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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-lineqi | Structured version Visualization version GIF version |
Description: Solution of a (scalar) linear equation. (Contributed by BJ, 6-Jun-2019.) |
Ref | Expression |
---|---|
bj-lineqi.a | โข (๐ โ ๐ด โ โ) |
bj-lineqi.b | โข (๐ โ ๐ต โ โ) |
bj-lineqi.x | โข (๐ โ ๐ โ โ) |
bj-lineqi.y | โข (๐ โ ๐ โ โ) |
bj-lineqi.n0 | โข (๐ โ ๐ด โ 0) |
bj-lineqi.1 | โข (๐ โ ((๐ด ยท ๐) + ๐ต) = ๐) |
Ref | Expression |
---|---|
bj-lineqi | โข (๐ โ ๐ = ((๐ โ ๐ต) / ๐ด)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bj-lineqi.1 | . 2 โข (๐ โ ((๐ด ยท ๐) + ๐ต) = ๐) | |
2 | bj-lineqi.a | . . 3 โข (๐ โ ๐ด โ โ) | |
3 | bj-lineqi.b | . . 3 โข (๐ โ ๐ต โ โ) | |
4 | bj-lineqi.x | . . 3 โข (๐ โ ๐ โ โ) | |
5 | bj-lineqi.y | . . 3 โข (๐ โ ๐ โ โ) | |
6 | bj-lineqi.n0 | . . 3 โข (๐ โ ๐ด โ 0) | |
7 | 2, 3, 4, 5, 6 | lineq 11914 | . 2 โข (๐ โ (((๐ด ยท ๐) + ๐ต) = ๐ โ ๐ = ((๐ โ ๐ต) / ๐ด))) |
8 | 1, 7 | mpbid 231 | 1 โข (๐ โ ๐ = ((๐ โ ๐ต) / ๐ด)) |
Colors of variables: wff setvar class |
Syntax hints: โ wi 4 = wceq 1540 โ wcel 2105 โ wne 2940 (class class class)co 7338 โcc 10971 0cc0 10973 + caddc 10976 ยท cmul 10978 โ cmin 11307 / cdiv 11734 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2707 ax-sep 5244 ax-nul 5251 ax-pow 5309 ax-pr 5373 ax-un 7651 ax-resscn 11030 ax-1cn 11031 ax-icn 11032 ax-addcl 11033 ax-addrcl 11034 ax-mulcl 11035 ax-mulrcl 11036 ax-mulcom 11037 ax-addass 11038 ax-mulass 11039 ax-distr 11040 ax-i2m1 11041 ax-1ne0 11042 ax-1rid 11043 ax-rnegex 11044 ax-rrecex 11045 ax-cnre 11046 ax-pre-lttri 11047 ax-pre-lttrn 11048 ax-pre-ltadd 11049 ax-pre-mulgt0 11050 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3349 df-reu 3350 df-rab 3404 df-v 3443 df-sbc 3728 df-csb 3844 df-dif 3901 df-un 3903 df-in 3905 df-ss 3915 df-nul 4271 df-if 4475 df-pw 4550 df-sn 4575 df-pr 4577 df-op 4581 df-uni 4854 df-br 5094 df-opab 5156 df-mpt 5177 df-id 5519 df-po 5533 df-so 5534 df-xp 5627 df-rel 5628 df-cnv 5629 df-co 5630 df-dm 5631 df-rn 5632 df-res 5633 df-ima 5634 df-iota 6432 df-fun 6482 df-fn 6483 df-f 6484 df-f1 6485 df-fo 6486 df-f1o 6487 df-fv 6488 df-riota 7294 df-ov 7341 df-oprab 7342 df-mpo 7343 df-er 8570 df-en 8806 df-dom 8807 df-sdom 8808 df-pnf 11113 df-mnf 11114 df-xr 11115 df-ltxr 11116 df-le 11117 df-sub 11309 df-neg 11310 df-div 11735 |
This theorem is referenced by: (None) |
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