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Theorem erdsze2lem1 30885
Description: Lemma for erdsze2 30887. (Contributed by Mario Carneiro, 22-Jan-2015.)
Hypotheses
Ref Expression
erdsze2.r (𝜑𝑅 ∈ ℕ)
erdsze2.s (𝜑𝑆 ∈ ℕ)
erdsze2.f (𝜑𝐹:𝐴1-1→ℝ)
erdsze2.a (𝜑𝐴 ⊆ ℝ)
erdsze2lem.n 𝑁 = ((𝑅 − 1) · (𝑆 − 1))
erdsze2lem.l (𝜑𝑁 < (#‘𝐴))
Assertion
Ref Expression
erdsze2lem1 (𝜑 → ∃𝑓(𝑓:(1...(𝑁 + 1))–1-1𝐴𝑓 Isom < , < ((1...(𝑁 + 1)), ran 𝑓)))
Distinct variable groups:   𝐴,𝑓   𝑓,𝐹   𝑅,𝑓   𝑆,𝑓   𝑓,𝑁   𝜑,𝑓

Proof of Theorem erdsze2lem1
Dummy variable 𝑠 is distinct from all other variables.
StepHypRef Expression
1 erdsze2lem.n . . . . . . . . 9 𝑁 = ((𝑅 − 1) · (𝑆 − 1))
2 erdsze2.r . . . . . . . . . . 11 (𝜑𝑅 ∈ ℕ)
3 nnm1nn0 11279 . . . . . . . . . . 11 (𝑅 ∈ ℕ → (𝑅 − 1) ∈ ℕ0)
42, 3syl 17 . . . . . . . . . 10 (𝜑 → (𝑅 − 1) ∈ ℕ0)
5 erdsze2.s . . . . . . . . . . 11 (𝜑𝑆 ∈ ℕ)
6 nnm1nn0 11279 . . . . . . . . . . 11 (𝑆 ∈ ℕ → (𝑆 − 1) ∈ ℕ0)
75, 6syl 17 . . . . . . . . . 10 (𝜑 → (𝑆 − 1) ∈ ℕ0)
84, 7nn0mulcld 11301 . . . . . . . . 9 (𝜑 → ((𝑅 − 1) · (𝑆 − 1)) ∈ ℕ0)
91, 8syl5eqel 2708 . . . . . . . 8 (𝜑𝑁 ∈ ℕ0)
10 peano2nn0 11278 . . . . . . . 8 (𝑁 ∈ ℕ0 → (𝑁 + 1) ∈ ℕ0)
11 hashfz1 13071 . . . . . . . 8 ((𝑁 + 1) ∈ ℕ0 → (#‘(1...(𝑁 + 1))) = (𝑁 + 1))
129, 10, 113syl 18 . . . . . . 7 (𝜑 → (#‘(1...(𝑁 + 1))) = (𝑁 + 1))
1312adantr 481 . . . . . 6 ((𝜑𝐴 ∈ Fin) → (#‘(1...(𝑁 + 1))) = (𝑁 + 1))
14 erdsze2lem.l . . . . . . . 8 (𝜑𝑁 < (#‘𝐴))
1514adantr 481 . . . . . . 7 ((𝜑𝐴 ∈ Fin) → 𝑁 < (#‘𝐴))
16 hashcl 13084 . . . . . . . 8 (𝐴 ∈ Fin → (#‘𝐴) ∈ ℕ0)
17 nn0ltp1le 11380 . . . . . . . 8 ((𝑁 ∈ ℕ0 ∧ (#‘𝐴) ∈ ℕ0) → (𝑁 < (#‘𝐴) ↔ (𝑁 + 1) ≤ (#‘𝐴)))
189, 16, 17syl2an 494 . . . . . . 7 ((𝜑𝐴 ∈ Fin) → (𝑁 < (#‘𝐴) ↔ (𝑁 + 1) ≤ (#‘𝐴)))
1915, 18mpbid 222 . . . . . 6 ((𝜑𝐴 ∈ Fin) → (𝑁 + 1) ≤ (#‘𝐴))
2013, 19eqbrtrd 4640 . . . . 5 ((𝜑𝐴 ∈ Fin) → (#‘(1...(𝑁 + 1))) ≤ (#‘𝐴))
21 fzfid 12709 . . . . . 6 ((𝜑𝐴 ∈ Fin) → (1...(𝑁 + 1)) ∈ Fin)
22 simpr 477 . . . . . 6 ((𝜑𝐴 ∈ Fin) → 𝐴 ∈ Fin)
