Users' Mathboxes Mathbox for Stefan O'Rear < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  diophrex Structured version   Visualization version   GIF version

Theorem diophrex 42337
Description: Projecting a Diophantine set by removing a coordinate results in a Diophantine set. (Contributed by Stefan O'Rear, 10-Oct-2014.)
Assertion
Ref Expression
diophrex ((𝑁 ∈ ℕ0𝑀 ∈ (ℤ𝑁) ∧ 𝑆 ∈ (Dioph‘𝑀)) → {𝑡 ∣ ∃𝑢𝑆 𝑡 = (𝑢 ↾ (1...𝑁))} ∈ (Dioph‘𝑁))
Distinct variable groups:   𝑡,𝑁,𝑢   𝑡,𝑆,𝑢
Allowed substitution hints:   𝑀(𝑢,𝑡)

Proof of Theorem diophrex
Dummy variables 𝑎 𝑏 𝑐 𝑑 𝑒 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqeq1 2729 . . . . 5 (𝑎 = 𝑡 → (𝑎 = (𝑏 ↾ (1...𝑁)) ↔ 𝑡 = (𝑏 ↾ (1...𝑁))))
21rexbidv 3168 . . . 4 (𝑎 = 𝑡 → (∃𝑏𝑆 𝑎 = (𝑏 ↾ (1...𝑁)) ↔ ∃𝑏𝑆 𝑡 = (𝑏 ↾ (1...𝑁))))
3 reseq1 5979 . . . . . 6 (𝑏 = 𝑢 → (𝑏 ↾ (1...𝑁)) = (𝑢 ↾ (1...𝑁)))
43eqeq2d 2736 . . . . 5 (𝑏 = 𝑢 → (𝑡 = (𝑏 ↾ (1...𝑁)) ↔ 𝑡 = (𝑢 ↾ (1...𝑁))))
54cbvrexvw 3225 . . . 4 (∃𝑏𝑆 𝑡 = (𝑏 ↾ (1...𝑁)) ↔ ∃𝑢𝑆 𝑡 = (𝑢 ↾ (1...𝑁)))
62, 5bitrdi 286 . . 3 (𝑎 = 𝑡 → (∃𝑏𝑆 𝑎 = (𝑏 ↾ (1...𝑁)) ↔ ∃𝑢𝑆 𝑡 = (𝑢 ↾ (1...𝑁))))
76cbvabv 2798 . 2 {𝑎 ∣ ∃𝑏𝑆 𝑎 = (𝑏 ↾ (1...𝑁))} = {𝑡 ∣ ∃𝑢𝑆 𝑡 = (𝑢 ↾ (1...𝑁))}
8 rexeq 3310 . . . . . 6 (𝑆 = {𝑑 ∣ ∃𝑒 ∈ (ℕ0m ℕ)(𝑑 = (𝑒 ↾ (1...𝑀)) ∧ (𝑐𝑒) = 0)} → (∃𝑏𝑆 𝑎 = (𝑏 ↾ (1...𝑁)) ↔ ∃𝑏 ∈ {𝑑 ∣ ∃𝑒 ∈ (ℕ0m ℕ)(𝑑 = (𝑒 ↾ (1...𝑀)) ∧ (𝑐𝑒) = 0)}𝑎 = (𝑏 ↾ (1...𝑁))))
98abbidv 2794 . . . . 5 (𝑆 = {𝑑 ∣ ∃𝑒 ∈ (ℕ0m ℕ)(𝑑 = (𝑒 ↾ (1...𝑀)) ∧ (𝑐𝑒) = 0)} → {𝑎 ∣ ∃𝑏𝑆 𝑎 = (𝑏 ↾ (1...𝑁))} = {𝑎 ∣ ∃𝑏 ∈ {𝑑 ∣ ∃𝑒 ∈ (ℕ0m ℕ)(𝑑 = (𝑒 ↾ (1...𝑀)) ∧ (𝑐𝑒) = 0)}𝑎 = (𝑏 ↾ (1...𝑁))})
