Users' Mathboxes Mathbox for Jeff Madsen < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  heiborlem4 Structured version   Visualization version   GIF version

Theorem heiborlem4 34979
Description: Lemma for heibor 34986. Using the function 𝑇 constructed in heiborlem3 34978, construct an infinite path in 𝐺. (Contributed by Jeff Madsen, 23-Jan-2014.)
Hypotheses
Ref Expression
heibor.1 𝐽 = (MetOpen‘𝐷)
heibor.3 𝐾 = {𝑢 ∣ ¬ ∃𝑣 ∈ (𝒫 𝑈 ∩ Fin)𝑢 𝑣}
heibor.4 𝐺 = {⟨𝑦, 𝑛⟩ ∣ (𝑛 ∈ ℕ0𝑦 ∈ (𝐹𝑛) ∧ (𝑦𝐵𝑛) ∈ 𝐾)}
heibor.5 𝐵 = (𝑧𝑋, 𝑚 ∈ ℕ0 ↦ (𝑧(ball‘𝐷)(1 / (2↑𝑚))))
heibor.6 (𝜑𝐷 ∈ (CMet‘𝑋))
heibor.7 (𝜑𝐹:ℕ0⟶(𝒫 𝑋 ∩ Fin))
heibor.8 (𝜑 → ∀𝑛 ∈ ℕ0 𝑋 = 𝑦 ∈ (𝐹𝑛)(𝑦𝐵𝑛))
heibor.9 (𝜑 → ∀𝑥𝐺 ((𝑇𝑥)𝐺((2nd𝑥) + 1) ∧ ((𝐵𝑥) ∩ ((𝑇𝑥)𝐵((2nd𝑥) + 1))) ∈ 𝐾))
heibor.10 (𝜑𝐶𝐺0)
heibor.11 𝑆 = seq0(𝑇, (𝑚 ∈ ℕ0 ↦ if(𝑚 = 0, 𝐶, (𝑚 − 1))))
Assertion
Ref Expression
heiborlem4 ((𝜑𝐴 ∈ ℕ0) → (𝑆𝐴)𝐺𝐴)
Distinct variable groups:   𝑥,𝑛,𝑦,𝐴   𝑢,𝑛,𝐹,𝑥,𝑦   𝑥,𝐺   𝜑,𝑥   𝑚,𝑛,𝑢,𝑣,𝑥,𝑦,𝑧,𝐷   𝑇,𝑚,𝑛,𝑥,𝑦,𝑧   𝐵,𝑛,𝑢,𝑣,𝑦   𝑚,𝐽,𝑛,𝑢,𝑣,𝑥,𝑦,𝑧   𝑈,𝑛,𝑢,𝑣,𝑥,𝑦,𝑧   𝑆,𝑚,𝑛,𝑢,𝑣,𝑥,𝑦,𝑧   𝑚,𝑋,𝑛,𝑢,𝑣,𝑥,𝑦,𝑧   𝐶,𝑚,𝑛,𝑢,𝑣,𝑦   𝑛,𝐾,𝑥,𝑦,𝑧   𝑥,𝐵
Allowed substitution hints:   𝜑(𝑦,𝑧,𝑣,𝑢,𝑚,𝑛)   𝐴(𝑧,𝑣,𝑢,𝑚)   𝐵(𝑧,𝑚)   𝐶(𝑥,𝑧)   𝑇(𝑣,𝑢)   𝑈(𝑚)   𝐹(𝑧,𝑣,𝑚)   𝐺(𝑦,𝑧,𝑣,𝑢,𝑚,𝑛)   𝐾(𝑣,𝑢,𝑚)

Proof of Theorem heiborlem4
Dummy variable 𝑘 is distinct from all other variables.
