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Theorem erdsze2lem1 31484
Description: Lemma for erdsze2 31486. (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 11518 . . . . . . . . . . 11 (𝑅 ∈ ℕ → (𝑅 − 1) ∈ ℕ0)
42, 3syl 17 . . . . . . . . . 10 (𝜑 → (𝑅 − 1) ∈ ℕ0)
5 erdsze2.s . . . . . . . . . . 11 (𝜑𝑆 ∈ ℕ)
6 nnm1nn0 11518 . . . . . . . . . . 11 (𝑆 ∈ ℕ → (𝑆 − 1) ∈ ℕ0)
75, 6syl 17 . . . . . . . . . 10 (𝜑 → (𝑆 − 1) ∈ ℕ0)
84, 7nn0mulcld 11540 . . . . . . . . 9 (𝜑 → ((𝑅 − 1) · (𝑆 − 1)) ∈ ℕ0)
91, 8syl5eqel 2835 . . . . . . . 8 (𝜑𝑁 ∈ ℕ0)
10 peano2nn0 11517 . . . . . . . 8 (𝑁 ∈ ℕ0 → (𝑁 + 1) ∈ ℕ0)
11 hashfz1 13320 . . . . . . . 8 ((𝑁 + 1) ∈ ℕ0 → (♯‘(1...(𝑁 + 1))) = (𝑁 + 1))
129, 10, 113syl 18 . . . . . . 7 (𝜑 → (♯‘(1...(𝑁 + 1))) = (𝑁 + 1))
1312adantr 472 . . . . . 6 ((𝜑𝐴 ∈ Fin) → (♯‘(1...(𝑁 + 1))) = (𝑁 + 1))
14 erdsze2lem.l . . . . . . . 8 (𝜑𝑁 < (♯‘𝐴))
1514adantr 472 . . . . . . 7 ((𝜑𝐴 ∈ Fin) → 𝑁 < (♯‘𝐴))
16 hashcl 13331 . . . . . . . 8 (𝐴 ∈ Fin → (♯‘𝐴) ∈ ℕ0)
17 nn0ltp1le 11619 . . . . . . . 8 ((𝑁 ∈ ℕ0 ∧ (♯‘𝐴) ∈ ℕ0) → (𝑁 < (♯‘𝐴) ↔ (𝑁 + 1) ≤ (♯‘𝐴)))
189, 16, 17syl2an 495 . . . . . . 7 ((𝜑𝐴 ∈ Fin) → (𝑁 < (♯‘𝐴) ↔ (𝑁 + 1) ≤ (♯‘𝐴)))
1915, 18mpbid 222 . . . . . 6 ((𝜑𝐴 ∈ Fin) → (𝑁 + 1) ≤ (♯‘𝐴))
2013, 19eqbrtrd 4818 . . . . 5 ((𝜑𝐴 ∈ Fin) → (♯‘(1...(𝑁 + 1))) ≤ (♯‘𝐴))
21 fzfid 12958 . . . . . 6 ((𝜑𝐴 ∈ Fin) → (1...(𝑁 + 1)) ∈ Fin)
22 simpr 479 . . . . . 6 ((𝜑𝐴 ∈ Fin) → 𝐴 ∈ Fin)
23 hashdom 13352 . . . . . 6 (((1...(𝑁 + 1)) ∈ Fin ∧ 𝐴 ∈ Fin) → ((♯‘(1...(𝑁 + 1))) ≤ (♯‘𝐴) ↔ (1...(𝑁 + 1)) ≼ 𝐴))
2421, 22, 23syl2anc 696 . . . . 5 ((𝜑𝐴 ∈ Fin) → ((♯‘(1...(𝑁 + 1))) ≤ (♯‘𝐴) ↔ (1...(𝑁 + 1)) ≼ 𝐴))
2520, 24mpbid 222 . . . 4 ((𝜑𝐴 ∈ Fin) → (1...(𝑁 + 1)) ≼ 𝐴)
26 simpr 479 . . . . . 6 ((𝜑 ∧ ¬ 𝐴 ∈ Fin) → ¬ 𝐴 ∈ Fin)
27 fzfid 12958 . . . . . 6 ((𝜑 ∧ ¬ 𝐴 ∈ Fin) → (1...(𝑁 + 1)) ∈ Fin)
28 isinffi 9000 . . . . . 6 ((¬ 𝐴 ∈ Fin ∧ (1...(𝑁 + 1)) ∈ Fin) → ∃𝑓 𝑓:(1...(𝑁 + 1))–1-1𝐴)
2926, 27, 28syl2anc 696 . . . . 5 ((𝜑 ∧ ¬ 𝐴 ∈ Fin) → ∃𝑓 𝑓:(1...(𝑁 + 1))–1-1𝐴)