109adantl 480 . . . 4 ((((𝑁 ∈ ℕ0𝑀 ∈ (ℤ𝑁) ∧ 𝑆 ∈ (Dioph‘𝑀)) ∧ 𝑐 ∈ (mzPoly‘ℕ)) ∧ 𝑆 = {𝑑 ∣ ∃𝑒 ∈ (ℕ0m ℕ)(𝑑 = (𝑒 ↾ (1...𝑀)) ∧ (𝑐𝑒) = 0)}) → {𝑎 ∣ ∃𝑏𝑆 𝑎 = (𝑏 ↾ (1...𝑁))} = {𝑎 ∣ ∃𝑏 ∈ {𝑑 ∣ ∃𝑒 ∈ (ℕ0m ℕ)(𝑑 = (𝑒 ↾ (1...𝑀)) ∧ (𝑐𝑒) = 0)}𝑎 = (𝑏 ↾ (1...𝑁))})
11 eqeq1 2729 . . . . . . . . . . 11 (𝑑 = 𝑏 → (𝑑 = (𝑒 ↾ (1...𝑀)) ↔ 𝑏 = (𝑒 ↾ (1...𝑀))))
1211anbi1d 629 . . . . . . . . . 10 (𝑑 = 𝑏 → ((𝑑 = (𝑒 ↾ (1...𝑀)) ∧ (𝑐𝑒) = 0) ↔ (𝑏 = (𝑒 ↾ (1...𝑀)) ∧ (𝑐𝑒) = 0)))
1312rexbidv 3168 . . . . . . . . 9 (𝑑 = 𝑏 → (∃𝑒 ∈ (ℕ0m ℕ)(𝑑 = (𝑒 ↾ (1...𝑀)) ∧ (𝑐𝑒) = 0) ↔ ∃𝑒 ∈ (ℕ0m ℕ)(𝑏 = (𝑒 ↾ (1...𝑀)) ∧ (𝑐𝑒) = 0)))
1413rexab 3686 . . . . . . . 8 (∃𝑏 ∈ {𝑑 ∣ ∃𝑒 ∈ (ℕ0m ℕ)(𝑑 = (𝑒 ↾ (1...𝑀)) ∧ (𝑐𝑒) = 0)}𝑎 = (𝑏 ↾ (1...𝑁)) ↔ ∃𝑏(∃𝑒 ∈ (ℕ0m ℕ)(𝑏 = (𝑒 ↾ (1...𝑀)) ∧ (𝑐𝑒) = 0) ∧ 𝑎 = (𝑏 ↾ (1...𝑁))))
15 r19.41v 3178 . . . . . . . . . 10 (∃𝑒 ∈ (ℕ0m ℕ)((𝑏 = (𝑒 ↾ (1...𝑀)) ∧ (𝑐𝑒) = 0) ∧ 𝑎 = (𝑏 ↾ (1...𝑁))) ↔ (∃𝑒 ∈ (ℕ0m ℕ)(𝑏 = (𝑒 ↾ (1...𝑀)) ∧ (𝑐𝑒) = 0) ∧ 𝑎 = (𝑏 ↾ (1...𝑁))))
1615exbii 1842 . . . . . . . . 9 (∃𝑏𝑒 ∈ (ℕ0m ℕ)((𝑏 = (𝑒 ↾ (1...𝑀)) ∧ (𝑐𝑒) = 0) ∧ 𝑎 = (𝑏 ↾ (1...𝑁))) ↔ ∃𝑏(∃𝑒 ∈ (ℕ0m ℕ)(𝑏 = (𝑒 ↾ (1...𝑀)) ∧ (𝑐𝑒) = 0) ∧ 𝑎 = (𝑏 ↾ (1...𝑁))))
17 rexcom4 3275 . . . . . . . . . 10 (∃𝑒 ∈ (ℕ0m ℕ)∃𝑏((𝑏 = (𝑒 ↾ (1...𝑀)) ∧ (𝑐𝑒) = 0) ∧ 𝑎 = (𝑏 ↾ (1...𝑁))) ↔ ∃𝑏𝑒 ∈ (ℕ0m ℕ)((𝑏 = (𝑒 ↾ (1...𝑀)) ∧ (𝑐𝑒) = 0) ∧ 𝑎 = (𝑏 ↾ (1...𝑁))))
18 anass 467 . . . . . . . . . . . . . 14 (((𝑏 = (𝑒 ↾ (1...𝑀)) ∧ (𝑐𝑒) = 0) ∧ 𝑎 = (𝑏 ↾ (1...𝑁))) ↔ (𝑏 = (𝑒 ↾ (1...𝑀)) ∧ ((𝑐𝑒) = 0 ∧ 𝑎 = (𝑏 ↾ (1...𝑁)))))
1918exbii 1842 . . . . . . . . . . . . 13 (∃𝑏((𝑏 = (𝑒 ↾ (1...𝑀)) ∧ (𝑐𝑒) = 0) ∧ 𝑎 = (𝑏 ↾ (1...𝑁))) ↔ ∃𝑏(𝑏 = (𝑒 ↾ (1...𝑀)) ∧ ((𝑐𝑒) = 0 ∧ 𝑎 = (𝑏 ↾ (1...𝑁)))))
20 vex 3465 . . . . . . . . . . . . . . 15 𝑒 ∈ V
2120resex 6034 . . . . . . . . . . . . . 14 (𝑒 ↾ (1...𝑀)) ∈ V
