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Theorem diophrex 37158
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 2624 . . . . 5 (𝑎 = 𝑡 → (𝑎 = (𝑏 ↾ (1...𝑁)) ↔ 𝑡 = (𝑏 ↾ (1...𝑁))))
21rexbidv 3048 . . . 4 (𝑎 = 𝑡 → (∃𝑏𝑆 𝑎 = (𝑏 ↾ (1...𝑁)) ↔ ∃𝑏𝑆 𝑡 = (𝑏 ↾ (1...𝑁))))
3 reseq1 5379 . . . . . 6 (𝑏 = 𝑢 → (𝑏 ↾ (1...𝑁)) = (𝑢 ↾ (1...𝑁)))
43eqeq2d 2630 . . . . 5 (𝑏 = 𝑢 → (𝑡 = (𝑏 ↾ (1...𝑁)) ↔ 𝑡 = (𝑢 ↾ (1...𝑁))))
54cbvrexv 3167 . . . 4 (∃𝑏𝑆 𝑡 = (𝑏 ↾ (1...𝑁)) ↔ ∃𝑢𝑆 𝑡 = (𝑢 ↾ (1...𝑁)))
62, 5syl6bb 276 . . 3 (𝑎 = 𝑡 → (∃𝑏𝑆 𝑎 = (𝑏 ↾ (1...𝑁)) ↔ ∃𝑢𝑆 𝑡 = (𝑢 ↾ (1...𝑁))))
76cbvabv 2745 . 2 {𝑎 ∣ ∃𝑏𝑆 𝑎 = (𝑏 ↾ (1...𝑁))} = {𝑡 ∣ ∃𝑢𝑆 𝑡 = (𝑢 ↾ (1...𝑁))}
8 rexeq 3134 . . . . . 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 2739 . . . . 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 482 . . . 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 2624 . . . . . . . . . . 11 (𝑑 = 𝑏 → (𝑑 = (𝑒 ↾ (1...𝑀)) ↔ 𝑏 = (𝑒 ↾ (1...𝑀))))
1211anbi1d 740 . . . . . . . . . 10 (𝑑 = 𝑏 → ((𝑑 = (𝑒 ↾ (1...𝑀)) ∧ (𝑐𝑒) = 0) ↔ (𝑏 = (𝑒 ↾ (1...𝑀)) ∧ (𝑐𝑒) = 0)))
1312rexbidv 3048 . . . . . . . . 9 (𝑑 = 𝑏 → (∃𝑒 ∈ (ℕ0𝑚 ℕ)(𝑑 = (𝑒 ↾ (1...𝑀)) ∧ (𝑐𝑒) = 0) ↔ ∃𝑒 ∈ (ℕ0𝑚 ℕ)(𝑏 = (𝑒 ↾ (1...𝑀)) ∧ (𝑐𝑒) = 0)))
1413rexab 3363 . . . . . . . 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 3084 . . . . . . . . . 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 1772 . . . . . . . . 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 3220 . . . . . . . . . 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 680 . . . . . . . . . . . . . 14 (((𝑏 = (𝑒 ↾ (1...𝑀)) ∧ (𝑐𝑒) = 0) ∧ 𝑎 = (𝑏 ↾ (1...𝑁))) ↔ (𝑏 = (𝑒 ↾ (1...𝑀)) ∧ ((𝑐𝑒) = 0 ∧ 𝑎 = (𝑏 ↾ (1...𝑁)))))
1918exbii 1772 . . . . . . . . . . . . 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 3198 . . . . . . . . . . . . . . 15 𝑒 ∈ V
2120resex 5431 . . . . . . . . . . . . . 14 (𝑒 ↾ (1...𝑀)) ∈ V
22 reseq1 5379 . . . . . . . . . . . . . . . 16 (𝑏 = (𝑒 ↾ (1...𝑀)) → (𝑏 ↾ (1...𝑁)) = ((𝑒 ↾ (1...𝑀)) ↾ (1...𝑁)))
2322eqeq2d 2630 . . . . . . . . . . . . . . 15 (𝑏 = (𝑒 ↾ (1...𝑀)) → (𝑎 = (𝑏 ↾ (1...𝑁)) ↔ 𝑎 = ((𝑒 ↾ (1...𝑀)) ↾ (1...𝑁))))
2423anbi2d 739 . . . . . . . . . . . . . 14 (𝑏 = (𝑒 ↾ (1...𝑀)) → (((𝑐𝑒) = 0 ∧ 𝑎 = (𝑏 ↾ (1...𝑁))) ↔ ((𝑐𝑒) = 0 ∧ 𝑎 = ((𝑒 ↾ (1...𝑀)) ↾ (1...𝑁)))))