23 hashdom 13105 . . . . . 6 (((1...(𝑁 + 1)) ∈ Fin ∧ 𝐴 ∈ Fin) → ((#‘(1...(𝑁 + 1))) ≤ (#‘𝐴) ↔ (1...(𝑁 + 1)) ≼ 𝐴))
2421, 22, 23syl2anc 692 . . . . 5 ((𝜑𝐴 ∈ Fin) → ((#‘(1...(𝑁 + 1))) ≤ (#‘𝐴) ↔ (1...(𝑁 + 1)) ≼ 𝐴))
2520, 24mpbid 222 . . . 4 ((𝜑𝐴 ∈ Fin) → (1...(𝑁 + 1)) ≼ 𝐴)
26 simpr 477 . . . . . 6 ((𝜑 ∧ ¬ 𝐴 ∈ Fin) → ¬ 𝐴 ∈ Fin)
27 fzfid 12709 . . . . . 6 ((𝜑 ∧ ¬ 𝐴 ∈ Fin) → (1...(𝑁 + 1)) ∈ Fin)
28 isinffi 8763 . . . . . 6 ((¬ 𝐴 ∈ Fin ∧ (1...(𝑁 + 1)) ∈ Fin) → ∃𝑓 𝑓:(1...(𝑁 + 1))–1-1𝐴)
2926, 27, 28syl2anc 692 . . . . 5 ((𝜑 ∧ ¬ 𝐴 ∈ Fin) → ∃𝑓 𝑓:(1...(𝑁 + 1))–1-1𝐴)
30 erdsze2.a . . . . . . . 8 (𝜑𝐴 ⊆ ℝ)
31 reex 9972 . . . . . . . 8 ℝ ∈ V
32 ssexg 4769 . . . . . . . 8 ((𝐴 ⊆ ℝ ∧ ℝ ∈ V) → 𝐴 ∈ V)
3330, 31, 32sylancl 693 . . . . . . 7 (𝜑𝐴 ∈ V)
3433adantr 481 . . . . . 6 ((𝜑 ∧ ¬ 𝐴 ∈ Fin) → 𝐴 ∈ V)
35 brdomg 7910 . . . . . 6 (𝐴 ∈ V → ((1...(𝑁 + 1)) ≼ 𝐴 ↔ ∃𝑓 𝑓:(1...(𝑁 + 1))–1-1𝐴))
3634, 35syl 17 . . . . 5 ((𝜑 ∧ ¬ 𝐴 ∈ Fin) → ((1...(𝑁 + 1)) ≼ 𝐴 ↔ ∃𝑓 𝑓:(1...(𝑁 + 1))–1-1𝐴))
3729, 36mpbird 247 . . . 4 ((𝜑 ∧ ¬ 𝐴 ∈ Fin) → (1...(𝑁 + 1)) ≼ 𝐴)
3825, 37pm2.61dan 831 . . 3 (𝜑 → (1...(𝑁 + 1)) ≼ 𝐴)
39 domeng 7914 . . . 4 (𝐴 ∈ V → ((1...(𝑁 + 1)) ≼ 𝐴 ↔ ∃𝑠((1...(𝑁 + 1)) ≈ 𝑠𝑠𝐴)))
4033, 39syl 17 . . 3 (𝜑 → ((1...(𝑁 + 1)) ≼ 𝐴 ↔ ∃𝑠((1...(𝑁 + 1)) ≈ 𝑠𝑠𝐴)))
4138, 40mpbid 222 . 2 (𝜑 → ∃𝑠((1...(𝑁 + 1)) ≈ 𝑠𝑠𝐴))
42 simprr 795 . . . . . 6 ((𝜑 ∧ ((1...(𝑁 + 1)) ≈ 𝑠𝑠𝐴)) → 𝑠𝐴)
4330adantr 481 . . . . . 6 ((𝜑 ∧ ((1...(𝑁 + 1)) ≈ 𝑠𝑠𝐴)) → 𝐴 ⊆ ℝ)
4442, 43sstrd 3598 . . . . 5 ((𝜑 ∧ ((1...(𝑁 + 1)) ≈ 𝑠𝑠𝐴)) → 𝑠 ⊆ ℝ)
45 ltso 10063 . . . . 5 < Or ℝ
46 soss 5018 . . . . 5 (𝑠 ⊆ ℝ → ( < Or ℝ → < Or 𝑠))
4744, 45, 46mpisyl 21 . . . 4 ((𝜑 ∧ ((1...(𝑁 + 1)) ≈ 𝑠𝑠𝐴)) → < Or 𝑠)
48 fzfid 12709 . . . . 5 ((𝜑 ∧ ((1...(𝑁 + 1)) ≈ 𝑠𝑠𝐴)) → (1...(𝑁 + 1)) ∈ Fin)
49 simprl 793 . . . . . 6 ((𝜑 ∧ ((1...(𝑁 + 1)) ≈ 𝑠𝑠𝐴)) → (1...(𝑁 + 1)) ≈ 𝑠)
50 enfi 8121 . . . . . 6 ((1...(𝑁 + 1)) ≈ 𝑠 → ((1...(𝑁 + 1)) ∈ Fin ↔ 𝑠 ∈ Fin))
5149, 50syl 17 . . . . 5 ((𝜑 ∧ ((1...(𝑁 + 1)) ≈ 𝑠𝑠𝐴)) → ((1...(𝑁 + 1)) ∈ Fin ↔ 𝑠 ∈ Fin))
