3rexfrabdioph - Mathbox for Stefan O'Rear

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Theorem 3rexfrabdioph 42139

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 ∈ (ℕ₀ ↑_m (1...𝐾)) ∣ [(𝑡 ↾ (1...𝑁)) / 𝑢][(𝑡‘𝑀) / 𝑣][(𝑡‘𝐿) / 𝑤][(𝑡‘𝐾) / 𝑥]𝜑} ∈ (Dioph‘𝐾)) → {𝑢 ∈ (ℕ₀ ↑_m (1...𝑁)) ∣ ∃𝑣 ∈ ℕ₀ ∃𝑤 ∈ ℕ₀ ∃𝑥 ∈ 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.

Step	Hyp	Ref	Expression
1		sbc2rex 42129	. . . . . 6 ⊢ ([(𝑎‘𝑀) / 𝑣]∃𝑤 ∈ ℕ₀ ∃𝑥 ∈ ℕ₀ 𝜑 ↔ ∃𝑤 ∈ ℕ₀ ∃𝑥 ∈ ℕ₀ [(𝑎‘𝑀) / 𝑣]𝜑)
2	1	sbcbii 3834	. . . . 5 ⊢ ([(𝑎 ↾ (1...𝑁)) / 𝑢][(𝑎‘𝑀) / 𝑣]∃𝑤 ∈ ℕ₀ ∃𝑥 ∈ ℕ₀ 𝜑 ↔ [(𝑎 ↾ (1...𝑁)) / 𝑢]∃𝑤 ∈ ℕ₀ ∃𝑥 ∈ ℕ₀ [(𝑎‘𝑀) / 𝑣]𝜑)
3		sbc2rex 42129	. . . . 5 ⊢ ([(𝑎 ↾ (1...𝑁)) / 𝑢]∃𝑤 ∈ ℕ₀ ∃𝑥 ∈ ℕ₀ [(𝑎‘𝑀) / 𝑣]𝜑 ↔ ∃𝑤 ∈ ℕ₀ ∃𝑥 ∈ ℕ₀ [(𝑎 ↾ (1...𝑁)) / 𝑢][(𝑎‘𝑀) / 𝑣]𝜑)
4	2, 3	bitri 275	. . . 4 ⊢ ([(𝑎 ↾ (1...𝑁)) / 𝑢][(𝑎‘𝑀) / 𝑣]∃𝑤 ∈ ℕ₀ ∃𝑥 ∈ ℕ₀ 𝜑 ↔ ∃𝑤 ∈ ℕ₀ ∃𝑥 ∈ ℕ₀ [(𝑎 ↾ (1...𝑁)) / 𝑢][(𝑎‘𝑀) / 𝑣]𝜑)
5	4	rabbii 3433	. . 3 ⊢ {𝑎 ∈ (ℕ₀ ↑_m (1...𝑀)) ∣ [(𝑎 ↾ (1...𝑁)) / 𝑢][(𝑎‘𝑀) / 𝑣]∃𝑤 ∈ ℕ₀ ∃𝑥 ∈ ℕ₀ 𝜑} = {𝑎 ∈ (ℕ₀ ↑_m (1...𝑀)) ∣ ∃𝑤 ∈ ℕ₀ ∃𝑥 ∈ ℕ₀ [(𝑎 ↾ (1...𝑁)) / 𝑢][(𝑎‘𝑀) / 𝑣]𝜑}
6		rexfrabdioph.1	. . . . . . 7 ⊢ 𝑀 = (𝑁 + 1)
7		nn0p1nn 12533	. . . . . . 7 ⊢ (𝑁 ∈ ℕ₀ → (𝑁 + 1) ∈ ℕ)
8	6, 7	eqeltrid 2832	. . . . . 6 ⊢ (𝑁 ∈ ℕ₀ → 𝑀 ∈ ℕ)
9	8	nnnn0d 12554	. . . . 5 ⊢ (𝑁 ∈ ℕ₀ → 𝑀 ∈ ℕ₀)
10	9	adantr 480	. . . 4 ⊢ ((𝑁 ∈ ℕ₀ ∧ {𝑡 ∈ (ℕ₀ ↑_m (1...𝐾)) ∣ [(𝑡 ↾ (1...𝑁)) / 𝑢][(𝑡‘𝑀) / 𝑣][(𝑡‘𝐿) / 𝑤][(𝑡‘𝐾) / 𝑥]𝜑} ∈ (Dioph‘𝐾)) → 𝑀 ∈ ℕ₀)