StepHypRef Expression
1 fveq2 6669 . . . . 5 (𝑥 = 0 → (𝑆𝑥) = (𝑆‘0))
2 id 22 . . . . 5 (𝑥 = 0 → 𝑥 = 0)
31, 2breq12d 5076 . . . 4 (𝑥 = 0 → ((𝑆𝑥)𝐺𝑥 ↔ (𝑆‘0)𝐺0))
43imbi2d 342 . . 3 (𝑥 = 0 → ((𝜑 → (𝑆𝑥)𝐺𝑥) ↔ (𝜑 → (𝑆‘0)𝐺0)))
5 fveq2 6669 . . . . 5 (𝑥 = 𝑘 → (𝑆𝑥) = (𝑆𝑘))
6 id 22 . . . . 5 (𝑥 = 𝑘𝑥 = 𝑘)
75, 6breq12d 5076 . . . 4 (𝑥 = 𝑘 → ((𝑆𝑥)𝐺𝑥 ↔ (𝑆𝑘)𝐺𝑘))
87imbi2d 342 . . 3 (𝑥 = 𝑘 → ((𝜑 → (𝑆𝑥)𝐺𝑥) ↔ (𝜑 → (𝑆𝑘)𝐺𝑘)))
9 fveq2 6669 . . . . 5 (𝑥 = (𝑘 + 1) → (𝑆𝑥) = (𝑆‘(𝑘 + 1)))
10 id 22 . . . . 5 (𝑥 = (𝑘 + 1) → 𝑥 = (𝑘 + 1))
119, 10breq12d 5076 . . . 4 (𝑥 = (𝑘 + 1) → ((𝑆𝑥)𝐺𝑥 ↔ (𝑆‘(𝑘 + 1))𝐺(𝑘 + 1)))
1211imbi2d 342 . . 3 (𝑥 = (𝑘 + 1) → ((𝜑 → (𝑆𝑥)𝐺𝑥) ↔ (𝜑 → (𝑆‘(𝑘 + 1))𝐺(𝑘 + 1))))
13 fveq2 6669 . . . . 5 (𝑥 = 𝐴 → (𝑆𝑥) = (𝑆𝐴))
14 id 22 . . . . 5 (𝑥 = 𝐴𝑥 = 𝐴)
1513, 14breq12d 5076 . . . 4 (𝑥 = 𝐴 → ((𝑆𝑥)𝐺𝑥 ↔ (𝑆𝐴)𝐺𝐴))
1615imbi2d 342 . . 3 (𝑥 = 𝐴 → ((𝜑 → (𝑆𝑥)𝐺𝑥) ↔ (𝜑 → (𝑆𝐴)𝐺𝐴)))
17 heibor.11 . . . . . . 7 𝑆 = seq0(𝑇, (𝑚 ∈ ℕ0 ↦ if(𝑚 = 0, 𝐶, (𝑚 − 1))))
1817fveq1i 6670 . . . . . 6 (𝑆‘0) = (seq0(𝑇, (𝑚 ∈ ℕ0 ↦ if(𝑚 = 0, 𝐶, (𝑚 − 1))))‘0)
19 0z 11986 . . . . . . 7 0 ∈ ℤ
20 seq1 13377 . . . . . . 7 (0 ∈ ℤ → (seq0(𝑇, (𝑚 ∈ ℕ0 ↦ if(𝑚 = 0, 𝐶, (𝑚 − 1))))‘0) = ((𝑚 ∈ ℕ0 ↦ if(𝑚 = 0, 𝐶, (𝑚 − 1)))‘0))
2119, 20ax-mp 5 . . . . . 6 (seq0(𝑇, (𝑚 ∈ ℕ0 ↦ if(𝑚 = 0, 𝐶, (𝑚 − 1))))‘0) = ((𝑚 ∈ ℕ0 ↦ if(𝑚 = 0, 𝐶, (𝑚 − 1)))‘0)
2218, 21eqtri 2849 . . . . 5 (𝑆‘0) = ((𝑚 ∈ ℕ0 ↦ if(𝑚 = 0, 𝐶, (𝑚 − 1)))‘0)
23 0nn0 11906 . . . . . 6 0 ∈ ℕ0
24 heibor.10 . . . . . . 7 (𝜑𝐶𝐺0)
25 heibor.4 . . . . . . . . 9 𝐺 = {⟨𝑦, 𝑛⟩ ∣ (𝑛 ∈ ℕ0𝑦 ∈ (𝐹𝑛) ∧ (𝑦𝐵𝑛) ∈ 𝐾)}
2625relopabi 5693 . . . . . . . 8 Rel 𝐺