30 erdsze2.a . . . . . . . 8 (𝜑𝐴 ⊆ ℝ)
31 reex 10211 . . . . . . . 8 ℝ ∈ V
32 ssexg 4948 . . . . . . . 8 ((𝐴 ⊆ ℝ ∧ ℝ ∈ V) → 𝐴 ∈ V)
3330, 31, 32sylancl 697 . . . . . . 7 (𝜑𝐴 ∈ V)
3433adantr 472 . . . . . 6 ((𝜑 ∧ ¬ 𝐴 ∈ Fin) → 𝐴 ∈ V)
35 brdomg 8123 . . . . . 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 867 . . 3 (𝜑 → (1...(𝑁 + 1)) ≼ 𝐴)
39 domeng 8127 . . . 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 813 . . . . . 6 ((𝜑 ∧ ((1...(𝑁 + 1)) ≈ 𝑠𝑠𝐴)) → 𝑠𝐴)
4330adantr 472 . . . . . 6 ((𝜑 ∧ ((1...(𝑁 + 1)) ≈ 𝑠𝑠𝐴)) → 𝐴 ⊆ ℝ)
4442, 43sstrd 3746 . . . . 5 ((𝜑 ∧ ((1...(𝑁 + 1)) ≈ 𝑠𝑠𝐴)) → 𝑠 ⊆ ℝ)
45 ltso 10302 . . . . 5 < Or ℝ
46 soss 5197 . . . . 5 (𝑠 ⊆ ℝ → ( < Or ℝ → < Or 𝑠))
4744, 45, 46mpisyl 21 . . . 4 ((𝜑 ∧ ((1...(𝑁 + 1)) ≈ 𝑠𝑠𝐴)) → < Or 𝑠)
48 fzfid 12958 . . . . 5 ((𝜑 ∧ ((1...(𝑁 + 1)) ≈ 𝑠𝑠𝐴)) → (1...(𝑁 + 1)) ∈ Fin)
49 simprl 811 . . . . . 6 ((𝜑 ∧ ((1...(𝑁 + 1)) ≈ 𝑠𝑠𝐴)) → (1...(𝑁 + 1)) ≈ 𝑠)
50 enfi 8333 . . . . . 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 13430 . . . 4 (( < Or 𝑠𝑠 ∈ Fin) → ∃𝑓 𝑓 Isom < , < ((1...(♯‘𝑠)), 𝑠))
5447, 52, 53syl2anc 696 . . 3 ((𝜑 ∧ ((1...(𝑁 + 1)) ≈ 𝑠𝑠𝐴)) → ∃𝑓 𝑓 Isom < , < ((1...(♯‘𝑠)), 𝑠))
55 isof1o 6728 . . . . . . . . . 10 (𝑓 Isom < , < ((1...(♯‘𝑠)), 𝑠) → 𝑓:(1...(♯‘𝑠))–1-1-onto𝑠)
5655adantl 473 . . . . . . . . 9 (((𝜑 ∧ ((1...(𝑁 + 1)) ≈ 𝑠𝑠𝐴)) ∧ 𝑓 Isom < , < ((1...(♯‘𝑠)), 𝑠)) → 𝑓:(1...(♯‘𝑠))–1-1-onto𝑠)
57 hashen 13321 . . . . . . . . . . . . . . 15 (((1...(𝑁 + 1)) ∈ Fin ∧ 𝑠 ∈ Fin) → ((♯‘(1...(𝑁 + 1))) = (♯‘𝑠) ↔ (1...(𝑁 + 1)) ≈ 𝑠))
5848, 52, 57syl2anc 696 . . . . . . . . . . . . . 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 472 . . . . . . . . . . . . 13 ((𝜑 ∧ ((1...(𝑁 + 1)) ≈ 𝑠𝑠𝐴)) → (♯‘(1...(𝑁 + 1))) = (𝑁 + 1))
6159, 60eqtr3d 2788 . . . . . . . . . . . 12 ((𝜑 ∧ ((1...(𝑁 + 1)) ≈ 𝑠𝑠𝐴)) → (♯‘𝑠) = (𝑁 + 1))
6261adantr 472 . . . . . . . . . . 11 (((𝜑 ∧ ((1...(𝑁 + 1)) ≈ 𝑠𝑠𝐴)) ∧ 𝑓 Isom < , < ((1...(♯‘𝑠)), 𝑠)) → (♯‘𝑠) = (𝑁 + 1))
