Theorem erlval 33518

Description: Value of the ring localization equivalence relation. (Contributed by Thierry Arnoux, 4-May-2025.)

Hypotheses

Ref	Expression
rlocval.1	⊢ 𝐵 = (Base‘𝑅)
rlocval.2	⊢ 0 = (0g‘𝑅)
rlocval.3	⊢ · = (.r‘𝑅)
rlocval.4	⊢ − = (-g‘𝑅)
erlval.w	⊢ 𝑊 = (𝐵 × 𝑆)
erlval.q	⊢ ∼ = {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ 𝑊 ∧ 𝑏 ∈ 𝑊) ∧ ∃𝑡 ∈ 𝑆 (𝑡 · (((1st ‘𝑎) · (2nd ‘𝑏)) − ((1st ‘𝑏) · (2nd ‘𝑎)))) = 0 )}
erlval.20	⊢ (𝜑 → 𝑆 ⊆ 𝐵)

Assertion

Ref	Expression
erlval	⊢ (𝜑 → (𝑅 ~RL 𝑆) = ∼ )

Distinct variable groups:   · ,𝑎,𝑏,𝑡   𝑅,𝑎,𝑏,𝑡   𝑆,𝑎,𝑏,𝑡   𝑊,𝑎,𝑏,𝑡   𝜑,𝑎,𝑏

Allowed substitution hints:   𝜑(𝑡)   𝐵(𝑡,𝑎,𝑏)   ∼ (𝑡,𝑎,𝑏)   − (𝑡,𝑎,𝑏)   0 (𝑡,𝑎,𝑏)