2726brrelex1i 5607 . . . . . . 7 (𝐶𝐺0 → 𝐶 ∈ V)
2824, 27syl 17 . . . . . 6 (𝜑𝐶 ∈ V)
29 iftrue 4476 . . . . . . 7 (𝑚 = 0 → if(𝑚 = 0, 𝐶, (𝑚 − 1)) = 𝐶)
30 eqid 2826 . . . . . . 7 (𝑚 ∈ ℕ0 ↦ if(𝑚 = 0, 𝐶, (𝑚 − 1))) = (𝑚 ∈ ℕ0 ↦ if(𝑚 = 0, 𝐶, (𝑚 − 1)))
3129, 30fvmptg 6765 . . . . . 6 ((0 ∈ ℕ0𝐶 ∈ V) → ((𝑚 ∈ ℕ0 ↦ if(𝑚 = 0, 𝐶, (𝑚 − 1)))‘0) = 𝐶)
3223, 28, 31sylancr 587 . . . . 5 (𝜑 → ((𝑚 ∈ ℕ0 ↦ if(𝑚 = 0, 𝐶, (𝑚 − 1)))‘0) = 𝐶)
3322, 32syl5eq 2873 . . . 4 (𝜑 → (𝑆‘0) = 𝐶)
3433, 24eqbrtrd 5085 . . 3 (𝜑 → (𝑆‘0)𝐺0)
35 df-br 5064 . . . . . 6 ((𝑆𝑘)𝐺𝑘 ↔ ⟨(𝑆𝑘), 𝑘⟩ ∈ 𝐺)
36 heibor.9 . . . . . . 7 (𝜑 → ∀𝑥𝐺 ((𝑇𝑥)𝐺((2nd𝑥) + 1) ∧ ((𝐵𝑥) ∩ ((𝑇𝑥)𝐵((2nd𝑥) + 1))) ∈ 𝐾))
37 fveq2 6669 . . . . . . . . . . 11 (𝑥 = ⟨(𝑆𝑘), 𝑘⟩ → (𝑇𝑥) = (𝑇‘⟨(𝑆𝑘), 𝑘⟩))
38 df-ov 7153 . . . . . . . . . . 11 ((𝑆𝑘)𝑇𝑘) = (𝑇‘⟨(𝑆𝑘), 𝑘⟩)
3937, 38syl6eqr 2879 . . . . . . . . . 10 (𝑥 = ⟨(𝑆𝑘), 𝑘⟩ → (𝑇𝑥) = ((𝑆𝑘)𝑇𝑘))
40 fvex 6682 . . . . . . . . . . . 12 (𝑆𝑘) ∈ V
41 vex 3503 . . . . . . . . . . . 12 𝑘 ∈ V
4240, 41op2ndd 7696 . . . . . . . . . . 11 (𝑥 = ⟨(𝑆𝑘), 𝑘⟩ → (2nd𝑥) = 𝑘)
4342oveq1d 7165 . . . . . . . . . 10 (𝑥 = ⟨(𝑆𝑘), 𝑘⟩ → ((2nd𝑥) + 1) = (𝑘 + 1))
4439, 43breq12d 5076 . . . . . . . . 9 (𝑥 = ⟨(𝑆𝑘), 𝑘⟩ → ((𝑇𝑥)𝐺((2nd𝑥) + 1) ↔ ((𝑆𝑘)𝑇𝑘)𝐺(𝑘 + 1)))
45 fveq2 6669 . . . . . . . . . . . 12 (𝑥 = ⟨(𝑆𝑘), 𝑘⟩ → (𝐵𝑥) = (𝐵‘⟨(𝑆𝑘), 𝑘⟩))
46 df-ov 7153 . . . . . . . . . . . 12 ((𝑆𝑘)𝐵𝑘) = (𝐵‘⟨(𝑆𝑘), 𝑘⟩)
4745, 46syl6eqr 2879 . . . . . . . . . . 11 (𝑥 = ⟨(𝑆𝑘), 𝑘⟩ → (𝐵𝑥) = ((𝑆𝑘)𝐵𝑘))
4839, 43oveq12d 7168 . . . . . . . . . . 11 (𝑥 = ⟨(𝑆𝑘), 𝑘⟩ → ((𝑇𝑥)𝐵((2nd𝑥) + 1)) = (((𝑆𝑘)𝑇𝑘)𝐵(𝑘 + 1)))
