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Theorem eupares 26502
 Description: The restriction of an Eulerian path to an initial segment of the path forms an Eulerian path on the subgraph consisting of the edges in the initial segment. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Mario Carneiro, 3-May-2015.)
Hypotheses
Ref Expression
eupares.g (𝜑𝐺(𝑉 EulPaths 𝐸)𝑃)
eupares.n (𝜑𝑁 ∈ (0...(#‘𝐺)))
eupares.f 𝐹 = (𝐸 ↾ (𝐺 “ (1...𝑁)))
eupares.h 𝐻 = (𝐺 ↾ (1...𝑁))
eupares.q 𝑄 = (𝑃 ↾ (0...𝑁))
Assertion
Ref Expression
eupares (𝜑𝐻(𝑉 EulPaths 𝐹)𝑄)

Proof of Theorem eupares
Dummy variables 𝑘 𝑛 𝑥 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eupares.g . . . . 5 (𝜑𝐺(𝑉 EulPaths 𝐸)𝑃)
2 eupagra 26493 . . . . 5 (𝐺(𝑉 EulPaths 𝐸)𝑃𝑉 UMGrph 𝐸)
31, 2syl 17 . . . 4 (𝜑𝑉 UMGrph 𝐸)
4 umgrares 25853 . . . 4 (𝑉 UMGrph 𝐸𝑉 UMGrph (𝐸 ↾ (𝐺 “ (1...𝑁))))
53, 4syl 17 . . 3 (𝜑𝑉 UMGrph (𝐸 ↾ (𝐺 “ (1...𝑁))))
6 eupares.f . . 3 𝐹 = (𝐸 ↾ (𝐺 “ (1...𝑁)))
75, 6syl6breqr 4625 . 2 (𝜑𝑉 UMGrph 𝐹)
8 eupares.n . . . 4 (𝜑𝑁 ∈ (0...(#‘𝐺)))
9 elfznn0 12302 . . . 4 (𝑁 ∈ (0...(#‘𝐺)) → 𝑁 ∈ ℕ0)
108, 9syl 17 . . 3 (𝜑𝑁 ∈ ℕ0)
11 umgraf2 25846 . . . . . . . . 9 (𝑉 UMGrph 𝐸𝐸:dom 𝐸⟶{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑥) ≤ 2})
123, 11syl 17 . . . . . . . 8 (𝜑𝐸:dom 𝐸⟶{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑥) ≤ 2})
13 ffn 5958 . . . . . . . 8 (𝐸:dom 𝐸⟶{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑥) ≤ 2} → 𝐸 Fn dom 𝐸)
1412, 13syl 17 . . . . . . 7 (𝜑𝐸 Fn dom 𝐸)
15 eupaf1o 26497 . . . . . . 7 ((𝐺(𝑉 EulPaths 𝐸)𝑃𝐸 Fn dom 𝐸) → 𝐺:(1...(#‘𝐺))–1-1-onto→dom 𝐸)
161, 14, 15syl2anc 691 . . . . . 6 (𝜑𝐺:(1...(#‘𝐺))–1-1-onto→dom 𝐸)
17 f1of1 6049 . . . . . 6 (𝐺:(1...(#‘𝐺))–1-1-onto→dom 𝐸𝐺:(1...(#‘𝐺))–1-1→dom 𝐸)
