Theorem 3rexfrabdioph 39414

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 39404	. . . . . 6 ⊢ ([(𝑎‘𝑀) / 𝑣]∃𝑤 ∈ ℕ0 ∃𝑥 ∈ ℕ0 𝜑 ↔ ∃𝑤 ∈ ℕ0 ∃𝑥 ∈ ℕ0 [(𝑎‘𝑀) / 𝑣]𝜑)
2	1	sbcbii 3829	. . . . 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 39404	. . . . 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 277	. . . 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 3473	. . 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 11937	. . . . . . 7 ⊢ (𝑁 ∈ ℕ0 → (𝑁 + 1) ∈ ℕ)
8	6, 7	eqeltrid 2917	. . . . . 6 ⊢ (𝑁 ∈ ℕ0 → 𝑀 ∈ ℕ)
9	8	nnnn0d 11956	. . . . 5 ⊢ (𝑁 ∈ ℕ0 → 𝑀 ∈ ℕ0)
10	9	adantr 483	. . . 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 39408	. . . . . . . . . . 11 ⊢ ([(𝑎‘𝑀) / 𝑣][(𝑡‘𝐿) / 𝑤][(𝑡‘𝐾) / 𝑥]𝜑 ↔ [(𝑡‘𝐿) / 𝑤][(𝑡‘𝐾) / 𝑥][(𝑎‘𝑀) / 𝑣]𝜑)
12	11	sbcbii 3829	. . . . . . . . . 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 39408	. . . . . . . . . 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 277	. . . . . . . . 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 3829	. . . . . . . 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 5847	. . . . . . . . . 10 ⊢ (𝑎 = (𝑡 ↾ (1...𝑀)) → (𝑎 ↾ (1...𝑁)) = ((𝑡 ↾ (1...𝑀)) ↾ (1...𝑁)))
17	16	sbccomieg 39410	. . . . . . . . 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 12951	. . . . . . . . . . . 12 ⊢ (1...𝑁) ⊆ (1...(𝑁 + 1))
19	6	oveq2i 7167	. . . . . . . . . . . 12 ⊢ (1...𝑀) = (1...(𝑁 + 1))
20	18, 19	sseqtrri 4004	. . . . . . . . . . 11 ⊢ (1...𝑁) ⊆ (1...𝑀)
21		resabs1 5883	. . . . . . . . . . 11 ⊢ ((1...𝑁) ⊆ (1...𝑀) → ((𝑡 ↾ (1...𝑀)) ↾ (1...𝑁)) = (𝑡 ↾ (1...𝑁)))
22		dfsbcq 3774	. . . . . . . . . . 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 3497	. . . . . . . . . . . . . 14 ⊢ 𝑡 ∈ V
25	24	resex 5899	. . . . . . . . . . . . 13 ⊢ (𝑡 ↾ (1...𝑀)) ∈ V
26		fveq1 6669	. . . . . . . . . . . . . 14 ⊢ (𝑎 = (𝑡 ↾ (1...𝑀)) → (𝑎‘𝑀) = ((𝑡 ↾ (1...𝑀))‘𝑀))
27	26	sbcco3gw 4374	. . . . . . . . . . . . 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 12938	. . . . . . . . . . . . . 14 ⊢ (𝑀 ∈ ℕ ↔ 𝑀 ∈ (1...𝑀))
30	8, 29	sylib 220	. . . . . . . . . . . . 13 ⊢ (𝑁 ∈ ℕ0 → 𝑀 ∈ (1...𝑀))
31		fvres 6689	. . . . . . . . . . . . 13 ⊢ (𝑀 ∈ (1...𝑀) → ((𝑡 ↾ (1...𝑀))‘𝑀) = (𝑡‘𝑀))
32		dfsbcq 3774	. . . . . . . . . . . . 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	syl5bb 285	. . . . . . . . . . 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 3827	. . . . . . . . . 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	syl5bb 285	. . . . . . . . 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	syl5bb 285	. . . . . . . 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	syl5bbr 287	. . . . . . 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 3480	. . . . . 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 2897	. . . . 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 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)) → {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 39413	. . . 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 586	. . 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 2917	. 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 39412	. 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 593	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))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   = wceq 1537   ∈ wcel 2114  ∃wrex 3139  {crab 3142  Vcvv 3494  [wsbc 3772   ⊆ wss 3936   ↾ cres 5557  ‘cfv 6355  (class class class)co 7156   ↑_m cmap 8406  1c1 10538   + caddc 10540  ℕcn 11638  ℕ₀cn0 11898  ...cfz 12893  Diophcdioph 39372

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-rep 5190  ax-sep 5203  ax-nul 5210  ax-pow 5266  ax-pr 5330  ax-un 7461  ax-inf2 9104  ax-cnex 10593  ax-resscn 10594  ax-1cn 10595  ax-icn 10596  ax-addcl 10597  ax-addrcl 10598  ax-mulcl 10599  ax-mulrcl 10600  ax-mulcom 10601  ax-addass 10602  ax-mulass 10603  ax-distr 10604  ax-i2m1 10605  ax-1ne0 10606  ax-1rid 10607  ax-rnegex 10608  ax-rrecex 10609  ax-cnre 10610  ax-pre-lttri 10611  ax-pre-lttrn 10612  ax-pre-ltadd 10613  ax-pre-mulgt0 10614

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-nel 3124  df-ral 3143  df-rex 3144  df-reu 3145  df-rmo 3146  df-rab 3147  df-v 3496  df-sbc 3773  df-csb 3884  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-pss 3954  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4568  df-pr 4570  df-tp 4572  df-op 4574  df-uni 4839  df-int 4877  df-iun 4921  df-br 5067  df-opab 5129  df-mpt 5147  df-tr 5173  df-id 5460  df-eprel 5465  df-po 5474  df-so 5475  df-fr 5514  df-we 5516  df-xp 5561  df-rel 5562  df-cnv 5563  df-co 5564  df-dm 5565  df-rn 5566  df-res 5567  df-ima 5568  df-pred 6148  df-ord 6194  df-on 6195  df-lim 6196  df-suc 6197  df-iota 6314  df-fun 6357  df-fn 6358  df-f 6359  df-f1 6360  df-fo 6361  df-f1o 6362  df-fv 6363  df-riota 7114  df-ov 7159  df-oprab 7160  df-mpo 7161  df-of 7409  df-om 7581  df-1st 7689  df-2nd 7690  df-wrecs 7947  df-recs 8008  df-rdg 8046  df-1o 8102  df-oadd 8106  df-er 8289  df-map 8408  df-en 8510  df-dom 8511  df-sdom 8512  df-fin 8513  df-dju 9330  df-card 9368  df-pnf 10677  df-mnf 10678  df-xr 10679  df-ltxr 10680  df-le 10681  df-sub 10872  df-neg 10873  df-nn 11639  df-n0 11899  df-z 11983  df-uz 12245  df-fz 12894  df-hash 13692  df-mzpcl 39340  df-mzp 39341  df-dioph 39373

This theorem is referenced by:  expdiophlem2  39639