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Mirrors > Home > MPE Home > Th. List > Mathboxes > rrx2pnedifcoorneor | Structured version Visualization version GIF version |
Description: If two different points 𝑋 and 𝑌 in a real Euclidean space of dimension 2 are different, then at least one difference of two corresponding coordinates is not 0. (Contributed by AV, 26-Feb-2023.) |
Ref | Expression |
---|---|
rrx2pnecoorneor.i | ⊢ 𝐼 = {1, 2} |
rrx2pnecoorneor.b | ⊢ 𝑃 = (ℝ ↑m 𝐼) |
rrx2pnedifcoorneor.a | ⊢ 𝐴 = ((𝑌‘1) − (𝑋‘1)) |
rrx2pnedifcoorneor.b | ⊢ 𝐵 = ((𝑌‘2) − (𝑋‘2)) |
Ref | Expression |
---|---|
rrx2pnedifcoorneor | ⊢ ((𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ 𝑋 ≠ 𝑌) → (𝐴 ≠ 0 ∨ 𝐵 ≠ 0)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rrx2pnecoorneor.i | . . 3 ⊢ 𝐼 = {1, 2} | |
2 | rrx2pnecoorneor.b | . . 3 ⊢ 𝑃 = (ℝ ↑m 𝐼) | |
3 | 1, 2 | rrx2pnecoorneor 46891 | . 2 ⊢ ((𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ 𝑋 ≠ 𝑌) → ((𝑋‘1) ≠ (𝑌‘1) ∨ (𝑋‘2) ≠ (𝑌‘2))) |
4 | rrx2pnedifcoorneor.a | . . . . . 6 ⊢ 𝐴 = ((𝑌‘1) − (𝑋‘1)) | |
5 | 4 | neeq1i 3005 | . . . . 5 ⊢ (𝐴 ≠ 0 ↔ ((𝑌‘1) − (𝑋‘1)) ≠ 0) |
6 | rrx2pnedifcoorneor.b | . . . . . 6 ⊢ 𝐵 = ((𝑌‘2) − (𝑋‘2)) | |
7 | 6 | neeq1i 3005 | . . . . 5 ⊢ (𝐵 ≠ 0 ↔ ((𝑌‘2) − (𝑋‘2)) ≠ 0) |
8 | 5, 7 | orbi12i 914 | . . . 4 ⊢ ((𝐴 ≠ 0 ∨ 𝐵 ≠ 0) ↔ (((𝑌‘1) − (𝑋‘1)) ≠ 0 ∨ ((𝑌‘2) − (𝑋‘2)) ≠ 0)) |
9 | 1, 2 | rrx2pxel 46887 | . . . . . . . . 9 ⊢ (𝑌 ∈ 𝑃 → (𝑌‘1) ∈ ℝ) |
10 | 9 | recnd 11191 | . . . . . . . 8 ⊢ (𝑌 ∈ 𝑃 → (𝑌‘1) ∈ ℂ) |
11 | 1, 2 | rrx2pxel 46887 | . . . . . . . . 9 ⊢ (𝑋 ∈ 𝑃 → (𝑋‘1) ∈ ℝ) |
12 | 11 | recnd 11191 | . . . . . . . 8 ⊢ (𝑋 ∈ 𝑃 → (𝑋‘1) ∈ ℂ) |
13 | subeq0 11435 | . . . . . . . 8 ⊢ (((𝑌‘1) ∈ ℂ ∧ (𝑋‘1) ∈ ℂ) → (((𝑌‘1) − (𝑋‘1)) = 0 ↔ (𝑌‘1) = (𝑋‘1))) | |
14 | 10, 12, 13 | syl2anr 598 | . . . . . . 7 ⊢ ((𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃) → (((𝑌‘1) − (𝑋‘1)) = 0 ↔ (𝑌‘1) = (𝑋‘1))) |
15 | 14 | necon3bid 2985 | . . . . . 6 ⊢ ((𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃) → (((𝑌‘1) − (𝑋‘1)) ≠ 0 ↔ (𝑌‘1) ≠ (𝑋‘1))) |
16 | 1, 2 | rrx2pyel 46888 | . . . . . . . . 9 ⊢ (𝑌 ∈ 𝑃 → (𝑌‘2) ∈ ℝ) |
17 | 16 | recnd 11191 | . . . . . . . 8 ⊢ (𝑌 ∈ 𝑃 → (𝑌‘2) ∈ ℂ) |
18 | 1, 2 | rrx2pyel 46888 | . . . . . . . . 9 ⊢ (𝑋 ∈ 𝑃 → (𝑋‘2) ∈ ℝ) |
19 | 18 | recnd 11191 | . . . . . . . 8 ⊢ (𝑋 ∈ 𝑃 → (𝑋‘2) ∈ ℂ) |
20 | subeq0 11435 | . . . . . . . 8 ⊢ (((𝑌‘2) ∈ ℂ ∧ (𝑋‘2) ∈ ℂ) → (((𝑌‘2) − (𝑋‘2)) = 0 ↔ (𝑌‘2) = (𝑋‘2))) | |
21 | 17, 19, 20 | syl2anr 598 | . . . . . . 7 ⊢ ((𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃) → (((𝑌‘2) − (𝑋‘2)) = 0 ↔ (𝑌‘2) = (𝑋‘2))) |
