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Mirrors > Home > MPE Home > Th. List > eleesubd | Structured version Visualization version GIF version |
Description: Membership of a subtraction mapping in a Euclidean space. Deduction form of eleesub 27289. (Contributed by Scott Fenton, 17-Jul-2013.) |
Ref | Expression |
---|---|
eleesubd.1 | ⊢ (𝜑 → 𝐶 = (𝑖 ∈ (1...𝑁) ↦ ((𝐴‘𝑖) − (𝐵‘𝑖)))) |
Ref | Expression |
---|---|
eleesubd | ⊢ ((𝜑 ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) → 𝐶 ∈ (𝔼‘𝑁)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eleesubd.1 | . . 3 ⊢ (𝜑 → 𝐶 = (𝑖 ∈ (1...𝑁) ↦ ((𝐴‘𝑖) − (𝐵‘𝑖)))) | |
2 | 1 | 3ad2ant1 1132 | . 2 ⊢ ((𝜑 ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) → 𝐶 = (𝑖 ∈ (1...𝑁) ↦ ((𝐴‘𝑖) − (𝐵‘𝑖)))) |
3 | fveere 27279 | . . . . . . 7 ⊢ ((𝐴 ∈ (𝔼‘𝑁) ∧ 𝑖 ∈ (1...𝑁)) → (𝐴‘𝑖) ∈ ℝ) | |
4 | fveere 27279 | . . . . . . 7 ⊢ ((𝐵 ∈ (𝔼‘𝑁) ∧ 𝑖 ∈ (1...𝑁)) → (𝐵‘𝑖) ∈ ℝ) | |
5 | resubcl 11295 | . . . . . . 7 ⊢ (((𝐴‘𝑖) ∈ ℝ ∧ (𝐵‘𝑖) ∈ ℝ) → ((𝐴‘𝑖) − (𝐵‘𝑖)) ∈ ℝ) | |
6 | 3, 4, 5 | syl2an 596 | . . . . . 6 ⊢ (((𝐴 ∈ (𝔼‘𝑁) ∧ 𝑖 ∈ (1...𝑁)) ∧ (𝐵 ∈ (𝔼‘𝑁) ∧ 𝑖 ∈ (1...𝑁))) → ((𝐴‘𝑖) − (𝐵‘𝑖)) ∈ ℝ) |
7 | 6 | anandirs 676 | . . . . 5 ⊢ (((𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ 𝑖 ∈ (1...𝑁)) → ((𝐴‘𝑖) − (𝐵‘𝑖)) ∈ ℝ) |
8 | 7 | ralrimiva 3108 | . . . 4 ⊢ ((𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) → ∀𝑖 ∈ (1...𝑁)((𝐴‘𝑖) − (𝐵‘𝑖)) ∈ ℝ) |
9 | eleenn 27274 | . . . . . 6 ⊢ (𝐴 ∈ (𝔼‘𝑁) → 𝑁 ∈ ℕ) | |
10 | mptelee 27273 | . . . . . 6 ⊢ (𝑁 ∈ ℕ → ((𝑖 ∈ (1...𝑁) ↦ ((𝐴‘𝑖) − (𝐵‘𝑖))) ∈ (𝔼‘𝑁) ↔ ∀𝑖 ∈ (1...𝑁)((𝐴‘𝑖) − (𝐵‘𝑖)) ∈ ℝ)) | |
11 | 9, 10 | syl 17 | . . . . 5 ⊢ (𝐴 ∈ (𝔼‘𝑁) → ((𝑖 ∈ (1...𝑁) ↦ ((𝐴‘𝑖) − (𝐵‘𝑖))) ∈ (𝔼‘𝑁) ↔ ∀𝑖 ∈ (1...𝑁)((𝐴‘𝑖) − (𝐵‘𝑖)) ∈ ℝ)) |
12 | 11 | adantr 481 | . . . 4 ⊢ ((𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) → ((𝑖 ∈ (1...𝑁) ↦ ((𝐴‘𝑖) − (𝐵‘𝑖))) ∈ (𝔼‘𝑁) ↔ ∀𝑖 ∈ (1...𝑁)((𝐴‘𝑖) − (𝐵‘𝑖)) ∈ ℝ)) |
13 | 8, 12 | mpbird 256 | . . 3 ⊢ ((𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) → (𝑖 ∈ (1...𝑁) ↦ ((𝐴‘𝑖) − (𝐵‘𝑖))) ∈ (𝔼‘𝑁)) |
14 | 13 | 3adant1 1129 | . 2 ⊢ ((𝜑 ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) → (𝑖 ∈ (1...𝑁) ↦ ((𝐴‘𝑖) − (𝐵‘𝑖))) ∈ (𝔼‘𝑁)) |
15 | 2, 14 | eqeltrd 2839 | 1 ⊢ ((𝜑 ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) → 𝐶 ∈ (𝔼‘𝑁)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 ∧ w3a 1086 = wceq 1539 ∈ wcel 2106 ∀wral 3064 ↦ cmpt 5156 ‘cfv 6426 (class class class)co 7267 ℝcr 10880 1c1 10882 − cmin 11215 ℕcn 11983 ...cfz 13249 𝔼cee 27266 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-sep 5221 ax-nul 5228 ax-pow 5286 ax-pr 5350 ax-un 7578 ax-cnex 10937 ax-resscn 10938 ax-1cn 10939 ax-icn 10940 ax-addcl 10941 ax-addrcl 10942 ax-mulcl 10943 ax-mulrcl 10944 ax-mulcom 10945 ax-addass 10946 ax-mulass 10947 ax-distr 10948 ax-i2m1 10949 ax-1ne0 10950 ax-1rid 10951 ax-rnegex 10952 ax-rrecex 10953 ax-cnre 10954 ax-pre-lttri 10955 ax-pre-lttrn 10956 ax-pre-ltadd 10957 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-reu 3071 df-rab 3073 df-v 3431 df-sbc 3716 df-csb 3832 df-dif 3889 df-un 3891 df-in 3893 df-ss 3903 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-br 5074 df-opab 5136 df-mpt 5157 df-id 5484 df-po 5498 df-so 5499 df-xp 5590 df-rel 5591 df-cnv 5592 df-co 5593 df-dm 5594 df-rn 5595 df-res 5596 df-ima 5597 df-iota 6384 df-fun 6428 df-fn 6429 df-f 6430 df-f1 6431 df-fo 6432 df-f1o 6433 df-fv 6434 df-riota 7224 df-ov 7270 df-oprab 7271 df-mpo 7272 df-er 8485 df-map 8604 df-en 8721 df-dom 8722 df-sdom 8723 df-pnf 11021 df-mnf 11022 df-ltxr 11024 df-sub 11217 df-neg 11218 df-ee 27269 |
This theorem is referenced by: (None) |
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