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Mirrors > Home > MPE Home > Th. List > mvllmuld | Structured version Visualization version GIF version |
Description: Move the left term in a product on the LHS to the RHS, deduction form. (Contributed by David A. Wheeler, 11-Oct-2018.) |
Ref | Expression |
---|---|
mvllmuld.1 | ⊢ (𝜑 → 𝐴 ∈ ℂ) |
mvllmuld.2 | ⊢ (𝜑 → 𝐵 ∈ ℂ) |
mvllmuld.3 | ⊢ (𝜑 → 𝐴 ≠ 0) |
mvllmuld.4 | ⊢ (𝜑 → (𝐴 · 𝐵) = 𝐶) |
Ref | Expression |
---|---|
mvllmuld | ⊢ (𝜑 → 𝐵 = (𝐶 / 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mvllmuld.2 | . . 3 ⊢ (𝜑 → 𝐵 ∈ ℂ) | |
2 | mvllmuld.1 | . . 3 ⊢ (𝜑 → 𝐴 ∈ ℂ) | |
3 | mvllmuld.3 | . . 3 ⊢ (𝜑 → 𝐴 ≠ 0) | |
4 | 1, 2, 3 | divcan4d 11687 | . 2 ⊢ (𝜑 → ((𝐵 · 𝐴) / 𝐴) = 𝐵) |
5 | 2, 1 | mulcomd 10927 | . . . 4 ⊢ (𝜑 → (𝐴 · 𝐵) = (𝐵 · 𝐴)) |
6 | mvllmuld.4 | . . . 4 ⊢ (𝜑 → (𝐴 · 𝐵) = 𝐶) | |
7 | 5, 6 | eqtr3d 2780 | . . 3 ⊢ (𝜑 → (𝐵 · 𝐴) = 𝐶) |
8 | 7 | oveq1d 7270 | . 2 ⊢ (𝜑 → ((𝐵 · 𝐴) / 𝐴) = (𝐶 / 𝐴)) |
9 | 4, 8 | eqtr3d 2780 | 1 ⊢ (𝜑 → 𝐵 = (𝐶 / 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1539 ∈ wcel 2108 ≠ wne 2942 (class class class)co 7255 ℂcc 10800 0cc0 10802 · cmul 10807 / cdiv 11562 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-sep 5218 ax-nul 5225 ax-pow 5283 ax-pr 5347 ax-un 7566 ax-resscn 10859 ax-1cn 10860 ax-icn 10861 ax-addcl 10862 ax-addrcl 10863 ax-mulcl 10864 ax-mulrcl 10865 ax-mulcom 10866 ax-addass 10867 ax-mulass 10868 ax-distr 10869 ax-i2m1 10870 ax-1ne0 10871 ax-1rid 10872 ax-rnegex 10873 ax-rrecex 10874 ax-cnre 10875 ax-pre-lttri 10876 ax-pre-lttrn 10877 ax-pre-ltadd 10878 ax-pre-mulgt0 10879 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3068 df-rex 3069 df-reu 3070 df-rmo 3071 df-rab 3072 df-v 3424 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-nul 4254 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-op 4565 df-uni 4837 df-br 5071 df-opab 5133 df-mpt 5154 df-id 5480 df-po 5494 df-so 5495 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-res 5592 df-ima 5593 df-iota 6376 df-fun 6420 df-fn 6421 df-f 6422 df-f1 6423 df-fo 6424 df-f1o 6425 df-fv 6426 df-riota 7212 df-ov 7258 df-oprab 7259 df-mpo 7260 df-er 8456 df-en 8692 df-dom 8693 df-sdom 8694 df-pnf 10942 df-mnf 10943 df-xr 10944 df-ltxr 10945 df-le 10946 df-sub 11137 df-neg 11138 df-div 11563 |
This theorem is referenced by: exprec 13752 abs1m 14975 fsumkthpow 15694 efneg 15735 abvrec 20011 nminvr 23739 itgpowd 25119 cxpneg 25741 cxprec 25746 atantan 25978 atantayl2 25993 irrdifflemf 35423 fltne 40397 flt4lem5e 40409 rmxyneg 40658 bccm1k 41849 stoweidlem13 43444 eenglngeehlnmlem2 45972 i2linesd 46369 |
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