4947, 48ineq12d 4194 . . . . . . . . . 10 (𝑥 = ⟨(𝑆𝑘), 𝑘⟩ → ((𝐵𝑥) ∩ ((𝑇𝑥)𝐵((2nd𝑥) + 1))) = (((𝑆𝑘)𝐵𝑘) ∩ (((𝑆𝑘)𝑇𝑘)𝐵(𝑘 + 1))))
5049eleq1d 2902 . . . . . . . . 9 (𝑥 = ⟨(𝑆𝑘), 𝑘⟩ → (((𝐵𝑥) ∩ ((𝑇𝑥)𝐵((2nd𝑥) + 1))) ∈ 𝐾 ↔ (((𝑆𝑘)𝐵𝑘) ∩ (((𝑆𝑘)𝑇𝑘)𝐵(𝑘 + 1))) ∈ 𝐾))
5144, 50anbi12d 630 . . . . . . . 8 (𝑥 = ⟨(𝑆𝑘), 𝑘⟩ → (((𝑇𝑥)𝐺((2nd𝑥) + 1) ∧ ((𝐵𝑥) ∩ ((𝑇𝑥)𝐵((2nd𝑥) + 1))) ∈ 𝐾) ↔ (((𝑆𝑘)𝑇𝑘)𝐺(𝑘 + 1) ∧ (((𝑆𝑘)𝐵𝑘) ∩ (((𝑆𝑘)𝑇𝑘)𝐵(𝑘 + 1))) ∈ 𝐾)))
5251rspccv 3624 . . . . . . 7 (∀𝑥𝐺 ((𝑇𝑥)𝐺((2nd𝑥) + 1) ∧ ((𝐵𝑥) ∩ ((𝑇𝑥)𝐵((2nd𝑥) + 1))) ∈ 𝐾) → (⟨(𝑆𝑘), 𝑘⟩ ∈ 𝐺 → (((𝑆𝑘)𝑇𝑘)𝐺(𝑘 + 1) ∧ (((𝑆𝑘)𝐵𝑘) ∩ (((𝑆𝑘)𝑇𝑘)𝐵(𝑘 + 1))) ∈ 𝐾)))
5336, 52syl 17 . . . . . 6 (𝜑 → (⟨(𝑆𝑘), 𝑘⟩ ∈ 𝐺 → (((𝑆𝑘)𝑇𝑘)𝐺(𝑘 + 1) ∧ (((𝑆𝑘)𝐵𝑘) ∩ (((𝑆𝑘)𝑇𝑘)𝐵(𝑘 + 1))) ∈ 𝐾)))
5435, 53syl5bi 243 . . . . 5 (𝜑 → ((𝑆𝑘)𝐺𝑘 → (((𝑆𝑘)𝑇𝑘)𝐺(𝑘 + 1) ∧ (((𝑆𝑘)𝐵𝑘) ∩ (((𝑆𝑘)𝑇𝑘)𝐵(𝑘 + 1))) ∈ 𝐾)))
55 seqp1 13379 . . . . . . . . . . 11 (𝑘 ∈ (ℤ‘0) → (seq0(𝑇, (𝑚 ∈ ℕ0 ↦ if(𝑚 = 0, 𝐶, (𝑚 − 1))))‘(𝑘 + 1)) = ((seq0(𝑇, (𝑚 ∈ ℕ0 ↦ if(𝑚 = 0, 𝐶, (𝑚 − 1))))‘𝑘)𝑇((𝑚 ∈ ℕ0 ↦ if(𝑚 = 0, 𝐶, (𝑚 − 1)))‘(𝑘 + 1))))
56 nn0uz 12274 . . . . . . . . . . 11 0 = (ℤ‘0)
5755, 56eleq2s 2936 . . . . . . . . . 10 (𝑘 ∈ ℕ0 → (seq0(𝑇, (𝑚 ∈ ℕ0 ↦ if(𝑚 = 0, 𝐶, (𝑚 − 1))))‘(𝑘 + 1)) = ((seq0(𝑇, (𝑚 ∈ ℕ0 ↦ if(𝑚 = 0, 𝐶, (𝑚 − 1))))‘𝑘)𝑇((𝑚 ∈ ℕ0 ↦ if(𝑚 = 0, 𝐶, (𝑚 − 1)))‘(𝑘 + 1))))
5817fveq1i 6670 . . . . . . . . . 10 (𝑆‘(𝑘 + 1)) = (seq0(𝑇, (𝑚 ∈ ℕ0 ↦ if(𝑚 = 0, 𝐶, (𝑚 − 1))))‘(𝑘 + 1))
5917fveq1i 6670 . . . . . . . . . . 11 (𝑆𝑘) = (seq0(𝑇, (𝑚 ∈ ℕ0 ↦ if(𝑚 = 0, 𝐶, (𝑚 − 1))))‘𝑘)