6362oveq2d 6821 . . . . . . . . . 10 (((𝜑 ∧ ((1...(𝑁 + 1)) ≈ 𝑠𝑠𝐴)) ∧ 𝑓 Isom < , < ((1...(♯‘𝑠)), 𝑠)) → (1...(♯‘𝑠)) = (1...(𝑁 + 1)))
64 f1oeq2 6281 . . . . . . . . . 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 6289 . . . . . . . 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 820 . . . . . . 7 (((𝜑 ∧ ((1...(𝑁 + 1)) ≈ 𝑠𝑠𝐴)) ∧ 𝑓 Isom < , < ((1...(♯‘𝑠)), 𝑠)) → 𝑠𝐴)
70 f1ss 6259 . . . . . . 7 ((𝑓:(1...(𝑁 + 1))–1-1𝑠𝑠𝐴) → 𝑓:(1...(𝑁 + 1))–1-1𝐴)
7168, 69, 70syl2anc 696 . . . . . 6 (((𝜑 ∧ ((1...(𝑁 + 1)) ≈ 𝑠𝑠𝐴)) ∧ 𝑓 Isom < , < ((1...(♯‘𝑠)), 𝑠)) → 𝑓:(1...(𝑁 + 1))–1-1𝐴)
72 simpr 479 . . . . . . . 8 (((𝜑 ∧ ((1...(𝑁 + 1)) ≈ 𝑠𝑠𝐴)) ∧ 𝑓 Isom < , < ((1...(♯‘𝑠)), 𝑠)) → 𝑓 Isom < , < ((1...(♯‘𝑠)), 𝑠))
73 f1ofo 6297 . . . . . . . . 9 (𝑓:(1...(♯‘𝑠))–1-1-onto𝑠𝑓:(1...(♯‘𝑠))–onto𝑠)
74 forn 6271 . . . . . . . . 9 (𝑓:(1...(♯‘𝑠))–onto𝑠 → ran 𝑓 = 𝑠)
75 isoeq5 6726 . . . . . . . . 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 6725 . . . . . . . 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 555 . . . . 5 (((𝜑 ∧ ((1...(𝑁 + 1)) ≈ 𝑠𝑠𝐴)) ∧ 𝑓 Isom < , < ((1...(♯‘𝑠)), 𝑠)) → (𝑓:(1...(𝑁 + 1))–1-1𝐴𝑓 Isom < , < ((1...(𝑁 + 1)), ran 𝑓)))
8281ex 449 . . . 4 ((𝜑 ∧ ((1...(𝑁 + 1)) ≈ 𝑠𝑠𝐴)) → (𝑓 Isom < , < ((1...(♯‘𝑠)), 𝑠) → (𝑓:(1...(𝑁 + 1))–1-1𝐴𝑓 Isom < , < ((1...(𝑁 + 1)), ran 𝑓))))
8382eximdv 1987 . . 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 2004 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 383   = wceq 1624  wex 1845  wcel 2131  Vcvv 3332  wss 3707   class class class wbr 4796   Or wor 5178  ran crn 5259  1-1wf1 6038  ontowfo 6039  1-1-ontowf1o 6040  cfv 6041   Isom wiso 6042  (class class class)co 6805  cen 8110  cdom 8111  Fincfn 8113  cr 10119  1c1 10121   + caddc 10123   · cmul 10125   < clt 10258  cle 10259  cmin 10450  cn 11204  0cn0 11476  ...cfz 12511  chash 13303
