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Theorem lecldbas 21071
Description: The set of closed intervals forms a closed subbasis for the topology on the extended reals. Since our definition of a basis is in terms of open sets, we express this by showing that the complements of closed intervals form an open subbasis for the topology. (Contributed by Mario Carneiro, 3-Sep-2015.)
Hypothesis
Ref Expression
lecldbas.1 𝐹 = (𝑥 ∈ ran [,] ↦ (ℝ*𝑥))
Assertion
Ref Expression
lecldbas (ordTop‘ ≤ ) = (topGen‘(fi‘ran 𝐹))

Proof of Theorem lecldbas
Dummy variables 𝑎 𝑏 𝑐 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2651 . . . 4 ran (𝑦 ∈ ℝ* ↦ (𝑦(,]+∞)) = ran (𝑦 ∈ ℝ* ↦ (𝑦(,]+∞))
2 eqid 2651 . . . 4 ran (𝑦 ∈ ℝ* ↦ (-∞[,)𝑦)) = ran (𝑦 ∈ ℝ* ↦ (-∞[,)𝑦))
31, 2leordtval2 21064 . . 3 (ordTop‘ ≤ ) = (topGen‘(fi‘(ran (𝑦 ∈ ℝ* ↦ (𝑦(,]+∞)) ∪ ran (𝑦 ∈ ℝ* ↦ (-∞[,)𝑦)))))
4 fvex 6239 . . . 4 (fi‘ran 𝐹) ∈ V
5 fvex 6239 . . . . . 6 (ordTop‘ ≤ ) ∈ V
6 lecldbas.1 . . . . . . . 8 𝐹 = (𝑥 ∈ ran [,] ↦ (ℝ*𝑥))
7 iccf 12310 . . . . . . . . . . 11 [,]:(ℝ* × ℝ*)⟶𝒫 ℝ*
8 ffn 6083 . . . . . . . . . . 11 ([,]:(ℝ* × ℝ*)⟶𝒫 ℝ* → [,] Fn (ℝ* × ℝ*))
97, 8ax-mp 5 . . . . . . . . . 10 [,] Fn (ℝ* × ℝ*)
10 ovelrn 6852 . . . . . . . . . 10 ([,] Fn (ℝ* × ℝ*) → (𝑥 ∈ ran [,] ↔ ∃𝑎 ∈ ℝ*𝑏 ∈ ℝ* 𝑥 = (𝑎[,]𝑏)))
119, 10ax-mp 5 . . . . . . . . 9 (𝑥 ∈ ran [,] ↔ ∃𝑎 ∈ ℝ*𝑏 ∈ ℝ* 𝑥 = (𝑎[,]𝑏))
12 difeq2 3755 . . . . . . . . . . . 12 (𝑥 = (𝑎[,]𝑏) → (ℝ*𝑥) = (ℝ* ∖ (𝑎[,]𝑏)))
13 iccordt 21066 . . . . . . . . . . . . 13 (𝑎[,]𝑏) ∈ (Clsd‘(ordTop‘ ≤ ))
14 letopuni 21059 . . . . . . . . . . . . . 14 * = (ordTop‘ ≤ )
1514cldopn 20883 . . . . . . . . . . . . 13 ((𝑎[,]𝑏) ∈ (Clsd‘(ordTop‘ ≤ )) → (ℝ* ∖ (𝑎[,]𝑏)) ∈ (ordTop‘ ≤ ))
1613, 15ax-mp 5 . . . . . . . . . . . 12 (ℝ* ∖ (𝑎[,]𝑏)) ∈ (ordTop‘ ≤ )
1712, 16syl6eqel 2738 . . . . . . . . . . 11 (𝑥 = (𝑎[,]𝑏) → (ℝ*𝑥) ∈ (ordTop‘ ≤ ))
