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Theorem iseqf1olemmo 10491
Description: Lemma for seq3f1o 10503. Showing that 𝑄 is one-to-one. (Contributed by Jim Kingdon, 27-Aug-2022.)
Hypotheses
Ref Expression
iseqf1olemqf.k (𝜑𝐾 ∈ (𝑀...𝑁))
iseqf1olemqf.j (𝜑𝐽:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁))
iseqf1olemqf.q 𝑄 = (𝑢 ∈ (𝑀...𝑁) ↦ if(𝑢 ∈ (𝐾...(𝐽𝐾)), if(𝑢 = 𝐾, 𝐾, (𝐽‘(𝑢 − 1))), (𝐽𝑢)))
iseqf1olemmo.a (𝜑𝐴 ∈ (𝑀...𝑁))
iseqf1olemmo.b (𝜑𝐵 ∈ (𝑀...𝑁))
iseqf1olemmo.eq (𝜑 → (𝑄𝐴) = (𝑄𝐵))
Assertion
Ref Expression
iseqf1olemmo (𝜑𝐴 = 𝐵)
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 iseqf1olemmo
StepHypRef Expression
1 iseqf1olemqf.k . . . . 5 (𝜑𝐾 ∈ (𝑀...𝑁))
21ad2antrr 488 . . . 4 (((𝜑𝐴 ∈ (𝐾...(𝐽𝐾))) ∧ 𝐵 ∈ (𝐾...(𝐽𝐾))) → 𝐾 ∈ (𝑀...𝑁))
3 iseqf1olemqf.j . . . . 5 (𝜑𝐽:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁))
43ad2antrr 488 . . . 4 (((𝜑𝐴 ∈ (𝐾...(𝐽𝐾))) ∧ 𝐵 ∈ (𝐾...(𝐽𝐾))) → 𝐽:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁))
5 iseqf1olemmo.a . . . . 5 (𝜑𝐴 ∈ (𝑀...𝑁))
65ad2antrr 488 . . . 4 (((𝜑𝐴 ∈ (𝐾...(𝐽𝐾))) ∧ 𝐵 ∈ (𝐾...(𝐽𝐾))) → 𝐴 ∈ (𝑀...𝑁))
7 iseqf1olemmo.b . . . . 5 (𝜑𝐵 ∈ (𝑀...𝑁))
87ad2antrr 488 . . . 4 (((𝜑𝐴 ∈ (𝐾...(𝐽𝐾))) ∧ 𝐵 ∈ (𝐾...(𝐽𝐾))) → 𝐵 ∈ (𝑀...𝑁))
9 iseqf1olemmo.eq . . . . 5 (𝜑 → (𝑄𝐴) = (𝑄𝐵))
109ad2antrr 488 . . . 4 (((𝜑𝐴 ∈ (𝐾...(𝐽𝐾))) ∧ 𝐵 ∈ (𝐾...(𝐽𝐾))) → (𝑄𝐴) = (𝑄𝐵))
11 iseqf1olemqf.q . . . 4 𝑄 = (𝑢 ∈ (𝑀...𝑁) ↦ if(𝑢 ∈ (𝐾...(𝐽𝐾)), if(𝑢 = 𝐾, 𝐾, (𝐽‘(𝑢 − 1))), (𝐽𝑢)))
12 simplr 528 . . . 4 (((𝜑𝐴 ∈ (𝐾...(𝐽𝐾))) ∧ 𝐵 ∈ (𝐾...(𝐽𝐾))) → 𝐴 ∈ (𝐾...(𝐽𝐾)))
13 simpr 110 . . . 4 (((𝜑𝐴 ∈ (𝐾...(𝐽𝐾))) ∧ 𝐵 ∈ (𝐾...(𝐽𝐾))) → 𝐵 ∈ (𝐾...(𝐽𝐾)))
142, 4, 6, 8, 10, 11, 12, 13iseqf1olemab 10488 . . 3 (((𝜑𝐴 ∈ (𝐾...(𝐽𝐾))) ∧ 𝐵 ∈ (𝐾...(𝐽𝐾))) → 𝐴 = 𝐵)
