Theorem erlval 33253

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 484	. . 3 ⊢ ((𝜑 ∧ 𝑅 ∈ V) → 𝑅 ∈ V)
2		rlocval.1	. . . . . 6 ⊢ 𝐵 = (Base‘𝑅)
3	2	fvexi 6890	. . . . 5 ⊢ 𝐵 ∈ V
4	3	a1i 11	. . . 4 ⊢ ((𝜑 ∧ 𝑅 ∈ V) → 𝐵 ∈ V)
5		erlval.20	. . . . 5 ⊢ (𝜑 → 𝑆 ⊆ 𝐵)
6	5	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑅 ∈ V) → 𝑆 ⊆ 𝐵)
7	4, 6	ssexd 5294	. . 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 7745	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑅 ∈ V) → (𝐵 × 𝑆) ∈ V)
11	9, 10	eqeltrid 2838	. . . . . 6 ⊢ ((𝜑 ∧ 𝑅 ∈ V) → 𝑊 ∈ V)
12	11, 11	xpexd 7745	. . . . 5 ⊢ ((𝜑 ∧ 𝑅 ∈ V) → (𝑊 × 𝑊) ∈ V)
13		simprll 778	. . . . . . 7 ⊢ ((𝜑 ∧ ((𝑎 ∈ 𝑊 ∧ 𝑏 ∈ 𝑊) ∧ ∃𝑡 ∈ 𝑆 (𝑡 · (((1st ‘𝑎) · (2nd ‘𝑏)) − ((1st ‘𝑏) · (2nd ‘𝑎)))) = 0 )) → 𝑎 ∈ 𝑊)
14		simprlr 779	. . . . . . 7 ⊢ ((𝜑 ∧ ((𝑎 ∈ 𝑊 ∧ 𝑏 ∈ 𝑊) ∧ ∃𝑡 ∈ 𝑆 (𝑡 · (((1st ‘𝑎) · (2nd ‘𝑏)) − ((1st ‘𝑏) · (2nd ‘𝑎)))) = 0 )) → 𝑏 ∈ 𝑊)
15	13, 14	opabssxpd 5701	. . . . . 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 480	. . . . 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 5294	. . . 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 2838	. . 3 ⊢ ((𝜑 ∧ 𝑅 ∈ V) → ∼ ∈ V)
19		fvexd 6891	. . . . 5 ⊢ ((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) → (.r‘𝑟) ∈ V)
20		fveq2 6876	. . . . . . 7 ⊢ (𝑟 = 𝑅 → (.r‘𝑟) = (.r‘𝑅))
21	20	adantr 480	. . . . . 6 ⊢ ((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) → (.r‘𝑟) = (.r‘𝑅))
22		rlocval.3	. . . . . 6 ⊢ · = (.r‘𝑅)
23	21, 22	eqtr4di 2788	. . . . 5 ⊢ ((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) → (.r‘𝑟) = · )
24		fvexd 6891	. . . . . . 7 ⊢ (((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) ∧ 𝑥 = · ) → (Base‘𝑟) ∈ V)
25		vex 3463	. . . . . . . 8 ⊢ 𝑠 ∈ V
26	25	a1i 11	. . . . . . 7 ⊢ (((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) ∧ 𝑥 = · ) → 𝑠 ∈ V)
27	24, 26	xpexd 7745	. . . . . 6 ⊢ (((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) ∧ 𝑥 = · ) → ((Base‘𝑟) × 𝑠) ∈ V)
28		fveq2 6876	. . . . . . . . . 10 ⊢ (𝑟 = 𝑅 → (Base‘𝑟) = (Base‘𝑅))
29	28	ad2antrr 726	. . . . . . . . 9 ⊢ (((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) ∧ 𝑥 = · ) → (Base‘𝑟) = (Base‘𝑅))
30	29, 2	eqtr4di 2788	. . . . . . . 8 ⊢ (((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) ∧ 𝑥 = · ) → (Base‘𝑟) = 𝐵)
31		simplr 768	. . . . . . . 8 ⊢ (((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) ∧ 𝑥 = · ) → 𝑠 = 𝑆)
32	30, 31	xpeq12d 5685	. . . . . . 7 ⊢ (((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) ∧ 𝑥 = · ) → ((Base‘𝑟) × 𝑠) = (𝐵 × 𝑆))
33	32, 9	eqtr4di 2788	. . . . . 6 ⊢ (((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) ∧ 𝑥 = · ) → ((Base‘𝑟) × 𝑠) = 𝑊)
