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Theorem 3rexfrabdioph 39570
 Description: Diophantine set builder for existential quantifier, explicit substitution, two variables. (Contributed by Stefan O'Rear, 17-Oct-2014.) (Revised by Stefan O'Rear, 6-May-2015.)
Hypotheses
Ref Expression
rexfrabdioph.1 𝑀 = (𝑁 + 1)
rexfrabdioph.2 𝐿 = (𝑀 + 1)
rexfrabdioph.3 𝐾 = (𝐿 + 1)
Assertion
Ref Expression
3rexfrabdioph ((𝑁 ∈ ℕ0 ∧ {𝑡 ∈ (ℕ0m (1...𝐾)) ∣ [(𝑡 ↾ (1...𝑁)) / 𝑢][(𝑡𝑀) / 𝑣][(𝑡𝐿) / 𝑤][(𝑡𝐾) / 𝑥]𝜑} ∈ (Dioph‘𝐾)) → {𝑢 ∈ (ℕ0m (1...𝑁)) ∣ ∃𝑣 ∈ ℕ0𝑤 ∈ ℕ0𝑥 ∈ ℕ0 𝜑} ∈ (Dioph‘𝑁))
Distinct variable groups:   𝑢,𝑡,𝑣,𝑤,𝑥,𝐾   𝑡,𝐿,𝑢,𝑣,𝑤,𝑥   𝑡,𝑀,𝑢,𝑣,𝑤,𝑥   𝑡,𝑁,𝑢,𝑣,𝑤,𝑥   𝜑,𝑡
Allowed substitution hints:   𝜑(𝑥,𝑤,𝑣,𝑢)

Proof of Theorem 3rexfrabdioph
Dummy variable 𝑎 is distinct from all other variables.
StepHypRef Expression
1 sbc2rex 39560 . . . . . 6 ([(𝑎𝑀) / 𝑣]𝑤 ∈ ℕ0𝑥 ∈ ℕ0 𝜑 ↔ ∃𝑤 ∈ ℕ0𝑥 ∈ ℕ0 [(𝑎𝑀) / 𝑣]𝜑)
21sbcbii 3813 . . . . 5 ([(𝑎 ↾ (1...𝑁)) / 𝑢][(𝑎𝑀) / 𝑣]𝑤 ∈ ℕ0𝑥 ∈ ℕ0 𝜑[(𝑎 ↾ (1...𝑁)) / 𝑢]𝑤 ∈ ℕ0𝑥 ∈ ℕ0 [(𝑎𝑀) / 𝑣]𝜑)
3 sbc2rex 39560 . . . . 5 ([(𝑎 ↾ (1...𝑁)) / 𝑢]𝑤 ∈ ℕ0𝑥 ∈ ℕ0 [(𝑎𝑀) / 𝑣]𝜑 ↔ ∃𝑤 ∈ ℕ0𝑥 ∈ ℕ0 [(𝑎 ↾ (1...𝑁)) / 𝑢][(𝑎𝑀) / 𝑣]𝜑)
42, 3bitri 278 . . . 4 ([(𝑎 ↾ (1...𝑁)) / 𝑢][(𝑎𝑀) / 𝑣]𝑤 ∈ ℕ0𝑥 ∈ ℕ0 𝜑 ↔ ∃𝑤 ∈ ℕ0𝑥 ∈ ℕ0 [(𝑎 ↾ (1...𝑁)) / 𝑢][(𝑎𝑀) / 𝑣]𝜑)
54rabbii 3458 . . 3 {𝑎 ∈ (ℕ0m (1...𝑀)) ∣ [(𝑎 ↾ (1...𝑁)) / 𝑢][(𝑎𝑀) / 𝑣]𝑤 ∈ ℕ0𝑥 ∈ ℕ0 𝜑} = {𝑎 ∈ (ℕ0m (1...𝑀)) ∣ ∃𝑤 ∈ ℕ0𝑥 ∈ ℕ0 [(𝑎 ↾ (1...𝑁)) / 𝑢][(𝑎𝑀) / 𝑣]𝜑}
6 rexfrabdioph.1 . . . . . . 7 𝑀 = (𝑁 + 1)
7 nn0p1nn 11922 . . . . . . 7 (𝑁 ∈ ℕ0 → (𝑁 + 1) ∈ ℕ)
86, 7eqeltrid 2920 . . . . . 6 (𝑁 ∈ ℕ0𝑀 ∈ ℕ)