11		sbcrot3 42133	. . . . . . . . . . 11 ⊢ ([(𝑎‘𝑀) / 𝑣][(𝑡‘𝐿) / 𝑤][(𝑡‘𝐾) / 𝑥]𝜑 ↔ [(𝑡‘𝐿) / 𝑤][(𝑡‘𝐾) / 𝑥][(𝑎‘𝑀) / 𝑣]𝜑)
12	11	sbcbii 3834	. . . . . . . . . 10 ⊢ ([(𝑎 ↾ (1...𝑁)) / 𝑢][(𝑎‘𝑀) / 𝑣][(𝑡‘𝐿) / 𝑤][(𝑡‘𝐾) / 𝑥]𝜑 ↔ [(𝑎 ↾ (1...𝑁)) / 𝑢][(𝑡‘𝐿) / 𝑤][(𝑡‘𝐾) / 𝑥][(𝑎‘𝑀) / 𝑣]𝜑)
13		sbcrot3 42133	. . . . . . . . . 10 ⊢ ([(𝑎 ↾ (1...𝑁)) / 𝑢][(𝑡‘𝐿) / 𝑤][(𝑡‘𝐾) / 𝑥][(𝑎‘𝑀) / 𝑣]𝜑 ↔ [(𝑡‘𝐿) / 𝑤][(𝑡‘𝐾) / 𝑥][(𝑎 ↾ (1...𝑁)) / 𝑢][(𝑎‘𝑀) / 𝑣]𝜑)
14	12, 13	bitri 275	. . . . . . . . 9 ⊢ ([(𝑎 ↾ (1...𝑁)) / 𝑢][(𝑎‘𝑀) / 𝑣][(𝑡‘𝐿) / 𝑤][(𝑡‘𝐾) / 𝑥]𝜑 ↔ [(𝑡‘𝐿) / 𝑤][(𝑡‘𝐾) / 𝑥][(𝑎 ↾ (1...𝑁)) / 𝑢][(𝑎‘𝑀) / 𝑣]𝜑)
15	14	sbcbii 3834	. . . . . . . 8 ⊢ ([(𝑡 ↾ (1...𝑀)) / 𝑎][(𝑎 ↾ (1...𝑁)) / 𝑢][(𝑎‘𝑀) / 𝑣][(𝑡‘𝐿) / 𝑤][(𝑡‘𝐾) / 𝑥]𝜑 ↔ [(𝑡 ↾ (1...𝑀)) / 𝑎][(𝑡‘𝐿) / 𝑤][(𝑡‘𝐾) / 𝑥][(𝑎 ↾ (1...𝑁)) / 𝑢][(𝑎‘𝑀) / 𝑣]𝜑)
16		reseq1 5973	. . . . . . . . . 10 ⊢ (𝑎 = (𝑡 ↾ (1...𝑀)) → (𝑎 ↾ (1...𝑁)) = ((𝑡 ↾ (1...𝑀)) ↾ (1...𝑁)))
17	16	sbccomieg 42135	. . . . . . . . 9 ⊢ ([(𝑡 ↾ (1...𝑀)) / 𝑎][(𝑎 ↾ (1...𝑁)) / 𝑢][(𝑎‘𝑀) / 𝑣][(𝑡‘𝐿) / 𝑤][(𝑡‘𝐾) / 𝑥]𝜑 ↔ [((𝑡 ↾ (1...𝑀)) ↾ (1...𝑁)) / 𝑢][(𝑡 ↾ (1...𝑀)) / 𝑎][(𝑎‘𝑀) / 𝑣][(𝑡‘𝐿) / 𝑤][(𝑡‘𝐾) / 𝑥]𝜑)
18		fzssp1 13568	. . . . . . . . . . . 12 ⊢ (1...𝑁) ⊆ (1...(𝑁 + 1))
19	6	oveq2i 7425	. . . . . . . . . . . 12 ⊢ (1...𝑀) = (1...(𝑁 + 1))
20	18, 19	sseqtrri 4015	. . . . . . . . . . 11 ⊢ (1...𝑁) ⊆ (1...𝑀)
21		resabs1 6009	. . . . . . . . . . 11 ⊢ ((1...𝑁) ⊆ (1...𝑀) → ((𝑡 ↾ (1...𝑀)) ↾ (1...𝑁)) = (𝑡 ↾ (1...𝑁)))
