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Theorem erdszelem10 31308
Description: Lemma for erdsze 31310. (Contributed by Mario Carneiro, 22-Jan-2015.)
Hypotheses
Ref Expression
erdsze.n (𝜑𝑁 ∈ ℕ)
erdsze.f (𝜑𝐹:(1...𝑁)–1-1→ℝ)
erdszelem.i 𝐼 = (𝑥 ∈ (1...𝑁) ↦ sup((# “ {𝑦 ∈ 𝒫 (1...𝑥) ∣ ((𝐹𝑦) Isom < , < (𝑦, (𝐹𝑦)) ∧ 𝑥𝑦)}), ℝ, < ))
erdszelem.j 𝐽 = (𝑥 ∈ (1...𝑁) ↦ sup((# “ {𝑦 ∈ 𝒫 (1...𝑥) ∣ ((𝐹𝑦) Isom < , < (𝑦, (𝐹𝑦)) ∧ 𝑥𝑦)}), ℝ, < ))
erdszelem.t 𝑇 = (𝑛 ∈ (1...𝑁) ↦ ⟨(𝐼𝑛), (𝐽𝑛)⟩)
erdszelem.r (𝜑𝑅 ∈ ℕ)
erdszelem.s (𝜑𝑆 ∈ ℕ)
erdszelem.m (𝜑 → ((𝑅 − 1) · (𝑆 − 1)) < 𝑁)
Assertion
Ref Expression
erdszelem10 (𝜑 → ∃𝑚 ∈ (1...𝑁)(¬ (𝐼𝑚) ∈ (1...(𝑅 − 1)) ∨ ¬ (𝐽𝑚) ∈ (1...(𝑆 − 1))))
Distinct variable groups:   𝑥,𝑦   𝑚,𝑛,𝑥,𝑦,𝐹   𝑛,𝐼,𝑥,𝑦   𝑛,𝐽,𝑥,𝑦   𝑅,𝑚,𝑥,𝑦   𝑚,𝑁,𝑛,𝑥,𝑦   𝜑,𝑚,𝑛,𝑥,𝑦   𝑆,𝑚,𝑥,𝑦   𝑇,𝑚
Allowed substitution hints:   𝑅(𝑛)   𝑆(𝑛)   𝑇(𝑥,𝑦,𝑛)   𝐼(𝑚)   𝐽(𝑚)

Proof of Theorem erdszelem10
Dummy variable 𝑠 is distinct from all other variables.
StepHypRef Expression
1 fzfi 12811 . . . . . . . 8 (1...(𝑅 − 1)) ∈ Fin
2 fzfi 12811 . . . . . . . 8 (1...(𝑆 − 1)) ∈ Fin
3 xpfi 8272 . . . . . . . 8 (((1...(𝑅 − 1)) ∈ Fin ∧ (1...(𝑆 − 1)) ∈ Fin) → ((1...(𝑅 − 1)) × (1...(𝑆 − 1))) ∈ Fin)
41, 2, 3mp2an 708 . . . . . . 7 ((1...(𝑅 − 1)) × (1...(𝑆 − 1))) ∈ Fin
5 ssdomg 8043 . . . . . . 7 (((1...(𝑅 − 1)) × (1...(𝑆 − 1))) ∈ Fin → (ran 𝑇 ⊆ ((1...(𝑅 − 1)) × (1...(𝑆 − 1))) → ran 𝑇 ≼ ((1...(𝑅 − 1)) × (1...(𝑆 − 1)))))
64, 5ax-mp 5 . . . . . 6 (ran 𝑇 ⊆ ((1...(𝑅 − 1)) × (1...(𝑆 − 1))) → ran 𝑇 ≼ ((1...(𝑅 − 1)) × (1...(𝑆 − 1))))
7 domnsym 8127 . . . . . 6 (ran 𝑇 ≼ ((1...(𝑅 − 1)) × (1...(𝑆 − 1))) → ¬ ((1...(𝑅 − 1)) × (1...(𝑆 − 1))) ≺ ran 𝑇)