98nnnn0d 11941 . . . . 5 (𝑁 ∈ ℕ0𝑀 ∈ ℕ0)
109adantr 484 . . . 4 ((𝑁 ∈ ℕ0 ∧ {𝑡 ∈ (ℕ0m (1...𝐾)) ∣ [(𝑡 ↾ (1...𝑁)) / 𝑢][(𝑡𝑀) / 𝑣][(𝑡𝐿) / 𝑤][(𝑡𝐾) / 𝑥]𝜑} ∈ (Dioph‘𝐾)) → 𝑀 ∈ ℕ0)
11 sbcrot3 39564 . . . . . . . . . . 11 ([(𝑎𝑀) / 𝑣][(𝑡𝐿) / 𝑤][(𝑡𝐾) / 𝑥]𝜑[(𝑡𝐿) / 𝑤][(𝑡𝐾) / 𝑥][(𝑎𝑀) / 𝑣]𝜑)
1211sbcbii 3813 . . . . . . . . . 10 ([(𝑎 ↾ (1...𝑁)) / 𝑢][(𝑎𝑀) / 𝑣][(𝑡𝐿) / 𝑤][(𝑡𝐾) / 𝑥]𝜑[(𝑎 ↾ (1...𝑁)) / 𝑢][(𝑡𝐿) / 𝑤][(𝑡𝐾) / 𝑥][(𝑎𝑀) / 𝑣]𝜑)
13 sbcrot3 39564 . . . . . . . . . 10 ([(𝑎 ↾ (1...𝑁)) / 𝑢][(𝑡𝐿) / 𝑤][(𝑡𝐾) / 𝑥][(𝑎𝑀) / 𝑣]𝜑[(𝑡𝐿) / 𝑤][(𝑡𝐾) / 𝑥][(𝑎 ↾ (1...𝑁)) / 𝑢][(𝑎𝑀) / 𝑣]𝜑)
1412, 13bitri 278 . . . . . . . . 9 ([(𝑎 ↾ (1...𝑁)) / 𝑢][(𝑎𝑀) / 𝑣][(𝑡𝐿) / 𝑤][(𝑡𝐾) / 𝑥]𝜑[(𝑡𝐿) / 𝑤][(𝑡𝐾) / 𝑥][(𝑎 ↾ (1...𝑁)) / 𝑢][(𝑎𝑀) / 𝑣]𝜑)
1514sbcbii 3813 . . . . . . . 8 ([(𝑡 ↾ (1...𝑀)) / 𝑎][(𝑎 ↾ (1...𝑁)) / 𝑢][(𝑎𝑀) / 𝑣][(𝑡𝐿) / 𝑤][(𝑡𝐾) / 𝑥]𝜑[(𝑡 ↾ (1...𝑀)) / 𝑎][(𝑡𝐿) / 𝑤][(𝑡𝐾) / 𝑥][(𝑎 ↾ (1...𝑁)) / 𝑢][(𝑎𝑀) / 𝑣]𝜑)
16 reseq1 5828 . . . . . . . . . 10 (𝑎 = (𝑡 ↾ (1...𝑀)) → (𝑎 ↾ (1...𝑁)) = ((𝑡 ↾ (1...𝑀)) ↾ (1...𝑁)))
1716sbccomieg 39566 . . . . . . . . 9 ([(𝑡 ↾ (1...𝑀)) / 𝑎][(𝑎 ↾ (1...𝑁)) / 𝑢][(𝑎𝑀) / 𝑣][(𝑡𝐿) / 𝑤][(𝑡𝐾) / 𝑥]𝜑[((𝑡 ↾ (1...𝑀)) ↾ (1...𝑁)) / 𝑢][(𝑡 ↾ (1...𝑀)) / 𝑎][(𝑎𝑀) / 𝑣][(𝑡𝐿) / 𝑤][(𝑡𝐾) / 𝑥]𝜑)
18 fzssp1 12943 . . . . . . . . . . . 12 (1...𝑁) ⊆ (1...(𝑁 + 1))
196oveq2i 7149 . . . . . . . . . . . 12 (1...𝑀) = (1...(𝑁 + 1))
2018, 19sseqtrri 3988 . . . . . . . . . . 11 (1...𝑁) ⊆ (1...𝑀)
21 resabs1 5864 . . . . . . . . . . 11 ((1...𝑁) ⊆ (1...𝑀) → ((𝑡 ↾ (1...𝑀)) ↾ (1...𝑁)) = (𝑡 ↾ (1...𝑁)))