34		simpr 484	. . . . . . . . . . 11 ⊢ ((((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) ∧ 𝑥 = · ) ∧ 𝑤 = 𝑊) → 𝑤 = 𝑊)
35	34	eleq2d 2820	. . . . . . . . . 10 ⊢ ((((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) ∧ 𝑥 = · ) ∧ 𝑤 = 𝑊) → (𝑎 ∈ 𝑤 ↔ 𝑎 ∈ 𝑊))
36	34	eleq2d 2820	. . . . . . . . . 10 ⊢ ((((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) ∧ 𝑥 = · ) ∧ 𝑤 = 𝑊) → (𝑏 ∈ 𝑤 ↔ 𝑏 ∈ 𝑊))
37	35, 36	anbi12d 632	. . . . . . . . 9 ⊢ ((((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) ∧ 𝑥 = · ) ∧ 𝑤 = 𝑊) → ((𝑎 ∈ 𝑤 ∧ 𝑏 ∈ 𝑤) ↔ (𝑎 ∈ 𝑊 ∧ 𝑏 ∈ 𝑊)))
38	31	adantr 480	. . . . . . . . . 10 ⊢ ((((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) ∧ 𝑥 = · ) ∧ 𝑤 = 𝑊) → 𝑠 = 𝑆)
39		simplr 768	. . . . . . . . . . . 12 ⊢ ((((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) ∧ 𝑥 = · ) ∧ 𝑤 = 𝑊) → 𝑥 = · )
40		eqidd 2736	. . . . . . . . . . . 12 ⊢ ((((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) ∧ 𝑥 = · ) ∧ 𝑤 = 𝑊) → 𝑡 = 𝑡)
41		fveq2 6876	. . . . . . . . . . . . . . 15 ⊢ (𝑟 = 𝑅 → (-g‘𝑟) = (-g‘𝑅))
42	41	ad3antrrr 730	. . . . . . . . . . . . . 14 ⊢ ((((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) ∧ 𝑥 = · ) ∧ 𝑤 = 𝑊) → (-g‘𝑟) = (-g‘𝑅))
43		rlocval.4	. . . . . . . . . . . . . 14 ⊢ − = (-g‘𝑅)
44	42, 43	eqtr4di 2788	. . . . . . . . . . . . 13 ⊢ ((((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) ∧ 𝑥 = · ) ∧ 𝑤 = 𝑊) → (-g‘𝑟) = − )
45	39	oveqd 7422	. . . . . . . . . . . . 13 ⊢ ((((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) ∧ 𝑥 = · ) ∧ 𝑤 = 𝑊) → ((1st ‘𝑎)𝑥(2nd ‘𝑏)) = ((1st ‘𝑎) · (2nd ‘𝑏)))
46	39	oveqd 7422	. . . . . . . . . . . . 13 ⊢ ((((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) ∧ 𝑥 = · ) ∧ 𝑤 = 𝑊) → ((1st ‘𝑏)𝑥(2nd ‘𝑎)) = ((1st ‘𝑏) · (2nd ‘𝑎)))
47	44, 45, 46	oveq123d 7426	. . . . . . . . . . . 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 7426	. . . . . . . . . . 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 6876	. . . . . . . . . . . . 13 ⊢ (𝑟 = 𝑅 → (0g‘𝑟) = (0g‘𝑅))
50	49	ad3antrrr 730	. . . . . . . . . . . 12 ⊢ ((((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) ∧ 𝑥 = · ) ∧ 𝑤 = 𝑊) → (0g‘𝑟) = (0g‘𝑅))
51		rlocval.2	. . . . . . . . . . . 12 ⊢ 0 = (0g‘𝑅)
52	50, 51	eqtr4di 2788	. . . . . . . . . . 11 ⊢ ((((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) ∧ 𝑥 = · ) ∧ 𝑤 = 𝑊) → (0g‘𝑟) = 0 )
