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Theorem clwlkclwwlkf1 27160
Description: 𝐹 is a one-to-one function from the nonempty closed walks into the closed walks as word in a simple pseudograph. (Contributed by Alexander van der Vekens, 5-Jul-2018.) (Revised by AV, 3-May-2021.) (Revised by AV, 24-May-2022.)
Hypotheses
Ref Expression
clwlkclwwlkf.c 𝐶 = {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1st𝑤))}
clwlkclwwlkf.f 𝐹 = (𝑐𝐶 ↦ ((2nd𝑐) substr ⟨0, ((♯‘(2nd𝑐)) − 1)⟩))
Assertion
Ref Expression
clwlkclwwlkf1 (𝐺 ∈ USPGraph → 𝐹:𝐶1-1→(ClWWalks‘𝐺))
Distinct variable groups:   𝑤,𝐺,𝑐   𝐶,𝑐,𝑤   𝐹,𝑐,𝑤

Proof of Theorem clwlkclwwlkf1
Dummy variables 𝑖 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 clwlkclwwlkf.c . . 3 𝐶 = {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1st𝑤))}
2 clwlkclwwlkf.f . . 3 𝐹 = (𝑐𝐶 ↦ ((2nd𝑐) substr ⟨0, ((♯‘(2nd𝑐)) − 1)⟩))
31, 2clwlkclwwlkf 27158 . 2 (𝐺 ∈ USPGraph → 𝐹:𝐶⟶(ClWWalks‘𝐺))
42a1i 11 . . . . . . 7 (𝑥𝐶𝐹 = (𝑐𝐶 ↦ ((2nd𝑐) substr ⟨0, ((♯‘(2nd𝑐)) − 1)⟩)))
5 fveq2 6332 . . . . . . . . 9 (𝑐 = 𝑥 → (2nd𝑐) = (2nd𝑥))
65fveq2d 6336 . . . . . . . . . . 11 (𝑐 = 𝑥 → (♯‘(2nd𝑐)) = (♯‘(2nd𝑥)))
76oveq1d 6808 . . . . . . . . . 10 (𝑐 = 𝑥 → ((♯‘(2nd𝑐)) − 1) = ((♯‘(2nd𝑥)) − 1))
87opeq2d 4546 . . . . . . . . 9 (𝑐 = 𝑥 → ⟨0, ((♯‘(2nd𝑐)) − 1)⟩ = ⟨0, ((♯‘(2nd𝑥)) − 1)⟩)
95, 8oveq12d 6811 . . . . . . . 8 (𝑐 = 𝑥 → ((2nd𝑐) substr ⟨0, ((♯‘(2nd𝑐)) − 1)⟩) = ((2nd𝑥) substr ⟨0, ((♯‘(2nd𝑥)) − 1)⟩))
109adantl 467 . . . . . . 7 ((𝑥𝐶𝑐 = 𝑥) → ((2nd𝑐) substr ⟨0, ((♯‘(2nd𝑐)) − 1)⟩) = ((2nd𝑥) substr ⟨0, ((♯‘(2nd𝑥)) − 1)⟩))
11 id 22 . . . . . . 7 (𝑥𝐶𝑥𝐶)
12 ovexd 6825 . . . . . . 7 (𝑥𝐶 → ((2nd𝑥) substr ⟨0, ((♯‘(2nd𝑥)) − 1)⟩) ∈ V)
134, 10, 11, 12fvmptd 6430 . . . . . 6 (𝑥𝐶 → (𝐹𝑥) = ((2nd𝑥) substr ⟨0, ((♯‘(2nd𝑥)) − 1)⟩))
142a1i 11 . . . . . . 7 (𝑦𝐶𝐹 = (𝑐𝐶 ↦ ((2nd𝑐) substr ⟨0, ((♯‘(2nd𝑐)) − 1)⟩)))
15 fveq2 6332 . . . . . . . . 9 (𝑐 = 𝑦 → (2nd𝑐) = (2nd𝑦))
1615fveq2d 6336 . . . . . . . . . . 11 (𝑐 = 𝑦 → (♯‘(2nd𝑐)) = (♯‘(2nd𝑦)))
