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Mirrors > Home > MPE Home > Th. List > ulmdvlem2 | Structured version Visualization version GIF version |
Description: Lemma for ulmdv 25322. (Contributed by Mario Carneiro, 8-May-2015.) |
Ref | Expression |
---|---|
ulmdv.z | ⊢ 𝑍 = (ℤ≥‘𝑀) |
ulmdv.s | ⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ}) |
ulmdv.m | ⊢ (𝜑 → 𝑀 ∈ ℤ) |
ulmdv.f | ⊢ (𝜑 → 𝐹:𝑍⟶(ℂ ↑m 𝑋)) |
ulmdv.g | ⊢ (𝜑 → 𝐺:𝑋⟶ℂ) |
ulmdv.l | ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑋) → (𝑘 ∈ 𝑍 ↦ ((𝐹‘𝑘)‘𝑧)) ⇝ (𝐺‘𝑧)) |
ulmdv.u | ⊢ (𝜑 → (𝑘 ∈ 𝑍 ↦ (𝑆 D (𝐹‘𝑘)))(⇝𝑢‘𝑋)𝐻) |
Ref | Expression |
---|---|
ulmdvlem2 | ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → dom (𝑆 D (𝐹‘𝑘)) = 𝑋) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ovex 7265 | . . . . . 6 ⊢ (𝑆 D (𝐹‘𝑘)) ∈ V | |
2 | 1 | rgenw 3074 | . . . . 5 ⊢ ∀𝑘 ∈ 𝑍 (𝑆 D (𝐹‘𝑘)) ∈ V |
3 | eqid 2738 | . . . . . 6 ⊢ (𝑘 ∈ 𝑍 ↦ (𝑆 D (𝐹‘𝑘))) = (𝑘 ∈ 𝑍 ↦ (𝑆 D (𝐹‘𝑘))) | |
4 | 3 | fnmpt 6537 | . . . . 5 ⊢ (∀𝑘 ∈ 𝑍 (𝑆 D (𝐹‘𝑘)) ∈ V → (𝑘 ∈ 𝑍 ↦ (𝑆 D (𝐹‘𝑘))) Fn 𝑍) |
5 | 2, 4 | mp1i 13 | . . . 4 ⊢ (𝜑 → (𝑘 ∈ 𝑍 ↦ (𝑆 D (𝐹‘𝑘))) Fn 𝑍) |
6 | ulmdv.u | . . . 4 ⊢ (𝜑 → (𝑘 ∈ 𝑍 ↦ (𝑆 D (𝐹‘𝑘)))(⇝𝑢‘𝑋)𝐻) | |
7 | ulmf2 25303 | . . . 4 ⊢ (((𝑘 ∈ 𝑍 ↦ (𝑆 D (𝐹‘𝑘))) Fn 𝑍 ∧ (𝑘 ∈ 𝑍 ↦ (𝑆 D (𝐹‘𝑘)))(⇝𝑢‘𝑋)𝐻) → (𝑘 ∈ 𝑍 ↦ (𝑆 D (𝐹‘𝑘))):𝑍⟶(ℂ ↑m 𝑋)) | |
8 | 5, 6, 7 | syl2anc 587 | . . 3 ⊢ (𝜑 → (𝑘 ∈ 𝑍 ↦ (𝑆 D (𝐹‘𝑘))):𝑍⟶(ℂ ↑m 𝑋)) |
9 | 8 | fvmptelrn 6949 | . 2 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝑆 D (𝐹‘𝑘)) ∈ (ℂ ↑m 𝑋)) |
10 | elmapi 8551 | . 2 ⊢ ((𝑆 D (𝐹‘𝑘)) ∈ (ℂ ↑m 𝑋) → (𝑆 D (𝐹‘𝑘)):𝑋⟶ℂ) | |
11 | fdm 6573 | . 2 ⊢ ((𝑆 D (𝐹‘𝑘)):𝑋⟶ℂ → dom (𝑆 D (𝐹‘𝑘)) = 𝑋) | |
12 | 9, 10, 11 | 3syl 18 | 1 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → dom (𝑆 D (𝐹‘𝑘)) = 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1543 ∈ wcel 2111 ∀wral 3062 Vcvv 3421 {cpr 4558 class class class wbr 5068 ↦ cmpt 5150 dom cdm 5566 Fn wfn 6393 ⟶wf 6394 ‘cfv 6398 (class class class)co 7232 ↑m cmap 8529 ℂcc 10752 ℝcr 10753 ℤcz 12201 ℤ≥cuz 12463 ⇝ cli 15073 D cdv 24787 ⇝𝑢culm 25295 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2159 ax-12 2176 ax-ext 2709 ax-rep 5194 ax-sep 5207 ax-nul 5214 ax-pow 5273 ax-pr 5337 ax-un 7542 ax-cnex 10810 ax-resscn 10811 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2072 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2887 df-ne 2942 df-ral 3067 df-rex 3068 df-reu 3069 df-rab 3071 df-v 3423 df-sbc 3710 df-csb 3827 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4253 df-if 4455 df-pw 4530 df-sn 4557 df-pr 4559 df-op 4563 df-uni 4835 df-iun 4921 df-br 5069 df-opab 5131 df-mpt 5151 df-id 5470 df-xp 5572 df-rel 5573 df-cnv 5574 df-co 5575 df-dm 5576 df-rn 5577 df-res 5578 df-ima 5579 df-iota 6356 df-fun 6400 df-fn 6401 df-f 6402 df-f1 6403 df-fo 6404 df-f1o 6405 df-fv 6406 df-ov 7235 df-oprab 7236 df-mpo 7237 df-1st 7780 df-2nd 7781 df-map 8531 df-pm 8532 df-neg 11090 df-z 12202 df-uz 12464 df-ulm 25296 |
This theorem is referenced by: ulmdvlem3 25321 ulmdv 25322 |
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