ 𝐵 ∈ (𝐾...(𝐽𝐾)))) ∧ ¬ 𝐴 = 𝐾) → (𝐽𝐾) ∈ ℤ)
455adantr 270 . . . . . . . . . . 11 ((𝜑 ∧ (𝐴 ∈ (𝐾...(𝐽𝐾)) ∧ ¬ 𝐵 ∈ (𝐾...(𝐽𝐾)))) → 𝐴 ∈ (𝑀...𝑁))
46 elfzelz 9409 . . . . . . . . . . 11 (𝐴 ∈ (𝑀...𝑁) → 𝐴 ∈ ℤ)
4745, 46syl 14 . . . . . . . . . 10 ((𝜑 ∧ (𝐴 ∈ (𝐾...(𝐽𝐾)) ∧ ¬ 𝐵 ∈ (𝐾...(𝐽𝐾)))) → 𝐴 ∈ ℤ)
4847adantr 270 . . . . . . . . 9 (((𝜑 ∧ (𝐴 ∈ (𝐾...(𝐽𝐾)) ∧ ¬ 𝐵 ∈ (𝐾...(𝐽𝐾)))) ∧ ¬ 𝐴 = 𝐾) → 𝐴 ∈ ℤ)
49 peano2zm 8758 . . . . . . . . 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 1123 . . . . . . 7 (((𝜑 ∧ (𝐴 ∈ (𝐾...(𝐽𝐾)) ∧ ¬ 𝐵 ∈ (𝐾...(𝐽𝐾)))) ∧ ¬ 𝐴 = 𝐾) → (𝐾 ∈ ℤ ∧ (𝐽𝐾) ∈ ℤ ∧ (𝐴 − 1) ∈ ℤ))
52 simpr 108 . . . . . . . . . . 11 (((𝜑 ∧ (𝐴 ∈ (𝐾...(𝐽𝐾)) ∧ ¬ 𝐵 ∈ (𝐾...(𝐽𝐾)))) ∧ ¬ 𝐴 = 𝐾) → ¬ 𝐴 = 𝐾)
53 eqcom 2090 . . . . . . . . . . 11 (𝐴 = 𝐾𝐾 = 𝐴)
5452, 53sylnib 636 . . . . . . . . . 10 (((𝜑 ∧ (𝐴 ∈ (𝐾...(𝐽𝐾)) ∧ ¬ 𝐵 ∈ (𝐾...(𝐽𝐾)))) ∧ ¬ 𝐴 = 𝐾) → ¬ 𝐾 = 𝐴)
559adantr 270 . . . . . . . . . . . 12 (((𝜑 ∧ (𝐴 ∈ (𝐾...(𝐽𝐾)) ∧ ¬ 𝐵 ∈ (𝐾...(𝐽𝐾)))) ∧ ¬ 𝐴 = 𝐾) → 𝐴 ∈ (𝐾...(𝐽𝐾)))
56 elfzle1 9410 . . . . . . . . . . . 12 (𝐴 ∈ (𝐾...(𝐽𝐾)) → 𝐾𝐴)
5755, 56syl 14 . . . . . . . . . . 11 (((𝜑 ∧ (𝐴 ∈ (𝐾...(𝐽𝐾)) ∧ ¬ 𝐵 ∈ (𝐾...(𝐽𝐾)))) ∧ ¬ 𝐴 = 𝐾) → 𝐾𝐴)
58 zleloe 8767 . . . . . . . . . . . 12 ((𝐾 ∈ ℤ ∧ 𝐴 ∈ ℤ) → (𝐾𝐴 ↔ (𝐾 < 𝐴𝐾 = 𝐴)))
5941, 48, 58syl2anc 403 . . . . . . . . . . 11 (((𝜑 ∧ (𝐴 ∈ (𝐾...(𝐽𝐾)) ∧ ¬ 𝐵 ∈ (𝐾...(𝐽𝐾)))) ∧ ¬ 𝐴 = 𝐾) → (𝐾𝐴 ↔ (𝐾 < 𝐴𝐾 = 𝐴)))
6057, 59mpbid 145 . . . . . . . . . 10 (((𝜑 ∧ (𝐴 ∈ (𝐾...(𝐽𝐾)) ∧ ¬ 𝐵 ∈ (𝐾...(𝐽𝐾)))) ∧ ¬ 𝐴 = 𝐾) → (𝐾 < 𝐴𝐾 = 𝐴))
6154, 60ecased 1285 . . . . . . . . 9 (((𝜑 ∧ (𝐴 ∈ (𝐾...(𝐽𝐾)) ∧ ¬ 𝐵 ∈ (𝐾...(𝐽𝐾)))) ∧ ¬ 𝐴 = 𝐾) → 𝐾 < 𝐴)
62 zltlem1 8777 . . . . . . . . . 10 ((𝐾 ∈ ℤ ∧ 𝐴 ∈ ℤ) → (𝐾 < 𝐴𝐾 ≤ (𝐴 − 1)))