86, 7syl 17 . . . . 5 (ran 𝑇 ⊆ ((1...(𝑅 − 1)) × (1...(𝑆 − 1))) → ¬ ((1...(𝑅 − 1)) × (1...(𝑆 − 1))) ≺ ran 𝑇)
9 erdszelem.m . . . . . . . 8 (𝜑 → ((𝑅 − 1) · (𝑆 − 1)) < 𝑁)
10 hashxp 13259 . . . . . . . . . 10 (((1...(𝑅 − 1)) ∈ Fin ∧ (1...(𝑆 − 1)) ∈ Fin) → (#‘((1...(𝑅 − 1)) × (1...(𝑆 − 1)))) = ((#‘(1...(𝑅 − 1))) · (#‘(1...(𝑆 − 1)))))
111, 2, 10mp2an 708 . . . . . . . . 9 (#‘((1...(𝑅 − 1)) × (1...(𝑆 − 1)))) = ((#‘(1...(𝑅 − 1))) · (#‘(1...(𝑆 − 1))))
12 erdszelem.r . . . . . . . . . . 11 (𝜑𝑅 ∈ ℕ)
13 nnm1nn0 11372 . . . . . . . . . . 11 (𝑅 ∈ ℕ → (𝑅 − 1) ∈ ℕ0)
14 hashfz1 13174 . . . . . . . . . . 11 ((𝑅 − 1) ∈ ℕ0 → (#‘(1...(𝑅 − 1))) = (𝑅 − 1))
1512, 13, 143syl 18 . . . . . . . . . 10 (𝜑 → (#‘(1...(𝑅 − 1))) = (𝑅 − 1))
16 erdszelem.s . . . . . . . . . . 11 (𝜑𝑆 ∈ ℕ)
17 nnm1nn0 11372 . . . . . . . . . . 11 (𝑆 ∈ ℕ → (𝑆 − 1) ∈ ℕ0)
18 hashfz1 13174 . . . . . . . . . . 11 ((𝑆 − 1) ∈ ℕ0 → (#‘(1...(𝑆 − 1))) = (𝑆 − 1))
1916, 17, 183syl 18 . . . . . . . . . 10 (𝜑 → (#‘(1...(𝑆 − 1))) = (𝑆 − 1))
2015, 19oveq12d 6708 . . . . . . . . 9 (𝜑 → ((#‘(1...(𝑅 − 1))) · (#‘(1...(𝑆 − 1)))) = ((𝑅 − 1) · (𝑆 − 1)))
2111, 20syl5eq 2697 . . . . . . . 8 (𝜑 → (#‘((1...(𝑅 − 1)) × (1...(𝑆 − 1)))) = ((𝑅 − 1) · (𝑆 − 1)))
22 erdsze.n . . . . . . . . . 10 (𝜑𝑁 ∈ ℕ)
2322nnnn0d 11389 . . . . . . . . 9 (𝜑𝑁 ∈ ℕ0)
24 hashfz1 13174 . . . . . . . . 9 (𝑁 ∈ ℕ0 → (#‘(1...𝑁)) = 𝑁)
2523, 24syl 17 . . . . . . . 8 (𝜑 → (#‘(1...𝑁)) = 𝑁)
269, 21, 253brtr4d 4717 . . . . . . 7 (𝜑 → (#‘((1...(𝑅 − 1)) × (1...(𝑆 − 1)))) < (#‘(1...𝑁)))
27 fzfid 12812 . . . . . . . 8 (𝜑 → (1...𝑁) ∈ Fin)
28 hashsdom 13208 . . . . . . . 8 ((((1...(𝑅 − 1)) × (1...(𝑆 − 1))) ∈ Fin ∧ (1...𝑁) ∈ Fin) → ((#‘((1...(𝑅 − 1)) × (1...(𝑆 − 1)))) < (#‘(1...𝑁)) ↔ ((1...(𝑅 − 1)) × (1...(𝑆 − 1))) ≺ (1...𝑁)))
