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Mirrors > Home > MPE Home > Th. List > eqeefv | Structured version Visualization version GIF version |
Description: Two points are equal iff they agree in all dimensions. (Contributed by Scott Fenton, 10-Jun-2013.) |
Ref | Expression |
---|---|
eqeefv | ⊢ ((𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) → (𝐴 = 𝐵 ↔ ∀𝑖 ∈ (1...𝑁)(𝐴‘𝑖) = (𝐵‘𝑖))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eleei 25777 | . . 3 ⊢ (𝐴 ∈ (𝔼‘𝑁) → 𝐴:(1...𝑁)⟶ℝ) | |
2 | ffn 6045 | . . 3 ⊢ (𝐴:(1...𝑁)⟶ℝ → 𝐴 Fn (1...𝑁)) | |
3 | 1, 2 | syl 17 | . 2 ⊢ (𝐴 ∈ (𝔼‘𝑁) → 𝐴 Fn (1...𝑁)) |
4 | eleei 25777 | . . 3 ⊢ (𝐵 ∈ (𝔼‘𝑁) → 𝐵:(1...𝑁)⟶ℝ) | |
5 | ffn 6045 | . . 3 ⊢ (𝐵:(1...𝑁)⟶ℝ → 𝐵 Fn (1...𝑁)) | |
6 | 4, 5 | syl 17 | . 2 ⊢ (𝐵 ∈ (𝔼‘𝑁) → 𝐵 Fn (1...𝑁)) |
7 | eqfnfv 6311 | . 2 ⊢ ((𝐴 Fn (1...𝑁) ∧ 𝐵 Fn (1...𝑁)) → (𝐴 = 𝐵 ↔ ∀𝑖 ∈ (1...𝑁)(𝐴‘𝑖) = (𝐵‘𝑖))) | |
8 | 3, 6, 7 | syl2an 494 | 1 ⊢ ((𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) → (𝐴 = 𝐵 ↔ ∀𝑖 ∈ (1...𝑁)(𝐴‘𝑖) = (𝐵‘𝑖))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ∀wral 2912 Fn wfn 5883 ⟶wf 5884 ‘cfv 5888 (class class class)co 6650 ℝcr 9935 1c1 9937 ...cfz 12326 𝔼cee 25768 |
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-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-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-ral 2917 df-rex 2918 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-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-fv 5896 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-map 7859 df-ee 25771 |
This theorem is referenced by: eqeelen 25784 brbtwn2 25785 colinearalg 25790 axcgrid 25796 ax5seglem4 25812 ax5seglem5 25813 axbtwnid 25819 axeuclid 25843 axcontlem2 25845 axcontlem4 25847 axcontlem7 25850 |
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