Proof of Theorem erlval

Dummy variables 𝑤 𝑟 𝑠 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpr 489	. . 3 ⊢ ((𝜑 ∧ 𝑅 ∈ V) → 𝑅 ∈ V)
2		rlocval.1	. . . . . 6 ⊢ 𝐵 = (Base‘𝑅)
3	2	fvexi 6896	. . . . 5 ⊢ 𝐵 ∈ V
4	3	a1i 11	. . . 4 ⊢ ((𝜑 ∧ 𝑅 ∈ V) → 𝐵 ∈ V)
5		erlval.20	. . . . 5 ⊢ (𝜑 → 𝑆 ⊆ 𝐵)
6	5	adantr 485	. . . 4 ⊢ ((𝜑 ∧ 𝑅 ∈ V) → 𝑆 ⊆ 𝐵)
7	4, 6	ssexd 5295	. . 3 ⊢ ((𝜑 ∧ 𝑅 ∈ V) → 𝑆 ∈ V)
8		erlval.q	. . . 4 ⊢ ∼ = {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ 𝑊 ∧ 𝑏 ∈ 𝑊) ∧ ∃𝑡 ∈ 𝑆 (𝑡 · (((1st ‘𝑎) · (2nd ‘𝑏)) − ((1st ‘𝑏) · (2nd ‘𝑎)))) = 0 )}
9		erlval.w	. . . . . . 7 ⊢ 𝑊 = (𝐵 × 𝑆)
10	4, 7	xpexd 7749	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑅 ∈ V) → (𝐵 × 𝑆) ∈ V)
11	9, 10	eqeltrid 2873	. . . . . 6 ⊢ ((𝜑 ∧ 𝑅 ∈ V) → 𝑊 ∈ V)
12	11, 11	xpexd 7749	. . . . 5 ⊢ ((𝜑 ∧ 𝑅 ∈ V) → (𝑊 × 𝑊) ∈ V)
13		simprll 790	. . . . . . 7 ⊢ ((𝜑 ∧ ((𝑎 ∈ 𝑊 ∧ 𝑏 ∈ 𝑊) ∧ ∃𝑡 ∈ 𝑆 (𝑡 · (((1st ‘𝑎) · (2nd ‘𝑏)) − ((1st ‘𝑏) · (2nd ‘𝑎)))) = 0 )) → 𝑎 ∈ 𝑊)
14		simprlr 791	. . . . . . 7 ⊢ ((𝜑 ∧ ((𝑎 ∈ 𝑊 ∧ 𝑏 ∈ 𝑊) ∧ ∃𝑡 ∈ 𝑆 (𝑡 · (((1st ‘𝑎) · (2nd ‘𝑏)) − ((1st ‘𝑏) · (2nd ‘𝑎)))) = 0 )) → 𝑏 ∈ 𝑊)
15	13, 14	opabssxpd 5709	. . . . . 6 ⊢ (𝜑 → {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ 𝑊 ∧ 𝑏 ∈ 𝑊) ∧ ∃𝑡 ∈ 𝑆 (𝑡 · (((1st ‘𝑎) · (2nd ‘𝑏)) − ((1st ‘𝑏) · (2nd ‘𝑎)))) = 0 )} ⊆ (𝑊 × 𝑊))
16	15	adantr 485	. . . . 5 ⊢ ((𝜑 ∧ 𝑅 ∈ V) → {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ 𝑊 ∧ 𝑏 ∈ 𝑊) ∧ ∃𝑡 ∈ 𝑆 (𝑡 · (((1st ‘𝑎) · (2nd ‘𝑏)) − ((1st ‘𝑏) · (2nd ‘𝑎)))) = 0 )} ⊆ (𝑊 × 𝑊))
17	12, 16	ssexd 5295	. . . 4 ⊢ ((𝜑 ∧ 𝑅 ∈ V) → {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ 𝑊 ∧ 𝑏 ∈ 𝑊) ∧ ∃𝑡 ∈ 𝑆 (𝑡 · (((1st ‘𝑎) · (2nd ‘𝑏)) − ((1st ‘𝑏) · (2nd ‘𝑎)))) = 0 )} ∈ V)
18	8, 17	eqeltrid 2873	. . 3 ⊢ ((𝜑 ∧ 𝑅 ∈ V) → ∼ ∈ V)
19		fvexd 6897	. . . . 5 ⊢ ((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) → (.r‘𝑟) ∈ V)
20		fveq2 6882	. . . . . . 7 ⊢ (𝑟 = 𝑅 → (.r‘𝑟) = (.r‘𝑅))
21	20	adantr 485	. . . . . 6 ⊢ ((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) → (.r‘𝑟) = (.r‘𝑅))
22		rlocval.3	. . . . . 6 ⊢ · = (.r‘𝑅)
23	21, 22	eqtr4di 2822	. . . . 5 ⊢ ((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) → (.r‘𝑟) = · )
24		fvexd 6897	. . . . . . 7 ⊢ (((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) ∧ 𝑥 = · ) → (Base‘𝑟) ∈ V)
25		vex 3467	. . . . . . . 8 ⊢ 𝑠 ∈ V
26	25	a1i 11	. . . . . . 7 ⊢ (((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) ∧ 𝑥 = · ) → 𝑠 ∈ V)
27	24, 26	xpexd 7749	. . . . . 6 ⊢ (((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) ∧ 𝑥 = · ) → ((Base‘𝑟) × 𝑠) ∈ V)
28		fveq2 6882	. . . . . . . . . 10 ⊢ (𝑟 = 𝑅 → (Base‘𝑟) = (Base‘𝑅))