294, 27, 28sylancr 696 . . . . . . 7 (𝜑 → ((#‘((1...(𝑅 − 1)) × (1...(𝑆 − 1)))) < (#‘(1...𝑁)) ↔ ((1...(𝑅 − 1)) × (1...(𝑆 − 1))) ≺ (1...𝑁)))
3026, 29mpbid 222 . . . . . 6 (𝜑 → ((1...(𝑅 − 1)) × (1...(𝑆 − 1))) ≺ (1...𝑁))
31 erdsze.f . . . . . . . 8 (𝜑𝐹:(1...𝑁)–1-1→ℝ)
32 erdszelem.i . . . . . . . 8 𝐼 = (𝑥 ∈ (1...𝑁) ↦ sup((# “ {𝑦 ∈ 𝒫 (1...𝑥) ∣ ((𝐹𝑦) Isom < , < (𝑦, (𝐹𝑦)) ∧ 𝑥𝑦)}), ℝ, < ))
33 erdszelem.j . . . . . . . 8 𝐽 = (𝑥 ∈ (1...𝑁) ↦ sup((# “ {𝑦 ∈ 𝒫 (1...𝑥) ∣ ((𝐹𝑦) Isom < , < (𝑦, (𝐹𝑦)) ∧ 𝑥𝑦)}), ℝ, < ))
34 erdszelem.t . . . . . . . 8 𝑇 = (𝑛 ∈ (1...𝑁) ↦ ⟨(𝐼𝑛), (𝐽𝑛)⟩)
3522, 31, 32, 33, 34erdszelem9 31307 . . . . . . 7 (𝜑𝑇:(1...𝑁)–1-1→(ℕ × ℕ))
36 f1f1orn 6186 . . . . . . 7 (𝑇:(1...𝑁)–1-1→(ℕ × ℕ) → 𝑇:(1...𝑁)–1-1-onto→ran 𝑇)
37 ovex 6718 . . . . . . . 8 (1...𝑁) ∈ V
3837f1oen 8018 . . . . . . 7 (𝑇:(1...𝑁)–1-1-onto→ran 𝑇 → (1...𝑁) ≈ ran 𝑇)
3935, 36, 383syl 18 . . . . . 6 (𝜑 → (1...𝑁) ≈ ran 𝑇)
40 sdomentr 8135 . . . . . 6 ((((1...(𝑅 − 1)) × (1...(𝑆 − 1))) ≺ (1...𝑁) ∧ (1...𝑁) ≈ ran 𝑇) → ((1...(𝑅 − 1)) × (1...(𝑆 − 1))) ≺ ran 𝑇)
4130, 39, 40syl2anc 694 . . . . 5 (𝜑 → ((1...(𝑅 − 1)) × (1...(𝑆 − 1))) ≺ ran 𝑇)
428, 41nsyl3 133 . . . 4 (𝜑 → ¬ ran 𝑇 ⊆ ((1...(𝑅 − 1)) × (1...(𝑆 − 1))))
43 nss 3696 . . . . 5 (¬ ran 𝑇 ⊆ ((1...(𝑅 − 1)) × (1...(𝑆 − 1))) ↔ ∃𝑠(𝑠 ∈ ran 𝑇 ∧ ¬ 𝑠 ∈ ((1...(𝑅 − 1)) × (1...(𝑆 − 1)))))
44 df-rex 2947 . . . . 5 (∃𝑠 ∈ ran 𝑇 ¬ 𝑠 ∈ ((1...(𝑅 − 1)) × (1...(𝑆 − 1))) ↔ ∃𝑠(𝑠 ∈ ran 𝑇 ∧ ¬ 𝑠 ∈ ((1...(𝑅 − 1)) × (1...(𝑆 − 1)))))
4543, 44bitr4i 267 . . . 4 (¬ ran 𝑇 ⊆ ((1...(𝑅 − 1)) × (1...(𝑆 − 1))) ↔ ∃𝑠 ∈ ran 𝑇 ¬ 𝑠 ∈ ((1...(𝑅 − 1)) × (1...(𝑆 − 1))))