22 | 21 | necon3bid 2985 | . . . . . 6 ⊢ ((𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃) → (((𝑌‘2) − (𝑋‘2)) ≠ 0 ↔ (𝑌‘2) ≠ (𝑋‘2))) |
23 | 15, 22 | orbi12d 918 | . . . . 5 ⊢ ((𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃) → ((((𝑌‘1) − (𝑋‘1)) ≠ 0 ∨ ((𝑌‘2) − (𝑋‘2)) ≠ 0) ↔ ((𝑌‘1) ≠ (𝑋‘1) ∨ (𝑌‘2) ≠ (𝑋‘2)))) |
24 | necom 2994 | . . . . . 6 ⊢ ((𝑌‘1) ≠ (𝑋‘1) ↔ (𝑋‘1) ≠ (𝑌‘1)) | |
25 | necom 2994 | . . . . . 6 ⊢ ((𝑌‘2) ≠ (𝑋‘2) ↔ (𝑋‘2) ≠ (𝑌‘2)) | |
26 | 24, 25 | orbi12i 914 | . . . . 5 ⊢ (((𝑌‘1) ≠ (𝑋‘1) ∨ (𝑌‘2) ≠ (𝑋‘2)) ↔ ((𝑋‘1) ≠ (𝑌‘1) ∨ (𝑋‘2) ≠ (𝑌‘2))) |
27 | 23, 26 | bitrdi 287 | . . . 4 ⊢ ((𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃) → ((((𝑌‘1) − (𝑋‘1)) ≠ 0 ∨ ((𝑌‘2) − (𝑋‘2)) ≠ 0) ↔ ((𝑋‘1) ≠ (𝑌‘1) ∨ (𝑋‘2) ≠ (𝑌‘2)))) |
28 | 8, 27 | bitrid 283 | . . 3 ⊢ ((𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃) → ((𝐴 ≠ 0 ∨ 𝐵 ≠ 0) ↔ ((𝑋‘1) ≠ (𝑌‘1) ∨ (𝑋‘2) ≠ (𝑌‘2)))) |
29 | 28 | 3adant3 1133 | . 2 ⊢ ((𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ 𝑋 ≠ 𝑌) → ((𝐴 ≠ 0 ∨ 𝐵 ≠ 0) ↔ ((𝑋‘1) ≠ (𝑌‘1) ∨ (𝑋‘2) ≠ (𝑌‘2)))) |
30 | 3, 29 | mpbird 257 | 1 ⊢ ((𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ 𝑋 ≠ 𝑌) → (𝐴 ≠ 0 ∨ 𝐵 ≠ 0)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 ∨ wo 846 ∧ w3a 1088 = wceq 1542 ∈ wcel 2107 ≠ wne 2940 {cpr 4592 ‘cfv 6500 (class class class)co 7361 ↑m cmap 8771 ℂcc 11057 ℝcr 11058 0cc0 11059 1c1 11060 − cmin 11393 2c2 12216 |
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 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5260 ax-nul 5267 ax-pow 5324 ax-pr 5388 ax-un 7676 ax-resscn 11116 ax-1cn 11117 ax-icn 11118 ax-addcl 11119 ax-addrcl 11120 ax-mulcl 11121 ax-mulrcl 11122 ax-mulcom 11123 ax-addass 11124 ax-mulass 11125 ax-distr 11126 ax-i2m1 11127 ax-1ne0 11128 ax-1rid 11129 ax-rnegex 11130 ax-rrecex 11131 ax-cnre 11132 ax-pre-lttri 11133 ax-pre-lttrn 11134 ax-pre-ltadd 11135 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3353 df-rab 3407 df-v 3449 df-sbc 3744 df-csb 3860 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-nul 4287 df-if 4491 df-pw 4566 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4870 df-iun 4960 df-br 5110 df-opab 5172 df-mpt 5193 df-id 5535 df-po 5549 df-so 5550 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-rn 5648 df-res 5649 df-ima 5650 df-iota 6452 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-riota 7317 df-ov 7364 df-oprab 7365 df-mpo 7366 df-1st 7925 df-2nd 7926 df-er 8654 df-map 8773 df-en 8890 df-dom 8891 df-sdom 8892 df-pnf 11199 df-mnf 11200 df-ltxr 11202 df-sub 11395 df-2 12224 |
This theorem is referenced by: rrx2pnedifcoorneorr 46893 |
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