29	28	ad2antrr 738	. . . . . . . . 9 ⊢ (((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) ∧ 𝑥 = · ) → (Base‘𝑟) = (Base‘𝑅))
30	29, 2	eqtr4di 2822	. . . . . . . 8 ⊢ (((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) ∧ 𝑥 = · ) → (Base‘𝑟) = 𝐵)
31		simplr 780	. . . . . . . 8 ⊢ (((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) ∧ 𝑥 = · ) → 𝑠 = 𝑆)
32	30, 31	xpeq12d 5693	. . . . . . 7 ⊢ (((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) ∧ 𝑥 = · ) → ((Base‘𝑟) × 𝑠) = (𝐵 × 𝑆))
33	32, 9	eqtr4di 2822	. . . . . 6 ⊢ (((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) ∧ 𝑥 = · ) → ((Base‘𝑟) × 𝑠) = 𝑊)
34		simpr 489	. . . . . . . . . . 11 ⊢ ((((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) ∧ 𝑥 = · ) ∧ 𝑤 = 𝑊) → 𝑤 = 𝑊)
35	34	eleq2d 2855	. . . . . . . . . 10 ⊢ ((((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) ∧ 𝑥 = · ) ∧ 𝑤 = 𝑊) → (𝑎 ∈ 𝑤 ↔ 𝑎 ∈ 𝑊))
36	34	eleq2d 2855	. . . . . . . . . 10 ⊢ ((((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) ∧ 𝑥 = · ) ∧ 𝑤 = 𝑊) → (𝑏 ∈ 𝑤 ↔ 𝑏 ∈ 𝑊))
37	35, 36	anbi12d 643	. . . . . . . . 9 ⊢ ((((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) ∧ 𝑥 = · ) ∧ 𝑤 = 𝑊) → ((𝑎 ∈ 𝑤 ∧ 𝑏 ∈ 𝑤) ↔ (𝑎 ∈ 𝑊 ∧ 𝑏 ∈ 𝑊)))
38	31	adantr 485	. . . . . . . . . 10 ⊢ ((((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) ∧ 𝑥 = · ) ∧ 𝑤 = 𝑊) → 𝑠 = 𝑆)
39		simplr 780	. . . . . . . . . . . 12 ⊢ ((((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) ∧ 𝑥 = · ) ∧ 𝑤 = 𝑊) → 𝑥 = · )
40		eqidd 2770	. . . . . . . . . . . 12 ⊢ ((((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) ∧ 𝑥 = · ) ∧ 𝑤 = 𝑊) → 𝑡 = 𝑡)
41		fveq2 6882	. . . . . . . . . . . . . . 15 ⊢ (𝑟 = 𝑅 → (-g‘𝑟) = (-g‘𝑅))
42	41	ad3antrrr 742	. . . . . . . . . . . . . 14 ⊢ ((((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) ∧ 𝑥 = · ) ∧ 𝑤 = 𝑊) → (-g‘𝑟) = (-g‘𝑅))
43		rlocval.4	. . . . . . . . . . . . . 14 ⊢ − = (-g‘𝑅)
44	42, 43	eqtr4di 2822	. . . . . . . . . . . . 13 ⊢ ((((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) ∧ 𝑥 = · ) ∧ 𝑤 = 𝑊) → (-g‘𝑟) = − )
45	39	oveqd 7428	. . . . . . . . . . . . 13 ⊢ ((((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) ∧ 𝑥 = · ) ∧ 𝑤 = 𝑊) → ((1st ‘𝑎)𝑥(2nd ‘𝑏)) = ((1st ‘𝑎) · (2nd ‘𝑏)))
46	39	oveqd 7428	. . . . . . . . . . . . 13 ⊢ ((((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) ∧ 𝑥 = · ) ∧ 𝑤 = 𝑊) → ((1st ‘𝑏)𝑥(2nd ‘𝑎)) = ((1st ‘𝑏) · (2nd ‘𝑎)))
47	44, 45, 46	oveq123d 7432	. . . . . . . . . . . 12 ⊢ ((((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) ∧ 𝑥 = · ) ∧ 𝑤 = 𝑊) → (((1st ‘𝑎)𝑥(2nd ‘𝑏))(-g‘𝑟)((1st ‘𝑏)𝑥(2nd ‘𝑎))) = (((1st ‘𝑎) · (2nd ‘𝑏)) − ((1st ‘𝑏) · (2nd ‘𝑎))))