4642, 45sylib 208 . . 3 (𝜑 → ∃𝑠 ∈ ran 𝑇 ¬ 𝑠 ∈ ((1...(𝑅 − 1)) × (1...(𝑆 − 1))))
47 f1fn 6140 . . . 4 (𝑇:(1...𝑁)–1-1→(ℕ × ℕ) → 𝑇 Fn (1...𝑁))
48 eleq1 2718 . . . . . 6 (𝑠 = (𝑇𝑚) → (𝑠 ∈ ((1...(𝑅 − 1)) × (1...(𝑆 − 1))) ↔ (𝑇𝑚) ∈ ((1...(𝑅 − 1)) × (1...(𝑆 − 1)))))
4948notbid 307 . . . . 5 (𝑠 = (𝑇𝑚) → (¬ 𝑠 ∈ ((1...(𝑅 − 1)) × (1...(𝑆 − 1))) ↔ ¬ (𝑇𝑚) ∈ ((1...(𝑅 − 1)) × (1...(𝑆 − 1)))))
5049rexrn 6401 . . . 4 (𝑇 Fn (1...𝑁) → (∃𝑠 ∈ ran 𝑇 ¬ 𝑠 ∈ ((1...(𝑅 − 1)) × (1...(𝑆 − 1))) ↔ ∃𝑚 ∈ (1...𝑁) ¬ (𝑇𝑚) ∈ ((1...(𝑅 − 1)) × (1...(𝑆 − 1)))))
5135, 47, 503syl 18 . . 3 (𝜑 → (∃𝑠 ∈ ran 𝑇 ¬ 𝑠 ∈ ((1...(𝑅 − 1)) × (1...(𝑆 − 1))) ↔ ∃𝑚 ∈ (1...𝑁) ¬ (𝑇𝑚) ∈ ((1...(𝑅 − 1)) × (1...(𝑆 − 1)))))
5246, 51mpbid 222 . 2 (𝜑 → ∃𝑚 ∈ (1...𝑁) ¬ (𝑇𝑚) ∈ ((1...(𝑅 − 1)) × (1...(𝑆 − 1))))
53 fveq2 6229 . . . . . . . . . 10 (𝑛 = 𝑚 → (𝐼𝑛) = (𝐼𝑚))
54 fveq2 6229 . . . . . . . . . 10 (𝑛 = 𝑚 → (𝐽𝑛) = (𝐽𝑚))
5553, 54opeq12d 4441 . . . . . . . . 9 (𝑛 = 𝑚 → ⟨(𝐼𝑛), (𝐽𝑛)⟩ = ⟨(𝐼𝑚), (𝐽𝑚)⟩)
56 opex 4962 . . . . . . . . 9 ⟨(𝐼𝑚), (𝐽𝑚)⟩ ∈ V
5755, 34, 56fvmpt 6321 . . . . . . . 8 (𝑚 ∈ (1...𝑁) → (𝑇𝑚) = ⟨(𝐼𝑚), (𝐽𝑚)⟩)
5857adantl 481 . . . . . . 7 ((𝜑𝑚 ∈ (1...𝑁)) → (𝑇𝑚) = ⟨(𝐼𝑚), (𝐽𝑚)⟩)
5958eleq1d 2715 . . . . . 6 ((𝜑𝑚 ∈ (1...𝑁)) → ((𝑇𝑚) ∈ ((1...(𝑅 − 1)) × (1...(𝑆 − 1))) ↔ ⟨(𝐼𝑚), (𝐽𝑚)⟩ ∈ ((1...(𝑅 − 1)) × (1...(𝑆 − 1)))))
60 opelxp 5180 . . . . . 6 (⟨(𝐼𝑚), (𝐽𝑚)⟩ ∈ ((1...(𝑅 − 1)) × (1...(𝑆 − 1))) ↔ ((𝐼𝑚) ∈ (1...(𝑅 − 1)) ∧ (𝐽𝑚) ∈ (1...(𝑆 − 1))))
6159, 60syl6bb 276 . . . . 5 ((𝜑𝑚 ∈ (1...𝑁)) → ((𝑇𝑚) ∈ ((1...(𝑅 − 1)) × (1...(𝑆 − 1))) ↔ ((𝐼𝑚) ∈ (1...(𝑅 − 1)) ∧ (𝐽𝑚) ∈ (1...(𝑆 − 1)))))
6261notbid 307 . . . 4 ((𝜑𝑚 ∈ (1...𝑁)) → (¬ (𝑇𝑚) ∈ ((1...(𝑅 − 1)) × (1...(𝑆 − 1))) ↔ ¬ ((𝐼𝑚) ∈ (1...(𝑅 − 1)) ∧ (𝐽𝑚) ∈ (1...(𝑆 − 1)))))
