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Theorem iseqf1olemmo 10316
 Description: Lemma for seq3f1o 10328. 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 480 . . . 4 (((𝜑𝐴 ∈ (𝐾...(𝐽𝐾))) ∧ 𝐵 ∈ (𝐾...(𝐽𝐾))) → 𝐾 ∈ (𝑀...𝑁))
3 iseqf1olemqf.j . . . . 5 (𝜑𝐽:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁))
43ad2antrr 480 . . . 4 (((𝜑𝐴 ∈ (𝐾...(𝐽𝐾))) ∧ 𝐵 ∈ (𝐾...(𝐽𝐾))) → 𝐽:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁))
5 iseqf1olemmo.a . . . . 5 (𝜑𝐴 ∈ (𝑀...𝑁))
65ad2antrr 480 . . . 4 (((𝜑𝐴 ∈ (𝐾...(𝐽𝐾))) ∧ 𝐵 ∈ (𝐾...(𝐽𝐾))) → 𝐴 ∈ (𝑀...𝑁))
7 iseqf1olemmo.b . . . . 5 (𝜑𝐵 ∈ (𝑀...𝑁))
87ad2antrr 480 . . . 4 (((𝜑𝐴 ∈ (𝐾...(𝐽𝐾))) ∧ 𝐵 ∈ (𝐾...(𝐽𝐾))) → 𝐵 ∈ (𝑀...𝑁))
9 iseqf1olemmo.eq . . . . 5 (𝜑 → (𝑄𝐴) = (𝑄𝐵))
109ad2antrr 480 . . . 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 520 . . . 4 (((𝜑𝐴 ∈ (𝐾...(𝐽𝐾))) ∧ 𝐵 ∈ (𝐾...(𝐽𝐾))) → 𝐴 ∈ (𝐾...(𝐽𝐾)))
13 simpr 109 . . . 4 (((𝜑𝐴 ∈ (𝐾...(𝐽𝐾))) ∧ 𝐵 ∈ (𝐾...(𝐽𝐾))) → 𝐵 ∈ (𝐾...(𝐽𝐾)))
142, 4, 6, 8, 10, 11, 12, 13iseqf1olemab 10313 . . 3 (((𝜑𝐴 ∈ (𝐾...(𝐽𝐾))) ∧ 𝐵 ∈ (𝐾...(𝐽𝐾))) → 𝐴 = 𝐵)
15 simplr 520 . . . . 5 (((𝜑𝐴 ∈ (𝐾...(𝐽𝐾))) ∧ ¬ 𝐵 ∈ (𝐾...(𝐽𝐾))) → 𝐴 ∈ (𝐾...(𝐽𝐾)))
16 simpr 109 . . . . 5 (((𝜑𝐴 ∈ (𝐾...(𝐽𝐾))) ∧ ¬ 𝐵 ∈ (𝐾...(𝐽𝐾))) → ¬ 𝐵 ∈ (𝐾...(𝐽𝐾)))
1715, 16jca 304 . . . 4 (((𝜑𝐴 ∈ (𝐾...(𝐽𝐾))) ∧ ¬ 𝐵 ∈ (𝐾...(𝐽𝐾))) → (𝐴 ∈ (𝐾...(𝐽𝐾)) ∧ ¬ 𝐵 ∈ (𝐾...(𝐽𝐾))))
181, 3, 5, 7, 9, 11iseqf1olemnab 10312 . . . . 5 (𝜑 → ¬ (𝐴 ∈ (𝐾...(𝐽𝐾)) ∧ ¬ 𝐵 ∈ (𝐾...(𝐽𝐾))))
1918ad2antrr 480 . . . 4 (((𝜑𝐴 ∈ (𝐾...(𝐽𝐾))) ∧ ¬ 𝐵 ∈ (𝐾...(𝐽𝐾))) → ¬ (𝐴 ∈ (𝐾...(𝐽𝐾)) ∧ ¬ 𝐵 ∈ (𝐾...(𝐽𝐾))))
2017, 19pm2.21dd 610 . . 3 (((𝜑𝐴 ∈ (𝐾...(𝐽𝐾))) ∧ ¬ 𝐵 ∈ (𝐾...(𝐽𝐾))) → 𝐴 = 𝐵)
21 elfzelz 9857 . . . . . . 7 (𝐵 ∈ (𝑀...𝑁) → 𝐵 ∈ ℤ)
227, 21syl 14 . . . . . 6 (𝜑𝐵 ∈ ℤ)
23 elfzelz 9857 . . . . . . 7 (𝐾 ∈ (𝑀...𝑁) → 𝐾 ∈ ℤ)
241, 23syl 14 . . . . . 6 (𝜑𝐾 ∈ ℤ)
25 f1ocnv 5389 . . . . . . . . 9 (𝐽:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) → 𝐽:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁))
26 f1of 5376 . . . . . . . . 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, 1ffvelrnd 5565 . . . . . . 7 (𝜑 → (𝐽𝐾) ∈ (𝑀...𝑁))
