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Mirrors > Home > MPE Home > Th. List > Mathboxes > rrx2pnedifcoorneorr | 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)) |
rrx2pnedifcoorneorr.b | ⊢ 𝐵 = ((𝑋‘2) − (𝑌‘2)) |
Ref | Expression |
---|---|
rrx2pnedifcoorneorr | ⊢ ((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 | rrx2pnedifcoorneor.a | . . 3 ⊢ 𝐴 = ((𝑌‘1) − (𝑋‘1)) | |
4 | eqid 2821 | . . 3 ⊢ ((𝑌‘2) − (𝑋‘2)) = ((𝑌‘2) − (𝑋‘2)) | |
5 | 1, 2, 3, 4 | rrx2pnedifcoorneor 44696 | . 2 ⊢ ((𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ 𝑋 ≠ 𝑌) → (𝐴 ≠ 0 ∨ ((𝑌‘2) − (𝑋‘2)) ≠ 0)) |
6 | eqcom 2828 | . . . . . . 7 ⊢ ((𝑌‘2) = (𝑋‘2) ↔ (𝑋‘2) = (𝑌‘2)) | |
7 | 6 | a1i 11 | . . . . . 6 ⊢ ((𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ 𝑋 ≠ 𝑌) → ((𝑌‘2) = (𝑋‘2) ↔ (𝑋‘2) = (𝑌‘2))) |
8 | 1, 2 | rrx2pyel 44692 | . . . . . . . . . . 11 ⊢ (𝑋 ∈ 𝑃 → (𝑋‘2) ∈ ℝ) |
9 | 8 | recnd 10663 | . . . . . . . . . 10 ⊢ (𝑋 ∈ 𝑃 → (𝑋‘2) ∈ ℂ) |
10 | 1, 2 | rrx2pyel 44692 | . . . . . . . . . . 11 ⊢ (𝑌 ∈ 𝑃 → (𝑌‘2) ∈ ℝ) |
11 | 10 | recnd 10663 | . . . . . . . . . 10 ⊢ (𝑌 ∈ 𝑃 → (𝑌‘2) ∈ ℂ) |
12 | 9, 11 | anim12i 614 | . . . . . . . . 9 ⊢ ((𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃) → ((𝑋‘2) ∈ ℂ ∧ (𝑌‘2) ∈ ℂ)) |
13 | 12 | ancomd 464 | . . . . . . . 8 ⊢ ((𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃) → ((𝑌‘2) ∈ ℂ ∧ (𝑋‘2) ∈ ℂ)) |
14 | 13 | 3adant3 1128 | . . . . . . 7 ⊢ ((𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ 𝑋 ≠ 𝑌) → ((𝑌‘2) ∈ ℂ ∧ (𝑋‘2) ∈ ℂ)) |
15 | subeq0 10906 | . . . . . . 7 ⊢ (((𝑌‘2) ∈ ℂ ∧ (𝑋‘2) ∈ ℂ) → (((𝑌‘2) − (𝑋‘2)) = 0 ↔ (𝑌‘2) = (𝑋‘2))) | |
16 | 14, 15 | syl 17 | . . . . . 6 ⊢ ((𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ 𝑋 ≠ 𝑌) → (((𝑌‘2) − (𝑋‘2)) = 0 ↔ (𝑌‘2) = (𝑋‘2))) |
17 | 12 | 3adant3 1128 | . . . . . . 7 ⊢ ((𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ 𝑋 ≠ 𝑌) → ((𝑋‘2) ∈ ℂ ∧ (𝑌‘2) ∈ ℂ)) |
18 | subeq0 10906 | . . . . . . 7 ⊢ (((𝑋‘2) ∈ ℂ ∧ (𝑌‘2) ∈ ℂ) → (((𝑋‘2) − (𝑌‘2)) = 0 ↔ (𝑋‘2) = (𝑌‘2))) | |
19 | 17, 18 | syl 17 | . . . . . 6 ⊢ ((𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ 𝑋 ≠ 𝑌) → (((𝑋‘2) − (𝑌‘2)) = 0 ↔ (𝑋‘2) = (𝑌‘2))) |
20 | 7, 16, 19 | 3bitr4d 313 | . . . . 5 ⊢ ((𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ 𝑋 ≠ 𝑌) → (((𝑌‘2) − (𝑋‘2)) = 0 ↔ ((𝑋‘2) − (𝑌‘2)) = 0)) |
21 | rrx2pnedifcoorneorr.b | . . . . . . 7 ⊢ 𝐵 = ((𝑋‘2) − (𝑌‘2)) | |
22 | 21 | eqcomi 2830 | . . . . . 6 ⊢ ((𝑋‘2) − (𝑌‘2)) = 𝐵 |
23 | 22 | eqeq1i 2826 | . . . . 5 ⊢ (((𝑋‘2) − (𝑌‘2)) = 0 ↔ 𝐵 = 0) |
24 | 20, 23 | syl6bb 289 | . . . 4 ⊢ ((𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ 𝑋 ≠ 𝑌) → (((𝑌‘2) − (𝑋‘2)) = 0 ↔ 𝐵 = 0)) |
25 | 24 | necon3bid 3060 | . . 3 ⊢ ((𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ 𝑋 ≠ 𝑌) → (((𝑌‘2) − (𝑋‘2)) ≠ 0 ↔ 𝐵 ≠ 0)) |
26 | 25 | orbi2d 912 | . 2 ⊢ ((𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ 𝑋 ≠ 𝑌) → ((𝐴 ≠ 0 ∨ ((𝑌‘2) − (𝑋‘2)) ≠ 0) ↔ (𝐴 ≠ 0 ∨ 𝐵 ≠ 0))) |
27 | 5, 26 | mpbid 234 | 1 ⊢ ((𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ 𝑋 ≠ 𝑌) → (𝐴 ≠ 0 ∨ 𝐵 ≠ 0)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∨ wo 843 ∧ w3a 1083 = wceq 1533 ∈ wcel 2110 ≠ wne 3016 {cpr 4563 ‘cfv 6350 (class class class)co 7150 ↑m cmap 8400 ℂcc 10529 ℝcr 10530 0cc0 10531 1c1 10532 − cmin 10864 2c2 11686 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 ax-un 7455 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 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 3497 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 4562 df-pr 4564 df-op 4568 df-uni 4833 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-id 5455 df-po 5469 df-so 5470 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-1st 7683 df-2nd 7684 df-er 8283 df-map 8402 df-en 8504 df-dom 8505 df-sdom 8506 df-pnf 10671 df-mnf 10672 df-ltxr 10674 df-sub 10866 df-2 11694 |
This theorem is referenced by: itsclinecirc0 44753 itsclinecirc0b 44754 itsclinecirc0in 44755 inlinecirc02plem 44766 |
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