22 dfsbcq 3759 . . . . . . . . . . 11 (((𝑡 ↾ (1...𝑀)) ↾ (1...𝑁)) = (𝑡 ↾ (1...𝑁)) → ([((𝑡 ↾ (1...𝑀)) ↾ (1...𝑁)) / 𝑢][(𝑡 ↾ (1...𝑀)) / 𝑎][(𝑎𝑀) / 𝑣][(𝑡𝐿) / 𝑤][(𝑡𝐾) / 𝑥]𝜑[(𝑡 ↾ (1...𝑁)) / 𝑢][(𝑡 ↾ (1...𝑀)) / 𝑎][(𝑎𝑀) / 𝑣][(𝑡𝐿) / 𝑤][(𝑡𝐾) / 𝑥]𝜑))
2320, 21, 22mp2b 10 . . . . . . . . . 10 ([((𝑡 ↾ (1...𝑀)) ↾ (1...𝑁)) / 𝑢][(𝑡 ↾ (1...𝑀)) / 𝑎][(𝑎𝑀) / 𝑣][(𝑡𝐿) / 𝑤][(𝑡𝐾) / 𝑥]𝜑[(𝑡 ↾ (1...𝑁)) / 𝑢][(𝑡 ↾ (1...𝑀)) / 𝑎][(𝑎𝑀) / 𝑣][(𝑡𝐿) / 𝑤][(𝑡𝐾) / 𝑥]𝜑)
24 vex 3482 . . . . . . . . . . . . . 14 𝑡 ∈ V
2524resex 5880 . . . . . . . . . . . . 13 (𝑡 ↾ (1...𝑀)) ∈ V
26 fveq1 6650 . . . . . . . . . . . . . 14 (𝑎 = (𝑡 ↾ (1...𝑀)) → (𝑎𝑀) = ((𝑡 ↾ (1...𝑀))‘𝑀))
2726sbcco3gw 4355 . . . . . . . . . . . . 13 ((𝑡 ↾ (1...𝑀)) ∈ V → ([(𝑡 ↾ (1...𝑀)) / 𝑎][(𝑎𝑀) / 𝑣][(𝑡𝐿) / 𝑤][(𝑡𝐾) / 𝑥]𝜑[((𝑡 ↾ (1...𝑀))‘𝑀) / 𝑣][(𝑡𝐿) / 𝑤][(𝑡𝐾) / 𝑥]𝜑))
2825, 27ax-mp 5 . . . . . . . . . . . 12 ([(𝑡 ↾ (1...𝑀)) / 𝑎][(𝑎𝑀) / 𝑣][(𝑡𝐿) / 𝑤][(𝑡𝐾) / 𝑥]𝜑[((𝑡 ↾ (1...𝑀))‘𝑀) / 𝑣][(𝑡𝐿) / 𝑤][(𝑡𝐾) / 𝑥]𝜑)
29 elfz1end 12930 . . . . . . . . . . . . . 14 (𝑀 ∈ ℕ ↔ 𝑀 ∈ (1...𝑀))
308, 29sylib 221 . . . . . . . . . . . . 13 (𝑁 ∈ ℕ0𝑀 ∈ (1...𝑀))
31 fvres 6670 . . . . . . . . . . . . 13 (𝑀 ∈ (1...𝑀) → ((𝑡 ↾ (1...𝑀))‘𝑀) = (𝑡𝑀))
32 dfsbcq 3759 . . . . . . . . . . . . 13 (((𝑡 ↾ (1...𝑀))‘𝑀) = (𝑡𝑀) → ([((𝑡 ↾ (1...𝑀))‘𝑀) / 𝑣][(𝑡𝐿) / 𝑤][(𝑡𝐾) / 𝑥]𝜑[(𝑡𝑀) / 𝑣][(𝑡𝐿) / 𝑤][(𝑡𝐾) / 𝑥]𝜑))
3330, 31, 323syl 18 . . . . . . . . . . . 12 (𝑁 ∈ ℕ0 → ([((𝑡 ↾ (1...𝑀))‘𝑀) / 𝑣][(𝑡𝐿) / 𝑤][(𝑡𝐾) / 𝑥]𝜑[(𝑡𝑀) / 𝑣][(𝑡𝐿) / 𝑤][(𝑡𝐾) / 𝑥]𝜑))
3428, 33syl5bb 286 . . . . . . . . . . 11 (𝑁 ∈ ℕ0 → ([(𝑡 ↾ (1...𝑀)) / 𝑎][(𝑎𝑀) / 𝑣][(𝑡𝐿) / 𝑤][(𝑡𝐾) / 𝑥]𝜑[(𝑡𝑀) / 𝑣][(𝑡𝐿) / 𝑤][(𝑡𝐾) / 𝑥]𝜑))
