Theorem erlval 33491

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 6885	. . . . 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 5285	. . 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 7738	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑅 ∈ V) → (𝐵 × 𝑆) ∈ V)
11	9, 10	eqeltrid 2869	. . . . . 6 ⊢ ((𝜑 ∧ 𝑅 ∈ V) → 𝑊 ∈ V)
12	11, 11	xpexd 7738	. . . . 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 5699	. . . . . 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 5285	. . . 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 2869	. . 3 ⊢ ((𝜑 ∧ 𝑅 ∈ V) → ∼ ∈ V)
19		fvexd 6886	. . . . 5 ⊢ ((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) → (.r‘𝑟) ∈ V)
20		fveq2 6871	. . . . . . 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 2818	. . . . 5 ⊢ ((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) → (.r‘𝑟) = · )
24		fvexd 6886	. . . . . . 7 ⊢ (((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) ∧ 𝑥 = · ) → (Base‘𝑟) ∈ V)
25		vex 3461	. . . . . . . 8 ⊢ 𝑠 ∈ V
26	25	a1i 11	. . . . . . 7 ⊢ (((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) ∧ 𝑥 = · ) → 𝑠 ∈ V)
27	24, 26	xpexd 7738	. . . . . 6 ⊢ (((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) ∧ 𝑥 = · ) → ((Base‘𝑟) × 𝑠) ∈ V)
28		fveq2 6871	. . . . . . . . . 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 2818	. . . . . . . 8 ⊢ (((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) ∧ 𝑥 = · ) → (Base‘𝑟) = 𝐵)
31		simplr 780	. . . . . . . 8 ⊢ (((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) ∧ 𝑥 = · ) → 𝑠 = 𝑆)
32	30, 31	xpeq12d 5683	. . . . . . 7 ⊢ (((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) ∧ 𝑥 = · ) → ((Base‘𝑟) × 𝑠) = (𝐵 × 𝑆))
33	32, 9	eqtr4di 2818	. . . . . 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 2851	. . . . . . . . . 10 ⊢ ((((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) ∧ 𝑥 = · ) ∧ 𝑤 = 𝑊) → (𝑎 ∈ 𝑤 ↔ 𝑎 ∈ 𝑊))
36	34	eleq2d 2851	. . . . . . . . . 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 2766	. . . . . . . . . . . 12 ⊢ ((((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) ∧ 𝑥 = · ) ∧ 𝑤 = 𝑊) → 𝑡 = 𝑡)
41		fveq2 6871	. . . . . . . . . . . . . . 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 2818	. . . . . . . . . . . . 13 ⊢ ((((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) ∧ 𝑥 = · ) ∧ 𝑤 = 𝑊) → (-g‘𝑟) = − )
45	39	oveqd 7417	. . . . . . . . . . . . 13 ⊢ ((((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) ∧ 𝑥 = · ) ∧ 𝑤 = 𝑊) → ((1st ‘𝑎)𝑥(2nd ‘𝑏)) = ((1st ‘𝑎) · (2nd ‘𝑏)))
46	39	oveqd 7417	. . . . . . . . . . . . 13 ⊢ ((((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) ∧ 𝑥 = · ) ∧ 𝑤 = 𝑊) → ((1st ‘𝑏)𝑥(2nd ‘𝑎)) = ((1st ‘𝑏) · (2nd ‘𝑎)))
47	44, 45, 46	oveq123d 7421	. . . . . . . . . . . 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 7421	. . . . . . . . . . 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 6871	. . . . . . . . . . . . 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 2818	. . . . . . . . . . 11 ⊢ ((((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) ∧ 𝑥 = · ) ∧ 𝑤 = 𝑊) → (0g‘𝑟) = 0 )