29 elfzelz 9857 . . . . . . 7 ((𝐽𝐾) ∈ (𝑀...𝑁) → (𝐽𝐾) ∈ ℤ)
3028, 29syl 14 . . . . . 6 (𝜑 → (𝐽𝐾) ∈ ℤ)
31 fzdcel 9871 . . . . . 6 ((𝐵 ∈ ℤ ∧ 𝐾 ∈ ℤ ∧ (𝐽𝐾) ∈ ℤ) → DECID 𝐵 ∈ (𝐾...(𝐽𝐾)))
3222, 24, 30, 31syl3anc 1217 . . . . 5 (𝜑DECID 𝐵 ∈ (𝐾...(𝐽𝐾)))
33 exmiddc 822 . . . . 5 (DECID 𝐵 ∈ (𝐾...(𝐽𝐾)) → (𝐵 ∈ (𝐾...(𝐽𝐾)) ∨ ¬ 𝐵 ∈ (𝐾...(𝐽𝐾))))
3432, 33syl 14 . . . 4 (𝜑 → (𝐵 ∈ (𝐾...(𝐽𝐾)) ∨ ¬ 𝐵 ∈ (𝐾...(𝐽𝐾))))
3534adantr 274 . . 3 ((𝜑𝐴 ∈ (𝐾...(𝐽𝐾))) → (𝐵 ∈ (𝐾...(𝐽𝐾)) ∨ ¬ 𝐵 ∈ (𝐾...(𝐽𝐾))))
3614, 20, 35mpjaodan 788 . 2 ((𝜑𝐴 ∈ (𝐾...(𝐽𝐾))) → 𝐴 = 𝐵)
37 simpr 109 . . . . 5 (((𝜑 ∧ ¬ 𝐴 ∈ (𝐾...(𝐽𝐾))) ∧ 𝐵 ∈ (𝐾...(𝐽𝐾))) → 𝐵 ∈ (𝐾...(𝐽𝐾)))
38 simplr 520 . . . . 5 (((𝜑 ∧ ¬ 𝐴 ∈ (𝐾...(𝐽𝐾))) ∧ 𝐵 ∈ (𝐾...(𝐽𝐾))) → ¬ 𝐴 ∈ (𝐾...(𝐽𝐾)))
3937, 38jca 304 . . . 4 (((𝜑 ∧ ¬ 𝐴 ∈ (𝐾...(𝐽𝐾))) ∧ 𝐵 ∈ (𝐾...(𝐽𝐾))) → (𝐵 ∈ (𝐾...(𝐽𝐾)) ∧ ¬ 𝐴 ∈ (𝐾...(𝐽𝐾))))
409eqcomd 2146 . . . . . 6 (𝜑 → (𝑄𝐵) = (𝑄𝐴))
411, 3, 7, 5, 40, 11iseqf1olemnab 10312 . . . . 5 (𝜑 → ¬ (𝐵 ∈ (𝐾...(𝐽𝐾)) ∧ ¬ 𝐴 ∈ (𝐾...(𝐽𝐾))))
4241ad2antrr 480 . . . 4 (((𝜑 ∧ ¬ 𝐴 ∈ (𝐾...(𝐽𝐾))) ∧ 𝐵 ∈ (𝐾...(𝐽𝐾))) → ¬ (𝐵 ∈ (𝐾...(𝐽𝐾)) ∧ ¬ 𝐴 ∈ (𝐾...(𝐽𝐾))))
4339, 42pm2.21dd 610 . . 3 (((𝜑 ∧ ¬ 𝐴 ∈ (𝐾...(𝐽𝐾))) ∧ 𝐵 ∈ (𝐾...(𝐽𝐾))) → 𝐴 = 𝐵)
441ad2antrr 480 . . . 4 (((𝜑 ∧ ¬ 𝐴 ∈ (𝐾...(𝐽𝐾))) ∧ ¬ 𝐵 ∈ (𝐾...(𝐽𝐾))) → 𝐾 ∈ (𝑀...𝑁))
453ad2antrr 480 . . . 4 (((𝜑 ∧ ¬ 𝐴 ∈ (𝐾...(𝐽𝐾))) ∧ ¬ 𝐵 ∈ (𝐾...(𝐽𝐾))) → 𝐽:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁))
465ad2antrr 480 . . . 4 (((𝜑 ∧ ¬ 𝐴 ∈ (𝐾...(𝐽𝐾))) ∧ ¬ 𝐵 ∈ (𝐾...(𝐽𝐾))) → 𝐴 ∈ (𝑀...𝑁))
477ad2antrr 480 . . . 4 (((𝜑 ∧ ¬ 𝐴 ∈ (𝐾...(𝐽𝐾))) ∧ ¬ 𝐵 ∈ (𝐾...(𝐽𝐾))) → 𝐵 ∈ (𝑀...𝑁))
489ad2antrr 480 . . . 4 (((𝜑 ∧ ¬ 𝐴 ∈ (𝐾...(𝐽𝐾))) ∧ ¬ 𝐵 ∈ (𝐾...(𝐽𝐾))) → (𝑄𝐴) = (𝑄𝐵))
49 simplr 520 . . . 4 (((𝜑 ∧ ¬ 𝐴 ∈ (𝐾...(𝐽𝐾))) ∧ ¬ 𝐵 ∈ (𝐾...(𝐽𝐾))) → ¬ 𝐴 ∈ (𝐾...(𝐽𝐾)))
50 simpr 109 . . . 4 (((𝜑 ∧ ¬ 𝐴 ∈ (𝐾...(𝐽𝐾))) ∧ ¬ 𝐵 ∈ (𝐾...(𝐽𝐾))) → ¬ 𝐵 ∈ (𝐾...(𝐽𝐾)))
5144, 45, 46, 47, 48, 11, 49, 50iseqf1olemnanb 10314 . . 3 (((𝜑 ∧ ¬ 𝐴 ∈ (𝐾...(𝐽𝐾))) ∧ ¬ 𝐵 ∈ (𝐾...(𝐽𝐾))) → 𝐴 = 𝐵)