2521, 24ceqsexv 3237 . . . . . . . . . . . . 13 (∃𝑏(𝑏 = (𝑒 ↾ (1...𝑀)) ∧ ((𝑐𝑒) = 0 ∧ 𝑎 = (𝑏 ↾ (1...𝑁)))) ↔ ((𝑐𝑒) = 0 ∧ 𝑎 = ((𝑒 ↾ (1...𝑀)) ↾ (1...𝑁))))
2619, 25bitri 264 . . . . . . . . . . . 12 (∃𝑏((𝑏 = (𝑒 ↾ (1...𝑀)) ∧ (𝑐𝑒) = 0) ∧ 𝑎 = (𝑏 ↾ (1...𝑁))) ↔ ((𝑐𝑒) = 0 ∧ 𝑎 = ((𝑒 ↾ (1...𝑀)) ↾ (1...𝑁))))
27 ancom 466 . . . . . . . . . . . . 13 (((𝑐𝑒) = 0 ∧ 𝑎 = ((𝑒 ↾ (1...𝑀)) ↾ (1...𝑁))) ↔ (𝑎 = ((𝑒 ↾ (1...𝑀)) ↾ (1...𝑁)) ∧ (𝑐𝑒) = 0))
28 simpl2 1063 . . . . . . . . . . . . . . . 16 (((𝑁 ∈ ℕ0𝑀 ∈ (ℤ𝑁) ∧ 𝑆 ∈ (Dioph‘𝑀)) ∧ 𝑐 ∈ (mzPoly‘ℕ)) → 𝑀 ∈ (ℤ𝑁))
29 fzss2 12366 . . . . . . . . . . . . . . . 16 (𝑀 ∈ (ℤ𝑁) → (1...𝑁) ⊆ (1...𝑀))
30 resabs1 5415 . . . . . . . . . . . . . . . 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 2630 . . . . . . . . . . . . . 14 (((𝑁 ∈ ℕ0𝑀 ∈ (ℤ𝑁) ∧ 𝑆 ∈ (Dioph‘𝑀)) ∧ 𝑐 ∈ (mzPoly‘ℕ)) → (𝑎 = ((𝑒 ↾ (1...𝑀)) ↾ (1...𝑁)) ↔ 𝑎 = (𝑒 ↾ (1...𝑁))))
3332anbi1d 740 . . . . . . . . . . . . 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, 33syl5bb 272 . . . . . . . . . . . 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, 34syl5bb 272 . . . . . . . . . . 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 3048 . . . . . . . . . 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, 36syl5bbr 274 . . . . . . . . 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, 37syl5bbr 274 . . . . . . . 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, 38syl5bb 272 . . . . . . 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 2739 . . . . . 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 37148 . . . . . . 7 ((𝑁 ∈ ℕ0𝑐 ∈ (mzPoly‘ℕ)) → {𝑎 ∣ ∃𝑒 ∈ (ℕ0𝑚 ℕ)(𝑎 = (𝑒 ↾ (1...𝑁)) ∧ (𝑐𝑒) = 0)} ∈ (Dioph‘𝑁))
42413ad2antl1 1221 . . . . . 6 (((𝑁 ∈ ℕ0𝑀 ∈ (ℤ𝑁) ∧ 𝑆 ∈ (Dioph‘𝑀)) ∧ 𝑐 ∈ (mzPoly‘ℕ)) → {𝑎 ∣ ∃𝑒 ∈ (ℕ0𝑚 ℕ)(𝑎 = (𝑒 ↾ (1...𝑁)) ∧ (𝑐𝑒) = 0)} ∈ (Dioph‘𝑁))
4340, 42eqeltrd 2699 . . . . 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 481 . . . 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 2699 . . 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 37147 . . . . 5 (𝑆 ∈ (Dioph‘𝑀) ↔ (𝑀 ∈ ℕ0 ∧ ∃𝑐 ∈ (mzPoly‘ℕ)𝑆 = {𝑑 ∣ ∃𝑒 ∈ (ℕ0𝑚 ℕ)(𝑑 = (𝑒 ↾ (1...𝑀)) ∧ (𝑐𝑒) = 0)}))
4746simprbi 480 . . . 4 (𝑆 ∈ (Dioph‘𝑀) → ∃𝑐 ∈ (mzPoly‘ℕ)𝑆 = {𝑑 ∣ ∃𝑒 ∈ (ℕ0𝑚 ℕ)(𝑑 = (𝑒 ↾ (1...𝑀)) ∧ (𝑐𝑒) = 0)})