53	48, 52	eqeq12d 2751	. . . . . . . . . 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 3326	. . . . . . . . 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 632	. . . . . . . 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 5185	. . . . . . 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 2788	. . . . . 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 3911	. . . . 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 3911	. . . 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 33250	. . . 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 7561	. . 3 ⊢ ((𝑅 ∈ V ∧ 𝑆 ∈ V ∧ ∼ ∈ V) → (𝑅 ~RL 𝑆) = ∼ )
62	1, 7, 18, 61	syl3anc 1373	. 2 ⊢ ((𝜑 ∧ 𝑅 ∈ V) → (𝑅 ~RL 𝑆) = ∼ )
63	60	reldmmpo 7541	. . . . 5 ⊢ Rel dom ~RL
64	63	ovprc1 7444	. . . 4 ⊢ (¬ 𝑅 ∈ V → (𝑅 ~RL 𝑆) = ∅)
65	64	adantl 481	. . 3 ⊢ ((𝜑 ∧ ¬ 𝑅 ∈ V) → (𝑅 ~RL 𝑆) = ∅)
66	8, 15	eqsstrid 3997	. . . . . 6 ⊢ (𝜑 → ∼ ⊆ (𝑊 × 𝑊))
67	66	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ ¬ 𝑅 ∈ V) → ∼ ⊆ (𝑊 × 𝑊))
68		fvprc 6868	. . . . . . . . . . 11 ⊢ (¬ 𝑅 ∈ V → (Base‘𝑅) = ∅)
69	2, 68	eqtrid 2782	. . . . . . . . . 10 ⊢ (¬ 𝑅 ∈ V → 𝐵 = ∅)
70	69	xpeq1d 5683	. . . . . . . . 9 ⊢ (¬ 𝑅 ∈ V → (𝐵 × 𝑆) = (∅ × 𝑆))
71		0xp 5753	. . . . . . . . 9 ⊢ (∅ × 𝑆) = ∅
72	70, 71	eqtrdi 2786	. . . . . . . 8 ⊢ (¬ 𝑅 ∈ V → (𝐵 × 𝑆) = ∅)
73	9, 72	eqtrid 2782	. . . . . . 7 ⊢ (¬ 𝑅 ∈ V → 𝑊 = ∅)
74		id 22	. . . . . . . . 9 ⊢ (𝑊 = ∅ → 𝑊 = ∅)
75	74, 74	xpeq12d 5685	. . . . . . . 8 ⊢ (𝑊 = ∅ → (𝑊 × 𝑊) = (∅ × ∅))
76		0xp 5753	. . . . . . . 8 ⊢ (∅ × ∅) = ∅
77	75, 76	eqtrdi 2786	. . . . . . 7 ⊢ (𝑊 = ∅ → (𝑊 × 𝑊) = ∅)
78	73, 77	syl 17	. . . . . 6 ⊢ (¬ 𝑅 ∈ V → (𝑊 × 𝑊) = ∅)
79	78	adantl 481	. . . . 5 ⊢ ((𝜑 ∧ ¬ 𝑅 ∈ V) → (𝑊 × 𝑊) = ∅)
80	67, 79	sseqtrd 3995	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝑅 ∈ V) → ∼ ⊆ ∅)
81		ss0 4377	. . . 4 ⊢ ( ∼ ⊆ ∅ → ∼ = ∅)
82	80, 81	syl 17	. . 3 ⊢ ((𝜑 ∧ ¬ 𝑅 ∈ V) → ∼ = ∅)
83	65, 82	eqtr4d 2773	. 2 ⊢ ((𝜑 ∧ ¬ 𝑅 ∈ V) → (𝑅 ~RL 𝑆) = ∼ )
84	62, 83	pm2.61dan 812	1 ⊢ (𝜑 → (𝑅 ~RL 𝑆) = ∼ )

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2108  ∃wrex 3060  Vcvv 3459  ⦋csb 3874   ⊆ wss 3926  ∅c0 4308  {copab 5181   × cxp 5652  ‘cfv 6531  (class class class)co 7405  1^st c1st 7986  2^nd c2nd 7987  Basecbs 17228  ._rcmulr 17272  0_gc0g 17453  -_gcsg 18918   ~_RL cerl 33248

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-sep 5266  ax-nul 5276  ax-pow 5335  ax-pr 5402  ax-un 7729

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rab 3416  df-v 3461  df-sbc 3766  df-csb 3875  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-nul 4309  df-if 4501  df-pw 4577  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-br 5120  df-opab 5182  df-id 5548  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-iota 6484  df-fun 6533  df-fv 6539  df-ov 7408  df-oprab 7409  df-mpo 7410  df-erl 33250

This theorem is referenced by:  erlcl1  33255  erlcl2  33256  erldi  33257  erlbrd  33258  erler  33260  fracerl  33300