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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 2733 | . . 3 ⊢ ((𝑌‘2) − (𝑋‘2)) = ((𝑌‘2) − (𝑋‘2)) | |
5 | 1, 2, 3, 4 | rrx2pnedifcoorneor 46888 | . 2 ⊢ ((𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ 𝑋 ≠ 𝑌) → (𝐴 ≠ 0 ∨ ((𝑌‘2) − (𝑋‘2)) ≠ 0)) |
6 | eqcom 2740 | . . . . . . 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 46884 | . . . . . . . . . . 11 ⊢ (𝑋 ∈ 𝑃 → (𝑋‘2) ∈ ℝ) |
9 | 8 | recnd 11188 | . . . . . . . . . 10 ⊢ (𝑋 ∈ 𝑃 → (𝑋‘2) ∈ ℂ) |
10 | 1, 2 | rrx2pyel 46884 | . . . . . . . . . . 11 ⊢ (𝑌 ∈ 𝑃 → (𝑌‘2) ∈ ℝ) |
11 | 10 | recnd 11188 | . . . . . . . . . 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 463 | . . . . . . . 8 ⊢ ((𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃) → ((𝑌‘2) ∈ ℂ ∧ (𝑋‘2) ∈ ℂ)) |
14 | 13 | 3adant3 1133 | . . . . . . 7 ⊢ ((𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ 𝑋 ≠ 𝑌) → ((𝑌‘2) ∈ ℂ ∧ (𝑋‘2) ∈ ℂ)) |
15 | subeq0 11432 | . . . . . . 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 1133 | . . . . . . 7 ⊢ ((𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ 𝑋 ≠ 𝑌) → ((𝑋‘2) ∈ ℂ ∧ (𝑌‘2) ∈ ℂ)) |
18 | subeq0 11432 | . . . . . . 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 311 | . . . . 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 2742 | . . . . . 6 ⊢ ((𝑋‘2) − (𝑌‘2)) = 𝐵 |
23 | 22 | eqeq1i 2738 | . . . . 5 ⊢ (((𝑋‘2) − (𝑌‘2)) = 0 ↔ 𝐵 = 0) |
24 | 20, 23 | bitrdi 287 | . . . 4 ⊢ ((𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ 𝑋 ≠ 𝑌) → (((𝑌‘2) − (𝑋‘2)) = 0 ↔ 𝐵 = 0)) |
25 | 24 | necon3bid 2985 | . . 3 ⊢ ((𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ 𝑋 ≠ 𝑌) → (((𝑌‘2) − (𝑋‘2)) ≠ 0 ↔ 𝐵 ≠ 0)) |
26 | 25 | orbi2d 915 | . 2 ⊢ ((𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ 𝑋 ≠ 𝑌) → ((𝐴 ≠ 0 ∨ ((𝑌‘2) − (𝑋‘2)) ≠ 0) ↔ (𝐴 ≠ 0 ∨ 𝐵 ≠ 0))) |
27 | 5, 26 | mpbid 231 | 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 4589 ‘cfv 6497 (class class class)co 7358 ↑m cmap 8768 ℂcc 11054 ℝcr 11055 0cc0 11056 1c1 11057 − cmin 11390 2c2 12213 |
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 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-resscn 11113 ax-1cn 11114 ax-icn 11115 ax-addcl 11116 ax-addrcl 11117 ax-mulcl 11118 ax-mulrcl 11119 ax-mulcom 11120 ax-addass 11121 ax-mulass 11122 ax-distr 11123 ax-i2m1 11124 ax-1ne0 11125 ax-1rid 11126 ax-rnegex 11127 ax-rrecex 11128 ax-cnre 11129 ax-pre-lttri 11130 ax-pre-lttrn 11131 ax-pre-ltadd 11132 |
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 3446 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-id 5532 df-po 5546 df-so 5547 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-riota 7314 df-ov 7361 df-oprab 7362 df-mpo 7363 df-1st 7922 df-2nd 7923 df-er 8651 df-map 8770 df-en 8887 df-dom 8888 df-sdom 8889 df-pnf 11196 df-mnf 11197 df-ltxr 11199 df-sub 11392 df-2 12221 |
This theorem is referenced by: itsclinecirc0 46945 itsclinecirc0b 46946 itsclinecirc0in 46947 inlinecirc02plem 46958 |
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