ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  iseqf1olemqf1o GIF version

Theorem iseqf1olemqf1o 10506
Description: Lemma for seq3f1o 10517. 𝑄 is a permutation of (𝑀...𝑁). 𝑄 is formed from the constant portion of 𝐽, followed by the single element 𝐾 (at position 𝐾), followed by the rest of J (with the 𝐾 deleted and the elements before 𝐾 moved one position later to fill the gap). (Contributed by Jim Kingdon, 21-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)))
Assertion
Ref Expression
iseqf1olemqf1o (𝜑𝑄:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁))
Distinct variable groups:   𝑢,𝐽   𝑢,𝐾   𝑢,𝑀   𝑢,𝑁   𝜑,𝑢
Allowed substitution hint:   𝑄(𝑢)

Proof of Theorem iseqf1olemqf1o
Dummy variables 𝑣 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 iseqf1olemqf.k . . . 4 (𝜑𝐾 ∈ (𝑀...𝑁))
2 iseqf1olemqf.j . . . 4 (𝜑𝐽:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁))
3 iseqf1olemqf.q . . . 4 𝑄 = (𝑢 ∈ (𝑀...𝑁) ↦ if(𝑢 ∈ (𝐾...(𝐽𝐾)), if(𝑢 = 𝐾, 𝐾, (𝐽‘(𝑢 − 1))), (𝐽𝑢)))
41, 2, 3iseqf1olemqf 10504 . . 3 (𝜑𝑄:(𝑀...𝑁)⟶(𝑀...𝑁))
51ad2antrr 488 . . . . . 6 (((𝜑 ∧ (𝑣 ∈ (𝑀...𝑁) ∧ 𝑤 ∈ (𝑀...𝑁))) ∧ (𝑄𝑣) = (𝑄𝑤)) → 𝐾 ∈ (𝑀...𝑁))
62ad2antrr 488 . . . . . 6 (((𝜑 ∧ (𝑣 ∈ (𝑀...𝑁) ∧ 𝑤 ∈ (𝑀...𝑁))) ∧ (𝑄𝑣) = (𝑄𝑤)) → 𝐽:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁))
7 simplrl 535 . . . . . 6 (((𝜑 ∧ (𝑣 ∈ (𝑀...𝑁) ∧ 𝑤 ∈ (𝑀...𝑁))) ∧ (𝑄𝑣) = (𝑄𝑤)) → 𝑣 ∈ (𝑀...𝑁))
8 simplrr 536 . . . . . 6 (((𝜑 ∧ (𝑣 ∈ (𝑀...𝑁) ∧ 𝑤 ∈ (𝑀...𝑁))) ∧ (𝑄𝑣) = (𝑄𝑤)) → 𝑤 ∈ (𝑀...𝑁))
9 simpr 110 . . . . . 6 (((𝜑 ∧ (𝑣 ∈ (𝑀...𝑁) ∧ 𝑤 ∈ (𝑀...𝑁))) ∧ (𝑄𝑣) = (𝑄𝑤)) → (𝑄𝑣) = (𝑄𝑤))
105, 6, 3, 7, 8, 9iseqf1olemmo 10505 . . . . 5 (((𝜑 ∧ (𝑣 ∈ (𝑀...𝑁) ∧ 𝑤 ∈ (𝑀...𝑁))) ∧ (𝑄𝑣) = (𝑄𝑤)) → 𝑣 = 𝑤)
1110ex 115 . . . 4 ((𝜑 ∧ (𝑣 ∈ (𝑀...𝑁) ∧ 𝑤 ∈ (𝑀...𝑁))) → ((𝑄𝑣) = (𝑄𝑤) → 𝑣 = 𝑤))
1211ralrimivva 2569 . . 3 (𝜑 → ∀𝑣 ∈ (𝑀...𝑁)∀𝑤 ∈ (𝑀...𝑁)((𝑄𝑣) = (𝑄𝑤) → 𝑣 = 𝑤))
13 dff13 5782 . . 3 (𝑄:(𝑀...𝑁)–1-1→(𝑀...𝑁) ↔ (𝑄:(𝑀...𝑁)⟶(𝑀...𝑁) ∧ ∀𝑣 ∈ (𝑀...𝑁)∀𝑤 ∈ (𝑀...𝑁)((𝑄𝑣) = (𝑄𝑤) → 𝑣 = 𝑤)))
144, 12, 13sylanbrc 417 . 2 (𝜑𝑄:(𝑀...𝑁)–1-1→(𝑀...𝑁))
15 elfzel1 10037 . . . . . 6 (𝐾 ∈ (𝑀...𝑁) → 𝑀 ∈ ℤ)
161, 15syl 14 . . . . 5 (𝜑𝑀 ∈ ℤ)
17 elfzel2 10036 . . . . . 6 (𝐾 ∈ (𝑀...𝑁) → 𝑁 ∈ ℤ)
181, 17syl 14 . . . . 5 (𝜑𝑁 ∈ ℤ)
1916, 18fzfigd 10444 . . . 4 (𝜑 → (𝑀...𝑁) ∈ Fin)
20 enrefg 6777 . . . 4 ((𝑀...𝑁) ∈ Fin → (𝑀...𝑁) ≈ (𝑀...𝑁))
2119, 20syl 14 . . 3 (𝜑 → (𝑀...𝑁) ≈ (𝑀...𝑁))