48	39, 40, 47	oveq123d 7432	. . . . . . . . . . 11 ⊢ ((((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) ∧ 𝑥 = · ) ∧ 𝑤 = 𝑊) → (𝑡𝑥(((1st ‘𝑎)𝑥(2nd ‘𝑏))(-g‘𝑟)((1st ‘𝑏)𝑥(2nd ‘𝑎)))) = (𝑡 · (((1st ‘𝑎) · (2nd ‘𝑏)) − ((1st ‘𝑏) · (2nd ‘𝑎)))))
49		fveq2 6882	. . . . . . . . . . . . 13 ⊢ (𝑟 = 𝑅 → (0g‘𝑟) = (0g‘𝑅))
50	49	ad3antrrr 742	. . . . . . . . . . . 12 ⊢ ((((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) ∧ 𝑥 = · ) ∧ 𝑤 = 𝑊) → (0g‘𝑟) = (0g‘𝑅))
51		rlocval.2	. . . . . . . . . . . 12 ⊢ 0 = (0g‘𝑅)
52	50, 51	eqtr4di 2822	. . . . . . . . . . 11 ⊢ ((((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) ∧ 𝑥 = · ) ∧ 𝑤 = 𝑊) → (0g‘𝑟) = 0 )
53	48, 52	eqeq12d 2785	. . . . . . . . . 10 ⊢ ((((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) ∧ 𝑥 = · ) ∧ 𝑤 = 𝑊) → ((𝑡𝑥(((1st ‘𝑎)𝑥(2nd ‘𝑏))(-g‘𝑟)((1st ‘𝑏)𝑥(2nd ‘𝑎)))) = (0g‘𝑟) ↔ (𝑡 · (((1st ‘𝑎) · (2nd ‘𝑏)) − ((1st ‘𝑏) · (2nd ‘𝑎)))) = 0 ))
54	38, 53	rexeqbidv 3346	. . . . . . . . 9 ⊢ ((((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) ∧ 𝑥 = · ) ∧ 𝑤 = 𝑊) → (∃𝑡 ∈ 𝑠 (𝑡𝑥(((1st ‘𝑎)𝑥(2nd ‘𝑏))(-g‘𝑟)((1st ‘𝑏)𝑥(2nd ‘𝑎)))) = (0g‘𝑟) ↔ ∃𝑡 ∈ 𝑆 (𝑡 · (((1st ‘𝑎) · (2nd ‘𝑏)) − ((1st ‘𝑏) · (2nd ‘𝑎)))) = 0 ))
55	37, 54	anbi12d 643	. . . . . . . 8 ⊢ ((((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) ∧ 𝑥 = · ) ∧ 𝑤 = 𝑊) → (((𝑎 ∈ 𝑤 ∧ 𝑏 ∈ 𝑤) ∧ ∃𝑡 ∈ 𝑠 (𝑡𝑥(((1st ‘𝑎)𝑥(2nd ‘𝑏))(-g‘𝑟)((1st ‘𝑏)𝑥(2nd ‘𝑎)))) = (0g‘𝑟)) ↔ ((𝑎 ∈ 𝑊 ∧ 𝑏 ∈ 𝑊) ∧ ∃𝑡 ∈ 𝑆 (𝑡 · (((1st ‘𝑎) · (2nd ‘𝑏)) − ((1st ‘𝑏) · (2nd ‘𝑎)))) = 0 )))
56	55	opabbidv 5181	. . . . . . 7 ⊢ ((((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) ∧ 𝑥 = · ) ∧ 𝑤 = 𝑊) → {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ 𝑤 ∧ 𝑏 ∈ 𝑤) ∧ ∃𝑡 ∈ 𝑠 (𝑡𝑥(((1st ‘𝑎)𝑥(2nd ‘𝑏))(-g‘𝑟)((1st ‘𝑏)𝑥(2nd ‘𝑎)))) = (0g‘𝑟))} = {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ 𝑊 ∧ 𝑏 ∈ 𝑊) ∧ ∃𝑡 ∈ 𝑆 (𝑡 · (((1st ‘𝑎) · (2nd ‘𝑏)) − ((1st ‘𝑏) · (2nd ‘𝑎)))) = 0 )})
57	56, 8	eqtr4di 2822	. . . . . 6 ⊢ ((((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) ∧ 𝑥 = · ) ∧ 𝑤 = 𝑊) → {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ 𝑤 ∧ 𝑏 ∈ 𝑤) ∧ ∃𝑡 ∈ 𝑠 (𝑡𝑥(((1st ‘𝑎)𝑥(2nd ‘𝑏))(-g‘𝑟)((1st ‘𝑏)𝑥(2nd ‘𝑎)))) = (0g‘𝑟))} = ∼ )
58	27, 33, 57	csbied2 3898	. . . . 5 ⊢ (((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) ∧ 𝑥 = · ) → ⦋((Base‘𝑟) × 𝑠) / 𝑤⦌{⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ 𝑤 ∧ 𝑏 ∈ 𝑤) ∧ ∃𝑡 ∈ 𝑠 (𝑡𝑥(((1st ‘𝑎)𝑥(2nd ‘𝑏))(-g‘𝑟)((1st ‘𝑏)𝑥(2nd ‘𝑎)))) = (0g‘𝑟))} = ∼ )