1716oveq1d 6808 . . . . . . . . . 10 (𝑐 = 𝑦 → ((♯‘(2nd𝑐)) − 1) = ((♯‘(2nd𝑦)) − 1))
1817opeq2d 4546 . . . . . . . . 9 (𝑐 = 𝑦 → ⟨0, ((♯‘(2nd𝑐)) − 1)⟩ = ⟨0, ((♯‘(2nd𝑦)) − 1)⟩)
1915, 18oveq12d 6811 . . . . . . . 8 (𝑐 = 𝑦 → ((2nd𝑐) substr ⟨0, ((♯‘(2nd𝑐)) − 1)⟩) = ((2nd𝑦) substr ⟨0, ((♯‘(2nd𝑦)) − 1)⟩))
2019adantl 467 . . . . . . 7 ((𝑦𝐶𝑐 = 𝑦) → ((2nd𝑐) substr ⟨0, ((♯‘(2nd𝑐)) − 1)⟩) = ((2nd𝑦) substr ⟨0, ((♯‘(2nd𝑦)) − 1)⟩))
21 id 22 . . . . . . 7 (𝑦𝐶𝑦𝐶)
22 ovexd 6825 . . . . . . 7 (𝑦𝐶 → ((2nd𝑦) substr ⟨0, ((♯‘(2nd𝑦)) − 1)⟩) ∈ V)
2314, 20, 21, 22fvmptd 6430 . . . . . 6 (𝑦𝐶 → (𝐹𝑦) = ((2nd𝑦) substr ⟨0, ((♯‘(2nd𝑦)) − 1)⟩))
2413, 23eqeqan12d 2787 . . . . 5 ((𝑥𝐶𝑦𝐶) → ((𝐹𝑥) = (𝐹𝑦) ↔ ((2nd𝑥) substr ⟨0, ((♯‘(2nd𝑥)) − 1)⟩) = ((2nd𝑦) substr ⟨0, ((♯‘(2nd𝑦)) − 1)⟩)))
2524adantl 467 . . . 4 ((𝐺 ∈ USPGraph ∧ (𝑥𝐶𝑦𝐶)) → ((𝐹𝑥) = (𝐹𝑦) ↔ ((2nd𝑥) substr ⟨0, ((♯‘(2nd𝑥)) − 1)⟩) = ((2nd𝑦) substr ⟨0, ((♯‘(2nd𝑦)) − 1)⟩)))
26 simpl 468 . . . . . . . . . 10 ((𝑥𝐶𝑦𝐶) → 𝑥𝐶)
2726adantl 467 . . . . . . . . 9 ((𝐺 ∈ USPGraph ∧ (𝑥𝐶𝑦𝐶)) → 𝑥𝐶)
2827adantr 466 . . . . . . . 8 (((𝐺 ∈ USPGraph ∧ (𝑥𝐶𝑦𝐶)) ∧ ((2nd𝑥) substr ⟨0, ((♯‘(2nd𝑥)) − 1)⟩) = ((2nd𝑦) substr ⟨0, ((♯‘(2nd𝑦)) − 1)⟩)) → 𝑥𝐶)
29 simplrr 763 . . . . . . . 8 (((𝐺 ∈ USPGraph ∧ (𝑥𝐶𝑦𝐶)) ∧ ((2nd𝑥) substr ⟨0, ((♯‘(2nd𝑥)) − 1)⟩) = ((2nd𝑦) substr ⟨0, ((♯‘(2nd𝑦)) − 1)⟩)) → 𝑦𝐶)
30 eqid 2771 . . . . . . . . . . . . . . . 16 (1st𝑥) = (1st𝑥)
31 eqid 2771 . . . . . . . . . . . . . . . 16 (2nd𝑥) = (2nd𝑥)
321, 30, 31clwlkclwwlkflem 27154 . . . . . . . . . . . . . . 15 (𝑥𝐶 → ((1st𝑥)(Walks‘𝐺)(2nd𝑥) ∧ ((2nd𝑥)‘0) = ((2nd𝑥)‘(♯‘(1st𝑥))) ∧ (♯‘(1st𝑥)) ∈ ℕ))
33 wlklenvm1 26752 . . . . . . . . . . . . . . . . 17 ((1st𝑥)(Walks‘𝐺)(2nd𝑥) → (♯‘(1st𝑥)) = ((♯‘(2nd𝑥)) − 1))
3433eqcomd 2777 . . . . . . . . . . . . . . . 16 ((1st𝑥)(Walks‘𝐺)(2nd𝑥) → ((♯‘(2nd𝑥)) − 1) = (♯‘(1st𝑥)))
35343ad2ant1 1127 . . . . . . . . . . . . . . 15 (((1st𝑥)(Walks‘𝐺)(2nd𝑥) ∧ ((2nd𝑥)‘0) = ((2nd𝑥)‘(♯‘(1st𝑥))) ∧ (♯‘(1st𝑥)) ∈ ℕ) → ((♯‘(2nd𝑥)) − 1) = (♯‘(1st𝑥)))
