Mathbox for Mario Carneiro < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  erdszelem10 Structured version   Visualization version   GIF version

Theorem erdszelem10 30436
 Description: Lemma for erdsze 30438. (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 12633 . . . . . . . 8 (1...(𝑅 − 1)) ∈ Fin
2 fzfi 12633 . . . . . . . 8 (1...(𝑆 − 1)) ∈ Fin
3 xpfi 8116 . . . . . . . 8 (((1...(𝑅 − 1)) ∈ Fin ∧ (1...(𝑆 − 1)) ∈ Fin) → ((1...(𝑅 − 1)) × (1...(𝑆 − 1))) ∈ Fin)
41, 2, 3mp2an 704 . . . . . . 7 ((1...(𝑅 − 1)) × (1...(𝑆 − 1))) ∈ Fin
5 ssdomg 7887 . . . . . . 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 7971 . . . . . 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 13081 . . . . . . . . . 10 (((1...(𝑅 − 1)) ∈ Fin ∧ (1...(𝑆 − 1)) ∈ Fin) → (#‘((1...(𝑅 − 1)) × (1...(𝑆 − 1)))) = ((#‘(1...(𝑅 − 1))) · (#‘(1...(𝑆 − 1)))))
111, 2, 10mp2an 704 . . . . . . . . 9 (#‘((1...(𝑅 − 1)) × (1...(𝑆 − 1)))) = ((#‘(1...(𝑅 − 1))) · (#‘(1...(𝑆 − 1))))
12 erdszelem.r . . . . . . . . . . 11 (𝜑𝑅 ∈ ℕ)
13 nnm1nn0 11211 . . . . . . . . . . 11 (𝑅 ∈ ℕ → (𝑅 − 1) ∈ ℕ0)
14 hashfz1 12996 . . . . . . . . . . 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 11211 . . . . . . . . . . 11 (𝑆 ∈ ℕ → (𝑆 − 1) ∈ ℕ0)
18 hashfz1 12996 . . . . . . . . . . 11 ((𝑆 − 1) ∈ ℕ0 → (#‘(1...(𝑆 − 1))) = (𝑆 − 1))
1916, 17, 183syl 18 . . . . . . . . . 10 (𝜑 → (#‘(1...(𝑆 − 1))) = (𝑆 − 1))
2015, 19oveq12d 6567 . . . . . . . . 9 (𝜑 → ((#‘(1...(𝑅 − 1))) · (#‘(1...(𝑆 − 1)))) = ((𝑅 − 1) · (𝑆 − 1)))
2111, 20syl5eq 2656 . . . . . . . 8 (𝜑 → (#‘((1...(𝑅 − 1)) × (1...(𝑆 − 1)))) = ((𝑅 − 1) · (𝑆 − 1)))
22 erdsze.n . . . . . . . . . 10 (𝜑𝑁 ∈ ℕ)
2322nnnn0d 11228 . . . . . . . . 9 (𝜑𝑁 ∈ ℕ0)
24 hashfz1 12996 . . . . . . . . 9 (𝑁 ∈ ℕ0 → (#‘(1...𝑁)) = 𝑁)
2523, 24syl 17 . . . . . . . 8 (𝜑 → (#‘(1...𝑁)) = 𝑁)
269, 21, 253brtr4d 4615 . . . . . . 7 (𝜑 → (#‘((1...(𝑅 − 1)) × (1...(𝑆 − 1)))) < (#‘(1...𝑁)))
27 fzfid 12634 . . . . . . . 8 (𝜑 → (1...𝑁) ∈ Fin)
28 hashsdom 13031 . . . . . . . 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 694 . . . . . . 7 (𝜑 → ((#‘((1...(𝑅 − 1)) × (1...(𝑆 − 1)))) < (#‘(1...𝑁)) ↔ ((1...(𝑅 − 1)) × (1...(𝑆 − 1))) ≺ (1...𝑁)))
3026, 29mpbid 221 . . . . . 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 30435 . . . . . . 7 (𝜑𝑇:(1...𝑁)–1-1→(ℕ × ℕ))
36 f1f1orn 6061 . . . . . . 7 (𝑇:(1...𝑁)–1-1→(ℕ × ℕ) → 𝑇:(1...𝑁)–1-1-onto→ran 𝑇)
37 ovex 6577 . . . . . . . 8 (1...𝑁) ∈ V
3837f1oen 7862 . . . . . . 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 7979 . . . . . 6 ((((1...(𝑅 − 1)) × (1...(𝑆 − 1))) ≺ (1...𝑁) ∧ (1...𝑁) ≈ ran 𝑇) → ((1...(𝑅 − 1)) × (1...(𝑆 − 1))) ≺ ran 𝑇)
4130, 39, 40syl2anc 691 . . . . 5 (𝜑 → ((1...(𝑅 − 1)) × (1...(𝑆 − 1))) ≺ ran 𝑇)
428, 41nsyl3 132 . . . 4 (𝜑 → ¬ ran 𝑇 ⊆ ((1...(𝑅 − 1)) × (1...(𝑆 − 1))))