1817rexlimivw 3058 . . . . . . . . . 10 (∃𝑏 ∈ ℝ* 𝑥 = (𝑎[,]𝑏) → (ℝ*𝑥) ∈ (ordTop‘ ≤ ))
1918rexlimivw 3058 . . . . . . . . 9 (∃𝑎 ∈ ℝ*𝑏 ∈ ℝ* 𝑥 = (𝑎[,]𝑏) → (ℝ*𝑥) ∈ (ordTop‘ ≤ ))
2011, 19sylbi 207 . . . . . . . 8 (𝑥 ∈ ran [,] → (ℝ*𝑥) ∈ (ordTop‘ ≤ ))
216, 20fmpti 6423 . . . . . . 7 𝐹:ran [,]⟶(ordTop‘ ≤ )
22 frn 6091 . . . . . . 7 (𝐹:ran [,]⟶(ordTop‘ ≤ ) → ran 𝐹 ⊆ (ordTop‘ ≤ ))
2321, 22ax-mp 5 . . . . . 6 ran 𝐹 ⊆ (ordTop‘ ≤ )
245, 23ssexi 4836 . . . . 5 ran 𝐹 ∈ V
25 eqid 2651 . . . . . . . 8 (𝑦 ∈ ℝ* ↦ (𝑦(,]+∞)) = (𝑦 ∈ ℝ* ↦ (𝑦(,]+∞))
26 mnfxr 10134 . . . . . . . . . . 11 -∞ ∈ ℝ*
27 fnovrn 6851 . . . . . . . . . . 11 (([,] Fn (ℝ* × ℝ*) ∧ -∞ ∈ ℝ*𝑦 ∈ ℝ*) → (-∞[,]𝑦) ∈ ran [,])
289, 26, 27mp3an12 1454 . . . . . . . . . 10 (𝑦 ∈ ℝ* → (-∞[,]𝑦) ∈ ran [,])
2926a1i 11 . . . . . . . . . . . . . 14 (𝑦 ∈ ℝ* → -∞ ∈ ℝ*)
30 id 22 . . . . . . . . . . . . . 14 (𝑦 ∈ ℝ*𝑦 ∈ ℝ*)
31 pnfxr 10130 . . . . . . . . . . . . . . 15 +∞ ∈ ℝ*
3231a1i 11 . . . . . . . . . . . . . 14 (𝑦 ∈ ℝ* → +∞ ∈ ℝ*)
33 mnfle 12007 . . . . . . . . . . . . . 14 (𝑦 ∈ ℝ* → -∞ ≤ 𝑦)
34 pnfge 12002 . . . . . . . . . . . . . 14 (𝑦 ∈ ℝ*𝑦 ≤ +∞)
35 df-icc 12220 . . . . . . . . . . . . . . 15 [,] = (𝑎 ∈ ℝ*, 𝑏 ∈ ℝ* ↦ {𝑐 ∈ ℝ* ∣ (𝑎𝑐𝑐𝑏)})
36 df-ioc 12218 . . . . . . . . . . . . . . 15 (,] = (𝑎 ∈ ℝ*, 𝑏 ∈ ℝ* ↦ {𝑐 ∈ ℝ* ∣ (𝑎 < 𝑐𝑐𝑏)})
37 xrltnle 10143 . . . . . . . . . . . . . . 15 ((𝑦 ∈ ℝ*𝑧 ∈ ℝ*) → (𝑦 < 𝑧 ↔ ¬ 𝑧𝑦))
38 xrletr 12027 . . . . . . . . . . . . . . 15 ((𝑧 ∈ ℝ*𝑦 ∈ ℝ* ∧ +∞ ∈ ℝ*) → ((𝑧𝑦𝑦 ≤ +∞) → 𝑧 ≤ +∞))
39 xrlelttr 12025 . . . . . . . . . . . . . . . 16 ((-∞ ∈ ℝ*𝑦 ∈ ℝ*𝑧 ∈ ℝ*) → ((-∞ ≤ 𝑦𝑦 < 𝑧) → -∞ < 𝑧))
40 xrltle 12020 . . . . . . . . . . . . . . . . 17 ((-∞ ∈ ℝ*𝑧 ∈ ℝ*) → (-∞ < 𝑧 → -∞ ≤ 𝑧))
41403adant2 1100 . . . . . . . . . . . . . . . 16 ((-∞ ∈ ℝ*𝑦 ∈ ℝ*𝑧 ∈ ℝ*) → (-∞ < 𝑧 → -∞ ≤ 𝑧))
