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Mathbox for David A. Wheeler |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > i2linesi | Structured version Visualization version GIF version |
Description: Solve for the intersection of two lines expressed in Y = MX+B form (note that the lines cannot be vertical). Here we use inference form. We just solve for X, since Y can be trivially found by using X. This is an example of how to use the algebra helpers. Notice that because this proof uses algebra helpers, the main steps of the proof are higher level and easier to follow by a human reader. (Contributed by David A. Wheeler, 11-Oct-2018.) |
Ref | Expression |
---|---|
i2linesi.1 | โข ๐ด โ โ |
i2linesi.2 | โข ๐ต โ โ |
i2linesi.3 | โข ๐ถ โ โ |
i2linesi.4 | โข ๐ท โ โ |
i2linesi.5 | โข ๐ โ โ |
i2linesi.6 | โข ๐ = ((๐ด ยท ๐) + ๐ต) |
i2linesi.7 | โข ๐ = ((๐ถ ยท ๐) + ๐ท) |
i2linesi.8 | โข (๐ด โ ๐ถ) โ 0 |
Ref | Expression |
---|---|
i2linesi | โข ๐ = ((๐ท โ ๐ต) / (๐ด โ ๐ถ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | i2linesi.1 | . . 3 โข ๐ด โ โ | |
2 | i2linesi.3 | . . 3 โข ๐ถ โ โ | |
3 | 1, 2 | subcli 11482 | . 2 โข (๐ด โ ๐ถ) โ โ |
4 | i2linesi.5 | . 2 โข ๐ โ โ | |
5 | i2linesi.8 | . 2 โข (๐ด โ ๐ถ) โ 0 | |
6 | 2, 4 | mulcli 11167 | . . . 4 โข (๐ถ ยท ๐) โ โ |
7 | i2linesi.4 | . . . . 5 โข ๐ท โ โ | |
8 | i2linesi.2 | . . . . 5 โข ๐ต โ โ | |
9 | 7, 8 | subcli 11482 | . . . 4 โข (๐ท โ ๐ต) โ โ |
10 | 1, 4 | mulcli 11167 | . . . . . 6 โข (๐ด ยท ๐) โ โ |
11 | i2linesi.6 | . . . . . . 7 โข ๐ = ((๐ด ยท ๐) + ๐ต) | |
12 | i2linesi.7 | . . . . . . 7 โข ๐ = ((๐ถ ยท ๐) + ๐ท) | |
13 | 11, 12 | eqtr3i 2763 | . . . . . 6 โข ((๐ด ยท ๐) + ๐ต) = ((๐ถ ยท ๐) + ๐ท) |
14 | 10, 8, 13 | mvlraddi 47303 | . . . . 5 โข (๐ด ยท ๐) = (((๐ถ ยท ๐) + ๐ท) โ ๐ต) |
15 | 6, 7, 8, 14 | assraddsubi 47305 | . . . 4 โข (๐ด ยท ๐) = ((๐ถ ยท ๐) + (๐ท โ ๐ต)) |
16 | 6, 9, 15 | mvrladdi 47304 | . . 3 โข ((๐ด ยท ๐) โ (๐ถ ยท ๐)) = (๐ท โ ๐ต) |
17 | 1, 4, 2, 16 | joinlmulsubmuli 47308 | . 2 โข ((๐ด โ ๐ถ) ยท ๐) = (๐ท โ ๐ต) |
18 | 3, 4, 5, 17 | mvllmuli 11993 | 1 โข ๐ = ((๐ท โ ๐ต) / (๐ด โ ๐ถ)) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1542 โ wcel 2107 โ wne 2940 (class class class)co 7358 โcc 11054 0cc0 11056 + caddc 11059 ยท cmul 11061 โ cmin 11390 / cdiv 11817 |
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 ax-pre-mulgt0 11133 |
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-rmo 3352 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-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-er 8651 df-en 8887 df-dom 8888 df-sdom 8889 df-pnf 11196 df-mnf 11197 df-xr 11198 df-ltxr 11199 df-le 11200 df-sub 11392 df-neg 11393 df-div 11818 |
This theorem is referenced by: (None) |
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