43 nss 3626 . . . . 5 (¬ ran 𝑇 ⊆ ((1...(𝑅 − 1)) × (1...(𝑆 − 1))) ↔ ∃𝑠(𝑠 ∈ ran 𝑇 ∧ ¬ 𝑠 ∈ ((1...(𝑅 − 1)) × (1...(𝑆 − 1)))))
44 df-rex 2902 . . . . 5 (∃𝑠 ∈ ran 𝑇 ¬ 𝑠 ∈ ((1...(𝑅 − 1)) × (1...(𝑆 − 1))) ↔ ∃𝑠(𝑠 ∈ ran 𝑇 ∧ ¬ 𝑠 ∈ ((1...(𝑅 − 1)) × (1...(𝑆 − 1)))))
4543, 44bitr4i 266 . . . 4 (¬ ran 𝑇 ⊆ ((1...(𝑅 − 1)) × (1...(𝑆 − 1))) ↔ ∃𝑠 ∈ ran 𝑇 ¬ 𝑠 ∈ ((1...(𝑅 − 1)) × (1...(𝑆 − 1))))
4642, 45sylib 207 . . 3 (𝜑 → ∃𝑠 ∈ ran 𝑇 ¬ 𝑠 ∈ ((1...(𝑅 − 1)) × (1...(𝑆 − 1))))
47 f1fn 6015 . . . 4 (𝑇:(1...𝑁)–1-1→(ℕ × ℕ) → 𝑇 Fn (1...𝑁))
48 eleq1 2676 . . . . . 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 6269 . . . 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 221 . 2 (𝜑 → ∃𝑚 ∈ (1...𝑁) ¬ (𝑇𝑚) ∈ ((1...(𝑅 − 1)) × (1...(𝑆 − 1))))
53 fveq2 6103 . . . . . . . . . 10 (𝑛 = 𝑚 → (𝐼𝑛) = (𝐼𝑚))
54 fveq2 6103 . . . . . . . . . 10 (𝑛 = 𝑚 → (𝐽𝑛) = (𝐽𝑚))
5553, 54opeq12d 4348 . . . . . . . . 9 (𝑛 = 𝑚 → ⟨(𝐼𝑛), (𝐽𝑛)⟩ = ⟨(𝐼𝑚), (𝐽𝑚)⟩)
56 opex 4859 . . . . . . . . 9 ⟨(𝐼𝑚), (𝐽𝑚)⟩ ∈ V
5755, 34, 56fvmpt 6191 . . . . . . . 8 (𝑚 ∈ (1...𝑁) → (𝑇𝑚) = ⟨(𝐼𝑚), (𝐽𝑚)⟩)
5857adantl 481 . . . . . . 7 ((𝜑𝑚 ∈ (1...𝑁)) → (𝑇𝑚) = ⟨(𝐼𝑚), (𝐽𝑚)⟩)
5958eleq1d 2672 . . . . . 6 ((𝜑𝑚 ∈ (1...𝑁)) → ((𝑇𝑚) ∈ ((1...(𝑅 − 1)) × (1...(𝑆 − 1))) ↔ ⟨(𝐼𝑚), (𝐽𝑚)⟩ ∈ ((1...(𝑅 − 1)) × (1...(𝑆 − 1)))))
60 opelxp 5070 . . . . . 6 (⟨(𝐼𝑚), (𝐽𝑚)⟩ ∈ ((1...(𝑅 − 1)) × (1...(𝑆 − 1))) ↔ ((𝐼𝑚) ∈ (1...(𝑅 − 1)) ∧ (𝐽𝑚) ∈ (1...(𝑆 − 1))))
6159, 60syl6bb 275 . . . . 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 275 . . 3 ((𝜑𝑚 ∈ (1...𝑁)) → (¬ (𝑇𝑚) ∈ ((1...(𝑅 − 1)) × (1...(𝑆 − 1))) ↔ (¬ (𝐼𝑚) ∈ (1...(𝑅 − 1)) ∨ ¬ (𝐽𝑚) ∈ (1...(𝑆 − 1)))))
6564rexbidva 3031 . 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 221 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 195   ∨ wo 382   ∧ wa 383   = wceq 1475  ∃wex 1695   ∈ wcel 1977  ∃wrex 2897  {crab 2900   ⊆ wss 3540  𝒫 cpw 4108  ⟨cop 4131   class class class wbr 4583   ↦ cmpt 4643   × cxp 5036  ◡ccnv 5037  ran crn 5039   ↾ cres 5040   “ cima 5041   Fn wfn 5799  –1-1→wf1 5801  –1-1-onto→wf1o 5803  ‘cfv 5804   Isom wiso 5805  (class class class)co 6549   ≈ cen 7838   ≼ cdom 7839   ≺ csdm 7840  Fincfn 7841  supcsup 8229  ℝcr 9814  1c1 9816   · cmul 9820   < clt 9953   − cmin 10145  ℕcn 10897  ℕ0cn0 11169  ...cfz 12197  #chash 12979 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892  ax-pre-sup 9893 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-isom 5813  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-2o 7448  df-oadd 7451  df-er 7629  df-map 7746  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-sup 8231  df-card 8648  df-cda 8873  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-n0 11170  df-xnn0 11241  df-z 11255  df-uz 11564  df-fz 12198  df-hash 12980 This theorem is referenced by:  erdszelem11  30437
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