4239, 41syld 47 . . . . . . . . . . . . . . 15 ((-∞ ∈ ℝ*𝑦 ∈ ℝ*𝑧 ∈ ℝ*) → ((-∞ ≤ 𝑦𝑦 < 𝑧) → -∞ ≤ 𝑧))
4335, 36, 37, 35, 38, 42ixxun 12229 . . . . . . . . . . . . . 14 (((-∞ ∈ ℝ*𝑦 ∈ ℝ* ∧ +∞ ∈ ℝ*) ∧ (-∞ ≤ 𝑦𝑦 ≤ +∞)) → ((-∞[,]𝑦) ∪ (𝑦(,]+∞)) = (-∞[,]+∞))
4429, 30, 32, 33, 34, 43syl32anc 1374 . . . . . . . . . . . . 13 (𝑦 ∈ ℝ* → ((-∞[,]𝑦) ∪ (𝑦(,]+∞)) = (-∞[,]+∞))
45 iccmax 12287 . . . . . . . . . . . . 13 (-∞[,]+∞) = ℝ*
4644, 45syl6eq 2701 . . . . . . . . . . . 12 (𝑦 ∈ ℝ* → ((-∞[,]𝑦) ∪ (𝑦(,]+∞)) = ℝ*)
47 iccssxr 12294 . . . . . . . . . . . . 13 (-∞[,]𝑦) ⊆ ℝ*
4835, 36, 37ixxdisj 12228 . . . . . . . . . . . . . 14 ((-∞ ∈ ℝ*𝑦 ∈ ℝ* ∧ +∞ ∈ ℝ*) → ((-∞[,]𝑦) ∩ (𝑦(,]+∞)) = ∅)
4926, 31, 48mp3an13 1455 . . . . . . . . . . . . 13 (𝑦 ∈ ℝ* → ((-∞[,]𝑦) ∩ (𝑦(,]+∞)) = ∅)
50 uneqdifeq 4090 . . . . . . . . . . . . 13 (((-∞[,]𝑦) ⊆ ℝ* ∧ ((-∞[,]𝑦) ∩ (𝑦(,]+∞)) = ∅) → (((-∞[,]𝑦) ∪ (𝑦(,]+∞)) = ℝ* ↔ (ℝ* ∖ (-∞[,]𝑦)) = (𝑦(,]+∞)))
5147, 49, 50sylancr 696 . . . . . . . . . . . 12 (𝑦 ∈ ℝ* → (((-∞[,]𝑦) ∪ (𝑦(,]+∞)) = ℝ* ↔ (ℝ* ∖ (-∞[,]𝑦)) = (𝑦(,]+∞)))
5246, 51mpbid 222 . . . . . . . . . . 11 (𝑦 ∈ ℝ* → (ℝ* ∖ (-∞[,]𝑦)) = (𝑦(,]+∞))
5352eqcomd 2657 . . . . . . . . . 10 (𝑦 ∈ ℝ* → (𝑦(,]+∞) = (ℝ* ∖ (-∞[,]𝑦)))
54 difeq2 3755 . . . . . . . . . . . 12 (𝑥 = (-∞[,]𝑦) → (ℝ*𝑥) = (ℝ* ∖ (-∞[,]𝑦)))
5554eqeq2d 2661 . . . . . . . . . . 11 (𝑥 = (-∞[,]𝑦) → ((𝑦(,]+∞) = (ℝ*𝑥) ↔ (𝑦(,]+∞) = (ℝ* ∖ (-∞[,]𝑦))))
5655rspcev 3340 . . . . . . . . . 10 (((-∞[,]𝑦) ∈ ran [,] ∧ (𝑦(,]+∞) = (ℝ* ∖ (-∞[,]𝑦))) → ∃𝑥 ∈ ran [,](𝑦(,]+∞) = (ℝ*𝑥))
5728, 53, 56syl2anc 694 . . . . . . . . 9 (𝑦 ∈ ℝ* → ∃𝑥 ∈ ran [,](𝑦(,]+∞) = (ℝ*𝑥))
58 xrex 11867 . . . . . . . . . . 11 * ∈ V
59 difexg 4841 . . . . . . . . . . 11 (ℝ* ∈ V → (ℝ*𝑥) ∈ V)
6058, 59ax-mp 5 . . . . . . . . . 10 (ℝ*𝑥) ∈ V
616, 60elrnmpti 5408 . . . . . . . . 9 ((𝑦(,]+∞) ∈ ran 𝐹 ↔ ∃𝑥 ∈ ran [,](𝑦(,]+∞) = (ℝ*𝑥))
6257, 61sylibr 224 . . . . . . . 8 (𝑦 ∈ ℝ* → (𝑦(,]+∞) ∈ ran 𝐹)
