Theorem ulm2 26429

Description: Simplify ulmval 26424 when 𝐹 and 𝐺 are known to be functions. (Contributed by Mario Carneiro, 26-Feb-2015.)

Hypotheses

Ref	Expression
ulm2.z	⊢ 𝑍 = (ℤ≥‘𝑀)
ulm2.m	⊢ (𝜑 → 𝑀 ∈ ℤ)
ulm2.f	⊢ (𝜑 → 𝐹:𝑍⟶(ℂ ↑m 𝑆))
ulm2.b	⊢ ((𝜑 ∧ (𝑘 ∈ 𝑍 ∧ 𝑧 ∈ 𝑆)) → ((𝐹‘𝑘)‘𝑧) = 𝐵)
ulm2.a	⊢ ((𝜑 ∧ 𝑧 ∈ 𝑆) → (𝐺‘𝑧) = 𝐴)
ulm2.g	⊢ (𝜑 → 𝐺:𝑆⟶ℂ)
ulm2.s	⊢ (𝜑 → 𝑆 ∈ 𝑉)

Assertion

Ref	Expression
ulm2	⊢ (𝜑 → (𝐹(⇝𝑢‘𝑆)𝐺 ↔ ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(𝐵 − 𝐴)) < 𝑥))

Distinct variable groups:   𝑗,𝑘,𝑥,𝑧,𝐹   𝑗,𝐺,𝑘,𝑥,𝑧   𝑗,𝑀,𝑘,𝑥,𝑧   𝜑,𝑗,𝑘,𝑥,𝑧   𝐴,𝑗,𝑘,𝑥   𝑥,𝐵   𝑆,𝑗,𝑘,𝑥,𝑧   𝑗,𝑍,𝑥

Allowed substitution hints:   𝐴(𝑧)   𝐵(𝑧,𝑗,𝑘)   𝑉(𝑥,𝑧,𝑗,𝑘)   𝑍(𝑧,𝑘)