1816, 17syl 17 . . . . 5 (𝜑𝐺:(1...(#‘𝐺))–1-1→dom 𝐸)
19 elfzuz3 12210 . . . . . . 7 (𝑁 ∈ (0...(#‘𝐺)) → (#‘𝐺) ∈ (ℤ𝑁))
208, 19syl 17 . . . . . 6 (𝜑 → (#‘𝐺) ∈ (ℤ𝑁))
21 fzss2 12252 . . . . . 6 ((#‘𝐺) ∈ (ℤ𝑁) → (1...𝑁) ⊆ (1...(#‘𝐺)))
2220, 21syl 17 . . . . 5 (𝜑 → (1...𝑁) ⊆ (1...(#‘𝐺)))
23 f1ores 6064 . . . . 5 ((𝐺:(1...(#‘𝐺))–1-1→dom 𝐸 ∧ (1...𝑁) ⊆ (1...(#‘𝐺))) → (𝐺 ↾ (1...𝑁)):(1...𝑁)–1-1-onto→(𝐺 “ (1...𝑁)))
2418, 22, 23syl2anc 691 . . . 4 (𝜑 → (𝐺 ↾ (1...𝑁)):(1...𝑁)–1-1-onto→(𝐺 “ (1...𝑁)))
25 eupares.h . . . . 5 𝐻 = (𝐺 ↾ (1...𝑁))
26 f1oeq1 6040 . . . . 5 (𝐻 = (𝐺 ↾ (1...𝑁)) → (𝐻:(1...𝑁)–1-1-onto→(𝐺 “ (1...𝑁)) ↔ (𝐺 ↾ (1...𝑁)):(1...𝑁)–1-1-onto→(𝐺 “ (1...𝑁))))
2725, 26ax-mp 5 . . . 4 (𝐻:(1...𝑁)–1-1-onto→(𝐺 “ (1...𝑁)) ↔ (𝐺 ↾ (1...𝑁)):(1...𝑁)–1-1-onto→(𝐺 “ (1...𝑁)))
2824, 27sylibr 223 . . 3 (𝜑𝐻:(1...𝑁)–1-1-onto→(𝐺 “ (1...𝑁)))
29 eupapf 26499 . . . . . 6 (𝐺(𝑉 EulPaths 𝐸)𝑃𝑃:(0...(#‘𝐺))⟶𝑉)
301, 29syl 17 . . . . 5 (𝜑𝑃:(0...(#‘𝐺))⟶𝑉)
31 fzss2 12252 . . . . . 6 ((#‘𝐺) ∈ (ℤ𝑁) → (0...𝑁) ⊆ (0...(#‘𝐺)))
3220, 31syl 17 . . . . 5 (𝜑 → (0...𝑁) ⊆ (0...(#‘𝐺)))
3330, 32fssresd 5984 . . . 4 (𝜑 → (𝑃 ↾ (0...𝑁)):(0...𝑁)⟶𝑉)
34 eupares.q . . . . 5 𝑄 = (𝑃 ↾ (0...𝑁))
3534feq1i 5949 . . . 4 (𝑄:(0...𝑁)⟶𝑉 ↔ (𝑃 ↾ (0...𝑁)):(0...𝑁)⟶𝑉)
3633, 35sylibr 223 . . 3 (𝜑𝑄:(0...𝑁)⟶𝑉)
371adantr 480 . . . . . 6 ((𝜑𝑘 ∈ (1...𝑁)) → 𝐺(𝑉 EulPaths 𝐸)𝑃)
3822sselda 3568 . . . . . 6 ((𝜑𝑘 ∈ (1...𝑁)) → 𝑘 ∈ (1...(#‘𝐺)))
39 eupaseg 26500 . . . . . 6 ((𝐺(𝑉 EulPaths 𝐸)𝑃𝑘 ∈ (1...(#‘𝐺))) → (𝐸‘(𝐺𝑘)) = {(𝑃‘(𝑘 − 1)), (𝑃𝑘)})
4037, 38, 39syl2anc 691 . . . . 5 ((𝜑𝑘 ∈ (1...𝑁)) → (𝐸‘(𝐺𝑘)) = {(𝑃‘(𝑘 − 1)), (𝑃𝑘)})
4125fveq1i 6104 . . . . . . . 8 (𝐻𝑘) = ((𝐺 ↾ (1...𝑁))‘𝑘)
42 fvres 6117 . . . . . . . . 9 (𝑘 ∈ (1...𝑁) → ((𝐺 ↾ (1...𝑁))‘𝑘) = (𝐺𝑘))