6341, 48, 62syl2anc 403 . . . . . . . . 9 (((𝜑 ∧ (𝐴 ∈ (𝐾...(𝐽𝐾)) ∧ ¬ 𝐵 ∈ (𝐾...(𝐽𝐾)))) ∧ ¬ 𝐴 = 𝐾) → (𝐾 < 𝐴𝐾 ≤ (𝐴 − 1)))
6461, 63mpbid 145 . . . . . . . 8 (((𝜑 ∧ (𝐴 ∈ (𝐾...(𝐽𝐾)) ∧ ¬ 𝐵 ∈ (𝐾...(𝐽𝐾)))) ∧ ¬ 𝐴 = 𝐾) → 𝐾 ≤ (𝐴 − 1))
6550zred 8838 . . . . . . . . 9 (((𝜑 ∧ (𝐴 ∈ (𝐾...(𝐽𝐾)) ∧ ¬ 𝐵 ∈ (𝐾...(𝐽𝐾)))) ∧ ¬ 𝐴 = 𝐾) → (𝐴 − 1) ∈ ℝ)
6648zred 8838 . . . . . . . . 9 (((𝜑 ∧ (𝐴 ∈ (𝐾...(𝐽𝐾)) ∧ ¬ 𝐵 ∈ (𝐾...(𝐽𝐾)))) ∧ ¬ 𝐴 = 𝐾) → 𝐴 ∈ ℝ)
6744zred 8838 . . . . . . . . 9 (((𝜑 ∧ (𝐴 ∈ (𝐾...(𝐽𝐾)) ∧ ¬ 𝐵 ∈ (𝐾...(𝐽𝐾)))) ∧ ¬ 𝐴 = 𝐾) → (𝐽𝐾) ∈ ℝ)
6866lem1d 8366 . . . . . . . . 9 (((𝜑 ∧ (𝐴 ∈ (𝐾...(𝐽𝐾)) ∧ ¬ 𝐵 ∈ (𝐾...(𝐽𝐾)))) ∧ ¬ 𝐴 = 𝐾) → (𝐴 − 1) ≤ 𝐴)
69 elfzle2 9411 . . . . . . . . . 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 7586 . . . . . . . 8 (((𝜑 ∧ (𝐴 ∈ (𝐾...(𝐽𝐾)) ∧ ¬ 𝐵 ∈ (𝐾...(𝐽𝐾)))) ∧ ¬ 𝐴 = 𝐾) → (𝐴 − 1) ≤ (𝐽𝐾))
7264, 71jca 300 . . . . . . 7 (((𝜑 ∧ (𝐴 ∈ (𝐾...(𝐽𝐾)) ∧ ¬ 𝐵 ∈ (𝐾...(𝐽𝐾)))) ∧ ¬ 𝐴 = 𝐾) → (𝐾 ≤ (𝐴 − 1) ∧ (𝐴 − 1) ≤ (𝐽𝐾)))
73 elfz2 9400 . . . . . . 7 ((𝐴 − 1) ∈ (𝐾...(𝐽𝐾)) ↔ ((𝐾 ∈ ℤ ∧ (𝐽𝐾) ∈ ℤ ∧ (𝐴 − 1) ∈ ℤ) ∧ (𝐾 ≤ (𝐴 − 1) ∧ (𝐴 − 1) ≤ (𝐽𝐾))))
7451, 72, 73sylanbrc 408 . . . . . 6 (((𝜑 ∧ (𝐴 ∈ (𝐾...(𝐽𝐾)) ∧ ¬ 𝐵 ∈ (𝐾...(𝐽𝐾)))) ∧ ¬ 𝐴 = 𝐾) → (𝐴 − 1) ∈ (𝐾...(𝐽𝐾)))
75 fnfvima 5511 . . . . . 6 ((𝐽 Fn (𝑀...𝑁) ∧ (𝐾...(𝐽𝐾)) ⊆ (𝑀...𝑁) ∧ (𝐴 − 1) ∈ (𝐾...(𝐽𝐾))) → (𝐽‘(𝐴 − 1)) ∈ (𝐽 “ (𝐾...(𝐽𝐾))))
7636, 37, 74, 75syl3anc 1174 . . . . 5 (((𝜑 ∧ (𝐴 ∈ (𝐾...(𝐽𝐾)) ∧ ¬ 𝐵 ∈ (𝐾...(𝐽𝐾)))) ∧ ¬ 𝐴 = 𝐾) → (𝐽‘(𝐴 − 1)) ∈ (𝐽 “ (𝐾...(𝐽𝐾))))
77 zdceq 8792 . . . . . 6 ((𝐴 ∈ ℤ ∧ 𝐾 ∈ ℤ) → DECID 𝐴 = 𝐾)
7847, 40, 77syl2anc 403 . . . . 5 ((𝜑 ∧ (𝐴 ∈ (𝐾...(𝐽𝐾)) ∧ ¬ 𝐵 ∈ (𝐾...(𝐽𝐾)))) → DECID 𝐴 = 𝐾)