Proof of Theorem ulm2

Dummy variable 𝑛 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ulm2.s	. . 3 ⊢ (𝜑 → 𝑆 ∈ 𝑉)
2		ulmval 26424	. . 3 ⊢ (𝑆 ∈ 𝑉 → (𝐹(⇝𝑢‘𝑆)𝐺 ↔ ∃𝑛 ∈ ℤ (𝐹:(ℤ≥‘𝑛)⟶(ℂ ↑m 𝑆) ∧ 𝐺:𝑆⟶ℂ ∧ ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ (ℤ≥‘𝑛)∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − (𝐺‘𝑧))) < 𝑥)))
3	1, 2	syl 17	. 2 ⊢ (𝜑 → (𝐹(⇝𝑢‘𝑆)𝐺 ↔ ∃𝑛 ∈ ℤ (𝐹:(ℤ≥‘𝑛)⟶(ℂ ↑m 𝑆) ∧ 𝐺:𝑆⟶ℂ ∧ ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ (ℤ≥‘𝑛)∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − (𝐺‘𝑧))) < 𝑥)))
4		3anan12 1095	. . . 4 ⊢ ((𝐹:(ℤ≥‘𝑛)⟶(ℂ ↑m 𝑆) ∧ 𝐺:𝑆⟶ℂ ∧ ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ (ℤ≥‘𝑛)∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − (𝐺‘𝑧))) < 𝑥) ↔ (𝐺:𝑆⟶ℂ ∧ (𝐹:(ℤ≥‘𝑛)⟶(ℂ ↑m 𝑆) ∧ ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ (ℤ≥‘𝑛)∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − (𝐺‘𝑧))) < 𝑥)))
5		ulm2.z	. . . . . . . . . 10 ⊢ 𝑍 = (ℤ≥‘𝑀)
6		ulm2.f	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐹:𝑍⟶(ℂ ↑m 𝑆))
7	6	fdmd 6745	. . . . . . . . . . 11 ⊢ (𝜑 → dom 𝐹 = 𝑍)
8		fdm 6744	. . . . . . . . . . 11 ⊢ (𝐹:(ℤ≥‘𝑛)⟶(ℂ ↑m 𝑆) → dom 𝐹 = (ℤ≥‘𝑛))
9	7, 8	sylan9req 2797	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐹:(ℤ≥‘𝑛)⟶(ℂ ↑m 𝑆)) → 𝑍 = (ℤ≥‘𝑛))
10	5, 9	eqtr3id 2790	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐹:(ℤ≥‘𝑛)⟶(ℂ ↑m 𝑆)) → (ℤ≥‘𝑀) = (ℤ≥‘𝑛))
11		ulm2.m	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑀 ∈ ℤ)
12	11	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐹:(ℤ≥‘𝑛)⟶(ℂ ↑m 𝑆)) → 𝑀 ∈ ℤ)
13		uz11 12904	. . . . . . . . . 10 ⊢ (𝑀 ∈ ℤ → ((ℤ≥‘𝑀) = (ℤ≥‘𝑛) ↔ 𝑀 = 𝑛))
14	12, 13	syl 17	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐹:(ℤ≥‘𝑛)⟶(ℂ ↑m 𝑆)) → ((ℤ≥‘𝑀) = (ℤ≥‘𝑛) ↔ 𝑀 = 𝑛))
15	10, 14	mpbid 232	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐹:(ℤ≥‘𝑛)⟶(ℂ ↑m 𝑆)) → 𝑀 = 𝑛)
16	15	eqcomd 2742	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐹:(ℤ≥‘𝑛)⟶(ℂ ↑m 𝑆)) → 𝑛 = 𝑀)
17		fveq2 6905	. . . . . . . . . . 11 ⊢ (𝑛 = 𝑀 → (ℤ≥‘𝑛) = (ℤ≥‘𝑀))
18	17, 5	eqtr4di 2794	. . . . . . . . . 10 ⊢ (𝑛 = 𝑀 → (ℤ≥‘𝑛) = 𝑍)
19	18	feq2d 6721	. . . . . . . . 9 ⊢ (𝑛 = 𝑀 → (𝐹:(ℤ≥‘𝑛)⟶(ℂ ↑m 𝑆) ↔ 𝐹:𝑍⟶(ℂ ↑m 𝑆)))
20	19	biimparc 479	. . . . . . . 8 ⊢ ((𝐹:𝑍⟶(ℂ ↑m 𝑆) ∧ 𝑛 = 𝑀) → 𝐹:(ℤ≥‘𝑛)⟶(ℂ ↑m 𝑆))
21	6, 20	sylan 580	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 = 𝑀) → 𝐹:(ℤ≥‘𝑛)⟶(ℂ ↑m 𝑆))
22	16, 21	impbida 800	. . . . . 6 ⊢ (𝜑 → (𝐹:(ℤ≥‘𝑛)⟶(ℂ ↑m 𝑆) ↔ 𝑛 = 𝑀))
23	22	anbi1d 631	. . . . 5 ⊢ (𝜑 → ((𝐹:(ℤ≥‘𝑛)⟶(ℂ ↑m 𝑆) ∧ ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ (ℤ≥‘𝑛)∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − (𝐺‘𝑧))) < 𝑥) ↔ (𝑛 = 𝑀 ∧ ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ (ℤ≥‘𝑛)∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − (𝐺‘𝑧))) < 𝑥)))
24		ulm2.g	. . . . . 6 ⊢ (𝜑 → 𝐺:𝑆⟶ℂ)
25	24	biantrurd 532	. . . . 5 ⊢ (𝜑 → ((𝐹:(ℤ≥‘𝑛)⟶(ℂ ↑m 𝑆) ∧ ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ (ℤ≥‘𝑛)∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − (𝐺‘𝑧))) < 𝑥) ↔ (𝐺:𝑆⟶ℂ ∧ (𝐹:(ℤ≥‘𝑛)⟶(ℂ ↑m 𝑆) ∧ ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ (ℤ≥‘𝑛)∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − (𝐺‘𝑧))) < 𝑥))))
26		simp-4l 782	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑛 = 𝑀) ∧ 𝑗 ∈ (ℤ≥‘𝑛)) ∧ 𝑘 ∈ (ℤ≥‘𝑗)) ∧ 𝑧 ∈ 𝑆) → 𝜑)