4342adantl 481 . . . . . . . 8 ((𝜑𝑘 ∈ (1...𝑁)) → ((𝐺 ↾ (1...𝑁))‘𝑘) = (𝐺𝑘))
4441, 43syl5eq 2656 . . . . . . 7 ((𝜑𝑘 ∈ (1...𝑁)) → (𝐻𝑘) = (𝐺𝑘))
4544fveq2d 6107 . . . . . 6 ((𝜑𝑘 ∈ (1...𝑁)) → (𝐹‘(𝐻𝑘)) = (𝐹‘(𝐺𝑘)))
466fveq1i 6104 . . . . . . 7 (𝐹‘(𝐺𝑘)) = ((𝐸 ↾ (𝐺 “ (1...𝑁)))‘(𝐺𝑘))
47 f1ofun 6052 . . . . . . . . . . 11 (𝐺:(1...(#‘𝐺))–1-1-onto→dom 𝐸 → Fun 𝐺)
4816, 47syl 17 . . . . . . . . . 10 (𝜑 → Fun 𝐺)
49 f1of 6050 . . . . . . . . . . . 12 (𝐺:(1...(#‘𝐺))–1-1-onto→dom 𝐸𝐺:(1...(#‘𝐺))⟶dom 𝐸)
50 fdm 5964 . . . . . . . . . . . 12 (𝐺:(1...(#‘𝐺))⟶dom 𝐸 → dom 𝐺 = (1...(#‘𝐺)))
5116, 49, 503syl 18 . . . . . . . . . . 11 (𝜑 → dom 𝐺 = (1...(#‘𝐺)))
5222, 51sseqtr4d 3605 . . . . . . . . . 10 (𝜑 → (1...𝑁) ⊆ dom 𝐺)
53 funfvima2 6397 . . . . . . . . . 10 ((Fun 𝐺 ∧ (1...𝑁) ⊆ dom 𝐺) → (𝑘 ∈ (1...𝑁) → (𝐺𝑘) ∈ (𝐺 “ (1...𝑁))))
5448, 52, 53syl2anc 691 . . . . . . . . 9 (𝜑 → (𝑘 ∈ (1...𝑁) → (𝐺𝑘) ∈ (𝐺 “ (1...𝑁))))
5554imp 444 . . . . . . . 8 ((𝜑𝑘 ∈ (1...𝑁)) → (𝐺𝑘) ∈ (𝐺 “ (1...𝑁)))
56 fvres 6117 . . . . . . . 8 ((𝐺𝑘) ∈ (𝐺 “ (1...𝑁)) → ((𝐸 ↾ (𝐺 “ (1...𝑁)))‘(𝐺𝑘)) = (𝐸‘(𝐺𝑘)))
5755, 56syl 17 . . . . . . 7 ((𝜑𝑘 ∈ (1...𝑁)) → ((𝐸 ↾ (𝐺 “ (1...𝑁)))‘(𝐺𝑘)) = (𝐸‘(𝐺𝑘)))
5846, 57syl5eq 2656 . . . . . 6 ((𝜑𝑘 ∈ (1...𝑁)) → (𝐹‘(𝐺𝑘)) = (𝐸‘(𝐺𝑘)))
5945, 58eqtrd 2644 . . . . 5 ((𝜑𝑘 ∈ (1...𝑁)) → (𝐹‘(𝐻𝑘)) = (𝐸‘(𝐺𝑘)))
60 elfznn 12241 . . . . . . . . . . 11 (𝑘 ∈ (1...𝑁) → 𝑘 ∈ ℕ)
6160adantl 481 . . . . . . . . . 10 ((𝜑𝑘 ∈ (1...𝑁)) → 𝑘 ∈ ℕ)
62 nnm1nn0 11211 . . . . . . . . . 10 (𝑘 ∈ ℕ → (𝑘 − 1) ∈ ℕ0)
6361, 62syl 17 . . . . . . . . 9 ((𝜑𝑘 ∈ (1...𝑁)) → (𝑘 − 1) ∈ ℕ0)
64 nn0uz 11598 . . . . . . . . 9 0 = (ℤ‘0)
6563, 64syl6eleq 2698 . . . . . . . 8 ((𝜑𝑘 ∈ (1...𝑁)) → (𝑘 − 1) ∈ (ℤ‘0))
6661nncnd 10913 . . . . . . . . . 10 ((𝜑𝑘 ∈ (1...𝑁)) → 𝑘 ∈ ℂ)