22 f1finf1o 6959 . . 3 (((𝑀...𝑁) ≈ (𝑀...𝑁) ∧ (𝑀...𝑁) ∈ Fin) → (𝑄:(𝑀...𝑁)–1-1→(𝑀...𝑁) ↔ 𝑄:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁)))
2321, 19, 22syl2anc 411 . 2 (𝜑 → (𝑄:(𝑀...𝑁)–1-1→(𝑀...𝑁) ↔ 𝑄:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁)))
2414, 23mpbid 147 1 (𝜑𝑄:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wb 105   = wceq 1363  wcel 2158  wral 2465  ifcif 3546   class class class wbr 4015  cmpt 4076  ccnv 4637  wf 5224  1-1wf1 5225  1-1-ontowf1o 5227  cfv 5228  (class class class)co 5888  cen 6751  Fincfn 6753  1c1 7825  cmin 8141  cz 9266  ...cfz 10021
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 615  ax-in2 616  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2160  ax-14 2161  ax-ext 2169  ax-coll 4130  ax-sep 4133  ax-nul 4141  ax-pow 4186  ax-pr 4221  ax-un 4445  ax-setind 4548  ax-iinf 4599  ax-cnex 7915  ax-resscn 7916  ax-1cn 7917  ax-1re 7918  ax-icn 7919  ax-addcl 7920  ax-addrcl 7921  ax-mulcl 7922  ax-addcom 7924  ax-addass 7926  ax-distr 7928  ax-i2m1 7929  ax-0lt1 7930  ax-0id 7932  ax-rnegex 7933  ax-cnre 7935  ax-pre-ltirr 7936  ax-pre-ltwlin 7937  ax-pre-lttrn 7938  ax-pre-apti 7939  ax-pre-ltadd 7940
This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 980  df-3an 981  df-tru 1366  df-fal 1369  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ne 2358  df-nel 2453  df-ral 2470  df-rex 2471  df-reu 2472  df-rab 2474  df-v 2751  df-sbc 2975  df-csb 3070  df-dif 3143  df-un 3145  df-in 3147  df-ss 3154  df-nul 3435  df-if 3547  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-int 3857  df-iun 3900  df-br 4016  df-opab 4077  df-mpt 4078  df-tr 4114  df-id 4305  df-iord 4378  df-on 4380  df-ilim 4381  df-suc 4383  df-iom 4602  df-xp 4644  df-rel 4645  df-cnv 4646  df-co 4647  df-dm 4648  df-rn 4649  df-res 4650  df-ima 4651  df-iota 5190  df-fun 5230  df-fn 5231  df-f 5232  df-f1 5233  df-fo 5234  df-f1o 5235  df-fv 5236  df-riota 5844  df-ov 5891  df-oprab 5892  df-mpo 5893  df-1st 6154  df-2nd 6155  df-recs 6319  df-frec 6405  df-1o 6430  df-er 6548  df-en 6754  df-fin 6756  df-pnf 8007  df-mnf 8008  df-xr 8009  df-ltxr 8010  df-le 8011  df-sub 8143  df-neg 8144  df-inn 8933  df-n0 9190  df-z 9267  df-uz 9542  df-fz 10022
This theorem is referenced by:  seq3f1olemqsumkj  10511  seq3f1olemqsumk  10512  seq3f1olemstep  10514
  Copyright terms: Public domain W3C validator