27		simpr 484	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑛 = 𝑀) → 𝑛 = 𝑀)
28		uzid 12894	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ (ℤ≥‘𝑀))
29	11, 28	syl 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → 𝑀 ∈ (ℤ≥‘𝑀))
30	29, 5	eleqtrrdi 2851	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝑀 ∈ 𝑍)
31	30	adantr 480	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑛 = 𝑀) → 𝑀 ∈ 𝑍)
32	27, 31	eqeltrd 2840	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑛 = 𝑀) → 𝑛 ∈ 𝑍)
33	5	uztrn2 12898	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑛 ∈ 𝑍 ∧ 𝑗 ∈ (ℤ≥‘𝑛)) → 𝑗 ∈ 𝑍)
34	32, 33	sylan 580	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑛 = 𝑀) ∧ 𝑗 ∈ (ℤ≥‘𝑛)) → 𝑗 ∈ 𝑍)
35	5	uztrn2 12898	. . . . . . . . . . . . . . . 16 ⊢ ((𝑗 ∈ 𝑍 ∧ 𝑘 ∈ (ℤ≥‘𝑗)) → 𝑘 ∈ 𝑍)
36	34, 35	sylan 580	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑛 = 𝑀) ∧ 𝑗 ∈ (ℤ≥‘𝑛)) ∧ 𝑘 ∈ (ℤ≥‘𝑗)) → 𝑘 ∈ 𝑍)
37	36	adantr 480	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑛 = 𝑀) ∧ 𝑗 ∈ (ℤ≥‘𝑛)) ∧ 𝑘 ∈ (ℤ≥‘𝑗)) ∧ 𝑧 ∈ 𝑆) → 𝑘 ∈ 𝑍)
38		simpr 484	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑛 = 𝑀) ∧ 𝑗 ∈ (ℤ≥‘𝑛)) ∧ 𝑘 ∈ (ℤ≥‘𝑗)) ∧ 𝑧 ∈ 𝑆) → 𝑧 ∈ 𝑆)
39		ulm2.b	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝑍 ∧ 𝑧 ∈ 𝑆)) → ((𝐹‘𝑘)‘𝑧) = 𝐵)
40	26, 37, 38, 39	syl12anc 836	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑛 = 𝑀) ∧ 𝑗 ∈ (ℤ≥‘𝑛)) ∧ 𝑘 ∈ (ℤ≥‘𝑗)) ∧ 𝑧 ∈ 𝑆) → ((𝐹‘𝑘)‘𝑧) = 𝐵)
41		ulm2.a	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑆) → (𝐺‘𝑧) = 𝐴)
42	26, 41	sylancom 588	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑛 = 𝑀) ∧ 𝑗 ∈ (ℤ≥‘𝑛)) ∧ 𝑘 ∈ (ℤ≥‘𝑗)) ∧ 𝑧 ∈ 𝑆) → (𝐺‘𝑧) = 𝐴)
43	40, 42	oveq12d 7450	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑛 = 𝑀) ∧ 𝑗 ∈ (ℤ≥‘𝑛)) ∧ 𝑘 ∈ (ℤ≥‘𝑗)) ∧ 𝑧 ∈ 𝑆) → (((𝐹‘𝑘)‘𝑧) − (𝐺‘𝑧)) = (𝐵 − 𝐴))
44	43	fveq2d 6909	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑛 = 𝑀) ∧ 𝑗 ∈ (ℤ≥‘𝑛)) ∧ 𝑘 ∈ (ℤ≥‘𝑗)) ∧ 𝑧 ∈ 𝑆) → (abs‘(((𝐹‘𝑘)‘𝑧) − (𝐺‘𝑧))) = (abs‘(𝐵 − 𝐴)))
45	44	breq1d 5152	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑛 = 𝑀) ∧ 𝑗 ∈ (ℤ≥‘𝑛)) ∧ 𝑘 ∈ (ℤ≥‘𝑗)) ∧ 𝑧 ∈ 𝑆) → ((abs‘(((𝐹‘𝑘)‘𝑧) − (𝐺‘𝑧))) < 𝑥 ↔ (abs‘(𝐵 − 𝐴)) < 𝑥))
46	45	ralbidva 3175	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑛 = 𝑀) ∧ 𝑗 ∈ (ℤ≥‘𝑛)) ∧ 𝑘 ∈ (ℤ≥‘𝑗)) → (∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − (𝐺‘𝑧))) < 𝑥 ↔ ∀𝑧 ∈ 𝑆 (abs‘(𝐵 − 𝐴)) < 𝑥))
47	46	ralbidva 3175	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 = 𝑀) ∧ 𝑗 ∈ (ℤ≥‘𝑛)) → (∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − (𝐺‘𝑧))) < 𝑥 ↔ ∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(𝐵 − 𝐴)) < 𝑥))
48	47	rexbidva 3176	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 = 𝑀) → (∃𝑗 ∈ (ℤ≥‘𝑛)∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − (𝐺‘𝑧))) < 𝑥 ↔ ∃𝑗 ∈ (ℤ≥‘𝑛)∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(𝐵 − 𝐴)) < 𝑥))
49	48	ralbidv 3177	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 = 𝑀) → (∀𝑥 ∈ ℝ+ ∃𝑗 ∈ (ℤ≥‘𝑛)∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − (𝐺‘𝑧))) < 𝑥 ↔ ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ (ℤ≥‘𝑛)∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(𝐵 − 𝐴)) < 𝑥))
50	49	pm5.32da 579	. . . . 5 ⊢ (𝜑 → ((𝑛 = 𝑀 ∧ ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ (ℤ≥‘𝑛)∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − (𝐺‘𝑧))) < 𝑥) ↔ (𝑛 = 𝑀 ∧ ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ (ℤ≥‘𝑛)∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(𝐵 − 𝐴)) < 𝑥)))