3632, 35syl 17 . . . . . . . . . . . . . 14 (𝑥𝐶 → ((♯‘(2nd𝑥)) − 1) = (♯‘(1st𝑥)))
3736adantr 466 . . . . . . . . . . . . 13 ((𝑥𝐶𝑦𝐶) → ((♯‘(2nd𝑥)) − 1) = (♯‘(1st𝑥)))
3837opeq2d 4546 . . . . . . . . . . . 12 ((𝑥𝐶𝑦𝐶) → ⟨0, ((♯‘(2nd𝑥)) − 1)⟩ = ⟨0, (♯‘(1st𝑥))⟩)
3938oveq2d 6809 . . . . . . . . . . 11 ((𝑥𝐶𝑦𝐶) → ((2nd𝑥) substr ⟨0, ((♯‘(2nd𝑥)) − 1)⟩) = ((2nd𝑥) substr ⟨0, (♯‘(1st𝑥))⟩))
40 eqid 2771 . . . . . . . . . . . . . . . 16 (1st𝑦) = (1st𝑦)
41 eqid 2771 . . . . . . . . . . . . . . . 16 (2nd𝑦) = (2nd𝑦)
421, 40, 41clwlkclwwlkflem 27154 . . . . . . . . . . . . . . 15 (𝑦𝐶 → ((1st𝑦)(Walks‘𝐺)(2nd𝑦) ∧ ((2nd𝑦)‘0) = ((2nd𝑦)‘(♯‘(1st𝑦))) ∧ (♯‘(1st𝑦)) ∈ ℕ))
43 wlklenvm1 26752 . . . . . . . . . . . . . . . . 17 ((1st𝑦)(Walks‘𝐺)(2nd𝑦) → (♯‘(1st𝑦)) = ((♯‘(2nd𝑦)) − 1))
4443eqcomd 2777 . . . . . . . . . . . . . . . 16 ((1st𝑦)(Walks‘𝐺)(2nd𝑦) → ((♯‘(2nd𝑦)) − 1) = (♯‘(1st𝑦)))
45443ad2ant1 1127 . . . . . . . . . . . . . . 15 (((1st𝑦)(Walks‘𝐺)(2nd𝑦) ∧ ((2nd𝑦)‘0) = ((2nd𝑦)‘(♯‘(1st𝑦))) ∧ (♯‘(1st𝑦)) ∈ ℕ) → ((♯‘(2nd𝑦)) − 1) = (♯‘(1st𝑦)))
4642, 45syl 17 . . . . . . . . . . . . . 14 (𝑦𝐶 → ((♯‘(2nd𝑦)) − 1) = (♯‘(1st𝑦)))
4746adantl 467 . . . . . . . . . . . . 13 ((𝑥𝐶𝑦𝐶) → ((♯‘(2nd𝑦)) − 1) = (♯‘(1st𝑦)))
4847opeq2d 4546 . . . . . . . . . . . 12 ((𝑥𝐶𝑦𝐶) → ⟨0, ((♯‘(2nd𝑦)) − 1)⟩ = ⟨0, (♯‘(1st𝑦))⟩)
4948oveq2d 6809 . . . . . . . . . . 11 ((𝑥𝐶𝑦𝐶) → ((2nd𝑦) substr ⟨0, ((♯‘(2nd𝑦)) − 1)⟩) = ((2nd𝑦) substr ⟨0, (♯‘(1st𝑦))⟩))
5039, 49eqeq12d 2786 . . . . . . . . . 10 ((𝑥𝐶𝑦𝐶) → (((2nd𝑥) substr ⟨0, ((♯‘(2nd𝑥)) − 1)⟩) = ((2nd𝑦) substr ⟨0, ((♯‘(2nd𝑦)) − 1)⟩) ↔ ((2nd𝑥) substr ⟨0, (♯‘(1st𝑥))⟩) = ((2nd𝑦) substr ⟨0, (♯‘(1st𝑦))⟩)))
5150adantl 467 . . . . . . . . 9 ((𝐺 ∈ USPGraph ∧ (𝑥𝐶𝑦𝐶)) → (((2nd𝑥) substr ⟨0, ((♯‘(2nd𝑥)) − 1)⟩) = ((2nd𝑦) substr ⟨0, ((♯‘(2nd𝑦)) − 1)⟩) ↔ ((2nd𝑥) substr ⟨0, (♯‘(1st𝑥))⟩) = ((2nd𝑦) substr ⟨0, (♯‘(1st𝑦))⟩)))
