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Mirrors > Home > MPE Home > Th. List > ulmdvlem2 | Structured version Visualization version GIF version |
Description: Lemma for ulmdv 24991. (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 7189 | . . . . . 6 ⊢ (𝑆 D (𝐹‘𝑘)) ∈ V | |
2 | 1 | rgenw 3150 | . . . . 5 ⊢ ∀𝑘 ∈ 𝑍 (𝑆 D (𝐹‘𝑘)) ∈ V |
3 | eqid 2821 | . . . . . 6 ⊢ (𝑘 ∈ 𝑍 ↦ (𝑆 D (𝐹‘𝑘))) = (𝑘 ∈ 𝑍 ↦ (𝑆 D (𝐹‘𝑘))) | |
4 | 3 | fnmpt 6488 | . . . . 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 24972 | . . . 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 586 | . . 3 ⊢ (𝜑 → (𝑘 ∈ 𝑍 ↦ (𝑆 D (𝐹‘𝑘))):𝑍⟶(ℂ ↑m 𝑋)) |
9 | 8 | fvmptelrn 6877 | . 2 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝑆 D (𝐹‘𝑘)) ∈ (ℂ ↑m 𝑋)) |
10 | elmapi 8428 | . 2 ⊢ ((𝑆 D (𝐹‘𝑘)) ∈ (ℂ ↑m 𝑋) → (𝑆 D (𝐹‘𝑘)):𝑋⟶ℂ) | |
11 | fdm 6522 | . 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 398 = wceq 1537 ∈ wcel 2114 ∀wral 3138 Vcvv 3494 {cpr 4569 class class class wbr 5066 ↦ cmpt 5146 dom cdm 5555 Fn wfn 6350 ⟶wf 6351 ‘cfv 6355 (class class class)co 7156 ↑m cmap 8406 ℂcc 10535 ℝcr 10536 ℤcz 11982 ℤ≥cuz 12244 ⇝ cli 14841 D cdv 24461 ⇝𝑢culm 24964 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-ov 7159 df-oprab 7160 df-mpo 7161 df-1st 7689 df-2nd 7690 df-map 8408 df-pm 8409 df-neg 10873 df-z 11983 df-uz 12245 df-ulm 24965 |
This theorem is referenced by: ulmdvlem3 24990 ulmdv 24991 |
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