Mathbox for Alexander van der Vekens |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > Mathboxes > elrrx2linest2 | Structured version Visualization version GIF version |
Description: The line passing through the two different points 𝑋 and 𝑌 in a real Euclidean space of dimension 2 in another "standard form" (usually with (𝑝‘1) = 𝑥 and (𝑝‘2) = 𝑦). (Contributed by AV, 23-Feb-2023.) |
Ref | Expression |
---|---|
rrx2linest2.i | ⊢ 𝐼 = {1, 2} |
rrx2linest2.e | ⊢ 𝐸 = (ℝ^‘𝐼) |
rrx2linest2.p | ⊢ 𝑃 = (ℝ ↑m 𝐼) |
rrx2linest2.l | ⊢ 𝐿 = (LineM‘𝐸) |
rrx2linest2.a | ⊢ 𝐴 = ((𝑋‘2) − (𝑌‘2)) |
rrx2linest2.b | ⊢ 𝐵 = ((𝑌‘1) − (𝑋‘1)) |
rrx2linest2.c | ⊢ 𝐶 = (((𝑋‘2) · (𝑌‘1)) − ((𝑋‘1) · (𝑌‘2))) |
Ref | Expression |
---|---|
elrrx2linest2 | ⊢ ((𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ 𝑋 ≠ 𝑌) → (𝐺 ∈ (𝑋𝐿𝑌) ↔ (𝐺 ∈ 𝑃 ∧ ((𝐴 · (𝐺‘1)) + (𝐵 · (𝐺‘2))) = 𝐶))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rrx2linest2.i | . . . 4 ⊢ 𝐼 = {1, 2} | |
2 | rrx2linest2.e | . . . 4 ⊢ 𝐸 = (ℝ^‘𝐼) | |
3 | rrx2linest2.p | . . . 4 ⊢ 𝑃 = (ℝ ↑m 𝐼) | |
4 | rrx2linest2.l | . . . 4 ⊢ 𝐿 = (LineM‘𝐸) | |
5 | rrx2linest2.a | . . . 4 ⊢ 𝐴 = ((𝑋‘2) − (𝑌‘2)) | |
6 | rrx2linest2.b | . . . 4 ⊢ 𝐵 = ((𝑌‘1) − (𝑋‘1)) | |
7 | rrx2linest2.c | . . . 4 ⊢ 𝐶 = (((𝑋‘2) · (𝑌‘1)) − ((𝑋‘1) · (𝑌‘2))) | |
8 | 1, 2, 3, 4, 5, 6, 7 | rrx2linest2 44801 | . . 3 ⊢ ((𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ 𝑋 ≠ 𝑌) → (𝑋𝐿𝑌) = {𝑝 ∈ 𝑃 ∣ ((𝐴 · (𝑝‘1)) + (𝐵 · (𝑝‘2))) = 𝐶}) |
9 | 8 | eleq2d 2897 | . 2 ⊢ ((𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ 𝑋 ≠ 𝑌) → (𝐺 ∈ (𝑋𝐿𝑌) ↔ 𝐺 ∈ {𝑝 ∈ 𝑃 ∣ ((𝐴 · (𝑝‘1)) + (𝐵 · (𝑝‘2))) = 𝐶})) |
10 | fveq1 6662 | . . . . . 6 ⊢ (𝑝 = 𝐺 → (𝑝‘1) = (𝐺‘1)) | |
11 | 10 | oveq2d 7165 | . . . . 5 ⊢ (𝑝 = 𝐺 → (𝐴 · (𝑝‘1)) = (𝐴 · (𝐺‘1))) |
12 | fveq1 6662 | . . . . . 6 ⊢ (𝑝 = 𝐺 → (𝑝‘2) = (𝐺‘2)) | |
13 | 12 | oveq2d 7165 | . . . . 5 ⊢ (𝑝 = 𝐺 → (𝐵 · (𝑝‘2)) = (𝐵 · (𝐺‘2))) |
14 | 11, 13 | oveq12d 7167 | . . . 4 ⊢ (𝑝 = 𝐺 → ((𝐴 · (𝑝‘1)) + (𝐵 · (𝑝‘2))) = ((𝐴 · (𝐺‘1)) + (𝐵 · (𝐺‘2)))) |
15 | 14 | eqeq1d 2822 | . . 3 ⊢ (𝑝 = 𝐺 → (((𝐴 · (𝑝‘1)) + (𝐵 · (𝑝‘2))) = 𝐶 ↔ ((𝐴 · (𝐺‘1)) + (𝐵 · (𝐺‘2))) = 𝐶)) |
16 | 15 | elrab 3676 | . 2 ⊢ (𝐺 ∈ {𝑝 ∈ 𝑃 ∣ ((𝐴 · (𝑝‘1)) + (𝐵 · (𝑝‘2))) = 𝐶} ↔ (𝐺 ∈ 𝑃 ∧ ((𝐴 · (𝐺‘1)) + (𝐵 · (𝐺‘2))) = 𝐶)) |