6059oveq1i 7160 . . . . . . . . . 10 ((𝑆𝑘)𝑇((𝑚 ∈ ℕ0 ↦ if(𝑚 = 0, 𝐶, (𝑚 − 1)))‘(𝑘 + 1))) = ((seq0(𝑇, (𝑚 ∈ ℕ0 ↦ if(𝑚 = 0, 𝐶, (𝑚 − 1))))‘𝑘)𝑇((𝑚 ∈ ℕ0 ↦ if(𝑚 = 0, 𝐶, (𝑚 − 1)))‘(𝑘 + 1)))
6157, 58, 603eqtr4g 2886 . . . . . . . . 9 (𝑘 ∈ ℕ0 → (𝑆‘(𝑘 + 1)) = ((𝑆𝑘)𝑇((𝑚 ∈ ℕ0 ↦ if(𝑚 = 0, 𝐶, (𝑚 − 1)))‘(𝑘 + 1))))
62 eqeq1 2830 . . . . . . . . . . . . 13 (𝑚 = (𝑘 + 1) → (𝑚 = 0 ↔ (𝑘 + 1) = 0))
63 oveq1 7157 . . . . . . . . . . . . 13 (𝑚 = (𝑘 + 1) → (𝑚 − 1) = ((𝑘 + 1) − 1))
6462, 63ifbieq2d 4495 . . . . . . . . . . . 12 (𝑚 = (𝑘 + 1) → if(𝑚 = 0, 𝐶, (𝑚 − 1)) = if((𝑘 + 1) = 0, 𝐶, ((𝑘 + 1) − 1)))
65 peano2nn0 11931 . . . . . . . . . . . 12 (𝑘 ∈ ℕ0 → (𝑘 + 1) ∈ ℕ0)
66 nn0p1nn 11930 . . . . . . . . . . . . . 14 (𝑘 ∈ ℕ0 → (𝑘 + 1) ∈ ℕ)
67 nnne0 11665 . . . . . . . . . . . . . . 15 ((𝑘 + 1) ∈ ℕ → (𝑘 + 1) ≠ 0)
6867neneqd 3026 . . . . . . . . . . . . . 14 ((𝑘 + 1) ∈ ℕ → ¬ (𝑘 + 1) = 0)
69 iffalse 4479 . . . . . . . . . . . . . 14 (¬ (𝑘 + 1) = 0 → if((𝑘 + 1) = 0, 𝐶, ((𝑘 + 1) − 1)) = ((𝑘 + 1) − 1))
7066, 68, 693syl 18 . . . . . . . . . . . . 13 (𝑘 ∈ ℕ0 → if((𝑘 + 1) = 0, 𝐶, ((𝑘 + 1) − 1)) = ((𝑘 + 1) − 1))
71 ovex 7183 . . . . . . . . . . . . 13 ((𝑘 + 1) − 1) ∈ V
7270, 71syl6eqel 2926 . . . . . . . . . . . 12 (𝑘 ∈ ℕ0 → if((𝑘 + 1) = 0, 𝐶, ((𝑘 + 1) − 1)) ∈ V)
7330, 64, 65, 72fvmptd3 6789 . . . . . . . . . . 11 (𝑘 ∈ ℕ0 → ((𝑚 ∈ ℕ0 ↦ if(𝑚 = 0, 𝐶, (𝑚 − 1)))‘(𝑘 + 1)) = if((𝑘 + 1) = 0, 𝐶, ((𝑘 + 1) − 1)))
74 nn0cn 11901 . . . . . . . . . . . 12 (𝑘 ∈ ℕ0𝑘 ∈ ℂ)
75 ax-1cn 10589 . . . . . . . . . . . 12 1 ∈ ℂ
76 pncan 10886 . . . . . . . . . . . 12 ((𝑘 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑘 + 1) − 1) = 𝑘)