22		dfsbcq 3776	. . . . . . . . . . 11 ⊢ (((𝑡 ↾ (1...𝑀)) ↾ (1...𝑁)) = (𝑡 ↾ (1...𝑁)) → ([((𝑡 ↾ (1...𝑀)) ↾ (1...𝑁)) / 𝑢][(𝑡 ↾ (1...𝑀)) / 𝑎][(𝑎‘𝑀) / 𝑣][(𝑡‘𝐿) / 𝑤][(𝑡‘𝐾) / 𝑥]𝜑 ↔ [(𝑡 ↾ (1...𝑁)) / 𝑢][(𝑡 ↾ (1...𝑀)) / 𝑎][(𝑎‘𝑀) / 𝑣][(𝑡‘𝐿) / 𝑤][(𝑡‘𝐾) / 𝑥]𝜑))
23	20, 21, 22	mp2b 10	. . . . . . . . . 10 ⊢ ([((𝑡 ↾ (1...𝑀)) ↾ (1...𝑁)) / 𝑢][(𝑡 ↾ (1...𝑀)) / 𝑎][(𝑎‘𝑀) / 𝑣][(𝑡‘𝐿) / 𝑤][(𝑡‘𝐾) / 𝑥]𝜑 ↔ [(𝑡 ↾ (1...𝑁)) / 𝑢][(𝑡 ↾ (1...𝑀)) / 𝑎][(𝑎‘𝑀) / 𝑣][(𝑡‘𝐿) / 𝑤][(𝑡‘𝐾) / 𝑥]𝜑)
24		vex 3473	. . . . . . . . . . . . . 14 ⊢ 𝑡 ∈ V
25	24	resex 6027	. . . . . . . . . . . . 13 ⊢ (𝑡 ↾ (1...𝑀)) ∈ V
26		fveq1 6890	. . . . . . . . . . . . . 14 ⊢ (𝑎 = (𝑡 ↾ (1...𝑀)) → (𝑎‘𝑀) = ((𝑡 ↾ (1...𝑀))‘𝑀))
27	26	sbcco3gw 4418	. . . . . . . . . . . . 13 ⊢ ((𝑡 ↾ (1...𝑀)) ∈ V → ([(𝑡 ↾ (1...𝑀)) / 𝑎][(𝑎‘𝑀) / 𝑣][(𝑡‘𝐿) / 𝑤][(𝑡‘𝐾) / 𝑥]𝜑 ↔ [((𝑡 ↾ (1...𝑀))‘𝑀) / 𝑣][(𝑡‘𝐿) / 𝑤][(𝑡‘𝐾) / 𝑥]𝜑))
28	25, 27	ax-mp 5	. . . . . . . . . . . 12 ⊢ ([(𝑡 ↾ (1...𝑀)) / 𝑎][(𝑎‘𝑀) / 𝑣][(𝑡‘𝐿) / 𝑤][(𝑡‘𝐾) / 𝑥]𝜑 ↔ [((𝑡 ↾ (1...𝑀))‘𝑀) / 𝑣][(𝑡‘𝐿) / 𝑤][(𝑡‘𝐾) / 𝑥]𝜑)
29		elfz1end 13555	. . . . . . . . . . . . . 14 ⊢ (𝑀 ∈ ℕ ↔ 𝑀 ∈ (1...𝑀))
30	8, 29	sylib 217	. . . . . . . . . . . . 13 ⊢ (𝑁 ∈ ℕ₀ → 𝑀 ∈ (1...𝑀))
31		fvres 6910	. . . . . . . . . . . . 13 ⊢ (𝑀 ∈ (1...𝑀) → ((𝑡 ↾ (1...𝑀))‘𝑀) = (𝑡‘𝑀))
32		dfsbcq 3776	. . . . . . . . . . . . 13 ⊢ (((𝑡 ↾ (1...𝑀))‘𝑀) = (𝑡‘𝑀) → ([((𝑡 ↾ (1...𝑀))‘𝑀) / 𝑣][(𝑡‘𝐿) / 𝑤][(𝑡‘𝐾) / 𝑥]𝜑 ↔ [(𝑡‘𝑀) / 𝑣][(𝑡‘𝐿) / 𝑤][(𝑡‘𝐾) / 𝑥]𝜑))
33	30, 31, 32	3syl 18	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ ℕ₀ → ([((𝑡 ↾ (1...𝑀))‘𝑀) / 𝑣][(𝑡‘𝐿) / 𝑤][(𝑡‘𝐾) / 𝑥]𝜑 ↔ [(𝑡‘𝑀) / 𝑣][(𝑡‘𝐿) / 𝑤][(𝑡‘𝐾) / 𝑥]𝜑))