5248, 51mpbid 222 . . . 4 ((𝜑 ∧ ((1...(𝑁 + 1)) ≈ 𝑠𝑠𝐴)) → 𝑠 ∈ Fin)
53 fz1iso 13181 . . . 4 (( < Or 𝑠𝑠 ∈ Fin) → ∃𝑓 𝑓 Isom < , < ((1...(#‘𝑠)), 𝑠))
5447, 52, 53syl2anc 692 . . 3 ((𝜑 ∧ ((1...(𝑁 + 1)) ≈ 𝑠𝑠𝐴)) → ∃𝑓 𝑓 Isom < , < ((1...(#‘𝑠)), 𝑠))
55 isof1o 6528 . . . . . . . . . 10 (𝑓 Isom < , < ((1...(#‘𝑠)), 𝑠) → 𝑓:(1...(#‘𝑠))–1-1-onto𝑠)
5655adantl 482 . . . . . . . . 9 (((𝜑 ∧ ((1...(𝑁 + 1)) ≈ 𝑠𝑠𝐴)) ∧ 𝑓 Isom < , < ((1...(#‘𝑠)), 𝑠)) → 𝑓:(1...(#‘𝑠))–1-1-onto𝑠)
57 hashen 13072 . . . . . . . . . . . . . . 15 (((1...(𝑁 + 1)) ∈ Fin ∧ 𝑠 ∈ Fin) → ((#‘(1...(𝑁 + 1))) = (#‘𝑠) ↔ (1...(𝑁 + 1)) ≈ 𝑠))
5848, 52, 57syl2anc 692 . . . . . . . . . . . . . 14 ((𝜑 ∧ ((1...(𝑁 + 1)) ≈ 𝑠𝑠𝐴)) → ((#‘(1...(𝑁 + 1))) = (#‘𝑠) ↔ (1...(𝑁 + 1)) ≈ 𝑠))
5949, 58mpbird 247 . . . . . . . . . . . . 13 ((𝜑 ∧ ((1...(𝑁 + 1)) ≈ 𝑠𝑠𝐴)) → (#‘(1...(𝑁 + 1))) = (#‘𝑠))
6012adantr 481 . . . . . . . . . . . . 13 ((𝜑 ∧ ((1...(𝑁 + 1)) ≈ 𝑠𝑠𝐴)) → (#‘(1...(𝑁 + 1))) = (𝑁 + 1))
6159, 60eqtr3d 2662 . . . . . . . . . . . 12 ((𝜑 ∧ ((1...(𝑁 + 1)) ≈ 𝑠𝑠𝐴)) → (#‘𝑠) = (𝑁 + 1))
6261adantr 481 . . . . . . . . . . 11 (((𝜑 ∧ ((1...(𝑁 + 1)) ≈ 𝑠𝑠𝐴)) ∧ 𝑓 Isom < , < ((1...(#‘𝑠)), 𝑠)) → (#‘𝑠) = (𝑁 + 1))
6362oveq2d 6621 . . . . . . . . . 10 (((𝜑 ∧ ((1...(𝑁 + 1)) ≈ 𝑠𝑠𝐴)) ∧ 𝑓 Isom < , < ((1...(#‘𝑠)), 𝑠)) → (1...(#‘𝑠)) = (1...(𝑁 + 1)))
64 f1oeq2 6087 . . . . . . . . . 10 ((1...(#‘𝑠)) = (1...(𝑁 + 1)) → (𝑓:(1...(#‘𝑠))–1-1-onto𝑠𝑓:(1...(𝑁 + 1))–1-1-onto𝑠))
6563, 64syl 17 . . . . . . . . 9 (((𝜑 ∧ ((1...(𝑁 + 1)) ≈ 𝑠𝑠𝐴)) ∧ 𝑓 Isom < , < ((1...(#‘𝑠)), 𝑠)) → (𝑓:(1...(#‘𝑠))–1-1-onto𝑠𝑓:(1...(𝑁 + 1))–1-1-onto𝑠))
6656, 65mpbid 222 . . . . . . . 8 (((𝜑 ∧ ((1...(𝑁 + 1)) ≈ 𝑠𝑠𝐴)) ∧ 𝑓 Isom < , < ((1...(#‘𝑠)), 𝑠)) → 𝑓:(1...(𝑁 + 1))–1-1-onto𝑠)
67 f1of1 6095 . . . . . . . 8 (𝑓:(1...(𝑁 + 1))–1-1-onto𝑠𝑓:(1...(𝑁 + 1))–1-1𝑠)
6866, 67syl 17 . . . . . . 7 (((𝜑 ∧ ((1...(𝑁 + 1)) ≈ 𝑠𝑠𝐴)) ∧ 𝑓 Isom < , < ((1...(#‘𝑠)), 𝑠)) → 𝑓:(1...(𝑁 + 1))–1-1𝑠)
69 simplrr 800 . . . . . . 7 (((𝜑 ∧ ((1...(𝑁 + 1)) ≈ 𝑠𝑠𝐴)) ∧ 𝑓 Isom < , < ((1...(#‘𝑠)), 𝑠)) → 𝑠𝐴)
