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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 26697. (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 1129 | . 2 ⊢ ((𝜑 ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) → 𝐶 = (𝑖 ∈ (1...𝑁) ↦ ((𝐴‘𝑖) − (𝐵‘𝑖)))) |
3 | fveere 26687 | . . . . . . 7 ⊢ ((𝐴 ∈ (𝔼‘𝑁) ∧ 𝑖 ∈ (1...𝑁)) → (𝐴‘𝑖) ∈ ℝ) | |
4 | fveere 26687 | . . . . . . 7 ⊢ ((𝐵 ∈ (𝔼‘𝑁) ∧ 𝑖 ∈ (1...𝑁)) → (𝐵‘𝑖) ∈ ℝ) | |
5 | resubcl 10950 | . . . . . . 7 ⊢ (((𝐴‘𝑖) ∈ ℝ ∧ (𝐵‘𝑖) ∈ ℝ) → ((𝐴‘𝑖) − (𝐵‘𝑖)) ∈ ℝ) | |
6 | 3, 4, 5 | syl2an 597 | . . . . . 6 ⊢ (((𝐴 ∈ (𝔼‘𝑁) ∧ 𝑖 ∈ (1...𝑁)) ∧ (𝐵 ∈ (𝔼‘𝑁) ∧ 𝑖 ∈ (1...𝑁))) → ((𝐴‘𝑖) − (𝐵‘𝑖)) ∈ ℝ) |
7 | 6 | anandirs 677 | . . . . 5 ⊢ (((𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ 𝑖 ∈ (1...𝑁)) → ((𝐴‘𝑖) − (𝐵‘𝑖)) ∈ ℝ) |
8 | 7 | ralrimiva 3182 | . . . 4 ⊢ ((𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) → ∀𝑖 ∈ (1...𝑁)((𝐴‘𝑖) − (𝐵‘𝑖)) ∈ ℝ) |
9 | eleenn 26682 | . . . . . 6 ⊢ (𝐴 ∈ (𝔼‘𝑁) → 𝑁 ∈ ℕ) | |
10 | mptelee 26681 | . . . . . 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 483 | . . . 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 259 | . . 3 ⊢ ((𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) → (𝑖 ∈ (1...𝑁) ↦ ((𝐴‘𝑖) − (𝐵‘𝑖))) ∈ (𝔼‘𝑁)) |
14 | 13 | 3adant1 1126 | . 2 ⊢ ((𝜑 ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) → (𝑖 ∈ (1...𝑁) ↦ ((𝐴‘𝑖) − (𝐵‘𝑖))) ∈ (𝔼‘𝑁)) |
15 | 2, 14 | eqeltrd 2913 | 1 ⊢ ((𝜑 ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) → 𝐶 ∈ (𝔼‘𝑁)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 ∀wral 3138 ↦ cmpt 5146 ‘cfv 6355 (class class class)co 7156 ℝcr 10536 1c1 10538 − cmin 10870 ℕcn 11638 ...cfz 12893 𝔼cee 26674 |
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-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 |
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-nel 3124 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-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-po 5474 df-so 5475 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-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-er 8289 df-map 8408 df-en 8510 df-dom 8511 df-sdom 8512 df-pnf 10677 df-mnf 10678 df-ltxr 10680 df-sub 10872 df-neg 10873 df-ee 26677 |
This theorem is referenced by: (None) |
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