5251biimpa 462 . . . . . . . 8 (((𝐺 ∈ USPGraph ∧ (𝑥𝐶𝑦𝐶)) ∧ ((2nd𝑥) substr ⟨0, ((♯‘(2nd𝑥)) − 1)⟩) = ((2nd𝑦) substr ⟨0, ((♯‘(2nd𝑦)) − 1)⟩)) → ((2nd𝑥) substr ⟨0, (♯‘(1st𝑥))⟩) = ((2nd𝑦) substr ⟨0, (♯‘(1st𝑦))⟩))
5328, 29, 523jca 1122 . . . . . . 7 (((𝐺 ∈ USPGraph ∧ (𝑥𝐶𝑦𝐶)) ∧ ((2nd𝑥) substr ⟨0, ((♯‘(2nd𝑥)) − 1)⟩) = ((2nd𝑦) substr ⟨0, ((♯‘(2nd𝑦)) − 1)⟩)) → (𝑥𝐶𝑦𝐶 ∧ ((2nd𝑥) substr ⟨0, (♯‘(1st𝑥))⟩) = ((2nd𝑦) substr ⟨0, (♯‘(1st𝑦))⟩)))
541, 30, 31, 40, 41clwlkclwwlkf1lem2 27155 . . . . . . 7 ((𝑥𝐶𝑦𝐶 ∧ ((2nd𝑥) substr ⟨0, (♯‘(1st𝑥))⟩) = ((2nd𝑦) substr ⟨0, (♯‘(1st𝑦))⟩)) → ((♯‘(1st𝑥)) = (♯‘(1st𝑦)) ∧ ∀𝑖 ∈ (0..^(♯‘(1st𝑥)))((2nd𝑥)‘𝑖) = ((2nd𝑦)‘𝑖)))
55 simpl 468 . . . . . . 7 (((♯‘(1st𝑥)) = (♯‘(1st𝑦)) ∧ ∀𝑖 ∈ (0..^(♯‘(1st𝑥)))((2nd𝑥)‘𝑖) = ((2nd𝑦)‘𝑖)) → (♯‘(1st𝑥)) = (♯‘(1st𝑦)))
5653, 54, 553syl 18 . . . . . 6 (((𝐺 ∈ USPGraph ∧ (𝑥𝐶𝑦𝐶)) ∧ ((2nd𝑥) substr ⟨0, ((♯‘(2nd𝑥)) − 1)⟩) = ((2nd𝑦) substr ⟨0, ((♯‘(2nd𝑦)) − 1)⟩)) → (♯‘(1st𝑥)) = (♯‘(1st𝑦)))
571, 30, 31, 40, 41clwlkclwwlkf1lem3 27156 . . . . . . 7 ((𝑥𝐶𝑦𝐶 ∧ ((2nd𝑥) substr ⟨0, (♯‘(1st𝑥))⟩) = ((2nd𝑦) substr ⟨0, (♯‘(1st𝑦))⟩)) → ∀𝑖 ∈ (0...(♯‘(1st𝑥)))((2nd𝑥)‘𝑖) = ((2nd𝑦)‘𝑖))
5853, 57syl 17 . . . . . 6 (((𝐺 ∈ USPGraph ∧ (𝑥𝐶𝑦𝐶)) ∧ ((2nd𝑥) substr ⟨0, ((♯‘(2nd𝑥)) − 1)⟩) = ((2nd𝑦) substr ⟨0, ((♯‘(2nd𝑦)) − 1)⟩)) → ∀𝑖 ∈ (0...(♯‘(1st𝑥)))((2nd𝑥)‘𝑖) = ((2nd𝑦)‘𝑖))
59 simpl 468 . . . . . . . . 9 ((𝐺 ∈ USPGraph ∧ (𝑥𝐶𝑦𝐶)) → 𝐺 ∈ USPGraph)
60 wlkcpr 26759 . . . . . . . . . . . . . 14 (𝑥 ∈ (Walks‘𝐺) ↔ (1st𝑥)(Walks‘𝐺)(2nd𝑥))
6160biimpri 218 . . . . . . . . . . . . 13 ((1st𝑥)(Walks‘𝐺)(2nd𝑥) → 𝑥 ∈ (Walks‘𝐺))
62613ad2ant1 1127 . . . . . . . . . . . 12 (((1st𝑥)(Walks‘𝐺)(2nd𝑥) ∧ ((2nd𝑥)‘0) = ((2nd𝑥)‘(♯‘(1st𝑥))) ∧ (♯‘(1st𝑥)) ∈ ℕ) → 𝑥 ∈ (Walks‘𝐺))
6332, 62syl 17 . . . . . . . . . . 11 (𝑥𝐶𝑥 ∈ (Walks‘𝐺))
64 wlkcpr 26759 . . . . . . . . . . . . . 14 (𝑦 ∈ (Walks‘𝐺) ↔ (1st𝑦)(Walks‘𝐺)(2nd𝑦))
6564biimpri 218 . . . . . . . . . . . . 13 ((1st𝑦)(Walks‘𝐺)(2nd𝑦) → 𝑦 ∈ (Walks‘𝐺))
66653ad2ant1 1127 . . . . . . . . . . . 12 (((1st𝑦)(Walks‘𝐺)(2nd𝑦) ∧ ((2nd𝑦)‘0) = ((2nd𝑦)‘(♯‘(1st𝑦))) ∧ (♯‘(1st𝑦)) ∈ ℕ) → 𝑦 ∈ (Walks‘𝐺))