70 f1ss 6065 . . . . . . 7 ((𝑓:(1...(𝑁 + 1))–1-1𝑠𝑠𝐴) → 𝑓:(1...(𝑁 + 1))–1-1𝐴)
7168, 69, 70syl2anc 692 . . . . . 6 (((𝜑 ∧ ((1...(𝑁 + 1)) ≈ 𝑠𝑠𝐴)) ∧ 𝑓 Isom < , < ((1...(#‘𝑠)), 𝑠)) → 𝑓:(1...(𝑁 + 1))–1-1𝐴)
72 simpr 477 . . . . . . . 8 (((𝜑 ∧ ((1...(𝑁 + 1)) ≈ 𝑠𝑠𝐴)) ∧ 𝑓 Isom < , < ((1...(#‘𝑠)), 𝑠)) → 𝑓 Isom < , < ((1...(#‘𝑠)), 𝑠))
73 f1ofo 6103 . . . . . . . . 9 (𝑓:(1...(#‘𝑠))–1-1-onto𝑠𝑓:(1...(#‘𝑠))–onto𝑠)
74 forn 6077 . . . . . . . . 9 (𝑓:(1...(#‘𝑠))–onto𝑠 → ran 𝑓 = 𝑠)
75 isoeq5 6526 . . . . . . . . 9 (ran 𝑓 = 𝑠 → (𝑓 Isom < , < ((1...(#‘𝑠)), ran 𝑓) ↔ 𝑓 Isom < , < ((1...(#‘𝑠)), 𝑠)))
7656, 73, 74, 754syl 19 . . . . . . . 8 (((𝜑 ∧ ((1...(𝑁 + 1)) ≈ 𝑠𝑠𝐴)) ∧ 𝑓 Isom < , < ((1...(#‘𝑠)), 𝑠)) → (𝑓 Isom < , < ((1...(#‘𝑠)), ran 𝑓) ↔ 𝑓 Isom < , < ((1...(#‘𝑠)), 𝑠)))
7772, 76mpbird 247 . . . . . . 7 (((𝜑 ∧ ((1...(𝑁 + 1)) ≈ 𝑠𝑠𝐴)) ∧ 𝑓 Isom < , < ((1...(#‘𝑠)), 𝑠)) → 𝑓 Isom < , < ((1...(#‘𝑠)), ran 𝑓))
78 isoeq4 6525 . . . . . . . 8 ((1...(#‘𝑠)) = (1...(𝑁 + 1)) → (𝑓 Isom < , < ((1...(#‘𝑠)), ran 𝑓) ↔ 𝑓 Isom < , < ((1...(𝑁 + 1)), ran 𝑓)))
7963, 78syl 17 . . . . . . 7 (((𝜑 ∧ ((1...(𝑁 + 1)) ≈ 𝑠𝑠𝐴)) ∧ 𝑓 Isom < , < ((1...(#‘𝑠)), 𝑠)) → (𝑓 Isom < , < ((1...(#‘𝑠)), ran 𝑓) ↔ 𝑓 Isom < , < ((1...(𝑁 + 1)), ran 𝑓)))
8077, 79mpbid 222 . . . . . 6 (((𝜑 ∧ ((1...(𝑁 + 1)) ≈ 𝑠𝑠𝐴)) ∧ 𝑓 Isom < , < ((1...(#‘𝑠)), 𝑠)) → 𝑓 Isom < , < ((1...(𝑁 + 1)), ran 𝑓))
8171, 80jca 554 . . . . 5 (((𝜑 ∧ ((1...(𝑁 + 1)) ≈ 𝑠𝑠𝐴)) ∧ 𝑓 Isom < , < ((1...(#‘𝑠)), 𝑠)) → (𝑓:(1...(𝑁 + 1))–1-1𝐴𝑓 Isom < , < ((1...(𝑁 + 1)), ran 𝑓)))
8281ex 450 . . . 4 ((𝜑 ∧ ((1...(𝑁 + 1)) ≈ 𝑠𝑠𝐴)) → (𝑓 Isom < , < ((1...(#‘𝑠)), 𝑠) → (𝑓:(1...(𝑁 + 1))–1-1𝐴𝑓 Isom < , < ((1...(𝑁 + 1)), ran 𝑓))))
8382eximdv 1848 . . 3 ((𝜑 ∧ ((1...(𝑁 + 1)) ≈ 𝑠𝑠𝐴)) → (∃𝑓 𝑓 Isom < , < ((1...(#‘𝑠)), 𝑠) → ∃𝑓(𝑓:(1...(𝑁 + 1))–1-1𝐴𝑓 Isom < , < ((1...(𝑁 + 1)), ran 𝑓))))
8454, 83mpd 15 . 2 ((𝜑 ∧ ((1...(𝑁 + 1)) ≈ 𝑠𝑠𝐴)) → ∃𝑓(𝑓:(1...(𝑁 + 1))–1-1𝐴𝑓 Isom < , < ((1...(𝑁 + 1)), ran 𝑓)))