15 simplr 528 . . . . 5 (((𝜑𝐴 ∈ (𝐾...(𝐽𝐾))) ∧ ¬ 𝐵 ∈ (𝐾...(𝐽𝐾))) → 𝐴 ∈ (𝐾...(𝐽𝐾)))
16 simpr 110 . . . . 5 (((𝜑𝐴 ∈ (𝐾...(𝐽𝐾))) ∧ ¬ 𝐵 ∈ (𝐾...(𝐽𝐾))) → ¬ 𝐵 ∈ (𝐾...(𝐽𝐾)))
1715, 16jca 306 . . . 4 (((𝜑𝐴 ∈ (𝐾...(𝐽𝐾))) ∧ ¬ 𝐵 ∈ (𝐾...(𝐽𝐾))) → (𝐴 ∈ (𝐾...(𝐽𝐾)) ∧ ¬ 𝐵 ∈ (𝐾...(𝐽𝐾))))
181, 3, 5, 7, 9, 11iseqf1olemnab 10487 . . . . 5 (𝜑 → ¬ (𝐴 ∈ (𝐾...(𝐽𝐾)) ∧ ¬ 𝐵 ∈ (𝐾...(𝐽𝐾))))
1918ad2antrr 488 . . . 4 (((𝜑𝐴 ∈ (𝐾...(𝐽𝐾))) ∧ ¬ 𝐵 ∈ (𝐾...(𝐽𝐾))) → ¬ (𝐴 ∈ (𝐾...(𝐽𝐾)) ∧ ¬ 𝐵 ∈ (𝐾...(𝐽𝐾))))
2017, 19pm2.21dd 620 . . 3 (((𝜑𝐴 ∈ (𝐾...(𝐽𝐾))) ∧ ¬ 𝐵 ∈ (𝐾...(𝐽𝐾))) → 𝐴 = 𝐵)
21 elfzelz 10024 . . . . . . 7 (𝐵 ∈ (𝑀...𝑁) → 𝐵 ∈ ℤ)
227, 21syl 14 . . . . . 6 (𝜑𝐵 ∈ ℤ)
23 elfzelz 10024 . . . . . . 7 (𝐾 ∈ (𝑀...𝑁) → 𝐾 ∈ ℤ)
241, 23syl 14 . . . . . 6 (𝜑𝐾 ∈ ℤ)
25 f1ocnv 5474 . . . . . . . . 9 (𝐽:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) → 𝐽:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁))
26 f1of 5461 . . . . . . . . 9 (𝐽:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) → 𝐽:(𝑀...𝑁)⟶(𝑀...𝑁))
273, 25, 263syl 17 . . . . . . . 8 (𝜑𝐽:(𝑀...𝑁)⟶(𝑀...𝑁))
2827, 1ffvelcdmd 5652 . . . . . . 7 (𝜑 → (𝐽𝐾) ∈ (𝑀...𝑁))
29 elfzelz 10024 . . . . . . 7 ((𝐽𝐾) ∈ (𝑀...𝑁) → (𝐽𝐾) ∈ ℤ)
3028, 29syl 14 . . . . . 6 (𝜑 → (𝐽𝐾) ∈ ℤ)
31 fzdcel 10039 . . . . . 6 ((𝐵 ∈ ℤ ∧ 𝐾 ∈ ℤ ∧ (𝐽𝐾) ∈ ℤ) → DECID 𝐵 ∈ (𝐾...(𝐽𝐾)))
3222, 24, 30, 31syl3anc 1238 . . . . 5 (𝜑DECID 𝐵 ∈ (𝐾...(𝐽𝐾)))
33 exmiddc 836 . . . . 5 (DECID 𝐵 ∈ (𝐾...(𝐽𝐾)) → (𝐵 ∈ (𝐾...(𝐽𝐾)) ∨ ¬ 𝐵 ∈ (𝐾...(𝐽𝐾))))
3432, 33syl 14 . . . 4 (𝜑 → (𝐵 ∈ (𝐾...(𝐽𝐾)) ∨ ¬ 𝐵 ∈ (𝐾...(𝐽𝐾))))
3534adantr 276 . . 3 ((𝜑𝐴 ∈ (𝐾...(𝐽𝐾))) → (𝐵 ∈ (𝐾...(𝐽𝐾)) ∨ ¬ 𝐵 ∈ (𝐾...(𝐽𝐾))))