34	28, 33	bitrid 283	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ℕ₀ → ([(𝑡 ↾ (1...𝑀)) / 𝑎][(𝑎‘𝑀) / 𝑣][(𝑡‘𝐿) / 𝑤][(𝑡‘𝐾) / 𝑥]𝜑 ↔ [(𝑡‘𝑀) / 𝑣][(𝑡‘𝐿) / 𝑤][(𝑡‘𝐾) / 𝑥]𝜑))
35	34	sbcbidv 3833	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℕ₀ → ([(𝑡 ↾ (1...𝑁)) / 𝑢][(𝑡 ↾ (1...𝑀)) / 𝑎][(𝑎‘𝑀) / 𝑣][(𝑡‘𝐿) / 𝑤][(𝑡‘𝐾) / 𝑥]𝜑 ↔ [(𝑡 ↾ (1...𝑁)) / 𝑢][(𝑡‘𝑀) / 𝑣][(𝑡‘𝐿) / 𝑤][(𝑡‘𝐾) / 𝑥]𝜑))
36	23, 35	bitrid 283	. . . . . . . . 9 ⊢ (𝑁 ∈ ℕ₀ → ([((𝑡 ↾ (1...𝑀)) ↾ (1...𝑁)) / 𝑢][(𝑡 ↾ (1...𝑀)) / 𝑎][(𝑎‘𝑀) / 𝑣][(𝑡‘𝐿) / 𝑤][(𝑡‘𝐾) / 𝑥]𝜑 ↔ [(𝑡 ↾ (1...𝑁)) / 𝑢][(𝑡‘𝑀) / 𝑣][(𝑡‘𝐿) / 𝑤][(𝑡‘𝐾) / 𝑥]𝜑))
37	17, 36	bitrid 283	. . . . . . . 8 ⊢ (𝑁 ∈ ℕ₀ → ([(𝑡 ↾ (1...𝑀)) / 𝑎][(𝑎 ↾ (1...𝑁)) / 𝑢][(𝑎‘𝑀) / 𝑣][(𝑡‘𝐿) / 𝑤][(𝑡‘𝐾) / 𝑥]𝜑 ↔ [(𝑡 ↾ (1...𝑁)) / 𝑢][(𝑡‘𝑀) / 𝑣][(𝑡‘𝐿) / 𝑤][(𝑡‘𝐾) / 𝑥]𝜑))
38	15, 37	bitr3id 285	. . . . . . 7 ⊢ (𝑁 ∈ ℕ₀ → ([(𝑡 ↾ (1...𝑀)) / 𝑎][(𝑡‘𝐿) / 𝑤][(𝑡‘𝐾) / 𝑥][(𝑎 ↾ (1...𝑁)) / 𝑢][(𝑎‘𝑀) / 𝑣]𝜑 ↔ [(𝑡 ↾ (1...𝑁)) / 𝑢][(𝑡‘𝑀) / 𝑣][(𝑡‘𝐿) / 𝑤][(𝑡‘𝐾) / 𝑥]𝜑))
39	38	rabbidv 3435	. . . . . 6 ⊢ (𝑁 ∈ ℕ₀ → {𝑡 ∈ (ℕ₀ ↑_m (1...𝐾)) ∣ [(𝑡 ↾ (1...𝑀)) / 𝑎][(𝑡‘𝐿) / 𝑤][(𝑡‘𝐾) / 𝑥][(𝑎 ↾ (1...𝑁)) / 𝑢][(𝑎‘𝑀) / 𝑣]𝜑} = {𝑡 ∈ (ℕ₀ ↑_m (1...𝐾)) ∣ [(𝑡 ↾ (1...𝑁)) / 𝑢][(𝑡‘𝑀) / 𝑣][(𝑡‘𝐿) / 𝑤][(𝑡‘𝐾) / 𝑥]𝜑})
40	39	eleq1d 2813	. . . . 5 ⊢ (𝑁 ∈ ℕ₀ → ({𝑡 ∈ (ℕ₀ ↑_m (1...𝐾)) ∣ [(𝑡 ↾ (1...𝑀)) / 𝑎][(𝑡‘𝐿) / 𝑤][(𝑡‘𝐾) / 𝑥][(𝑎 ↾ (1...𝑁)) / 𝑢][(𝑎‘𝑀) / 𝑣]𝜑} ∈ (Dioph‘𝐾) ↔ {𝑡 ∈ (ℕ₀ ↑_m (1...𝐾)) ∣ [(𝑡 ↾ (1...𝑁)) / 𝑢][(𝑡‘𝑀) / 𝑣][(𝑡‘𝐿) / 𝑤][(𝑡‘𝐾) / 𝑥]𝜑} ∈ (Dioph‘𝐾)))