8541, 84exlimddv 1865 1 (𝜑 → ∃𝑓(𝑓:(1...(𝑁 + 1))–1-1𝐴𝑓 Isom < , < ((1...(𝑁 + 1)), ran 𝑓)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 384   = wceq 1480  wex 1701  wcel 1992  Vcvv 3191  wss 3560   class class class wbr 4618   Or wor 4999  ran crn 5080  1-1wf1 5847  ontowfo 5848  1-1-ontowf1o 5849  cfv 5850   Isom wiso 5851  (class class class)co 6605  cen 7897  cdom 7898  Fincfn 7900  cr 9880  1c1 9882   + caddc 9884   · cmul 9886   < clt 10019  cle 10020  cmin 10211  cn 10965  0cn0 11237  ...cfz 12265  #chash 13054
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6903  ax-cnex 9937  ax-resscn 9938  ax-1cn 9939  ax-icn 9940  ax-addcl 9941  ax-addrcl 9942  ax-mulcl 9943  ax-mulrcl 9944  ax-mulcom 9945  ax-addass 9946  ax-mulass 9947  ax-distr 9948  ax-i2m1 9949  ax-1ne0 9950  ax-1rid 9951  ax-rnegex 9952  ax-rrecex 9953  ax-cnre 9954  ax-pre-lttri 9955  ax-pre-lttrn 9956  ax-pre-ltadd 9957  ax-pre-mulgt0 9958
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-nel 2900  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3193  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-int 4446  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-se 5039  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5642  df-ord 5688  df-on 5689  df-lim 5690  df-suc 5691  df-iota 5813  df-fun 5852  df-fn 5853  df-f 5854  df-f1 5855  df-fo 5856  df-f1o 5857  df-fv 5858  df-isom 5859  df-riota 6566  df-ov 6608  df-oprab 6609  df-mpt2 6610  df-om 7014  df-1st 7116  df-2nd 7117  df-wrecs 7353  df-recs 7414  df-rdg 7452  df-1o 7506  df-oadd 7510  df-er 7688  df-en 7901  df-dom 7902  df-sdom 7903  df-fin 7904  df-oi 8360  df-card 8710  df-pnf 10021  df-mnf 10022  df-xr 10023  df-ltxr 10024  df-le 10025  df-sub 10213  df-neg 10214  df-nn 10966  df-n0 11238  df-xnn0 11309  df-z 11323  df-uz 11632  df-fz 12266  df-hash 13055
This theorem is referenced by:  erdsze2  30887
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