3614, 20, 35mpjaodan 798 . 2 ((𝜑𝐴 ∈ (𝐾...(𝐽𝐾))) → 𝐴 = 𝐵)
37 simpr 110 . . . . 5 (((𝜑 ∧ ¬ 𝐴 ∈ (𝐾...(𝐽𝐾))) ∧ 𝐵 ∈ (𝐾...(𝐽𝐾))) → 𝐵 ∈ (𝐾...(𝐽𝐾)))
38 simplr 528 . . . . 5 (((𝜑 ∧ ¬ 𝐴 ∈ (𝐾...(𝐽𝐾))) ∧ 𝐵 ∈ (𝐾...(𝐽𝐾))) → ¬ 𝐴 ∈ (𝐾...(𝐽𝐾)))
3937, 38jca 306 . . . 4 (((𝜑 ∧ ¬ 𝐴 ∈ (𝐾...(𝐽𝐾))) ∧ 𝐵 ∈ (𝐾...(𝐽𝐾))) → (𝐵 ∈ (𝐾...(𝐽𝐾)) ∧ ¬ 𝐴 ∈ (𝐾...(𝐽𝐾))))
409eqcomd 2183 . . . . . 6 (𝜑 → (𝑄𝐵) = (𝑄𝐴))
411, 3, 7, 5, 40, 11iseqf1olemnab 10487 . . . . 5 (𝜑 → ¬ (𝐵 ∈ (𝐾...(𝐽𝐾)) ∧ ¬ 𝐴 ∈ (𝐾...(𝐽𝐾))))
4241ad2antrr 488 . . . 4 (((𝜑 ∧ ¬ 𝐴 ∈ (𝐾...(𝐽𝐾))) ∧ 𝐵 ∈ (𝐾...(𝐽𝐾))) → ¬ (𝐵 ∈ (𝐾...(𝐽𝐾)) ∧ ¬ 𝐴 ∈ (𝐾...(𝐽𝐾))))
4339, 42pm2.21dd 620 . . 3 (((𝜑 ∧ ¬ 𝐴 ∈ (𝐾...(𝐽𝐾))) ∧ 𝐵 ∈ (𝐾...(𝐽𝐾))) → 𝐴 = 𝐵)
441ad2antrr 488 . . . 4 (((𝜑 ∧ ¬ 𝐴 ∈ (𝐾...(𝐽𝐾))) ∧ ¬ 𝐵 ∈ (𝐾...(𝐽𝐾))) → 𝐾 ∈ (𝑀...𝑁))
453ad2antrr 488 . . . 4 (((𝜑 ∧ ¬ 𝐴 ∈ (𝐾...(𝐽𝐾))) ∧ ¬ 𝐵 ∈ (𝐾...(𝐽𝐾))) → 𝐽:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁))
465ad2antrr 488 . . . 4 (((𝜑 ∧ ¬ 𝐴 ∈ (𝐾...(𝐽𝐾))) ∧ ¬ 𝐵 ∈ (𝐾...(𝐽𝐾))) → 𝐴 ∈ (𝑀...𝑁))
477ad2antrr 488 . . . 4 (((𝜑 ∧ ¬ 𝐴 ∈ (𝐾...(𝐽𝐾))) ∧ ¬ 𝐵 ∈ (𝐾...(𝐽𝐾))) → 𝐵 ∈ (𝑀...𝑁))
489ad2antrr 488 . . . 4 (((𝜑 ∧ ¬ 𝐴 ∈ (𝐾...(𝐽𝐾))) ∧ ¬ 𝐵 ∈ (𝐾...(𝐽𝐾))) → (𝑄𝐴) = (𝑄𝐵))
49 simplr 528 . . . 4 (((𝜑 ∧ ¬ 𝐴 ∈ (𝐾...(𝐽𝐾))) ∧ ¬ 𝐵 ∈ (𝐾...(𝐽𝐾))) → ¬ 𝐴 ∈ (𝐾...(𝐽𝐾)))
50 simpr 110 . . . 4 (((𝜑 ∧ ¬ 𝐴 ∈ (𝐾...(𝐽𝐾))) ∧ ¬ 𝐵 ∈ (𝐾...(𝐽𝐾))) → ¬ 𝐵 ∈ (𝐾...(𝐽𝐾)))
5144, 45, 46, 47, 48, 11, 49, 50iseqf1olemnanb 10489 . . 3 (((𝜑 ∧ ¬ 𝐴 ∈ (𝐾...(𝐽𝐾))) ∧ ¬ 𝐵 ∈ (𝐾...(𝐽𝐾))) → 𝐴 = 𝐵)
5234adantr 276 . . 3 ((𝜑 ∧ ¬ 𝐴 ∈ (𝐾...(𝐽𝐾))) → (𝐵 ∈ (𝐾...(𝐽𝐾)) ∨ ¬ 𝐵 ∈ (𝐾...(𝐽𝐾))))
5343, 51, 52mpjaodan 798 . 2 ((𝜑 ∧ ¬ 𝐴 ∈ (𝐾...(𝐽𝐾))) → 𝐴 = 𝐵)
54 elfzelz 10024 . . . . 5 (𝐴 ∈ (𝑀...𝑁) → 𝐴 ∈ ℤ)
555, 54syl 14 . . . 4 (𝜑𝐴 ∈ ℤ)