51	23, 25, 50	3bitr3d 309	. . . 4 ⊢ (𝜑 → ((𝐺:𝑆⟶ℂ ∧ (𝐹:(ℤ≥‘𝑛)⟶(ℂ ↑m 𝑆) ∧ ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ (ℤ≥‘𝑛)∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − (𝐺‘𝑧))) < 𝑥)) ↔ (𝑛 = 𝑀 ∧ ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ (ℤ≥‘𝑛)∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(𝐵 − 𝐴)) < 𝑥)))
52	4, 51	bitrid 283	. . 3 ⊢ (𝜑 → ((𝐹:(ℤ≥‘𝑛)⟶(ℂ ↑m 𝑆) ∧ 𝐺:𝑆⟶ℂ ∧ ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ (ℤ≥‘𝑛)∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − (𝐺‘𝑧))) < 𝑥) ↔ (𝑛 = 𝑀 ∧ ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ (ℤ≥‘𝑛)∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(𝐵 − 𝐴)) < 𝑥)))
53	52	rexbidv 3178	. 2 ⊢ (𝜑 → (∃𝑛 ∈ ℤ (𝐹:(ℤ≥‘𝑛)⟶(ℂ ↑m 𝑆) ∧ 𝐺:𝑆⟶ℂ ∧ ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ (ℤ≥‘𝑛)∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − (𝐺‘𝑧))) < 𝑥) ↔ ∃𝑛 ∈ ℤ (𝑛 = 𝑀 ∧ ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ (ℤ≥‘𝑛)∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(𝐵 − 𝐴)) < 𝑥)))
54	18	rexeqdv 3326	. . . . 5 ⊢ (𝑛 = 𝑀 → (∃𝑗 ∈ (ℤ≥‘𝑛)∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(𝐵 − 𝐴)) < 𝑥 ↔ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(𝐵 − 𝐴)) < 𝑥))
55	54	ralbidv 3177	. . . 4 ⊢ (𝑛 = 𝑀 → (∀𝑥 ∈ ℝ+ ∃𝑗 ∈ (ℤ≥‘𝑛)∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(𝐵 − 𝐴)) < 𝑥 ↔ ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(𝐵 − 𝐴)) < 𝑥))
56	55	ceqsrexv 3654	. . 3 ⊢ (𝑀 ∈ ℤ → (∃𝑛 ∈ ℤ (𝑛 = 𝑀 ∧ ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ (ℤ≥‘𝑛)∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(𝐵 − 𝐴)) < 𝑥) ↔ ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(𝐵 − 𝐴)) < 𝑥))
57	11, 56	syl 17	. 2 ⊢ (𝜑 → (∃𝑛 ∈ ℤ (𝑛 = 𝑀 ∧ ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ (ℤ≥‘𝑛)∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(𝐵 − 𝐴)) < 𝑥) ↔ ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(𝐵 − 𝐴)) < 𝑥))
58	3, 53, 57	3bitrd 305	1 ⊢ (𝜑 → (𝐹(⇝𝑢‘𝑆)𝐺 ↔ ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(𝐵 − 𝐴)) < 𝑥))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1539   ∈ wcel 2107  ∀wral 3060  ∃wrex 3069   class class class wbr 5142  dom cdm 5684  ⟶wf 6556  ‘cfv 6560  (class class class)co 7432   ↑_m cmap 8867  ℂcc 11154   < clt 11296   − cmin 11493  ℤcz 12615  ℤ_≥cuz 12879  ℝ⁺crp 13035  abscabs 15274  ⇝_𝑢culm 26420

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-rep 5278  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756  ax-cnex 11212  ax-resscn 11213  ax-pre-lttri 11230  ax-pre-lttrn 11231

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-iun 4992  df-br 5143  df-opab 5205  df-mpt 5225  df-id 5577  df-po 5591  df-so 5592  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-ov 7435  df-oprab 7436  df-mpo 7437  df-er 8746  df-map 8869  df-pm 8870  df-en 8987  df-dom 8988  df-sdom 8989  df-pnf 11298  df-mnf 11299  df-xr 11300  df-ltxr 11301  df-le 11302  df-neg 11496  df-z 12616  df-uz 12880  df-ulm 26421

This theorem is referenced by:  ulmi  26430  ulmclm  26431  ulmres  26432  ulmshftlem  26433  ulm0  26435  ulmcau  26439  ulmss  26441