6325, 62fmpti 6423 . . . . . . 7 (𝑦 ∈ ℝ* ↦ (𝑦(,]+∞)):ℝ*⟶ran 𝐹
64 frn 6091 . . . . . . 7 ((𝑦 ∈ ℝ* ↦ (𝑦(,]+∞)):ℝ*⟶ran 𝐹 → ran (𝑦 ∈ ℝ* ↦ (𝑦(,]+∞)) ⊆ ran 𝐹)
6563, 64ax-mp 5 . . . . . 6 ran (𝑦 ∈ ℝ* ↦ (𝑦(,]+∞)) ⊆ ran 𝐹
66 eqid 2651 . . . . . . . 8 (𝑦 ∈ ℝ* ↦ (-∞[,)𝑦)) = (𝑦 ∈ ℝ* ↦ (-∞[,)𝑦))
67 fnovrn 6851 . . . . . . . . . . 11 (([,] Fn (ℝ* × ℝ*) ∧ 𝑦 ∈ ℝ* ∧ +∞ ∈ ℝ*) → (𝑦[,]+∞) ∈ ran [,])
689, 31, 67mp3an13 1455 . . . . . . . . . 10 (𝑦 ∈ ℝ* → (𝑦[,]+∞) ∈ ran [,])
69 df-ico 12219 . . . . . . . . . . . . . . 15 [,) = (𝑎 ∈ ℝ*, 𝑏 ∈ ℝ* ↦ {𝑐 ∈ ℝ* ∣ (𝑎𝑐𝑐 < 𝑏)})
70 xrlenlt 10141 . . . . . . . . . . . . . . 15 ((𝑦 ∈ ℝ*𝑧 ∈ ℝ*) → (𝑦𝑧 ↔ ¬ 𝑧 < 𝑦))
71 xrltletr 12026 . . . . . . . . . . . . . . . 16 ((𝑧 ∈ ℝ*𝑦 ∈ ℝ* ∧ +∞ ∈ ℝ*) → ((𝑧 < 𝑦𝑦 ≤ +∞) → 𝑧 < +∞))
72 xrltle 12020 . . . . . . . . . . . . . . . . 17 ((𝑧 ∈ ℝ* ∧ +∞ ∈ ℝ*) → (𝑧 < +∞ → 𝑧 ≤ +∞))
73723adant2 1100 . . . . . . . . . . . . . . . 16 ((𝑧 ∈ ℝ*𝑦 ∈ ℝ* ∧ +∞ ∈ ℝ*) → (𝑧 < +∞ → 𝑧 ≤ +∞))
7471, 73syld 47 . . . . . . . . . . . . . . 15 ((𝑧 ∈ ℝ*𝑦 ∈ ℝ* ∧ +∞ ∈ ℝ*) → ((𝑧 < 𝑦𝑦 ≤ +∞) → 𝑧 ≤ +∞))
75 xrletr 12027 . . . . . . . . . . . . . . 15 ((-∞ ∈ ℝ*𝑦 ∈ ℝ*𝑧 ∈ ℝ*) → ((-∞ ≤ 𝑦𝑦𝑧) → -∞ ≤ 𝑧))
7669, 35, 70, 35, 74, 75ixxun 12229 . . . . . . . . . . . . . 14 (((-∞ ∈ ℝ*𝑦 ∈ ℝ* ∧ +∞ ∈ ℝ*) ∧ (-∞ ≤ 𝑦𝑦 ≤ +∞)) → ((-∞[,)𝑦) ∪ (𝑦[,]+∞)) = (-∞[,]+∞))
7729, 30, 32, 33, 34, 76syl32anc 1374 . . . . . . . . . . . . 13 (𝑦 ∈ ℝ* → ((-∞[,)𝑦) ∪ (𝑦[,]+∞)) = (-∞[,]+∞))
78 uncom 3790 . . . . . . . . . . . . 13 ((-∞[,)𝑦) ∪ (𝑦[,]+∞)) = ((𝑦[,]+∞) ∪ (-∞[,)𝑦))
7977, 78, 453eqtr3g 2708 . . . . . . . . . . . 12 (𝑦 ∈ ℝ* → ((𝑦[,]+∞) ∪ (-∞[,)𝑦)) = ℝ*)
80 iccssxr 12294 . . . . . . . . . . . . 13 (𝑦[,]+∞) ⊆ ℝ*
81 incom 3838 . . . . . . . . . . . . . 14 ((𝑦[,]+∞) ∩ (-∞[,)𝑦)) = ((-∞[,)𝑦) ∩ (𝑦[,]+∞))
8269, 35, 70ixxdisj 12228 . . . . . . . . . . . . . . 15 ((-∞ ∈ ℝ*𝑦 ∈ ℝ* ∧ +∞ ∈ ℝ*) → ((-∞[,)𝑦) ∩ (𝑦[,]+∞)) = ∅)
