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Mirrors > Home > MPE Home > Th. List > ulmdvlem2 | Structured version Visualization version GIF version |
Description: Lemma for ulmdv 24157. (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 6678 | . . . . . . 7 ⊢ (𝑆 D (𝐹‘𝑘)) ∈ V | |
2 | 1 | rgenw 2924 | . . . . . 6 ⊢ ∀𝑘 ∈ 𝑍 (𝑆 D (𝐹‘𝑘)) ∈ V |
3 | eqid 2622 | . . . . . . 7 ⊢ (𝑘 ∈ 𝑍 ↦ (𝑆 D (𝐹‘𝑘))) = (𝑘 ∈ 𝑍 ↦ (𝑆 D (𝐹‘𝑘))) | |
4 | 3 | fnmpt 6020 | . . . . . 6 ⊢ (∀𝑘 ∈ 𝑍 (𝑆 D (𝐹‘𝑘)) ∈ V → (𝑘 ∈ 𝑍 ↦ (𝑆 D (𝐹‘𝑘))) Fn 𝑍) |
5 | 2, 4 | mp1i 13 | . . . . 5 ⊢ (𝜑 → (𝑘 ∈ 𝑍 ↦ (𝑆 D (𝐹‘𝑘))) Fn 𝑍) |
6 | ulmdv.u | . . . . 5 ⊢ (𝜑 → (𝑘 ∈ 𝑍 ↦ (𝑆 D (𝐹‘𝑘)))(⇝𝑢‘𝑋)𝐻) | |
7 | ulmf2 24138 | . . . . 5 ⊢ (((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 693 | . . . 4 ⊢ (𝜑 → (𝑘 ∈ 𝑍 ↦ (𝑆 D (𝐹‘𝑘))):𝑍⟶(ℂ ↑𝑚 𝑋)) |
9 | 3 | fmpt 6381 | . . . 4 ⊢ (∀𝑘 ∈ 𝑍 (𝑆 D (𝐹‘𝑘)) ∈ (ℂ ↑𝑚 𝑋) ↔ (𝑘 ∈ 𝑍 ↦ (𝑆 D (𝐹‘𝑘))):𝑍⟶(ℂ ↑𝑚 𝑋)) |
10 | 8, 9 | sylibr 224 | . . 3 ⊢ (𝜑 → ∀𝑘 ∈ 𝑍 (𝑆 D (𝐹‘𝑘)) ∈ (ℂ ↑𝑚 𝑋)) |
11 | 10 | r19.21bi 2932 | . 2 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝑆 D (𝐹‘𝑘)) ∈ (ℂ ↑𝑚 𝑋)) |
12 | elmapi 7879 | . 2 ⊢ ((𝑆 D (𝐹‘𝑘)) ∈ (ℂ ↑𝑚 𝑋) → (𝑆 D (𝐹‘𝑘)):𝑋⟶ℂ) | |
13 | fdm 6051 | . 2 ⊢ ((𝑆 D (𝐹‘𝑘)):𝑋⟶ℂ → dom (𝑆 D (𝐹‘𝑘)) = 𝑋) | |
14 | 11, 12, 13 | 3syl 18 | 1 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → dom (𝑆 D (𝐹‘𝑘)) = 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ∀wral 2912 Vcvv 3200 {cpr 4179 class class class wbr 4653 ↦ cmpt 4729 dom cdm 5114 Fn wfn 5883 ⟶wf 5884 ‘cfv 5888 (class class class)co 6650 ↑𝑚 cmap 7857 ℂcc 9934 ℝcr 9935 ℤcz 11377 ℤ≥cuz 11687 ⇝ cli 14215 D cdv 23627 ⇝𝑢culm 24130 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-1st 7168 df-2nd 7169 df-map 7859 df-pm 7860 df-neg 10269 df-z 11378 df-uz 11688 df-ulm 24131 |
This theorem is referenced by: ulmdvlem3 24156 ulmdv 24157 |
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