56 fzdcel 10039 . . . 4 ((𝐴 ∈ ℤ ∧ 𝐾 ∈ ℤ ∧ (𝐽𝐾) ∈ ℤ) → DECID 𝐴 ∈ (𝐾...(𝐽𝐾)))
5755, 24, 30, 56syl3anc 1238 . . 3 (𝜑DECID 𝐴 ∈ (𝐾...(𝐽𝐾)))
58 exmiddc 836 . . 3 (DECID 𝐴 ∈ (𝐾...(𝐽𝐾)) → (𝐴 ∈ (𝐾...(𝐽𝐾)) ∨ ¬ 𝐴 ∈ (𝐾...(𝐽𝐾))))
5957, 58syl 14 . 2 (𝜑 → (𝐴 ∈ (𝐾...(𝐽𝐾)) ∨ ¬ 𝐴 ∈ (𝐾...(𝐽𝐾))))
6036, 53, 59mpjaodan 798 1 (𝜑𝐴 = 𝐵)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 104  wo 708  DECID wdc 834   = wceq 1353  wcel 2148  ifcif 3534  cmpt 4064  ccnv 4625  wf 5212  1-1-ontowf1o 5215  cfv 5216  (class class class)co 5874  1c1 7811  cmin 8127  cz 9252  ...cfz 10007
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 4121  ax-pow 4174  ax-pr 4209  ax-un 4433  ax-setind 4536  ax-cnex 7901  ax-resscn 7902  ax-1cn 7903  ax-1re 7904  ax-icn 7905  ax-addcl 7906  ax-addrcl 7907  ax-mulcl 7908  ax-addcom 7910  ax-addass 7912  ax-distr 7914  ax-i2m1 7915  ax-0lt1 7916  ax-0id 7918  ax-rnegex 7919  ax-cnre 7921  ax-pre-ltirr 7922  ax-pre-ltwlin 7923  ax-pre-lttrn 7924  ax-pre-ltadd 7926
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 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-int 3845  df-br 4004  df-opab 4065  df-mpt 4066  df-id 4293  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-rn 4637  df-res 4638  df-ima 4639  df-iota 5178  df-fun 5218  df-fn 5219  df-f 5220  df-f1 5221  df-fo 5222  df-f1o 5223  df-fv 5224  df-riota 5830  df-ov 5877  df-oprab 5878  df-mpo 5879  df-pnf 7993  df-mnf 7994  df-xr 7995  df-ltxr 7996  df-le 7997  df-sub 8129  df-neg 8130  df-inn 8919  df-n0 9176  df-z 9253  df-uz 9528  df-fz 10008
This theorem is referenced by:  iseqf1olemqf1o  10492
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