59	19, 23, 58	csbied2 3898	. . . 4 ⊢ ((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) → ⦋(.r‘𝑟) / 𝑥⦌⦋((Base‘𝑟) × 𝑠) / 𝑤⦌{⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ 𝑤 ∧ 𝑏 ∈ 𝑤) ∧ ∃𝑡 ∈ 𝑠 (𝑡𝑥(((1st ‘𝑎)𝑥(2nd ‘𝑏))(-g‘𝑟)((1st ‘𝑏)𝑥(2nd ‘𝑎)))) = (0g‘𝑟))} = ∼ )
60		df-erl 33515	. . . 4 ⊢ ~RL = (𝑟 ∈ V, 𝑠 ∈ V ↦ ⦋(.r‘𝑟) / 𝑥⦌⦋((Base‘𝑟) × 𝑠) / 𝑤⦌{⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ 𝑤 ∧ 𝑏 ∈ 𝑤) ∧ ∃𝑡 ∈ 𝑠 (𝑡𝑥(((1st ‘𝑎)𝑥(2nd ‘𝑏))(-g‘𝑟)((1st ‘𝑏)𝑥(2nd ‘𝑎)))) = (0g‘𝑟))})
61	59, 60	ovmpoga 7565	. . 3 ⊢ ((𝑅 ∈ V ∧ 𝑆 ∈ V ∧ ∼ ∈ V) → (𝑅 ~RL 𝑆) = ∼ )
62	1, 7, 18, 61	syl3anc 1396	. 2 ⊢ ((𝜑 ∧ 𝑅 ∈ V) → (𝑅 ~RL 𝑆) = ∼ )
63	60	reldmmpo 7545	. . . . 5 ⊢ Rel dom ~RL
64	63	ovprc1 7450	. . . 4 ⊢ (¬ 𝑅 ∈ V → (𝑅 ~RL 𝑆) = ∅)
65	64	adantl 486	. . 3 ⊢ ((𝜑 ∧ ¬ 𝑅 ∈ V) → (𝑅 ~RL 𝑆) = ∅)
66	8, 15	eqsstrid 3983	. . . . . 6 ⊢ (𝜑 → ∼ ⊆ (𝑊 × 𝑊))
67	66	adantr 485	. . . . 5 ⊢ ((𝜑 ∧ ¬ 𝑅 ∈ V) → ∼ ⊆ (𝑊 × 𝑊))
68		fvprc 6874	. . . . . . . . . . 11 ⊢ (¬ 𝑅 ∈ V → (Base‘𝑅) = ∅)
69	2, 68	eqtrid 2816	. . . . . . . . . 10 ⊢ (¬ 𝑅 ∈ V → 𝐵 = ∅)
70	69	xpeq1d 5691	. . . . . . . . 9 ⊢ (¬ 𝑅 ∈ V → (𝐵 × 𝑆) = (∅ × 𝑆))
71		0xp 5761	. . . . . . . . 9 ⊢ (∅ × 𝑆) = ∅
72	70, 71	eqtrdi 2820	. . . . . . . 8 ⊢ (¬ 𝑅 ∈ V → (𝐵 × 𝑆) = ∅)
73	9, 72	eqtrid 2816	. . . . . . 7 ⊢ (¬ 𝑅 ∈ V → 𝑊 = ∅)
74		id 23	. . . . . . . . 9 ⊢ (𝑊 = ∅ → 𝑊 = ∅)
75	74, 74	xpeq12d 5693	. . . . . . . 8 ⊢ (𝑊 = ∅ → (𝑊 × 𝑊) = (∅ × ∅))
76		0xp 5761	. . . . . . . 8 ⊢ (∅ × ∅) = ∅
77	75, 76	eqtrdi 2820	. . . . . . 7 ⊢ (𝑊 = ∅ → (𝑊 × 𝑊) = ∅)
78	73, 77	syl 18	. . . . . 6 ⊢ (¬ 𝑅 ∈ V → (𝑊 × 𝑊) = ∅)
79	78	adantl 486	. . . . 5 ⊢ ((𝜑 ∧ ¬ 𝑅 ∈ V) → (𝑊 × 𝑊) = ∅)
80	67, 79	sseqtrd 3981	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝑅 ∈ V) → ∼ ⊆ ∅)
81		ss0 4366	. . . 4 ⊢ ( ∼ ⊆ ∅ → ∼ = ∅)
82	80, 81	syl 18	. . 3 ⊢ ((𝜑 ∧ ¬ 𝑅 ∈ V) → ∼ = ∅)
83	65, 82	eqtr4d 2807	. 2 ⊢ ((𝜑 ∧ ¬ 𝑅 ∈ V) → (𝑅 ~RL 𝑆) = ∼ )
84	62, 83	pm2.61dan 824	1 ⊢ (𝜑 → (𝑅 ~RL 𝑆) = ∼ )

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 400   = wceq 1567   ∈ wcel 2149  ∃wrex 3095  Vcvv 3463  ⦋csb 3861   ⊆ wss 3913  ∅c0 4294  {copab 5177   × cxp 5660  ‘cfv 6537  (class class class)co 7411  1^st c1st 7983  2^nd c2nd 7984  Basecbs 17268  ._rcmulr 17310  0_gc0g 17491  -_gcsg 19001   ~_RL cerl 33513

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-sep 5261  ax-nul 5271  ax-pow 5337  ax-pr 5405  ax-un 7733

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-br 5114  df-opab 5178  df-id 5557  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-iota 6493  df-fun 6539  df-fv 6545  df-ov 7414  df-oprab 7415  df-mpo 7416  df-erl 33515

This theorem is referenced by:  erlcl1  33520  erlcl2  33521  erldi  33522  erlbrd  33523  erler  33525  fracerl  33569