8326, 31, 82mp3an13 1455 . . . . . . . . . . . . . 14 (𝑦 ∈ ℝ* → ((-∞[,)𝑦) ∩ (𝑦[,]+∞)) = ∅)
8481, 83syl5eq 2697 . . . . . . . . . . . . 13 (𝑦 ∈ ℝ* → ((𝑦[,]+∞) ∩ (-∞[,)𝑦)) = ∅)
85 uneqdifeq 4090 . . . . . . . . . . . . 13 (((𝑦[,]+∞) ⊆ ℝ* ∧ ((𝑦[,]+∞) ∩ (-∞[,)𝑦)) = ∅) → (((𝑦[,]+∞) ∪ (-∞[,)𝑦)) = ℝ* ↔ (ℝ* ∖ (𝑦[,]+∞)) = (-∞[,)𝑦)))
8680, 84, 85sylancr 696 . . . . . . . . . . . 12 (𝑦 ∈ ℝ* → (((𝑦[,]+∞) ∪ (-∞[,)𝑦)) = ℝ* ↔ (ℝ* ∖ (𝑦[,]+∞)) = (-∞[,)𝑦)))
8779, 86mpbid 222 . . . . . . . . . . 11 (𝑦 ∈ ℝ* → (ℝ* ∖ (𝑦[,]+∞)) = (-∞[,)𝑦))
8887eqcomd 2657 . . . . . . . . . 10 (𝑦 ∈ ℝ* → (-∞[,)𝑦) = (ℝ* ∖ (𝑦[,]+∞)))
89 difeq2 3755 . . . . . . . . . . . 12 (𝑥 = (𝑦[,]+∞) → (ℝ*𝑥) = (ℝ* ∖ (𝑦[,]+∞)))
9089eqeq2d 2661 . . . . . . . . . . 11 (𝑥 = (𝑦[,]+∞) → ((-∞[,)𝑦) = (ℝ*𝑥) ↔ (-∞[,)𝑦) = (ℝ* ∖ (𝑦[,]+∞))))
9190rspcev 3340 . . . . . . . . . 10 (((𝑦[,]+∞) ∈ ran [,] ∧ (-∞[,)𝑦) = (ℝ* ∖ (𝑦[,]+∞))) → ∃𝑥 ∈ ran [,](-∞[,)𝑦) = (ℝ*𝑥))
9268, 88, 91syl2anc 694 . . . . . . . . 9 (𝑦 ∈ ℝ* → ∃𝑥 ∈ ran [,](-∞[,)𝑦) = (ℝ*𝑥))
936, 60elrnmpti 5408 . . . . . . . . 9 ((-∞[,)𝑦) ∈ ran 𝐹 ↔ ∃𝑥 ∈ ran [,](-∞[,)𝑦) = (ℝ*𝑥))
9492, 93sylibr 224 . . . . . . . 8 (𝑦 ∈ ℝ* → (-∞[,)𝑦) ∈ ran 𝐹)
9566, 94fmpti 6423 . . . . . . 7 (𝑦 ∈ ℝ* ↦ (-∞[,)𝑦)):ℝ*⟶ran 𝐹
96 frn 6091 . . . . . . 7 ((𝑦 ∈ ℝ* ↦ (-∞[,)𝑦)):ℝ*⟶ran 𝐹 → ran (𝑦 ∈ ℝ* ↦ (-∞[,)𝑦)) ⊆ ran 𝐹)
9795, 96ax-mp 5 . . . . . 6 ran (𝑦 ∈ ℝ* ↦ (-∞[,)𝑦)) ⊆ ran 𝐹
9865, 97unssi 3821 . . . . 5 (ran (𝑦 ∈ ℝ* ↦ (𝑦(,]+∞)) ∪ ran (𝑦 ∈ ℝ* ↦ (-∞[,)𝑦))) ⊆ ran 𝐹
99 fiss 8371 . . . . 5 ((ran 𝐹 ∈ V ∧ (ran (𝑦 ∈ ℝ* ↦ (𝑦(,]+∞)) ∪ ran (𝑦 ∈ ℝ* ↦ (-∞[,)𝑦))) ⊆ ran 𝐹) → (fi‘(ran (𝑦 ∈ ℝ* ↦ (𝑦(,]+∞)) ∪ ran (𝑦 ∈ ℝ* ↦ (-∞[,)𝑦)))) ⊆ (fi‘ran 𝐹))
10024, 98, 99mp2an 708 . . . 4 (fi‘(ran (𝑦 ∈ ℝ* ↦ (𝑦(,]+∞)) ∪ ran (𝑦 ∈ ℝ* ↦ (-∞[,)𝑦)))) ⊆ (fi‘ran 𝐹)