67 ax-1cn 9873 . . . . . . . . . 10 1 ∈ ℂ
68 npcan 10169 . . . . . . . . . 10 ((𝑘 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑘 − 1) + 1) = 𝑘)
6966, 67, 68sylancl 693 . . . . . . . . 9 ((𝜑𝑘 ∈ (1...𝑁)) → ((𝑘 − 1) + 1) = 𝑘)
70 1e0p1 11428 . . . . . . . . . . . 12 1 = (0 + 1)
7170oveq1i 6559 . . . . . . . . . . 11 (1...𝑁) = ((0 + 1)...𝑁)
72 0z 11265 . . . . . . . . . . . 12 0 ∈ ℤ
73 fzp1ss 12262 . . . . . . . . . . . 12 (0 ∈ ℤ → ((0 + 1)...𝑁) ⊆ (0...𝑁))
7472, 73mp1i 13 . . . . . . . . . . 11 (𝜑 → ((0 + 1)...𝑁) ⊆ (0...𝑁))
7571, 74syl5eqss 3612 . . . . . . . . . 10 (𝜑 → (1...𝑁) ⊆ (0...𝑁))
7675sselda 3568 . . . . . . . . 9 ((𝜑𝑘 ∈ (1...𝑁)) → 𝑘 ∈ (0...𝑁))
7769, 76eqeltrd 2688 . . . . . . . 8 ((𝜑𝑘 ∈ (1...𝑁)) → ((𝑘 − 1) + 1) ∈ (0...𝑁))
78 peano2fzr 12225 . . . . . . . 8 (((𝑘 − 1) ∈ (ℤ‘0) ∧ ((𝑘 − 1) + 1) ∈ (0...𝑁)) → (𝑘 − 1) ∈ (0...𝑁))
7965, 77, 78syl2anc 691 . . . . . . 7 ((𝜑𝑘 ∈ (1...𝑁)) → (𝑘 − 1) ∈ (0...𝑁))
8034fveq1i 6104 . . . . . . . 8 (𝑄‘(𝑘 − 1)) = ((𝑃 ↾ (0...𝑁))‘(𝑘 − 1))
81 fvres 6117 . . . . . . . 8 ((𝑘 − 1) ∈ (0...𝑁) → ((𝑃 ↾ (0...𝑁))‘(𝑘 − 1)) = (𝑃‘(𝑘 − 1)))
8280, 81syl5eq 2656 . . . . . . 7 ((𝑘 − 1) ∈ (0...𝑁) → (𝑄‘(𝑘 − 1)) = (𝑃‘(𝑘 − 1)))
8379, 82syl 17 . . . . . 6 ((𝜑𝑘 ∈ (1...𝑁)) → (𝑄‘(𝑘 − 1)) = (𝑃‘(𝑘 − 1)))
8434fveq1i 6104 . . . . . . . 8 (𝑄𝑘) = ((𝑃 ↾ (0...𝑁))‘𝑘)
85 fvres 6117 . . . . . . . 8 (𝑘 ∈ (0...𝑁) → ((𝑃 ↾ (0...𝑁))‘𝑘) = (𝑃𝑘))
8684, 85syl5eq 2656 . . . . . . 7 (𝑘 ∈ (0...𝑁) → (𝑄𝑘) = (𝑃𝑘))
8776, 86syl 17 . . . . . 6 ((𝜑𝑘 ∈ (1...𝑁)) → (𝑄𝑘) = (𝑃𝑘))
8883, 87preq12d 4220 . . . . 5 ((𝜑𝑘 ∈ (1...𝑁)) → {(𝑄‘(𝑘 − 1)), (𝑄𝑘)} = {(𝑃‘(𝑘 − 1)), (𝑃𝑘)})
8940, 59, 883eqtr4d 2654 . . . 4 ((𝜑𝑘 ∈ (1...𝑁)) → (𝐹‘(𝐻𝑘)) = {(𝑄‘(𝑘 − 1)), (𝑄𝑘)})
9089ralrimiva 2949 . . 3 (𝜑 → ∀𝑘 ∈ (1...𝑁)(𝐹‘(𝐻𝑘)) = {(𝑄‘(𝑘 − 1)), (𝑄𝑘)})
