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Theorem ulmi 26367

Description: The uniform limit property. (Contributed by Mario Carneiro, 27-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)
ulmi.u	⊢ (𝜑 → 𝐹(⇝_𝑢‘𝑆)𝐺)
ulmi.c	⊢ (𝜑 → 𝐶 ∈ ℝ⁺)

**Assertion**
Ref	Expression
ulmi	⊢ (𝜑 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(𝐵 − 𝐴)) < 𝐶)

Distinct variable groups: 𝑗,𝑘,𝑧,𝐹 𝑗,𝐺,𝑘,𝑧 𝑗,𝑀,𝑘,𝑧 𝜑,𝑗,𝑘,𝑧 𝐴,𝑗,𝑘 𝐶,𝑗,𝑘,𝑧 𝑆,𝑗,𝑘,𝑧 𝑗,𝑍 Allowed substitution hints: 𝐴(𝑧) 𝐵(𝑧,𝑗,𝑘) 𝑍(𝑧,𝑘)

Proof of Theorem ulmi Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		breq2 5153	. . . 4 ⊢ (𝑥 = 𝐶 → ((abs‘(𝐵 − 𝐴)) < 𝑥 ↔ (abs‘(𝐵 − 𝐴)) < 𝐶))
2	1	ralbidv 3167	. . 3 ⊢ (𝑥 = 𝐶 → (∀𝑧 ∈ 𝑆 (abs‘(𝐵 − 𝐴)) < 𝑥 ↔ ∀𝑧 ∈ 𝑆 (abs‘(𝐵 − 𝐴)) < 𝐶))
3	2	rexralbidv 3210	. 2 ⊢ (𝑥 = 𝐶 → (∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(𝐵 − 𝐴)) < 𝑥 ↔ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(𝐵 − 𝐴)) < 𝐶))
4		ulmi.u	. . 3 ⊢ (𝜑 → 𝐹(⇝_𝑢‘𝑆)𝐺)
5		ulm2.z	. . . 4 ⊢ 𝑍 = (ℤ_≥‘𝑀)
6		ulm2.m	. . . 4 ⊢ (𝜑 → 𝑀 ∈ ℤ)
7		ulm2.f	. . . 4 ⊢ (𝜑 → 𝐹:𝑍⟶(ℂ ↑_m 𝑆))
8		ulm2.b	. . . 4 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝑍 ∧ 𝑧 ∈ 𝑆)) → ((𝐹‘𝑘)‘𝑧) = 𝐵)
9		ulm2.a	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑆) → (𝐺‘𝑧) = 𝐴)
10		ulmcl 26362	. . . . 5 ⊢ (𝐹(⇝_𝑢‘𝑆)𝐺 → 𝐺:𝑆⟶ℂ)
11	4, 10	syl 17	. . . 4 ⊢ (𝜑 → 𝐺:𝑆⟶ℂ)
12		ulmscl 26360	. . . . 5 ⊢ (𝐹(⇝_𝑢‘𝑆)𝐺 → 𝑆 ∈ V)
13	4, 12	syl 17	. . . 4 ⊢ (𝜑 → 𝑆 ∈ V)
14	5, 6, 7, 8, 9, 11, 13	ulm2 26366	. . 3 ⊢ (𝜑 → (𝐹(⇝_𝑢‘𝑆)𝐺 ↔ ∀𝑥 ∈ ℝ⁺ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(𝐵 − 𝐴)) < 𝑥))
15	4, 14	mpbid 231	. 2 ⊢ (𝜑 → ∀𝑥 ∈ ℝ⁺ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(𝐵 − 𝐴)) < 𝑥)
16		ulmi.c	. 2 ⊢ (𝜑 → 𝐶 ∈ ℝ⁺)
17	3, 15, 16	rspcdva 3607	1 ⊢ (𝜑 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(𝐵 − 𝐴)) < 𝐶)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 394 = wceq 1533 ∈ wcel 2098 ∀wral 3050 ∃wrex 3059 Vcvv 3461 class class class wbr 5149 ⟶wf 6545 ‘cfv 6549 (class class class)co 7419 ↑_m cmap 8845 ℂcc 11138 < clt 11280 − cmin 11476 ℤcz 12591 ℤ_≥cuz 12855 ℝ⁺crp 13009 abscabs 15217 ⇝_𝑢culm 26357

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 ax-cnex 11196 ax-resscn 11197 ax-pre-lttri 11214 ax-pre-lttrn 11215

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-nel 3036 df-ral 3051 df-rex 3060 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-iun 4999 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5576 df-po 5590 df-so 5591 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-ov 7422 df-oprab 7423 df-mpo 7424 df-er 8725 df-map 8847 df-pm 8848 df-en 8965 df-dom 8966 df-sdom 8967 df-pnf 11282 df-mnf 11283 df-xr 11284 df-ltxr 11285 df-le 11286 df-neg 11479 df-z 12592 df-uz 12856 df-ulm 26358

This theorem is referenced by: ulmshftlem 26370 ulmcau 26376 ulmbdd 26379 ulmcn 26380 iblulm 26388 itgulm 26389

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