7935, 76, 78ifcldadc 3416 . . . 4 ((𝜑 ∧ (𝐴 ∈ (𝐾...(𝐽𝐾)) ∧ ¬ 𝐵 ∈ (𝐾...(𝐽𝐾)))) → if(𝐴 = 𝐾, 𝐾, (𝐽‘(𝐴 − 1))) ∈ (𝐽 “ (𝐾...(𝐽𝐾))))
8011, 79eqeltrd 2164 . . 3 ((𝜑 ∧ (𝐴 ∈ (𝐾...(𝐽𝐾)) ∧ ¬ 𝐵 ∈ (𝐾...(𝐽𝐾)))) → (𝑄𝐴) ∈ (𝐽 “ (𝐾...(𝐽𝐾))))
812, 80eqeltrrd 2165 . 2 ((𝜑 ∧ (𝐴 ∈ (𝐾...(𝐽𝐾)) ∧ ¬ 𝐵 ∈ (𝐾...(𝐽𝐾)))) → (𝑄𝐵) ∈ (𝐽 “ (𝐾...(𝐽𝐾))))
82 iseqf1olemnab.b . . . . . 6 (𝜑𝐵 ∈ (𝑀...𝑁))
833, 4, 82, 6iseqf1olemqval 9881 . . . . 5 (𝜑 → (𝑄𝐵) = if(𝐵 ∈ (𝐾...(𝐽𝐾)), if(𝐵 = 𝐾, 𝐾, (𝐽‘(𝐵 − 1))), (𝐽𝐵)))
8483adantr 270 . . . 4 ((𝜑 ∧ (𝐴 ∈ (𝐾...(𝐽𝐾)) ∧ ¬ 𝐵 ∈ (𝐾...(𝐽𝐾)))) → (𝑄𝐵) = if(𝐵 ∈ (𝐾...(𝐽𝐾)), if(𝐵 = 𝐾, 𝐾, (𝐽‘(𝐵 − 1))), (𝐽𝐵)))
85 simprr 499 . . . . 5 ((𝜑 ∧ (𝐴 ∈ (𝐾...(𝐽𝐾)) ∧ ¬ 𝐵 ∈ (𝐾...(𝐽𝐾)))) → ¬ 𝐵 ∈ (𝐾...(𝐽𝐾)))
8685iffalsed 3399 . . . 4 ((𝜑 ∧ (𝐴 ∈ (𝐾...(𝐽𝐾)) ∧ ¬ 𝐵 ∈ (𝐾...(𝐽𝐾)))) → if(𝐵 ∈ (𝐾...(𝐽𝐾)), if(𝐵 = 𝐾, 𝐾, (𝐽‘(𝐵 − 1))), (𝐽𝐵)) = (𝐽𝐵))
8784, 86eqtrd 2120 . . 3 ((𝜑 ∧ (𝐴 ∈ (𝐾...(𝐽𝐾)) ∧ ¬ 𝐵 ∈ (𝐾...(𝐽𝐾)))) → (𝑄𝐵) = (𝐽𝐵))
88 f1of1 5236 . . . . . . 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 5534 . . . . . 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 1174 . . . . 5 (𝜑 → ((𝐽𝐵) ∈ (𝐽 “ (𝐾...(𝐽𝐾))) ↔ 𝐵 ∈ (𝐾...(𝐽𝐾))))
9291adantr 270 . . . 4 ((𝜑 ∧ (𝐴 ∈ (𝐾...(𝐽𝐾)) ∧ ¬ 𝐵 ∈ (𝐾...(𝐽𝐾)))) → ((𝐽𝐵) ∈ (𝐽 “ (𝐾...(𝐽𝐾))) ↔ 𝐵 ∈ (𝐾...(𝐽𝐾))))
9385, 92mtbird 633 . . 3 ((𝜑 ∧ (𝐴 ∈ (𝐾...(𝐽𝐾)) ∧ ¬ 𝐵 ∈ (𝐾...(𝐽𝐾)))) → ¬ (𝐽𝐵) ∈ (𝐽 “ (𝐾...(𝐽𝐾))))
9487, 93eqneltrd 2183 . 2 ((𝜑 ∧ (𝐴 ∈ (𝐾...(𝐽𝐾)) ∧ ¬ 𝐵 ∈ (𝐾...(𝐽𝐾)))) → ¬ (𝑄𝐵) ∈ (𝐽 “ (𝐾...(𝐽𝐾))))