7774, 75, 76sylancl 586 . . . . . . . . . . 11 (𝑘 ∈ ℕ0 → ((𝑘 + 1) − 1) = 𝑘)
7873, 70, 773eqtrd 2865 . . . . . . . . . 10 (𝑘 ∈ ℕ0 → ((𝑚 ∈ ℕ0 ↦ if(𝑚 = 0, 𝐶, (𝑚 − 1)))‘(𝑘 + 1)) = 𝑘)
7978oveq2d 7166 . . . . . . . . 9 (𝑘 ∈ ℕ0 → ((𝑆𝑘)𝑇((𝑚 ∈ ℕ0 ↦ if(𝑚 = 0, 𝐶, (𝑚 − 1)))‘(𝑘 + 1))) = ((𝑆𝑘)𝑇𝑘))
8061, 79eqtrd 2861 . . . . . . . 8 (𝑘 ∈ ℕ0 → (𝑆‘(𝑘 + 1)) = ((𝑆𝑘)𝑇𝑘))
8180breq1d 5073 . . . . . . 7 (𝑘 ∈ ℕ0 → ((𝑆‘(𝑘 + 1))𝐺(𝑘 + 1) ↔ ((𝑆𝑘)𝑇𝑘)𝐺(𝑘 + 1)))
8281biimprd 249 . . . . . 6 (𝑘 ∈ ℕ0 → (((𝑆𝑘)𝑇𝑘)𝐺(𝑘 + 1) → (𝑆‘(𝑘 + 1))𝐺(𝑘 + 1)))
8382adantrd 492 . . . . 5 (𝑘 ∈ ℕ0 → ((((𝑆𝑘)𝑇𝑘)𝐺(𝑘 + 1) ∧ (((𝑆𝑘)𝐵𝑘) ∩ (((𝑆𝑘)𝑇𝑘)𝐵(𝑘 + 1))) ∈ 𝐾) → (𝑆‘(𝑘 + 1))𝐺(𝑘 + 1)))
8454, 83syl9r 78 . . . 4 (𝑘 ∈ ℕ0 → (𝜑 → ((𝑆𝑘)𝐺𝑘 → (𝑆‘(𝑘 + 1))𝐺(𝑘 + 1))))
8584a2d 29 . . 3 (𝑘 ∈ ℕ0 → ((𝜑 → (𝑆𝑘)𝐺𝑘) → (𝜑 → (𝑆‘(𝑘 + 1))𝐺(𝑘 + 1))))
864, 8, 12, 16, 34, 85nn0ind 12071 . 2 (𝐴 ∈ ℕ0 → (𝜑 → (𝑆𝐴)𝐺𝐴))
8786impcom 408 1 ((𝜑𝐴 ∈ ℕ0) → (𝑆𝐴)𝐺𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 396  w3a 1081   = wceq 1530  wcel 2107  {cab 2804  wral 3143  wrex 3144  Vcvv 3500  cin 3939  wss 3940  ifcif 4470  𝒫 cpw 4542  cop 4570   cuni 4837   ciun 4917   class class class wbr 5063  {copab 5125  cmpt 5143  wf 6350  cfv 6354  (class class class)co 7150  cmpo 7152  2nd c2nd 7684  Fincfn 8503  cc 10529  0cc0 10531  1c1 10532   + caddc 10534  cmin 10864   / cdiv 11291  cn 11632  2c2 11686  0cn0 11891  cz 11975  cuz 12237  seqcseq 13364  cexp 13424  ballcbl 20467  MetOpencmopn 20470  CMetccmet 23791