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1863  ax-4 1878  ax-5 1980  ax-6 2046  ax-7 2082  ax-8 2133  ax-9 2140  ax-10 2160  ax-11 2175  ax-12 2188  ax-13 2383  ax-ext 2732  ax-rep 4915  ax-sep 4925  ax-nul 4933  ax-pow 4984  ax-pr 5047  ax-un 7106  ax-cnex 10176  ax-resscn 10177  ax-1cn 10178  ax-icn 10179  ax-addcl 10180  ax-addrcl 10181  ax-mulcl 10182  ax-mulrcl 10183  ax-mulcom 10184  ax-addass 10185  ax-mulass 10186  ax-distr 10187  ax-i2m1 10188  ax-1ne0 10189  ax-1rid 10190  ax-rnegex 10191  ax-rrecex 10192  ax-cnre 10193  ax-pre-lttri 10194  ax-pre-lttrn 10195  ax-pre-ltadd 10196  ax-pre-mulgt0 10197
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1627  df-ex 1846  df-nf 1851  df-sb 2039  df-eu 2603  df-mo 2604  df-clab 2739  df-cleq 2745  df-clel 2748  df-nfc 2883  df-ne 2925  df-nel 3028  df-ral 3047  df-rex 3048  df-reu 3049  df-rmo 3050  df-rab 3051  df-v 3334  df-sbc 3569  df-csb 3667  df-dif 3710  df-un 3712  df-in 3714  df-ss 3721  df-pss 3723  df-nul 4051  df-if 4223  df-pw 4296  df-sn 4314  df-pr 4316  df-tp 4318  df-op 4320  df-uni 4581  df-int 4620  df-iun 4666  df-br 4797  df-opab 4857  df-mpt 4874  df-tr 4897  df-id 5166  df-eprel 5171  df-po 5179  df-so 5180  df-fr 5217  df-se 5218  df-we 5219  df-xp 5264  df-rel 5265  df-cnv 5266  df-co 5267  df-dm 5268  df-rn 5269  df-res 5270  df-ima 5271  df-pred 5833  df-ord 5879  df-on 5880  df-lim 5881  df-suc 5882  df-iota 6004  df-fun 6043  df-fn 6044  df-f 6045  df-f1 6046  df-fo 6047  df-f1o 6048  df-fv 6049  df-isom 6050  df-riota 6766  df-ov 6808  df-oprab 6809  df-mpt2 6810  df-om 7223  df-1st 7325  df-2nd 7326  df-wrecs 7568  df-recs 7629  df-rdg 7667  df-1o 7721  df-oadd 7725  df-er 7903  df-en 8114  df-dom 8115  df-sdom 8116  df-fin 8117  df-oi 8572  df-card 8947  df-pnf 10260  df-mnf 10261  df-xr 10262  df-ltxr 10263  df-le 10264  df-sub 10452  df-neg 10453  df-nn 11205  df-n0 11477  df-xnn0 11548  df-z 11562  df-uz 11872  df-fz 12512  df-hash 13304
This theorem is referenced by:  erdsze2  31486
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