22 reseq1 5979 . . . . . . . . . . . . . . . 16 (𝑏 = (𝑒 ↾ (1...𝑀)) → (𝑏 ↾ (1...𝑁)) = ((𝑒 ↾ (1...𝑀)) ↾ (1...𝑁)))
2322eqeq2d 2736 . . . . . . . . . . . . . . 15 (𝑏 = (𝑒 ↾ (1...𝑀)) → (𝑎 = (𝑏 ↾ (1...𝑁)) ↔ 𝑎 = ((𝑒 ↾ (1...𝑀)) ↾ (1...𝑁))))
2423anbi2d 628 . . . . . . . . . . . . . 14 (𝑏 = (𝑒 ↾ (1...𝑀)) → (((𝑐𝑒) = 0 ∧ 𝑎 = (𝑏 ↾ (1...𝑁))) ↔ ((𝑐𝑒) = 0 ∧ 𝑎 = ((𝑒 ↾ (1...𝑀)) ↾ (1...𝑁)))))
2521, 24ceqsexv 3514 . . . . . . . . . . . . 13 (∃𝑏(𝑏 = (𝑒 ↾ (1...𝑀)) ∧ ((𝑐𝑒) = 0 ∧ 𝑎 = (𝑏 ↾ (1...𝑁)))) ↔ ((𝑐𝑒) = 0 ∧ 𝑎 = ((𝑒 ↾ (1...𝑀)) ↾ (1...𝑁))))
2619, 25bitri 274 . . . . . . . . . . . 12 (∃𝑏((𝑏 = (𝑒 ↾ (1...𝑀)) ∧ (𝑐𝑒) = 0) ∧ 𝑎 = (𝑏 ↾ (1...𝑁))) ↔ ((𝑐𝑒) = 0 ∧ 𝑎 = ((𝑒 ↾ (1...𝑀)) ↾ (1...𝑁))))
27 ancom 459 . . . . . . . . . . . . 13 (((𝑐𝑒) = 0 ∧ 𝑎 = ((𝑒 ↾ (1...𝑀)) ↾ (1...𝑁))) ↔ (𝑎 = ((𝑒 ↾ (1...𝑀)) ↾ (1...𝑁)) ∧ (𝑐𝑒) = 0))
28 simpl2 1189 . . . . . . . . . . . . . . . 16 (((𝑁 ∈ ℕ0𝑀 ∈ (ℤ𝑁) ∧ 𝑆 ∈ (Dioph‘𝑀)) ∧ 𝑐 ∈ (mzPoly‘ℕ)) → 𝑀 ∈ (ℤ𝑁))
29 fzss2 13576 . . . . . . . . . . . . . . . 16 (𝑀 ∈ (ℤ𝑁) → (1...𝑁) ⊆ (1...𝑀))
30 resabs1 6012 . . . . . . . . . . . . . . . 16 ((1...𝑁) ⊆ (1...𝑀) → ((𝑒 ↾ (1...𝑀)) ↾ (1...𝑁)) = (𝑒 ↾ (1...𝑁)))
3128, 29, 303syl 18 . . . . . . . . . . . . . . 15 (((𝑁 ∈ ℕ0𝑀 ∈ (ℤ𝑁) ∧ 𝑆 ∈ (Dioph‘𝑀)) ∧ 𝑐 ∈ (mzPoly‘ℕ)) → ((𝑒 ↾ (1...𝑀)) ↾ (1...𝑁)) = (𝑒 ↾ (1...𝑁)))
3231eqeq2d 2736 . . . . . . . . . . . . . 14 (((𝑁 ∈ ℕ0𝑀 ∈ (ℤ𝑁) ∧ 𝑆 ∈ (Dioph‘𝑀)) ∧ 𝑐 ∈ (mzPoly‘ℕ)) → (𝑎 = ((𝑒 ↾ (1...𝑀)) ↾ (1...𝑁)) ↔ 𝑎 = (𝑒 ↾ (1...𝑁))))
3332anbi1d 629 . . . . . . . . . . . . 13 (((𝑁 ∈ ℕ0𝑀 ∈ (ℤ𝑁) ∧ 𝑆 ∈ (Dioph‘𝑀)) ∧ 𝑐 ∈ (mzPoly‘ℕ)) → ((𝑎 = ((𝑒 ↾ (1...𝑀)) ↾ (1...𝑁)) ∧ (𝑐𝑒) = 0) ↔ (𝑎 = (𝑒 ↾ (1...𝑁)) ∧ (𝑐𝑒) = 0)))