41	40	biimpar 477	. . . 4 ⊢ ((𝑁 ∈ ℕ₀ ∧ {𝑡 ∈ (ℕ₀ ↑_m (1...𝐾)) ∣ [(𝑡 ↾ (1...𝑁)) / 𝑢][(𝑡‘𝑀) / 𝑣][(𝑡‘𝐿) / 𝑤][(𝑡‘𝐾) / 𝑥]𝜑} ∈ (Dioph‘𝐾)) → {𝑡 ∈ (ℕ₀ ↑_m (1...𝐾)) ∣ [(𝑡 ↾ (1...𝑀)) / 𝑎][(𝑡‘𝐿) / 𝑤][(𝑡‘𝐾) / 𝑥][(𝑎 ↾ (1...𝑁)) / 𝑢][(𝑎‘𝑀) / 𝑣]𝜑} ∈ (Dioph‘𝐾))
42		rexfrabdioph.2	. . . . 5 ⊢ 𝐿 = (𝑀 + 1)
43		rexfrabdioph.3	. . . . 5 ⊢ 𝐾 = (𝐿 + 1)
44	42, 43	2rexfrabdioph 42138	. . . 4 ⊢ ((𝑀 ∈ ℕ₀ ∧ {𝑡 ∈ (ℕ₀ ↑_m (1...𝐾)) ∣ [(𝑡 ↾ (1...𝑀)) / 𝑎][(𝑡‘𝐿) / 𝑤][(𝑡‘𝐾) / 𝑥][(𝑎 ↾ (1...𝑁)) / 𝑢][(𝑎‘𝑀) / 𝑣]𝜑} ∈ (Dioph‘𝐾)) → {𝑎 ∈ (ℕ₀ ↑_m (1...𝑀)) ∣ ∃𝑤 ∈ ℕ₀ ∃𝑥 ∈ ℕ₀ [(𝑎 ↾ (1...𝑁)) / 𝑢][(𝑎‘𝑀) / 𝑣]𝜑} ∈ (Dioph‘𝑀))
45	10, 41, 44	syl2anc 583	. . 3 ⊢ ((𝑁 ∈ ℕ₀ ∧ {𝑡 ∈ (ℕ₀ ↑_m (1...𝐾)) ∣ [(𝑡 ↾ (1...𝑁)) / 𝑢][(𝑡‘𝑀) / 𝑣][(𝑡‘𝐿) / 𝑤][(𝑡‘𝐾) / 𝑥]𝜑} ∈ (Dioph‘𝐾)) → {𝑎 ∈ (ℕ₀ ↑_m (1...𝑀)) ∣ ∃𝑤 ∈ ℕ₀ ∃𝑥 ∈ ℕ₀ [(𝑎 ↾ (1...𝑁)) / 𝑢][(𝑎‘𝑀) / 𝑣]𝜑} ∈ (Dioph‘𝑀))
46	5, 45	eqeltrid 2832	. 2 ⊢ ((𝑁 ∈ ℕ₀ ∧ {𝑡 ∈ (ℕ₀ ↑_m (1...𝐾)) ∣ [(𝑡 ↾ (1...𝑁)) / 𝑢][(𝑡‘𝑀) / 𝑣][(𝑡‘𝐿) / 𝑤][(𝑡‘𝐾) / 𝑥]𝜑} ∈ (Dioph‘𝐾)) → {𝑎 ∈ (ℕ₀ ↑_m (1...𝑀)) ∣ [(𝑎 ↾ (1...𝑁)) / 𝑢][(𝑎‘𝑀) / 𝑣]∃𝑤 ∈ ℕ₀ ∃𝑥 ∈ ℕ₀ 𝜑} ∈ (Dioph‘𝑀))
47	6	rexfrabdioph 42137	. 2 ⊢ ((𝑁 ∈ ℕ₀ ∧ {𝑎 ∈ (ℕ₀ ↑_m (1...𝑀)) ∣ [(𝑎 ↾ (1...𝑁)) / 𝑢][(𝑎‘𝑀) / 𝑣]∃𝑤 ∈ ℕ₀ ∃𝑥 ∈ ℕ₀ 𝜑} ∈ (Dioph‘𝑀)) → {𝑢 ∈ (ℕ₀ ↑_m (1...𝑁)) ∣ ∃𝑣 ∈ ℕ₀ ∃𝑤 ∈ ℕ₀ ∃𝑥 ∈ ℕ₀ 𝜑} ∈ (Dioph‘𝑁))