91 oveq2 6557 . . . . . 6 (𝑛 = 𝑁 → (1...𝑛) = (1...𝑁))
92 f1oeq2 6041 . . . . . 6 ((1...𝑛) = (1...𝑁) → (𝐻:(1...𝑛)–1-1-onto→(𝐺 “ (1...𝑁)) ↔ 𝐻:(1...𝑁)–1-1-onto→(𝐺 “ (1...𝑁))))
9391, 92syl 17 . . . . 5 (𝑛 = 𝑁 → (𝐻:(1...𝑛)–1-1-onto→(𝐺 “ (1...𝑁)) ↔ 𝐻:(1...𝑁)–1-1-onto→(𝐺 “ (1...𝑁))))
94 oveq2 6557 . . . . . 6 (𝑛 = 𝑁 → (0...𝑛) = (0...𝑁))
9594feq2d 5944 . . . . 5 (𝑛 = 𝑁 → (𝑄:(0...𝑛)⟶𝑉𝑄:(0...𝑁)⟶𝑉))
9691raleqdv 3121 . . . . 5 (𝑛 = 𝑁 → (∀𝑘 ∈ (1...𝑛)(𝐹‘(𝐻𝑘)) = {(𝑄‘(𝑘 − 1)), (𝑄𝑘)} ↔ ∀𝑘 ∈ (1...𝑁)(𝐹‘(𝐻𝑘)) = {(𝑄‘(𝑘 − 1)), (𝑄𝑘)}))
9793, 95, 963anbi123d 1391 . . . 4 (𝑛 = 𝑁 → ((𝐻:(1...𝑛)–1-1-onto→(𝐺 “ (1...𝑁)) ∧ 𝑄:(0...𝑛)⟶𝑉 ∧ ∀𝑘 ∈ (1...𝑛)(𝐹‘(𝐻𝑘)) = {(𝑄‘(𝑘 − 1)), (𝑄𝑘)}) ↔ (𝐻:(1...𝑁)–1-1-onto→(𝐺 “ (1...𝑁)) ∧ 𝑄:(0...𝑁)⟶𝑉 ∧ ∀𝑘 ∈ (1...𝑁)(𝐹‘(𝐻𝑘)) = {(𝑄‘(𝑘 − 1)), (𝑄𝑘)})))
9897rspcev 3282 . . 3 ((𝑁 ∈ ℕ0 ∧ (𝐻:(1...𝑁)–1-1-onto→(𝐺 “ (1...𝑁)) ∧ 𝑄:(0...𝑁)⟶𝑉 ∧ ∀𝑘 ∈ (1...𝑁)(𝐹‘(𝐻𝑘)) = {(𝑄‘(𝑘 − 1)), (𝑄𝑘)})) → ∃𝑛 ∈ ℕ0 (𝐻:(1...𝑛)–1-1-onto→(𝐺 “ (1...𝑁)) ∧ 𝑄:(0...𝑛)⟶𝑉 ∧ ∀𝑘 ∈ (1...𝑛)(𝐹‘(𝐻𝑘)) = {(𝑄‘(𝑘 − 1)), (𝑄𝑘)}))
9910, 28, 36, 90, 98syl13anc 1320 . 2 (𝜑 → ∃𝑛 ∈ ℕ0 (𝐻:(1...𝑛)–1-1-onto→(𝐺 “ (1...𝑁)) ∧ 𝑄:(0...𝑛)⟶𝑉 ∧ ∀𝑘 ∈ (1...𝑛)(𝐹‘(𝐻𝑘)) = {(𝑄‘(𝑘 − 1)), (𝑄𝑘)}))
1006dmeqi 5247 . . . . 5 dom 𝐹 = dom (𝐸 ↾ (𝐺 “ (1...𝑁)))
101 dmres 5339 . . . . 5 dom (𝐸 ↾ (𝐺 “ (1...𝑁))) = ((𝐺 “ (1...𝑁)) ∩ dom 𝐸)
102100, 101eqtri 2632 . . . 4 dom 𝐹 = ((𝐺 “ (1...𝑁)) ∩ dom 𝐸)
103 imassrn 5396 . . . . . 6 (𝐺 “ (1...𝑁)) ⊆ ran 𝐺
104 f1ofo 6057 . . . . . . 7 (𝐺:(1...(#‘𝐺))–1-1-onto→dom 𝐸𝐺:(1...(#‘𝐺))–onto→dom 𝐸)
105 forn 6031 . . . . . . 7 (𝐺:(1...(#‘𝐺))–onto→dom 𝐸 → ran 𝐺 = dom 𝐸)
10616, 104, 1053syl 18 . . . . . 6 (𝜑 → ran 𝐺 = dom 𝐸)
107103, 106syl5sseq 3616 . . . . 5 (𝜑 → (𝐺 “ (1...𝑁)) ⊆ dom 𝐸)