3427, 33bitrid 282 . . . . . . . . . . . 12 (((𝑁 ∈ ℕ0𝑀 ∈ (ℤ𝑁) ∧ 𝑆 ∈ (Dioph‘𝑀)) ∧ 𝑐 ∈ (mzPoly‘ℕ)) → (((𝑐𝑒) = 0 ∧ 𝑎 = ((𝑒 ↾ (1...𝑀)) ↾ (1...𝑁))) ↔ (𝑎 = (𝑒 ↾ (1...𝑁)) ∧ (𝑐𝑒) = 0)))
3526, 34bitrid 282 . . . . . . . . . . 11 (((𝑁 ∈ ℕ0𝑀 ∈ (ℤ𝑁) ∧ 𝑆 ∈ (Dioph‘𝑀)) ∧ 𝑐 ∈ (mzPoly‘ℕ)) → (∃𝑏((𝑏 = (𝑒 ↾ (1...𝑀)) ∧ (𝑐𝑒) = 0) ∧ 𝑎 = (𝑏 ↾ (1...𝑁))) ↔ (𝑎 = (𝑒 ↾ (1...𝑁)) ∧ (𝑐𝑒) = 0)))
3635rexbidv 3168 . . . . . . . . . 10 (((𝑁 ∈ ℕ0𝑀 ∈ (ℤ𝑁) ∧ 𝑆 ∈ (Dioph‘𝑀)) ∧ 𝑐 ∈ (mzPoly‘ℕ)) → (∃𝑒 ∈ (ℕ0m ℕ)∃𝑏((𝑏 = (𝑒 ↾ (1...𝑀)) ∧ (𝑐𝑒) = 0) ∧ 𝑎 = (𝑏 ↾ (1...𝑁))) ↔ ∃𝑒 ∈ (ℕ0m ℕ)(𝑎 = (𝑒 ↾ (1...𝑁)) ∧ (𝑐𝑒) = 0)))
3717, 36bitr3id 284 . . . . . . . . 9 (((𝑁 ∈ ℕ0𝑀 ∈ (ℤ𝑁) ∧ 𝑆 ∈ (Dioph‘𝑀)) ∧ 𝑐 ∈ (mzPoly‘ℕ)) → (∃𝑏𝑒 ∈ (ℕ0m ℕ)((𝑏 = (𝑒 ↾ (1...𝑀)) ∧ (𝑐𝑒) = 0) ∧ 𝑎 = (𝑏 ↾ (1...𝑁))) ↔ ∃𝑒 ∈ (ℕ0m ℕ)(𝑎 = (𝑒 ↾ (1...𝑁)) ∧ (𝑐𝑒) = 0)))
3816, 37bitr3id 284 . . . . . . . 8 (((𝑁 ∈ ℕ0𝑀 ∈ (ℤ𝑁) ∧ 𝑆 ∈ (Dioph‘𝑀)) ∧ 𝑐 ∈ (mzPoly‘ℕ)) → (∃𝑏(∃𝑒 ∈ (ℕ0m ℕ)(𝑏 = (𝑒 ↾ (1...𝑀)) ∧ (𝑐𝑒) = 0) ∧ 𝑎 = (𝑏 ↾ (1...𝑁))) ↔ ∃𝑒 ∈ (ℕ0m ℕ)(𝑎 = (𝑒 ↾ (1...𝑁)) ∧ (𝑐𝑒) = 0)))
3914, 38bitrid 282 . . . . . . 7 (((𝑁 ∈ ℕ0𝑀 ∈ (ℤ𝑁) ∧ 𝑆 ∈ (Dioph‘𝑀)) ∧ 𝑐 ∈ (mzPoly‘ℕ)) → (∃𝑏 ∈ {𝑑 ∣ ∃𝑒 ∈ (ℕ0m ℕ)(𝑑 = (𝑒 ↾ (1...𝑀)) ∧ (𝑐𝑒) = 0)}𝑎 = (𝑏 ↾ (1...𝑁)) ↔ ∃𝑒 ∈ (ℕ0m ℕ)(𝑎 = (𝑒 ↾ (1...𝑁)) ∧ (𝑐𝑒) = 0)))
4039abbidv 2794 . . . . . 6 (((𝑁 ∈ ℕ0𝑀 ∈ (ℤ𝑁) ∧ 𝑆 ∈ (Dioph‘𝑀)) ∧ 𝑐 ∈ (mzPoly‘ℕ)) → {𝑎 ∣ ∃𝑏 ∈ {𝑑 ∣ ∃𝑒 ∈ (ℕ0m ℕ)(𝑑 = (𝑒 ↾ (1...𝑀)) ∧ (𝑐𝑒) = 0)}𝑎 = (𝑏 ↾ (1...𝑁))} = {𝑎 ∣ ∃𝑒 ∈ (ℕ0m ℕ)(𝑎 = (𝑒 ↾ (1...𝑁)) ∧ (𝑐𝑒) = 0)})
41 eldioph3 42328 . . . . . . 7 ((𝑁 ∈ ℕ0𝑐 ∈ (mzPoly‘ℕ)) → {𝑎 ∣ ∃𝑒 ∈ (ℕ0m ℕ)(𝑎 = (𝑒 ↾ (1...𝑁)) ∧ (𝑐𝑒) = 0)} ∈ (Dioph‘𝑁))
42413ad2antl1 1182 . . . . . 6 (((𝑁 ∈ ℕ0𝑀 ∈ (ℤ𝑁) ∧ 𝑆 ∈ (Dioph‘𝑀)) ∧ 𝑐 ∈ (mzPoly‘ℕ)) → {𝑎 ∣ ∃𝑒 ∈ (ℕ0m ℕ)(𝑎 = (𝑒 ↾ (1...𝑁)) ∧ (𝑐𝑒) = 0)} ∈ (Dioph‘𝑁))