3534sbcbidv 3811 . . . . . . . . . 10 (𝑁 ∈ ℕ0 → ([(𝑡 ↾ (1...𝑁)) / 𝑢][(𝑡 ↾ (1...𝑀)) / 𝑎][(𝑎𝑀) / 𝑣][(𝑡𝐿) / 𝑤][(𝑡𝐾) / 𝑥]𝜑[(𝑡 ↾ (1...𝑁)) / 𝑢][(𝑡𝑀) / 𝑣][(𝑡𝐿) / 𝑤][(𝑡𝐾) / 𝑥]𝜑))
3623, 35syl5bb 286 . . . . . . . . 9 (𝑁 ∈ ℕ0 → ([((𝑡 ↾ (1...𝑀)) ↾ (1...𝑁)) / 𝑢][(𝑡 ↾ (1...𝑀)) / 𝑎][(𝑎𝑀) / 𝑣][(𝑡𝐿) / 𝑤][(𝑡𝐾) / 𝑥]𝜑[(𝑡 ↾ (1...𝑁)) / 𝑢][(𝑡𝑀) / 𝑣][(𝑡𝐿) / 𝑤][(𝑡𝐾) / 𝑥]𝜑))
3717, 36syl5bb 286 . . . . . . . 8 (𝑁 ∈ ℕ0 → ([(𝑡 ↾ (1...𝑀)) / 𝑎][(𝑎 ↾ (1...𝑁)) / 𝑢][(𝑎𝑀) / 𝑣][(𝑡𝐿) / 𝑤][(𝑡𝐾) / 𝑥]𝜑[(𝑡 ↾ (1...𝑁)) / 𝑢][(𝑡𝑀) / 𝑣][(𝑡𝐿) / 𝑤][(𝑡𝐾) / 𝑥]𝜑))
3815, 37bitr3id 288 . . . . . . 7 (𝑁 ∈ ℕ0 → ([(𝑡 ↾ (1...𝑀)) / 𝑎][(𝑡𝐿) / 𝑤][(𝑡𝐾) / 𝑥][(𝑎 ↾ (1...𝑁)) / 𝑢][(𝑎𝑀) / 𝑣]𝜑[(𝑡 ↾ (1...𝑁)) / 𝑢][(𝑡𝑀) / 𝑣][(𝑡𝐿) / 𝑤][(𝑡𝐾) / 𝑥]𝜑))
3938rabbidv 3465 . . . . . 6 (𝑁 ∈ ℕ0 → {𝑡 ∈ (ℕ0m (1...𝐾)) ∣ [(𝑡 ↾ (1...𝑀)) / 𝑎][(𝑡𝐿) / 𝑤][(𝑡𝐾) / 𝑥][(𝑎 ↾ (1...𝑁)) / 𝑢][(𝑎𝑀) / 𝑣]𝜑} = {𝑡 ∈ (ℕ0m (1...𝐾)) ∣ [(𝑡 ↾ (1...𝑁)) / 𝑢][(𝑡𝑀) / 𝑣][(𝑡𝐿) / 𝑤][(𝑡𝐾) / 𝑥]𝜑})
4039eleq1d 2900 . . . . 5 (𝑁 ∈ ℕ0 → ({𝑡 ∈ (ℕ0m (1...𝐾)) ∣ [(𝑡 ↾ (1...𝑀)) / 𝑎][(𝑡𝐿) / 𝑤][(𝑡𝐾) / 𝑥][(𝑎 ↾ (1...𝑁)) / 𝑢][(𝑎𝑀) / 𝑣]𝜑} ∈ (Dioph‘𝐾) ↔ {𝑡 ∈ (ℕ0m (1...𝐾)) ∣ [(𝑡 ↾ (1...𝑁)) / 𝑢][(𝑡𝑀) / 𝑣][(𝑡𝐿) / 𝑤][(𝑡𝐾) / 𝑥]𝜑} ∈ (Dioph‘𝐾)))
4140biimpar 481 . . . 4 ((𝑁 ∈ ℕ0 ∧ {𝑡 ∈ (ℕ0m (1...𝐾)) ∣ [(𝑡 ↾ (1...𝑁)) / 𝑢][(𝑡𝑀) / 𝑣][(𝑡𝐿) / 𝑤][(𝑡𝐾) / 𝑥]𝜑} ∈ (Dioph‘𝐾)) → {𝑡 ∈ (ℕ0m (1...𝐾)) ∣ [(𝑡 ↾ (1...𝑀)) / 𝑎][(𝑡𝐿) / 𝑤][(𝑡𝐾) / 𝑥][(𝑎 ↾ (1...𝑁)) / 𝑢][(𝑎𝑀) / 𝑣]𝜑} ∈ (Dioph‘𝐾))
42 rexfrabdioph.2 . . . . 5 𝐿 = (𝑀 + 1)
43 rexfrabdioph.3 . . . . 5 𝐾 = (𝐿 + 1)