108 df-ss 3554 . . . . 5 ((𝐺 “ (1...𝑁)) ⊆ dom 𝐸 ↔ ((𝐺 “ (1...𝑁)) ∩ dom 𝐸) = (𝐺 “ (1...𝑁)))
109107, 108sylib 207 . . . 4 (𝜑 → ((𝐺 “ (1...𝑁)) ∩ dom 𝐸) = (𝐺 “ (1...𝑁)))
110102, 109syl5eq 2656 . . 3 (𝜑 → dom 𝐹 = (𝐺 “ (1...𝑁)))
111 iseupa 26492 . . 3 (dom 𝐹 = (𝐺 “ (1...𝑁)) → (𝐻(𝑉 EulPaths 𝐹)𝑄 ↔ (𝑉 UMGrph 𝐹 ∧ ∃𝑛 ∈ ℕ0 (𝐻:(1...𝑛)–1-1-onto→(𝐺 “ (1...𝑁)) ∧ 𝑄:(0...𝑛)⟶𝑉 ∧ ∀𝑘 ∈ (1...𝑛)(𝐹‘(𝐻𝑘)) = {(𝑄‘(𝑘 − 1)), (𝑄𝑘)}))))
112110, 111syl 17 . 2 (𝜑 → (𝐻(𝑉 EulPaths 𝐹)𝑄 ↔ (𝑉 UMGrph 𝐹 ∧ ∃𝑛 ∈ ℕ0 (𝐻:(1...𝑛)–1-1-onto→(𝐺 “ (1...𝑁)) ∧ 𝑄:(0...𝑛)⟶𝑉 ∧ ∀𝑘 ∈ (1...𝑛)(𝐹‘(𝐻𝑘)) = {(𝑄‘(𝑘 − 1)), (𝑄𝑘)}))))
1137, 99, 112mpbir2and 959 1 (𝜑𝐻(𝑉 EulPaths 𝐹)𝑄)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ∀wral 2896  ∃wrex 2897  {crab 2900   ∖ cdif 3537   ∩ cin 3539   ⊆ wss 3540  ∅c0 3874  𝒫 cpw 4108  {csn 4125  {cpr 4127   class class class wbr 4583  dom cdm 5038  ran crn 5039   ↾ cres 5040   “ cima 5041  Fun wfun 5798   Fn wfn 5799  ⟶wf 5800  –1-1→wf1 5801  –onto→wfo 5802  –1-1-onto→wf1o 5803  ‘cfv 5804  (class class class)co 6549  ℂcc 9813  0cc0 9815  1c1 9816   + caddc 9818   ≤ cle 9954   − cmin 10145  ℕcn 10897  2c2 10947  ℕ0cn0 11169  ℤcz 11254  ℤ≥cuz 11563  ...cfz 12197  #chash 12979   UMGrph cumg 25841   EulPaths ceup 26489 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 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-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-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-er 7629  df-pm 7747  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-card 8648  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-z 11255  df-uz 11564  df-fz 12198  df-hash 12980  df-umgra 25842  df-eupa 26490 This theorem is referenced by: (None)
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