6742, 66syl 17 . . . . . . . . . . 11 (𝑦𝐶𝑦 ∈ (Walks‘𝐺))
6863, 67anim12i 600 . . . . . . . . . 10 ((𝑥𝐶𝑦𝐶) → (𝑥 ∈ (Walks‘𝐺) ∧ 𝑦 ∈ (Walks‘𝐺)))
6968adantl 467 . . . . . . . . 9 ((𝐺 ∈ USPGraph ∧ (𝑥𝐶𝑦𝐶)) → (𝑥 ∈ (Walks‘𝐺) ∧ 𝑦 ∈ (Walks‘𝐺)))
70 eqidd 2772 . . . . . . . . 9 ((𝐺 ∈ USPGraph ∧ (𝑥𝐶𝑦𝐶)) → (♯‘(1st𝑥)) = (♯‘(1st𝑥)))
7159, 69, 703jca 1122 . . . . . . . 8 ((𝐺 ∈ USPGraph ∧ (𝑥𝐶𝑦𝐶)) → (𝐺 ∈ USPGraph ∧ (𝑥 ∈ (Walks‘𝐺) ∧ 𝑦 ∈ (Walks‘𝐺)) ∧ (♯‘(1st𝑥)) = (♯‘(1st𝑥))))
7271adantr 466 . . . . . . 7 (((𝐺 ∈ USPGraph ∧ (𝑥𝐶𝑦𝐶)) ∧ ((2nd𝑥) substr ⟨0, ((♯‘(2nd𝑥)) − 1)⟩) = ((2nd𝑦) substr ⟨0, ((♯‘(2nd𝑦)) − 1)⟩)) → (𝐺 ∈ USPGraph ∧ (𝑥 ∈ (Walks‘𝐺) ∧ 𝑦 ∈ (Walks‘𝐺)) ∧ (♯‘(1st𝑥)) = (♯‘(1st𝑥))))
73 uspgr2wlkeq 26777 . . . . . . 7 ((𝐺 ∈ USPGraph ∧ (𝑥 ∈ (Walks‘𝐺) ∧ 𝑦 ∈ (Walks‘𝐺)) ∧ (♯‘(1st𝑥)) = (♯‘(1st𝑥))) → (𝑥 = 𝑦 ↔ ((♯‘(1st𝑥)) = (♯‘(1st𝑦)) ∧ ∀𝑖 ∈ (0...(♯‘(1st𝑥)))((2nd𝑥)‘𝑖) = ((2nd𝑦)‘𝑖))))
7472, 73syl 17 . . . . . 6 (((𝐺 ∈ USPGraph ∧ (𝑥𝐶𝑦𝐶)) ∧ ((2nd𝑥) substr ⟨0, ((♯‘(2nd𝑥)) − 1)⟩) = ((2nd𝑦) substr ⟨0, ((♯‘(2nd𝑦)) − 1)⟩)) → (𝑥 = 𝑦 ↔ ((♯‘(1st𝑥)) = (♯‘(1st𝑦)) ∧ ∀𝑖 ∈ (0...(♯‘(1st𝑥)))((2nd𝑥)‘𝑖) = ((2nd𝑦)‘𝑖))))
7556, 58, 74mpbir2and 692 . . . . 5 (((𝐺 ∈ USPGraph ∧ (𝑥𝐶𝑦𝐶)) ∧ ((2nd𝑥) substr ⟨0, ((♯‘(2nd𝑥)) − 1)⟩) = ((2nd𝑦) substr ⟨0, ((♯‘(2nd𝑦)) − 1)⟩)) → 𝑥 = 𝑦)
7675ex 397 . . . 4 ((𝐺 ∈ USPGraph ∧ (𝑥𝐶𝑦𝐶)) → (((2nd𝑥) substr ⟨0, ((♯‘(2nd𝑥)) − 1)⟩) = ((2nd𝑦) substr ⟨0, ((♯‘(2nd𝑦)) − 1)⟩) → 𝑥 = 𝑦))
7725, 76sylbid 230 . . 3 ((𝐺 ∈ USPGraph ∧ (𝑥𝐶𝑦𝐶)) → ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦))
7877ralrimivva 3120 . 2 (𝐺 ∈ USPGraph → ∀𝑥𝐶𝑦𝐶 ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦))
79 dff13 6655 . 2 (𝐹:𝐶1-1→(ClWWalks‘𝐺) ↔ (𝐹:𝐶⟶(ClWWalks‘𝐺) ∧ ∀𝑥𝐶𝑦𝐶 ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦)))
803, 78, 79sylanbrc 572 1 (𝐺 ∈ USPGraph → 𝐹:𝐶1-1→(ClWWalks‘𝐺))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 382  w3a 1071   = wceq 1631  wcel 2145  wral 3061  {crab 3065  Vcvv 3351  cop 4322   class class class wbr 4786  cmpt 4863  wf 6027  1-1wf1 6028  cfv 6031  (class class class)co 6793  1st c1st 7313  2nd c2nd 7314  0cc0 10138  1c1 10139  cle 10277  cmin 10468  cn 11222  ...cfz 12533  ..^cfzo 12673  chash 13321   substr csubstr 13491  USPGraphcuspgr 26265  Walkscwlks 26727  ClWalkscclwlks 26901  ClWWalkscclwwlk 27131
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096  ax-cnex 10194  ax-resscn 10195  ax-1cn 10196  ax-icn 10197  ax-addcl 10198  ax-addrcl 10199  ax-mulcl 10200  ax-mulrcl 10201  ax-mulcom 10202  ax-addass 10203  ax-mulass 10204  ax-distr 10205  ax-i2m1 10206  ax-1ne0 10207  ax-1rid 10208  ax-rnegex 10209  ax-rrecex 10210  ax-cnre 10211  ax-pre-lttri 10212  ax-pre-lttrn 10213  ax-pre-ltadd 10214  ax-pre-mulgt0 10215
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-ifp 1050  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-nel 3047  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-tp 4321  df-op 4323  df-uni 4575  df-int 4612  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-tr 4887  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5823  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-riota 6754  df-ov 6796  df-oprab 6797  df-mpt2 6798  df-om 7213  df-1st 7315  df-2nd 7316  df-wrecs 7559  df-recs 7621  df-rdg 7659  df-1o 7713  df-2o 7714  df-oadd 7717  df-er 7896  df-map 8011  df-pm 8012  df-en 8110  df-dom 8111  df-sdom 8112  df-fin 8113  df-card 8965  df-cda 9192  df-pnf 10278  df-mnf 10279  df-xr 10280  df-ltxr 10281  df-le 10282  df-sub 10470  df-neg 10471  df-nn 11223  df-2 11281  df-n0 11495  df-xnn0 11566  df-z 11580  df-uz 11889  df-fz 12534  df-fzo 12674  df-hash 13322  df-word 13495  df-lsw 13496  df-substr 13499  df-edg 26161  df-uhgr 26174  df-upgr 26198  df-uspgr 26267  df-wlks 26730  df-clwlks 26902  df-clwwlk 27132
This theorem is referenced by:  clwlkclwwlkf1o  27161
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