9581, 94pm2.65da 622 1 (𝜑 → ¬ (𝐴 ∈ (𝐾...(𝐽𝐾)) ∧ ¬ 𝐵 ∈ (𝐾...(𝐽𝐾))))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 102  wb 103  wo 664  DECID wdc 780  w3a 924   = wceq 1289  wcel 1438  wss 2997  ifcif 3389   class class class wbr 3837  cmpt 3891  ccnv 4427  cima 4431   Fn wfn 4997  wf 4998  1-1wf1 4999  1-1-ontowf1o 5001  cfv 5002  (class class class)co 5634  1c1 7330   < clt 7501  cle 7502  cmin 7632  cz 8720  cuz 8988  ...cfz 9393
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-pow 4001  ax-pr 4027  ax-un 4251  ax-setind 4343  ax-cnex 7415  ax-resscn 7416  ax-1cn 7417  ax-1re 7418  ax-icn 7419  ax-addcl 7420  ax-addrcl 7421  ax-mulcl 7422  ax-addcom 7424  ax-addass 7426  ax-distr 7428  ax-i2m1 7429  ax-0lt1 7430  ax-0id 7432  ax-rnegex 7433  ax-cnre 7435  ax-pre-ltirr 7436  ax-pre-ltwlin 7437  ax-pre-lttrn 7438  ax-pre-ltadd 7440
This theorem depends on definitions:  df-bi 115  df-dc 781  df-3or 925  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-nel 2351  df-ral 2364  df-rex 2365  df-reu 2366  df-rab 2368  df-v 2621  df-sbc 2839  df-dif 2999  df-un 3001  df-in 3003  df-ss 3010  df-if 3390  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-uni 3649  df-int 3684  df-br 3838  df-opab 3892  df-mpt 3893  df-id 4111  df-xp 4434  df-rel 4435  df-cnv 4436  df-co 4437  df-dm 4438  df-rn 4439  df-res 4440  df-ima 4441  df-iota 4967  df-fun 5004  df-fn 5005  df-f 5006  df-f1 5007  df-fo 5008  df-f1o 5009  df-fv 5010  df-riota 5590  df-ov 5637  df-oprab 5638  df-mpt2 5639  df-pnf 7503  df-mnf 7504  df-xr 7505  df-ltxr 7506  df-le 7507  df-sub 7634  df-neg 7635  df-inn 8395  df-n0 8644  df-z 8721  df-uz 8989  df-fz 9394
This theorem is referenced by:  iseqf1olemmo  9886
  Copyright terms: Public domain W3C validator