4340, 42eqeltrd 2825 . . . . 5 (((𝑁 ∈ ℕ0𝑀 ∈ (ℤ𝑁) ∧ 𝑆 ∈ (Dioph‘𝑀)) ∧ 𝑐 ∈ (mzPoly‘ℕ)) → {𝑎 ∣ ∃𝑏 ∈ {𝑑 ∣ ∃𝑒 ∈ (ℕ0m ℕ)(𝑑 = (𝑒 ↾ (1...𝑀)) ∧ (𝑐𝑒) = 0)}𝑎 = (𝑏 ↾ (1...𝑁))} ∈ (Dioph‘𝑁))
4443adantr 479 . . . 4 ((((𝑁 ∈ ℕ0𝑀 ∈ (ℤ𝑁) ∧ 𝑆 ∈ (Dioph‘𝑀)) ∧ 𝑐 ∈ (mzPoly‘ℕ)) ∧ 𝑆 = {𝑑 ∣ ∃𝑒 ∈ (ℕ0m ℕ)(𝑑 = (𝑒 ↾ (1...𝑀)) ∧ (𝑐𝑒) = 0)}) → {𝑎 ∣ ∃𝑏 ∈ {𝑑 ∣ ∃𝑒 ∈ (ℕ0m ℕ)(𝑑 = (𝑒 ↾ (1...𝑀)) ∧ (𝑐𝑒) = 0)}𝑎 = (𝑏 ↾ (1...𝑁))} ∈ (Dioph‘𝑁))
4510, 44eqeltrd 2825 . . 3 ((((𝑁 ∈ ℕ0𝑀 ∈ (ℤ𝑁) ∧ 𝑆 ∈ (Dioph‘𝑀)) ∧ 𝑐 ∈ (mzPoly‘ℕ)) ∧ 𝑆 = {𝑑 ∣ ∃𝑒 ∈ (ℕ0m ℕ)(𝑑 = (𝑒 ↾ (1...𝑀)) ∧ (𝑐𝑒) = 0)}) → {𝑎 ∣ ∃𝑏𝑆 𝑎 = (𝑏 ↾ (1...𝑁))} ∈ (Dioph‘𝑁))
46 eldioph3b 42327 . . . . 5 (𝑆 ∈ (Dioph‘𝑀) ↔ (𝑀 ∈ ℕ0 ∧ ∃𝑐 ∈ (mzPoly‘ℕ)𝑆 = {𝑑 ∣ ∃𝑒 ∈ (ℕ0m ℕ)(𝑑 = (𝑒 ↾ (1...𝑀)) ∧ (𝑐𝑒) = 0)}))
4746simprbi 495 . . . 4 (𝑆 ∈ (Dioph‘𝑀) → ∃𝑐 ∈ (mzPoly‘ℕ)𝑆 = {𝑑 ∣ ∃𝑒 ∈ (ℕ0m ℕ)(𝑑 = (𝑒 ↾ (1...𝑀)) ∧ (𝑐𝑒) = 0)})
48473ad2ant3 1132 . . 3 ((𝑁 ∈ ℕ0𝑀 ∈ (ℤ𝑁) ∧ 𝑆 ∈ (Dioph‘𝑀)) → ∃𝑐 ∈ (mzPoly‘ℕ)𝑆 = {𝑑 ∣ ∃𝑒 ∈ (ℕ0m ℕ)(𝑑 = (𝑒 ↾ (1...𝑀)) ∧ (𝑐𝑒) = 0)})
4945, 48r19.29a 3151 . 2 ((𝑁 ∈ ℕ0𝑀 ∈ (ℤ𝑁) ∧ 𝑆 ∈ (Dioph‘𝑀)) → {𝑎 ∣ ∃𝑏𝑆 𝑎 = (𝑏 ↾ (1...𝑁))} ∈ (Dioph‘𝑁))
507, 49eqeltrrid 2830 1 ((𝑁 ∈ ℕ0𝑀 ∈ (ℤ𝑁) ∧ 𝑆 ∈ (Dioph‘𝑀)) → {𝑡 ∣ ∃𝑢𝑆 𝑡 = (𝑢 ↾ (1...𝑁))} ∈ (Dioph‘𝑁))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394  w3a 1084   = wceq 1533  wex 1773  wcel 2098  {cab 2702  wrex 3059  wss 3944  cres 5680  cfv 6549  (class class class)co 7419  m cmap 8845  0cc0 11140  1c1 11141  cn 12245  0cn0 12505  cuz 12855  ...cfz 13519  mzPolycmzp 42284  Diophcdioph 42317