48	46, 47	syldan 590	1 ⊢ ((𝑁 ∈ ℕ₀ ∧ {𝑡 ∈ (ℕ₀ ↑_m (1...𝐾)) ∣ [(𝑡 ↾ (1...𝑁)) / 𝑢][(𝑡‘𝑀) / 𝑣][(𝑡‘𝐿) / 𝑤][(𝑡‘𝐾) / 𝑥]𝜑} ∈ (Dioph‘𝐾)) → {𝑢 ∈ (ℕ₀ ↑_m (1...𝑁)) ∣ ∃𝑣 ∈ ℕ₀ ∃𝑤 ∈ ℕ₀ ∃𝑥 ∈ ℕ₀ 𝜑} ∈ (Dioph‘𝑁))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1534 ∈ wcel 2099 ∃wrex 3065 {crab 3427 Vcvv 3469 [wsbc 3774 ⊆ wss 3944 ↾ cres 5674 ‘cfv 6542 (class class class)co 7414 ↑_m cmap 8836 1c1 11131 + caddc 11133 ℕcn 12234 ℕ₀cn0 12494 ...cfz 13508 Diophcdioph 42097

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 ax-inf2 9656 ax-cnex 11186 ax-resscn 11187 ax-1cn 11188 ax-icn 11189 ax-addcl 11190 ax-addrcl 11191 ax-mulcl 11192 ax-mulrcl 11193 ax-mulcom 11194 ax-addass 11195 ax-mulass 11196 ax-distr 11197 ax-i2m1 11198 ax-1ne0 11199 ax-1rid 11200 ax-rnegex 11201 ax-rrecex 11202 ax-cnre 11203 ax-pre-lttri 11204 ax-pre-lttrn 11205 ax-pre-ltadd 11206 ax-pre-mulgt0 11207

This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-nel 3042 df-ral 3057 df-rex 3066 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-int 4945 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-of 7679 df-om 7865 df-1st 7987 df-2nd 7988 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-1o 8480 df-oadd 8484 df-er 8718 df-map 8838 df-en 8956 df-dom 8957 df-sdom 8958 df-fin 8959 df-dju 9916 df-card 9954 df-pnf 11272 df-mnf 11273 df-xr 11274 df-ltxr 11275 df-le 11276 df-sub 11468 df-neg 11469 df-nn 12235 df-n0 12495 df-z 12581 df-uz 12845 df-fz 13509 df-hash 14314 df-mzpcl 42065 df-mzp 42066 df-dioph 42098

This theorem is referenced by: expdiophlem2 42365

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