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Theorem vdwmc2 15602
Description: Expand out the definition of an arithmetic progression. (Contributed by Mario Carneiro, 18-Aug-2014.)
Hypotheses
Ref Expression
vdwmc.1 𝑋 ∈ V
vdwmc.2 (𝜑𝐾 ∈ ℕ0)
vdwmc.3 (𝜑𝐹:𝑋𝑅)
vdwmc2.4 (𝜑𝐴𝑋)
Assertion
Ref Expression
vdwmc2 (𝜑 → (𝐾 MonoAP 𝐹 ↔ ∃𝑐𝑅𝑎 ∈ ℕ ∃𝑑 ∈ ℕ ∀𝑚 ∈ (0...(𝐾 − 1))(𝑎 + (𝑚 · 𝑑)) ∈ (𝐹 “ {𝑐})))
Distinct variable groups:   𝑎,𝑐,𝑑,𝑚,𝐹   𝐾,𝑎,𝑐,𝑑,𝑚   𝜑,𝑐   𝑅,𝑎,𝑐,𝑑   𝜑,𝑎,𝑑
Allowed substitution hints:   𝜑(𝑚)   𝐴(𝑚,𝑎,𝑐,𝑑)   𝑅(𝑚)   𝑋(𝑚,𝑎,𝑐,𝑑)

Proof of Theorem vdwmc2
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 vdwmc.1 . . 3 𝑋 ∈ V
2 vdwmc.2 . . 3 (𝜑𝐾 ∈ ℕ0)
3 vdwmc.3 . . 3 (𝜑𝐹:𝑋𝑅)
41, 2, 3vdwmc 15601 . 2 (𝜑 → (𝐾 MonoAP 𝐹 ↔ ∃𝑐𝑎 ∈ ℕ ∃𝑑 ∈ ℕ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑐})))
5 vdwapid1 15598 . . . . . . . . . . . 12 ((𝐾 ∈ ℕ ∧ 𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) → 𝑎 ∈ (𝑎(AP‘𝐾)𝑑))
6 ne0i 3902 . . . . . . . . . . . 12 (𝑎 ∈ (𝑎(AP‘𝐾)𝑑) → (𝑎(AP‘𝐾)𝑑) ≠ ∅)
75, 6syl 17 . . . . . . . . . . 11 ((𝐾 ∈ ℕ ∧ 𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) → (𝑎(AP‘𝐾)𝑑) ≠ ∅)
873expb 1263 . . . . . . . . . 10 ((𝐾 ∈ ℕ ∧ (𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ)) → (𝑎(AP‘𝐾)𝑑) ≠ ∅)
98adantll 749 . . . . . . . . 9 (((𝜑𝐾 ∈ ℕ) ∧ (𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ)) → (𝑎(AP‘𝐾)𝑑) ≠ ∅)
10 ssn0 3953 . . . . . . . . . 10 (((𝑎(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑐}) ∧ (𝑎(AP‘𝐾)𝑑) ≠ ∅) → (𝐹 “ {𝑐}) ≠ ∅)
1110expcom 451 . . . . . . . . 9 ((𝑎(AP‘𝐾)𝑑) ≠ ∅ → ((𝑎(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑐}) → (𝐹 “ {𝑐}) ≠ ∅))
129, 11syl 17 . . . . . . . 8 (((𝜑𝐾 ∈ ℕ) ∧ (𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ)) → ((𝑎(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑐}) → (𝐹 “ {𝑐}) ≠ ∅))
13 disjsn 4221 . . . . . . . . . 10 ((𝑅 ∩ {𝑐}) = ∅ ↔ ¬ 𝑐𝑅)
143adantr 481 . . . . . . . . . . . 12 ((𝜑𝐾 ∈ ℕ) → 𝐹:𝑋𝑅)
15 fimacnvdisj 6042 . . . . . . . . . . . . 13 ((𝐹:𝑋𝑅 ∧ (𝑅 ∩ {𝑐}) = ∅) → (𝐹 “ {𝑐}) = ∅)
1615ex 450 . . . . . . . . . . . 12 (𝐹:𝑋𝑅 → ((𝑅 ∩ {𝑐}) = ∅ → (𝐹 “ {𝑐}) = ∅))
1714, 16syl 17 . . . . . . . . . . 11 ((𝜑𝐾 ∈ ℕ) → ((𝑅 ∩ {𝑐}) = ∅ → (𝐹 “ {𝑐}) = ∅))