17 | 9, 16 | syl6bb 289 | 1 ⊢ ((𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ 𝑋 ≠ 𝑌) → (𝐺 ∈ (𝑋𝐿𝑌) ↔ (𝐺 ∈ 𝑃 ∧ ((𝐴 · (𝐺‘1)) + (𝐵 · (𝐺‘2))) = 𝐶))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∧ w3a 1082 = wceq 1536 ∈ wcel 2113 ≠ wne 3015 {crab 3141 {cpr 4562 ‘cfv 6348 (class class class)co 7149 ↑m cmap 8399 ℝcr 10529 1c1 10531 + caddc 10533 · cmul 10535 − cmin 10863 2c2 11686 ℝ^crrx 23979 LineMcline 44784 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 ax-rep 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5323 ax-un 7454 ax-cnex 10586 ax-resscn 10587 ax-1cn 10588 ax-icn 10589 ax-addcl 10590 ax-addrcl 10591 ax-mulcl 10592 ax-mulrcl 10593 ax-mulcom 10594 ax-addass 10595 ax-mulass 10596 ax-distr 10597 ax-i2m1 10598 ax-1ne0 10599 ax-1rid 10600 ax-rnegex 10601 ax-rrecex 10602 ax-cnre 10603 ax-pre-lttri 10604 ax-pre-lttrn 10605 ax-pre-ltadd 10606 ax-pre-mulgt0 10607 ax-pre-sup 10608 ax-addf 10609 ax-mulf 10610 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1083 df-3an 1084 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-ne 3016 df-nel 3123 df-ral 3142 df-rex 3143 df-reu 3144 df-rmo 3145 df-rab 3146 df-v 3493 df-sbc 3769 df-csb 3877 df-dif 3932 df-un 3934 df-in 3936 df-ss 3945 df-pss 3947 df-nul 4285 df-if 4461 df-pw 4534 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-int 4870 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7107 df-ov 7152 df-oprab 7153 df-mpo 7154 df-of 7402 df-om 7574 df-1st 7682 df-2nd 7683 df-supp 7824 df-tpos 7885 df-wrecs 7940 df-recs 8001 df-rdg 8039 df-1o 8095 df-oadd 8099 df-er 8282 df-map 8401 df-ixp 8455 df-en 8503 df-dom 8504 df-sdom 8505 df-fin 8506 df-fsupp 8827 df-sup 8899 df-pnf 10670 df-mnf 10671 df-xr 10672 df-ltxr 10673 df-le 10674 df-sub 10865 df-neg 10866 df-div 11291 df-nn 11632 df-2 11694 df-3 11695 df-4 11696 df-5 11697 df-6 11698 df-7 11699 df-8 11700 df-9 11701 df-n0 11892 df-z 11976 df-dec 12093 df-uz 12238 df-rp 12384 df-fz 12890 df-seq 13367 df-exp 13427 df-cj 14451 df-re 14452 df-im 14453 df-sqrt 14587 df-abs 14588 df-struct 16478 df-ndx 16479 df-slot 16480 df-base 16482 df-sets 16483 df-ress 16484 df-plusg 16571 df-mulr 16572 df-starv 16573 df-sca 16574 df-vsca 16575 df-ip 16576 df-tset 16577 df-ple 16578 df-ds 16580 df-unif 16581 df-hom 16582 df-cco 16583 df-0g 16708 df-prds 16714 df-pws 16716 df-mgm 17845 df-sgrp 17894 df-mnd 17905 df-mhm 17949 df-grp 18099 df-minusg 18100 df-sbg 18101 df-subg 18269 df-ghm 18349 df-cmn 18901 df-mgp 19233 df-ur 19245 df-ring 19292 df-cring 19293 df-oppr 19366 df-dvdsr 19384 df-unit 19385 df-invr 19415 df-dvr 19426 df-rnghom 19460 df-drng 19497 df-field 19498 df-subrg 19526 df-staf 19609 df-srng 19610 df-lmod 19629 df-lss 19697 df-sra 19937 df-rgmod 19938 df-cnfld 20539 df-refld 20742 df-dsmm 20869 df-frlm 20884 df-tng 23187 df-tcph 23766 df-rrx 23981 df-line 44786 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |