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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 45129 | . 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 3051 | . . . . 5 ⊢ (𝐴 ≠ 0 ↔ ((𝑌‘1) − (𝑋‘1)) ≠ 0) |
6 | rrx2pnedifcoorneor.b | . . . . . 6 ⊢ 𝐵 = ((𝑌‘2) − (𝑋‘2)) | |
7 | 6 | neeq1i 3051 | . . . . 5 ⊢ (𝐵 ≠ 0 ↔ ((𝑌‘2) − (𝑋‘2)) ≠ 0) |
8 | 5, 7 | orbi12i 912 | . . . 4 ⊢ ((𝐴 ≠ 0 ∨ 𝐵 ≠ 0) ↔ (((𝑌‘1) − (𝑋‘1)) ≠ 0 ∨ ((𝑌‘2) − (𝑋‘2)) ≠ 0)) |
9 | 1, 2 | rrx2pxel 45125 | . . . . . . . . 9 ⊢ (𝑌 ∈ 𝑃 → (𝑌‘1) ∈ ℝ) |
10 | 9 | recnd 10658 | . . . . . . . 8 ⊢ (𝑌 ∈ 𝑃 → (𝑌‘1) ∈ ℂ) |
11 | 1, 2 | rrx2pxel 45125 | . . . . . . . . 9 ⊢ (𝑋 ∈ 𝑃 → (𝑋‘1) ∈ ℝ) |
12 | 11 | recnd 10658 | . . . . . . . 8 ⊢ (𝑋 ∈ 𝑃 → (𝑋‘1) ∈ ℂ) |
13 | subeq0 10901 | . . . . . . . 8 ⊢ (((𝑌‘1) ∈ ℂ ∧ (𝑋‘1) ∈ ℂ) → (((𝑌‘1) − (𝑋‘1)) = 0 ↔ (𝑌‘1) = (𝑋‘1))) | |
14 | 10, 12, 13 | syl2anr 599 | . . . . . . 7 ⊢ ((𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃) → (((𝑌‘1) − (𝑋‘1)) = 0 ↔ (𝑌‘1) = (𝑋‘1))) |
15 | 14 | necon3bid 3031 | . . . . . 6 ⊢ ((𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃) → (((𝑌‘1) − (𝑋‘1)) ≠ 0 ↔ (𝑌‘1) ≠ (𝑋‘1))) |
16 | 1, 2 | rrx2pyel 45126 | . . . . . . . . 9 ⊢ (𝑌 ∈ 𝑃 → (𝑌‘2) ∈ ℝ) |
17 | 16 | recnd 10658 | . . . . . . . 8 ⊢ (𝑌 ∈ 𝑃 → (𝑌‘2) ∈ ℂ) |
18 | 1, 2 | rrx2pyel 45126 | . . . . . . . . 9 ⊢ (𝑋 ∈ 𝑃 → (𝑋‘2) ∈ ℝ) |
19 | 18 | recnd 10658 | . . . . . . . 8 ⊢ (𝑋 ∈ 𝑃 → (𝑋‘2) ∈ ℂ) |
20 | subeq0 10901 | . . . . . . . 8 ⊢ (((𝑌‘2) ∈ ℂ ∧ (𝑋‘2) ∈ ℂ) → (((𝑌‘2) − (𝑋‘2)) = 0 ↔ (𝑌‘2) = (𝑋‘2))) | |
21 | 17, 19, 20 | syl2anr 599 | . . . . . . 7 ⊢ ((𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃) → (((𝑌‘2) − (𝑋‘2)) = 0 ↔ (𝑌‘2) = (𝑋‘2))) |
22 | 21 | necon3bid 3031 | . . . . . 6 ⊢ ((𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃) → (((𝑌‘2) − (𝑋‘2)) ≠ 0 ↔ (𝑌‘2) ≠ (𝑋‘2))) |
23 | 15, 22 | orbi12d 916 | . . . . 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 3040 | . . . . . 6 ⊢ ((𝑌‘1) ≠ (𝑋‘1) ↔ (𝑋‘1) ≠ (𝑌‘1)) | |
25 | necom 3040 | . . . . . 6 ⊢ ((𝑌‘2) ≠ (𝑋‘2) ↔ (𝑋‘2) ≠ (𝑌‘2)) | |
26 | 24, 25 | orbi12i 912 | . . . . 5 ⊢ (((𝑌‘1) ≠ (𝑋‘1) ∨ (𝑌‘2) ≠ (𝑋‘2)) ↔ ((𝑋‘1) ≠ (𝑌‘1) ∨ (𝑋‘2) ≠ (𝑌‘2))) |
27 | 23, 26 | syl6bb 290 | . . . 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 | syl5bb 286 | . . 3 ⊢ ((𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃) → ((𝐴 ≠ 0 ∨ 𝐵 ≠ 0) ↔ ((𝑋‘1) ≠ (𝑌‘1) ∨ (𝑋‘2) ≠ (𝑌‘2)))) |
29 | 28 | 3adant3 1129 | . 2 ⊢ ((𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ 𝑋 ≠ 𝑌) → ((𝐴 ≠ 0 ∨ 𝐵 ≠ 0) ↔ ((𝑋‘1) ≠ (𝑌‘1) ∨ (𝑋‘2) ≠ (𝑌‘2)))) |
30 | 3, 29 | mpbird 260 | 1 ⊢ ((𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ 𝑋 ≠ 𝑌) → (𝐴 ≠ 0 ∨ 𝐵 ≠ 0)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 ∨ wo 844 ∧ w3a 1084 = wceq 1538 ∈ wcel 2111 ≠ wne 2987 {cpr 4527 ‘cfv 6324 (class class class)co 7135 ↑m cmap 8389 ℂcc 10524 ℝcr 10525 0cc0 10526 1c1 10527 − cmin 10859 2c2 11680 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-po 5438 df-so 5439 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-1st 7671 df-2nd 7672 df-er 8272 df-map 8391 df-en 8493 df-dom 8494 df-sdom 8495 df-pnf 10666 df-mnf 10667 df-ltxr 10669 df-sub 10861 df-2 11688 |
This theorem is referenced by: rrx2pnedifcoorneorr 45131 |
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