1817adantr 481 . . . . . . . . . 10 (((𝜑𝐾 ∈ ℕ) ∧ (𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ)) → ((𝑅 ∩ {𝑐}) = ∅ → (𝐹 “ {𝑐}) = ∅))
1913, 18syl5bir 233 . . . . . . . . 9 (((𝜑𝐾 ∈ ℕ) ∧ (𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ)) → (¬ 𝑐𝑅 → (𝐹 “ {𝑐}) = ∅))
2019necon1ad 2813 . . . . . . . 8 (((𝜑𝐾 ∈ ℕ) ∧ (𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ)) → ((𝐹 “ {𝑐}) ≠ ∅ → 𝑐𝑅))
2112, 20syld 47 . . . . . . 7 (((𝜑𝐾 ∈ ℕ) ∧ (𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ)) → ((𝑎(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑐}) → 𝑐𝑅))
2221rexlimdvva 3036 . . . . . 6 ((𝜑𝐾 ∈ ℕ) → (∃𝑎 ∈ ℕ ∃𝑑 ∈ ℕ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑐}) → 𝑐𝑅))
2322pm4.71rd 666 . . . . 5 ((𝜑𝐾 ∈ ℕ) → (∃𝑎 ∈ ℕ ∃𝑑 ∈ ℕ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑐}) ↔ (𝑐𝑅 ∧ ∃𝑎 ∈ ℕ ∃𝑑 ∈ ℕ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑐}))))
2423exbidv 1852 . . . 4 ((𝜑𝐾 ∈ ℕ) → (∃𝑐𝑎 ∈ ℕ ∃𝑑 ∈ ℕ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑐}) ↔ ∃𝑐(𝑐𝑅 ∧ ∃𝑎 ∈ ℕ ∃𝑑 ∈ ℕ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑐}))))
25 df-rex 2918 . . . 4 (∃𝑐𝑅𝑎 ∈ ℕ ∃𝑑 ∈ ℕ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑐}) ↔ ∃𝑐(𝑐𝑅 ∧ ∃𝑎 ∈ ℕ ∃𝑑 ∈ ℕ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑐})))
2624, 25syl6bbr 278 . . 3 ((𝜑𝐾 ∈ ℕ) → (∃𝑐𝑎 ∈ ℕ ∃𝑑 ∈ ℕ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑐}) ↔ ∃𝑐𝑅𝑎 ∈ ℕ ∃𝑑 ∈ ℕ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑐})))
27 vdwmc2.4 . . . . . . . . 9 (𝜑𝐴𝑋)
283, 27ffvelrnd 6317 . . . . . . . 8 (𝜑 → (𝐹𝐴) ∈ 𝑅)
29 ne0i 3902 . . . . . . . 8 ((𝐹𝐴) ∈ 𝑅𝑅 ≠ ∅)
3028, 29syl 17 . . . . . . 7 (𝜑𝑅 ≠ ∅)
3130adantr 481 . . . . . 6 ((𝜑𝐾 = 0) → 𝑅 ≠ ∅)
32 1nn 10976 . . . . . . . . 9 1 ∈ ℕ
3332ne0ii 3904 . . . . . . . 8 ℕ ≠ ∅
34 simpllr 798 . . . . . . . . . . . . . . 15 ((((𝜑𝐾 = 0) ∧ 𝑎 ∈ ℕ) ∧ 𝑑 ∈ ℕ) → 𝐾 = 0)
3534fveq2d 6154 . . . . . . . . . . . . . 14 ((((𝜑𝐾 = 0) ∧ 𝑎 ∈ ℕ) ∧ 𝑑 ∈ ℕ) → (AP‘𝐾) = (AP‘0))
3635oveqd 6622 . . . . . . . . . . . . 13 ((((𝜑𝐾 = 0) ∧ 𝑎 ∈ ℕ) ∧ 𝑑 ∈ ℕ) → (𝑎(AP‘𝐾)𝑑) = (𝑎(AP‘0)𝑑))
37 vdwap0 15599 . . . . . . . . . . . . . 14 ((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) → (𝑎(AP‘0)𝑑) = ∅)
3837adantll 749 . . . . . . . . . . . . 13 ((((𝜑𝐾 = 0) ∧ 𝑎 ∈ ℕ) ∧ 𝑑 ∈ ℕ) → (𝑎(AP‘0)𝑑) = ∅)
3936, 38eqtrd 2660 . . . . . . . . . . . 12 ((((𝜑𝐾 = 0) ∧ 𝑎 ∈ ℕ) ∧ 𝑑 ∈ ℕ) → (𝑎(AP‘𝐾)𝑑) = ∅)
40 0ss 3949 . . . . . . . . . . . 12 ∅ ⊆ (𝐹 “ {𝑐})
4139, 40syl6eqss 3639 . . . . . . . . . . 11 ((((𝜑𝐾 = 0) ∧ 𝑎 ∈ ℕ) ∧ 𝑑 ∈ ℕ) → (𝑎(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑐}))
4241ralrimiva 2965 . . . . . . . . . 10 (((𝜑𝐾 = 0) ∧ 𝑎 ∈ ℕ) → ∀𝑑 ∈ ℕ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑐}))
43 r19.2z 4037 . . . . . . . . . 10 ((ℕ ≠ ∅ ∧ ∀𝑑 ∈ ℕ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑐})) → ∃𝑑 ∈ ℕ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑐}))
4433, 42, 43sylancr 694 . . . . . . . . 9 (((𝜑𝐾 = 0) ∧ 𝑎 ∈ ℕ) → ∃𝑑 ∈ ℕ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑐}))
4544ralrimiva 2965 . . . . . . . 8 ((𝜑𝐾 = 0) → ∀𝑎 ∈ ℕ ∃𝑑 ∈ ℕ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑐}))
46 r19.2z 4037 . . . . . . . 8 ((ℕ ≠ ∅ ∧ ∀𝑎 ∈ ℕ ∃𝑑 ∈ ℕ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑐})) → ∃𝑎 ∈ ℕ ∃𝑑 ∈ ℕ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑐}))
4733, 45, 46sylancr 694 . . . . . . 7 ((𝜑𝐾 = 0) → ∃𝑎 ∈ ℕ ∃𝑑 ∈ ℕ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑐}))
4847ralrimivw 2966 . . . . . 6 ((𝜑𝐾 = 0) → ∀𝑐𝑅𝑎 ∈ ℕ ∃𝑑 ∈ ℕ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑐}))
49 r19.2z 4037 . . . . . 6 ((𝑅 ≠ ∅ ∧ ∀𝑐𝑅𝑎 ∈ ℕ ∃𝑑 ∈ ℕ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑐})) → ∃𝑐𝑅𝑎 ∈ ℕ ∃𝑑 ∈ ℕ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑐}))
5031, 48, 49syl2anc 692 . . . . 5 ((𝜑𝐾 = 0) → ∃𝑐𝑅𝑎 ∈ ℕ ∃𝑑 ∈ ℕ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑐}))
51 rexex 3001 . . . . 5 (∃𝑐𝑅𝑎 ∈ ℕ ∃𝑑 ∈ ℕ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑐}) → ∃𝑐𝑎 ∈ ℕ ∃𝑑 ∈ ℕ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑐}))
5250, 51syl 17 . . . 4 ((𝜑𝐾 = 0) → ∃𝑐𝑎 ∈ ℕ ∃𝑑 ∈ ℕ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑐}))
5352, 502thd 255 . . 3 ((𝜑𝐾 = 0) → (∃𝑐𝑎 ∈ ℕ ∃𝑑 ∈ ℕ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑐}) ↔ ∃𝑐𝑅𝑎 ∈ ℕ ∃𝑑 ∈ ℕ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑐})))
54 elnn0 11239 . . . 4 (𝐾 ∈ ℕ0 ↔ (𝐾 ∈ ℕ ∨ 𝐾 = 0))
552, 54sylib 208 . . 3 (𝜑 → (𝐾 ∈ ℕ ∨ 𝐾 = 0))
5626, 53, 55mpjaodan 826 . 2 (𝜑 → (∃𝑐𝑎 ∈ ℕ ∃𝑑 ∈ ℕ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑐}) ↔ ∃𝑐𝑅𝑎 ∈ ℕ ∃𝑑 ∈ ℕ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑐})))
57 vdwapval 15596 . . . . . . . . 9 ((𝐾 ∈ ℕ0𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) → (𝑥 ∈ (𝑎(AP‘𝐾)𝑑) ↔ ∃𝑚 ∈ (0...(𝐾 − 1))𝑥 = (𝑎 + (𝑚 · 𝑑))))
58573expb 1263 . . . . . . . 8 ((𝐾 ∈ ℕ0 ∧ (𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ)) → (𝑥 ∈ (𝑎(AP‘𝐾)𝑑) ↔ ∃𝑚 ∈ (0...(𝐾 − 1))𝑥 = (𝑎 + (𝑚 · 𝑑))))
592, 58sylan 488 . . . . . . 7 ((𝜑 ∧ (𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ)) → (𝑥 ∈ (𝑎(AP‘𝐾)𝑑) ↔ ∃𝑚 ∈ (0...(𝐾 − 1))𝑥 = (𝑎 + (𝑚 · 𝑑))))
6059imbi1d 331 . . . . . 6 ((𝜑 ∧ (𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ)) → ((𝑥 ∈ (𝑎(AP‘𝐾)𝑑) → 𝑥 ∈ (𝐹 “ {𝑐})) ↔ (∃𝑚 ∈ (0...(𝐾 − 1))𝑥 = (𝑎 + (𝑚 · 𝑑)) → 𝑥 ∈ (𝐹 “ {𝑐}))))
6160albidv 1851 . . . . 5 ((𝜑 ∧ (𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ)) → (∀𝑥(𝑥 ∈ (𝑎(AP‘𝐾)𝑑) → 𝑥 ∈ (𝐹 “ {𝑐})) ↔ ∀𝑥(∃𝑚 ∈ (0...(𝐾 − 1))𝑥 = (𝑎 + (𝑚 · 𝑑)) → 𝑥 ∈ (𝐹 “ {𝑐}))))
62 dfss2 3577 . . . . 5 ((𝑎(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑐}) ↔ ∀𝑥(𝑥 ∈ (𝑎(AP‘𝐾)𝑑) → 𝑥 ∈ (𝐹 “ {𝑐})))
63 ralcom4 3215 . . . . . 6 (∀𝑚 ∈ (0...(𝐾 − 1))∀𝑥(𝑥 = (𝑎 + (𝑚 · 𝑑)) → 𝑥 ∈ (𝐹 “ {𝑐})) ↔ ∀𝑥𝑚 ∈ (0...(𝐾 − 1))(𝑥 = (𝑎 + (𝑚 · 𝑑)) → 𝑥 ∈ (𝐹 “ {𝑐})))
64 ovex 6633 . . . . . . . 8 (𝑎 + (𝑚 · 𝑑)) ∈ V
65 eleq1 2692 . . . . . . . 8 (𝑥 = (𝑎 + (𝑚 · 𝑑)) → (𝑥 ∈ (𝐹 “ {𝑐}) ↔ (𝑎 + (𝑚 · 𝑑)) ∈ (𝐹 “ {𝑐})))
6664, 65ceqsalv 3224 . . . . . . 7 (∀𝑥(𝑥 = (𝑎 + (𝑚 · 𝑑)) → 𝑥 ∈ (𝐹 “ {𝑐})) ↔ (𝑎 + (𝑚 · 𝑑)) ∈ (𝐹 “ {𝑐}))
6766ralbii 2979 . . . . . 6 (∀𝑚 ∈ (0...(𝐾 − 1))∀𝑥(𝑥 = (𝑎 + (𝑚 · 𝑑)) → 𝑥 ∈ (𝐹 “ {𝑐})) ↔ ∀𝑚 ∈ (0...(𝐾 − 1))(𝑎 + (𝑚 · 𝑑)) ∈ (𝐹 “ {𝑐}))
68 r19.23v 3021 . . . . . . 7 (∀𝑚 ∈ (0...(𝐾 − 1))(𝑥 = (𝑎 + (𝑚 · 𝑑)) → 𝑥 ∈ (𝐹 “ {𝑐})) ↔ (∃𝑚 ∈ (0...(𝐾 − 1))𝑥 = (𝑎 + (𝑚 · 𝑑)) → 𝑥 ∈ (𝐹 “ {𝑐})))
6968albii 1744 . . . . . 6 (∀𝑥𝑚 ∈ (0...(𝐾 − 1))(𝑥 = (𝑎 + (𝑚 · 𝑑)) → 𝑥 ∈ (𝐹 “ {𝑐})) ↔ ∀𝑥(∃𝑚 ∈ (0...(𝐾 − 1))𝑥 = (𝑎 + (𝑚 · 𝑑)) → 𝑥 ∈ (𝐹 “ {𝑐})))
7063, 67, 693bitr3i 290 . . . . 5 (∀𝑚 ∈ (0...(𝐾 − 1))(𝑎 + (𝑚 · 𝑑)) ∈ (𝐹 “ {𝑐}) ↔ ∀𝑥(∃𝑚 ∈ (0...(𝐾 − 1))𝑥 = (𝑎 + (𝑚 · 𝑑)) → 𝑥 ∈ (𝐹 “ {𝑐})))
7161, 62, 703bitr4g 303 . . . 4 ((𝜑 ∧ (𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ)) → ((𝑎(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑐}) ↔ ∀𝑚 ∈ (0...(𝐾 − 1))(𝑎 + (𝑚 · 𝑑)) ∈ (𝐹 “ {𝑐})))
72712rexbidva 3054 . . 3 (𝜑 → (∃𝑎 ∈ ℕ ∃𝑑 ∈ ℕ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑐}) ↔ ∃𝑎 ∈ ℕ ∃𝑑 ∈ ℕ ∀𝑚 ∈ (0...(𝐾 − 1))(𝑎 + (𝑚 · 𝑑)) ∈ (𝐹 “ {𝑐})))
7372rexbidv 3050 . 2 (𝜑 → (∃𝑐𝑅𝑎 ∈ ℕ ∃𝑑 ∈ ℕ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑐}) ↔ ∃𝑐𝑅𝑎 ∈ ℕ ∃𝑑 ∈ ℕ ∀𝑚 ∈ (0...(𝐾 − 1))(𝑎 + (𝑚 · 𝑑)) ∈ (𝐹 “ {𝑐})))
744, 56, 733bitrd 294 1 (𝜑 → (𝐾 MonoAP 𝐹 ↔ ∃𝑐𝑅𝑎 ∈ ℕ ∃𝑑 ∈ ℕ ∀𝑚 ∈ (0...(𝐾 − 1))(𝑎 + (𝑚 · 𝑑)) ∈ (𝐹 “ {𝑐})))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wo 383  wa 384  w3a 1036  wal 1478   = wceq 1480  wex 1701  wcel 1992  wne 2796  wral 2912  wrex 2913  Vcvv 3191  cin 3559  wss 3560  c0 3896  {csn 4153   class class class wbr 4618  ccnv 5078  cima 5082  wf 5846  cfv 5850  (class class class)co 6605  0cc0 9881  1c1 9882   + caddc 9884   · cmul 9886  cmin 10211  cn 10965  0cn0 11237  ...cfz 12265  APcvdwa 15588   MonoAP cvdwm 15589
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6903  ax-cnex 9937  ax-resscn 9938  ax-1cn 9939  ax-icn 9940  ax-addcl 9941  ax-addrcl 9942  ax-mulcl 9943  ax-mulrcl 9944  ax-mulcom 9945  ax-addass 9946  ax-mulass 9947  ax-distr 9948  ax-i2m1 9949  ax-1ne0 9950  ax-1rid 9951  ax-rnegex 9952  ax-rrecex 9953  ax-cnre 9954  ax-pre-lttri 9955  ax-pre-lttrn 9956  ax-pre-ltadd 9957  ax-pre-mulgt0 9958
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-nel 2900  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3193  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5642  df-ord 5688  df-on 5689  df-lim 5690  df-suc 5691  df-iota 5813  df-fun 5852  df-fn 5853  df-f 5854  df-f1 5855  df-fo 5856  df-f1o 5857  df-fv 5858  df-riota 6566  df-ov 6608  df-oprab 6609  df-mpt2 6610  df-om 7014  df-1st 7116  df-2nd 7117  df-wrecs 7353  df-recs 7414  df-rdg 7452  df-er 7688  df-en 7901  df-dom 7902  df-sdom 7903  df-pnf 10021  df-mnf 10022  df-xr 10023  df-ltxr 10024  df-le 10025  df-sub 10213  df-neg 10214  df-nn 10966  df-n0 11238  df-z 11323  df-uz 11632  df-fz 12266  df-vdwap 15591  df-vdwmc 15592
This theorem is referenced by:  vdw  15617
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