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-sep 5200  ax-nul 5207  ax-pow 5263  ax-pr 5326  ax-un 7455  ax-cnex 10587  ax-resscn 10588  ax-1cn 10589  ax-icn 10590  ax-addcl 10591  ax-addrcl 10592  ax-mulcl 10593  ax-mulrcl 10594  ax-mulcom 10595  ax-addass 10596  ax-mulass 10597  ax-distr 10598  ax-i2m1 10599  ax-1ne0 10600  ax-1rid 10601  ax-rnegex 10602  ax-rrecex 10603  ax-cnre 10604  ax-pre-lttri 10605  ax-pre-lttrn 10606  ax-pre-ltadd 10607  ax-pre-mulgt0 10608
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-nel 3129  df-ral 3148  df-rex 3149  df-reu 3150  df-rab 3152  df-v 3502  df-sbc 3777  df-csb 3888  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-pss 3958  df-nul 4296  df-if 4471  df-pw 4544  df-sn 4565  df-pr 4567  df-tp 4569  df-op 4571  df-uni 4838  df-iun 4919  df-br 5064  df-opab 5126  df-mpt 5144  df-tr 5170  df-id 5459  df-eprel 5464  df-po 5473  df-so 5474  df-fr 5513  df-we 5515  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-pred 6147  df-ord 6193  df-on 6194  df-lim 6195  df-suc 6196  df-iota 6313  df-fun 6356  df-fn 6357  df-f 6358  df-f1 6359  df-fo 6360  df-f1o 6361  df-fv 6362  df-riota 7108  df-ov 7153  df-oprab 7154  df-mpo 7155  df-om 7574  df-2nd 7686  df-wrecs 7943  df-recs 8004  df-rdg 8042  df-er 8284  df-en 8504  df-dom 8505  df-sdom 8506  df-pnf 10671  df-mnf 10672  df-xr 10673  df-ltxr 10674  df-le 10675  df-sub 10866  df-neg 10867  df-nn 11633  df-n0 11892  df-z 11976  df-uz 12238  df-seq 13365
This theorem is referenced by:  heiborlem5  34980  heiborlem6  34981  heiborlem8  34983
  Copyright terms: Public domain W3C validator