48473ad2ant3 1082 . . 3 ((𝑁 ∈ ℕ0𝑀 ∈ (ℤ𝑁) ∧ 𝑆 ∈ (Dioph‘𝑀)) → ∃𝑐 ∈ (mzPoly‘ℕ)𝑆 = {𝑑 ∣ ∃𝑒 ∈ (ℕ0𝑚 ℕ)(𝑑 = (𝑒 ↾ (1...𝑀)) ∧ (𝑐𝑒) = 0)})
4945, 48r19.29a 3074 . 2 ((𝑁 ∈ ℕ0𝑀 ∈ (ℤ𝑁) ∧ 𝑆 ∈ (Dioph‘𝑀)) → {𝑎 ∣ ∃𝑏𝑆 𝑎 = (𝑏 ↾ (1...𝑁))} ∈ (Dioph‘𝑁))
507, 49syl5eqelr 2704 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 384  w3a 1036   = wceq 1481  wex 1702  wcel 1988  {cab 2606  wrex 2910  wss 3567  cres 5106  cfv 5876  (class class class)co 6635  𝑚 cmap 7842  0cc0 9921  1c1 9922  cn 11005  0cn0 11277  cuz 11672  ...cfz 12311  mzPolycmzp 37104  Diophcdioph 37137
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-rep 4762  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934  ax-inf2 8523  ax-cnex 9977  ax-resscn 9978  ax-1cn 9979  ax-icn 9980  ax-addcl 9981  ax-addrcl 9982  ax-mulcl 9983  ax-mulrcl 9984  ax-mulcom 9985  ax-addass 9986  ax-mulass 9987  ax-distr 9988  ax-i2m1 9989  ax-1ne0 9990  ax-1rid 9991  ax-rnegex 9992  ax-rrecex 9993  ax-cnre 9994  ax-pre-lttri 9995  ax-pre-lttrn 9996  ax-pre-ltadd 9997  ax-pre-mulgt0 9998
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-nel 2895  df-ral 2914  df-rex 2915  df-reu 2916  df-rmo 2917  df-rab 2918  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-pss 3583  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-tp 4173  df-op 4175  df-uni 4428  df-int 4467  df-iun 4513  df-br 4645  df-opab 4704  df-mpt 4721  df-tr 4744  df-id 5014  df-eprel 5019  df-po 5025  df-so 5026  df-fr 5063  df-we 5065  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-pred 5668  df-ord 5714  df-on 5715  df-lim 5716  df-suc 5717  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-f1 5881  df-fo 5882  df-f1o 5883  df-fv 5884  df-riota 6596  df-ov 6638  df-oprab 6639  df-mpt2 6640  df-of 6882  df-om 7051  df-1st 7153  df-2nd 7154  df-wrecs 7392  df-recs 7453  df-rdg 7491  df-1o 7545  df-oadd 7549  df-er 7727  df-map 7844  df-en 7941  df-dom 7942  df-sdom 7943  df-fin 7944  df-card 8750  df-cda 8975  df-pnf 10061  df-mnf 10062  df-xr 10063  df-ltxr 10064  df-le 10065  df-sub 10253  df-neg 10254  df-nn 11006  df-n0 11278  df-z 11363  df-uz 11673  df-fz 12312  df-hash 13101  df-mzpcl 37105  df-mzp 37106  df-dioph 37138
This theorem is referenced by:  rexrabdioph  37177
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