5234adantr 274 . . 3 ((𝜑 ∧ ¬ 𝐴 ∈ (𝐾...(𝐽𝐾))) → (𝐵 ∈ (𝐾...(𝐽𝐾)) ∨ ¬ 𝐵 ∈ (𝐾...(𝐽𝐾))))
5343, 51, 52mpjaodan 788 . 2 ((𝜑 ∧ ¬ 𝐴 ∈ (𝐾...(𝐽𝐾))) → 𝐴 = 𝐵)
54 elfzelz 9857 . . . . 5 (𝐴 ∈ (𝑀...𝑁) → 𝐴 ∈ ℤ)
555, 54syl 14 . . . 4 (𝜑𝐴 ∈ ℤ)
56 fzdcel 9871 . . . 4 ((𝐴 ∈ ℤ ∧ 𝐾 ∈ ℤ ∧ (𝐽𝐾) ∈ ℤ) → DECID 𝐴 ∈ (𝐾...(𝐽𝐾)))
5755, 24, 30, 56syl3anc 1217 . . 3 (𝜑DECID 𝐴 ∈ (𝐾...(𝐽𝐾)))
58 exmiddc 822 . . 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 788 1 (𝜑𝐴 = 𝐵)
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 103   ∨ wo 698  DECID wdc 820   = wceq 1332   ∈ wcel 1481  ifcif 3480   ↦ cmpt 3998  ◡ccnv 4547  ⟶wf 5128  –1-1-onto→wf1o 5131  ‘cfv 5132  (class class class)co 5783  1c1 7665   − cmin 7977  ℤcz 9098  ...cfz 9841 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 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4055  ax-pow 4107  ax-pr 4140  ax-un 4364  ax-setind 4461  ax-cnex 7755  ax-resscn 7756  ax-1cn 7757  ax-1re 7758  ax-icn 7759  ax-addcl 7760  ax-addrcl 7761  ax-mulcl 7762  ax-addcom 7764  ax-addass 7766  ax-distr 7768  ax-i2m1 7769  ax-0lt1 7770  ax-0id 7772  ax-rnegex 7773  ax-cnre 7775  ax-pre-ltirr 7776  ax-pre-ltwlin 7777  ax-pre-lttrn 7778  ax-pre-ltadd 7780 This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-nel 2405  df-ral 2422  df-rex 2423  df-reu 2424  df-rab 2426  df-v 2692  df-sbc 2915  df-dif 3079  df-un 3081  df-in 3083  df-ss 3090  df-if 3481  df-pw 3518  df-sn 3539  df-pr 3540  df-op 3542  df-uni 3746  df-int 3781  df-br 3939  df-opab 3999  df-mpt 4000  df-id 4224  df-xp 4554  df-rel 4555  df-cnv 4556  df-co 4557  df-dm 4558  df-rn 4559  df-res 4560  df-ima 4561  df-iota 5097  df-fun 5134  df-fn 5135  df-f 5136  df-f1 5137  df-fo 5138  df-f1o 5139  df-fv 5140  df-riota 5739  df-ov 5786  df-oprab 5787  df-mpo 5788  df-pnf 7846  df-mnf 7847  df-xr 7848  df-ltxr 7849  df-le 7850  df-sub 7979  df-neg 7980  df-inn 8765  df-n0 9022  df-z 9099  df-uz 9371  df-fz 9842 This theorem is referenced by:  iseqf1olemqf1o  10317
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