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Theorem ulmi 26312

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 5099	. . . 4 ⊢ (𝑥 = 𝐶 → ((abs‘(𝐵 − 𝐴)) < 𝑥 ↔ (abs‘(𝐵 − 𝐴)) < 𝐶))
2	1	ralbidv 3152	. . 3 ⊢ (𝑥 = 𝐶 → (∀𝑧 ∈ 𝑆 (abs‘(𝐵 − 𝐴)) < 𝑥 ↔ ∀𝑧 ∈ 𝑆 (abs‘(𝐵 − 𝐴)) < 𝐶))
3	2	rexralbidv 3195	. 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 26307	. . . . 5 ⊢ (𝐹(⇝_𝑢‘𝑆)𝐺 → 𝐺:𝑆⟶ℂ)
11	4, 10	syl 17	. . . 4 ⊢ (𝜑 → 𝐺:𝑆⟶ℂ)
12		ulmscl 26305	. . . . 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 26311	. . 3 ⊢ (𝜑 → (𝐹(⇝_𝑢‘𝑆)𝐺 ↔ ∀𝑥 ∈ ℝ⁺ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(𝐵 − 𝐴)) < 𝑥))
15	4, 14	mpbid 232	. 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 3580	1 ⊢ (𝜑 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(𝐵 − 𝐴)) < 𝐶)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2109 ∀wral 3044 ∃wrex 3053 Vcvv 3438 class class class wbr 5095 ⟶wf 6482 ‘cfv 6486 (class class class)co 7353 ↑_m cmap 8760 ℂcc 11026 < clt 11168 − cmin 11366 ℤcz 12490 ℤ_≥cuz 12754 ℝ⁺crp 12912 abscabs 15160 ⇝_𝑢culm 26302

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 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5221 ax-sep 5238 ax-nul 5248 ax-pow 5307 ax-pr 5374 ax-un 7675 ax-cnex 11084 ax-resscn 11085 ax-pre-lttri 11102 ax-pre-lttrn 11103

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-reu 3346 df-rab 3397 df-v 3440 df-sbc 3745 df-csb 3854 df-dif 3908 df-un 3910 df-in 3912 df-ss 3922 df-nul 4287 df-if 4479 df-pw 4555 df-sn 4580 df-pr 4582 df-op 4586 df-uni 4862 df-iun 4946 df-br 5096 df-opab 5158 df-mpt 5177 df-id 5518 df-po 5531 df-so 5532 df-xp 5629 df-rel 5630 df-cnv 5631 df-co 5632 df-dm 5633 df-rn 5634 df-res 5635 df-ima 5636 df-iota 6442 df-fun 6488 df-fn 6489 df-f 6490 df-f1 6491 df-fo 6492 df-f1o 6493 df-fv 6494 df-ov 7356 df-oprab 7357 df-mpo 7358 df-er 8632 df-map 8762 df-pm 8763 df-en 8880 df-dom 8881 df-sdom 8882 df-pnf 11170 df-mnf 11171 df-xr 11172 df-ltxr 11173 df-le 11174 df-neg 11369 df-z 12491 df-uz 12755 df-ulm 26303

This theorem is referenced by: ulmshftlem 26315 ulmcau 26321 ulmbdd 26324 ulmcn 26325 iblulm 26333 itgulm 26334

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