63 ianor 508 . . . 4 (¬ ((𝐼𝑚) ∈ (1...(𝑅 − 1)) ∧ (𝐽𝑚) ∈ (1...(𝑆 − 1))) ↔ (¬ (𝐼𝑚) ∈ (1...(𝑅 − 1)) ∨ ¬ (𝐽𝑚) ∈ (1...(𝑆 − 1))))
6462, 63syl6bb 276 . . 3 ((𝜑𝑚 ∈ (1...𝑁)) → (¬ (𝑇𝑚) ∈ ((1...(𝑅 − 1)) × (1...(𝑆 − 1))) ↔ (¬ (𝐼𝑚) ∈ (1...(𝑅 − 1)) ∨ ¬ (𝐽𝑚) ∈ (1...(𝑆 − 1)))))
6564rexbidva 3078 . 2 (𝜑 → (∃𝑚 ∈ (1...𝑁) ¬ (𝑇𝑚) ∈ ((1...(𝑅 − 1)) × (1...(𝑆 − 1))) ↔ ∃𝑚 ∈ (1...𝑁)(¬ (𝐼𝑚) ∈ (1...(𝑅 − 1)) ∨ ¬ (𝐽𝑚) ∈ (1...(𝑆 − 1)))))
6652, 65mpbid 222 1 (𝜑 → ∃𝑚 ∈ (1...𝑁)(¬ (𝐼𝑚) ∈ (1...(𝑅 − 1)) ∨ ¬ (𝐽𝑚) ∈ (1...(𝑆 − 1))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wo 382  wa 383   = wceq 1523  wex 1744  wcel 2030  wrex 2942  {crab 2945  wss 3607  𝒫 cpw 4191  cop 4216   class class class wbr 4685  cmpt 4762   × cxp 5141  ccnv 5142  ran crn 5144  cres 5145  cima 5146   Fn wfn 5921  1-1wf1 5923  1-1-ontowf1o 5925  cfv 5926   Isom wiso 5927  (class class class)co 6690  cen 7994  cdom 7995  csdm 7996  Fincfn 7997  supcsup 8387  cr 9973  1c1 9975   · cmul 9979   < clt 10112  cmin 10304  cn 11058  0cn0 11330  ...cfz 12364  #chash 13157
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051  ax-pre-sup 10052
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-isom 5935  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-2o 7606  df-oadd 7609  df-er 7787  df-map 7901  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-sup 8389  df-card 8803  df-cda 9028  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-nn 11059  df-n0 11331  df-xnn0 11402  df-z 11416  df-uz 11726  df-fz 12365  df-hash 13158
This theorem is referenced by:  erdszelem11  31309
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