4442, 432rexfrabdioph 39569 . . . 4 ((𝑀 ∈ ℕ0 ∧ {𝑡 ∈ (ℕ0m (1...𝐾)) ∣ [(𝑡 ↾ (1...𝑀)) / 𝑎][(𝑡𝐿) / 𝑤][(𝑡𝐾) / 𝑥][(𝑎 ↾ (1...𝑁)) / 𝑢][(𝑎𝑀) / 𝑣]𝜑} ∈ (Dioph‘𝐾)) → {𝑎 ∈ (ℕ0m (1...𝑀)) ∣ ∃𝑤 ∈ ℕ0𝑥 ∈ ℕ0 [(𝑎 ↾ (1...𝑁)) / 𝑢][(𝑎𝑀) / 𝑣]𝜑} ∈ (Dioph‘𝑀))
4510, 41, 44syl2anc 587 . . 3 ((𝑁 ∈ ℕ0 ∧ {𝑡 ∈ (ℕ0m (1...𝐾)) ∣ [(𝑡 ↾ (1...𝑁)) / 𝑢][(𝑡𝑀) / 𝑣][(𝑡𝐿) / 𝑤][(𝑡𝐾) / 𝑥]𝜑} ∈ (Dioph‘𝐾)) → {𝑎 ∈ (ℕ0m (1...𝑀)) ∣ ∃𝑤 ∈ ℕ0𝑥 ∈ ℕ0 [(𝑎 ↾ (1...𝑁)) / 𝑢][(𝑎𝑀) / 𝑣]𝜑} ∈ (Dioph‘𝑀))
465, 45eqeltrid 2920 . 2 ((𝑁 ∈ ℕ0 ∧ {𝑡 ∈ (ℕ0m (1...𝐾)) ∣ [(𝑡 ↾ (1...𝑁)) / 𝑢][(𝑡𝑀) / 𝑣][(𝑡𝐿) / 𝑤][(𝑡𝐾) / 𝑥]𝜑} ∈ (Dioph‘𝐾)) → {𝑎 ∈ (ℕ0m (1...𝑀)) ∣ [(𝑎 ↾ (1...𝑁)) / 𝑢][(𝑎𝑀) / 𝑣]𝑤 ∈ ℕ0𝑥 ∈ ℕ0 𝜑} ∈ (Dioph‘𝑀))
476rexfrabdioph 39568 . 2 ((𝑁 ∈ ℕ0 ∧ {𝑎 ∈ (ℕ0m (1...𝑀)) ∣ [(𝑎 ↾ (1...𝑁)) / 𝑢][(𝑎𝑀) / 𝑣]𝑤 ∈ ℕ0𝑥 ∈ ℕ0 𝜑} ∈ (Dioph‘𝑀)) → {𝑢 ∈ (ℕ0m (1...𝑁)) ∣ ∃𝑣 ∈ ℕ0𝑤 ∈ ℕ0𝑥 ∈ ℕ0 𝜑} ∈ (Dioph‘𝑁))
4846, 47syldan 594 1 ((𝑁 ∈ ℕ0 ∧ {𝑡 ∈ (ℕ0m (1...𝐾)) ∣ [(𝑡 ↾ (1...𝑁)) / 𝑢][(𝑡𝑀) / 𝑣][(𝑡𝐿) / 𝑤][(𝑡𝐾) / 𝑥]𝜑} ∈ (Dioph‘𝐾)) → {𝑢 ∈ (ℕ0m (1...𝑁)) ∣ ∃𝑣 ∈ ℕ0𝑤 ∈ ℕ0𝑥 ∈ ℕ0 𝜑} ∈ (Dioph‘𝑁))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538   ∈ wcel 2115  ∃wrex 3133  {crab 3136  Vcvv 3479  [wsbc 3757   ⊆ wss 3918   ↾ cres 5538  ‘cfv 6336  (class class class)co 7138   ↑m cmap 8389  1c1 10523   + caddc 10525  ℕcn 11623  ℕ0cn0 11883  ...cfz 12883  Diophcdioph 39528 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5171  ax-sep 5184  ax-nul 5191  ax-pow 5247  ax-pr 5311  ax-un 7444  ax-inf2 9088  ax-cnex 10578  ax-resscn 10579  ax-1cn 10580  ax-icn 10581  ax-addcl 10582  ax-addrcl 10583  ax-mulcl 10584  ax-mulrcl 10585  ax-mulcom 10586  ax-addass 10587  ax-mulass 10588  ax-distr 10589  ax-i2m1 