101 tgss 20820 . . . 4 (((fi‘ran 𝐹) ∈ V ∧ (fi‘(ran (𝑦 ∈ ℝ* ↦ (𝑦(,]+∞)) ∪ ran (𝑦 ∈ ℝ* ↦ (-∞[,)𝑦)))) ⊆ (fi‘ran 𝐹)) → (topGen‘(fi‘(ran (𝑦 ∈ ℝ* ↦ (𝑦(,]+∞)) ∪ ran (𝑦 ∈ ℝ* ↦ (-∞[,)𝑦))))) ⊆ (topGen‘(fi‘ran 𝐹)))
1024, 100, 101mp2an 708 . . 3 (topGen‘(fi‘(ran (𝑦 ∈ ℝ* ↦ (𝑦(,]+∞)) ∪ ran (𝑦 ∈ ℝ* ↦ (-∞[,)𝑦))))) ⊆ (topGen‘(fi‘ran 𝐹))
1033, 102eqsstri 3668 . 2 (ordTop‘ ≤ ) ⊆ (topGen‘(fi‘ran 𝐹))
104 letop 21058 . . 3 (ordTop‘ ≤ ) ∈ Top
105 tgfiss 20843 . . 3 (((ordTop‘ ≤ ) ∈ Top ∧ ran 𝐹 ⊆ (ordTop‘ ≤ )) → (topGen‘(fi‘ran 𝐹)) ⊆ (ordTop‘ ≤ ))
106104, 23, 105mp2an 708 . 2 (topGen‘(fi‘ran 𝐹)) ⊆ (ordTop‘ ≤ )
107103, 106eqssi 3652 1 (ordTop‘ ≤ ) = (topGen‘(fi‘ran 𝐹))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  w3a 1054   = wceq 1523  wcel 2030  wrex 2942  Vcvv 3231  cdif 3604  cun 3605  cin 3606  wss 3607  c0 3948  𝒫 cpw 4191   class class class wbr 4685  cmpt 4762   × cxp 5141  ran crn 5144   Fn wfn 5921  wf 5922  cfv 5926  (class class class)co 6690  ficfi 8357  +∞cpnf 10109  -∞cmnf 10110  *cxr 10111   < clt 10112  cle 10113  (,]cioc 12214  [,)cico 12215  [,]cicc 12216  topGenctg 16145  ordTopcordt 16206  Topctop 20746  Clsdccld 20868
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-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
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-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-iin 4555  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-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-oadd 7609  df-er 7787  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-fi 8358  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-ioc 12218  df-ico 12219  df-icc 12220  df-topgen 16151  df-ordt 16208  df-ps 17247  df-tsr 17248  df-top 20747  df-topon 20764  df-bases 20798  df-cld 20871
This theorem is referenced by: (None)
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