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 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5365  ax-pr 5429  ax-un 7741  ax-inf2 9666  ax-cnex 11196  ax-resscn 11197  ax-1cn 11198  ax-icn 11199  ax-addcl 11200  ax-addrcl 11201  ax-mulcl 11202  ax-mulrcl 11203  ax-mulcom 11204  ax-addass 11205  ax-mulass 11206  ax-distr 11207  ax-i2m1 11208  ax-1ne0 11209  ax-1rid 11210  ax-rnegex 11211  ax-rrecex 11212  ax-cnre 11213  ax-pre-lttri 11214  ax-pre-lttrn 11215  ax-pre-ltadd 11216  ax-pre-mulgt0 11217
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-nel 3036  df-ral 3051  df-rex 3060  df-reu 3364  df-rab 3419  df-v 3463  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3964  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-int 4951  df-iun 4999  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5576  df-eprel 5582  df-po 5590  df-so 5591  df-fr 5633  df-we 5635  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-pred 6307  df-ord 6374  df-on 6375  df-lim 6376  df-suc 6377  df-iota 6501  df-fun 6551  df-fn 6552  df-f 6553  df-f1 6554  df-fo 6555  df-f1o 6556  df-fv 6557  df-riota 7375  df-ov 7422  df-oprab 7423  df-mpo 7424  df-of 7685  df-om 7872  df-1st 7994  df-2nd 7995  df-frecs 8287  df-wrecs 8318  df-recs 8392  df-rdg 8431  df-1o 8487  df-oadd 8491  df-er 8725  df-map 8847  df-en 8965  df-dom 8966  df-sdom 8967  df-fin 8968  df-dju 9926  df-card 9964  df-pnf 11282  df-mnf 11283  df-xr 11284  df-ltxr 11285  df-le 11286  df-sub 11478  df-neg 11479  df-nn 12246  df-n0 12506  df-z 12592  df-uz 12856  df-fz 13520  df-hash 14326  df-mzpcl 42285  df-mzp 42286  df-dioph 42318
This theorem is referenced by:  rexrabdioph  42356
  Copyright terms: Public domain W3C validator