10590  ax-1ne0 10591  ax-1rid 10592  ax-rnegex 10593  ax-rrecex 10594  ax-cnre 10595  ax-pre-lttri 10596  ax-pre-lttrn 10597  ax-pre-ltadd 10598  ax-pre-mulgt0 10599 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3014  df-nel 3118  df-ral 3137  df-rex 3138  df-reu 3139  df-rmo 3140  df-rab 3141  df-v 3481  df-sbc 3758  df-csb 3866  df-dif 3921  df-un 3923  df-in 3925  df-ss 3935  df-pss 3937  df-nul 4275  df-if 4449  df-pw 4522  df-sn 4549  df-pr 4551  df-tp 4553  df-op 4555  df-uni 4820  df-int 4858  df-iun 4902  df-br 5048  df-opab 5110  df-mpt 5128  df-tr 5154  df-id 5441  df-eprel 5446  df-po 5455  df-so 5456  df-fr 5495  df-we 5497  df-xp 5542  df-rel 5543  df-cnv 5544  df-co 5545  df-dm 5546  df-rn 5547  df-res 5548  df-ima 5549  df-pred 6129  df-ord 6175  df-on 6176  df-lim 6177  df-suc 6178  df-iota 6295  df-fun 6338  df-fn 6339  df-f 6340  df-f1 6341  df-fo 6342  df-f1o 6343  df-fv 6344  df-riota 7096  df-ov 7141  df-oprab 7142  df-mpo 7143  df-of 7392  df-om 7564  df-1st 7672  df-2nd 7673  df-wrecs 7930  df-recs 7991  df-rdg 8029  df-1o 8085  df-oadd 8089  df-er 8272  df-map 8391  df-en 8493  df-dom 8494  df-sdom 8495  df-fin 8496  df-dju 9314  df-card 9352  df-pnf 10662  df-mnf 10663  df-xr 10664  df-ltxr 10665  df-le 10666  df-sub 10857  df-neg 10858  df-nn 11624  df-n0 11884  df-z 11968  df-uz 12230  df-fz 12884  df-hash 13685  df-mzpcl 39496  df-mzp 39497  df-dioph 39529 This theorem is referenced by:  expdiophlem2  39795
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