53	48, 52	eqeq12d 2781	. . . . . . . . . 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 3340	. . . . . . . . 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 5171	. . . . . . 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 2818	. . . . . 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 3892	. . . . 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 3892	. . . 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 33488	. . . 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 7554	. . 3 ⊢ ((𝑅 ∈ V ∧ 𝑆 ∈ V ∧ ∼ ∈ V) → (𝑅 ~RL 𝑆) = ∼ )
62	1, 7, 18, 61	syl3anc 1394	. 2 ⊢ ((𝜑 ∧ 𝑅 ∈ V) → (𝑅 ~RL 𝑆) = ∼ )
63	60	reldmmpo 7534	. . . . 5 ⊢ Rel dom ~RL
64	63	ovprc1 7439	. . . 4 ⊢ (¬ 𝑅 ∈ V → (𝑅 ~RL 𝑆) = ∅)
65	64	adantl 486	. . 3 ⊢ ((𝜑 ∧ ¬ 𝑅 ∈ V) → (𝑅 ~RL 𝑆) = ∅)
66	8, 15	eqsstrid 3977	. . . . . 6 ⊢ (𝜑 → ∼ ⊆ (𝑊 × 𝑊))
67	66	adantr 485	. . . . 5 ⊢ ((𝜑 ∧ ¬ 𝑅 ∈ V) → ∼ ⊆ (𝑊 × 𝑊))
68		fvprc 6863	. . . . . . . . . . 11 ⊢ (¬ 𝑅 ∈ V → (Base‘𝑅) = ∅)
69	2, 68	eqtrid 2812	. . . . . . . . . 10 ⊢ (¬ 𝑅 ∈ V → 𝐵 = ∅)
70	69	xpeq1d 5681	. . . . . . . . 9 ⊢ (¬ 𝑅 ∈ V → (𝐵 × 𝑆) = (∅ × 𝑆))
71		0xp 5751	. . . . . . . . 9 ⊢ (∅ × 𝑆) = ∅
72	70, 71	eqtrdi 2816	. . . . . . . 8 ⊢ (¬ 𝑅 ∈ V → (𝐵 × 𝑆) = ∅)
73	9, 72	eqtrid 2812	. . . . . . 7 ⊢ (¬ 𝑅 ∈ V → 𝑊 = ∅)
74		id 23	. . . . . . . . 9 ⊢ (𝑊 = ∅ → 𝑊 = ∅)
75	74, 74	xpeq12d 5683	. . . . . . . 8 ⊢ (𝑊 = ∅ → (𝑊 × 𝑊) = (∅ × ∅))
76		0xp 5751	. . . . . . . 8 ⊢ (∅ × ∅) = ∅
77	75, 76	eqtrdi 2816	. . . . . . 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 3975	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝑅 ∈ V) → ∼ ⊆ ∅)
81		ss0 4359	. . . 4 ⊢ ( ∼ ⊆ ∅ → ∼ = ∅)
82	80, 81	syl 18	. . 3 ⊢ ((𝜑 ∧ ¬ 𝑅 ∈ V) → ∼ = ∅)
83	65, 82	eqtr4d 2803	. 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 1563   ∈ wcel 2145  ∃wrex 3089  Vcvv 3457  ⦋csb 3855   ⊆ wss 3907  ∅c0 4288  {copab 5167   × cxp 5650  ‘cfv 6525  (class class class)co 7400  1^st c1st 7972  2^nd c2nd 7973  Basecbs 17259  ._rcmulr 17301  0_gc0g 17482  -_gcsg 18992   ~_RL cerl 33486

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-sep 5251  ax-nul 5261  ax-pow 5327  ax-pr 5395  ax-un 7722

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3080  df-rex 3090  df-rab 3418  df-v 3459  df-sbc 3748  df-csb 3856  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-nul 4289  df-if 4484  df-pw 4560  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4869  df-br 5106  df-opab 5168  df-id 5547  df-xp 5658  df-rel 5659  df-cnv 5660  df-co 5661  df-dm 5662  df-iota 6481  df-fun 6527  df-fv 6533  df-ov 7403  df-oprab 7404  df-mpo 7405  df-erl 33488

This theorem is referenced by:  erlcl1  33493  erlcl2  33494  erldi  33495  erlbrd  33496  erler  33498  fracerl  33542