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Mirrors > Home > MPE Home > Th. List > mulcanad | Structured version Visualization version GIF version |
Description: Cancellation of a nonzero factor on the left in an equation. One-way deduction form of mulcand 11886. (Contributed by David Moews, 28-Feb-2017.) |
Ref | Expression |
---|---|
mulcanad.1 | ⊢ (𝜑 → 𝐴 ∈ ℂ) |
mulcanad.2 | ⊢ (𝜑 → 𝐵 ∈ ℂ) |
mulcanad.3 | ⊢ (𝜑 → 𝐶 ∈ ℂ) |
mulcanad.4 | ⊢ (𝜑 → 𝐶 ≠ 0) |
mulcanad.5 | ⊢ (𝜑 → (𝐶 · 𝐴) = (𝐶 · 𝐵)) |
Ref | Expression |
---|---|
mulcanad | ⊢ (𝜑 → 𝐴 = 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mulcanad.5 | . 2 ⊢ (𝜑 → (𝐶 · 𝐴) = (𝐶 · 𝐵)) | |
2 | mulcanad.1 | . . 3 ⊢ (𝜑 → 𝐴 ∈ ℂ) | |
3 | mulcanad.2 | . . 3 ⊢ (𝜑 → 𝐵 ∈ ℂ) | |
4 | mulcanad.3 | . . 3 ⊢ (𝜑 → 𝐶 ∈ ℂ) | |
5 | mulcanad.4 | . . 3 ⊢ (𝜑 → 𝐶 ≠ 0) | |
6 | 2, 3, 4, 5 | mulcand 11886 | . 2 ⊢ (𝜑 → ((𝐶 · 𝐴) = (𝐶 · 𝐵) ↔ 𝐴 = 𝐵)) |
7 | 1, 6 | mpbid 231 | 1 ⊢ (𝜑 → 𝐴 = 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1534 ∈ wcel 2099 ≠ wne 2930 (class class class)co 7414 ℂcc 11145 0cc0 11147 · cmul 11152 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-sep 5295 ax-nul 5302 ax-pow 5360 ax-pr 5424 ax-un 7736 ax-resscn 11204 ax-1cn 11205 ax-icn 11206 ax-addcl 11207 ax-addrcl 11208 ax-mulcl 11209 ax-mulrcl 11210 ax-mulcom 11211 ax-addass 11212 ax-mulass 11213 ax-distr 11214 ax-i2m1 11215 ax-1ne0 11216 ax-1rid 11217 ax-rnegex 11218 ax-rrecex 11219 ax-cnre 11220 ax-pre-lttri 11221 ax-pre-lttrn 11222 ax-pre-ltadd 11223 ax-pre-mulgt0 11224 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-reu 3366 df-rab 3421 df-v 3465 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4907 df-br 5145 df-opab 5207 df-mpt 5228 df-id 5571 df-po 5585 df-so 5586 df-xp 5679 df-rel 5680 df-cnv 5681 df-co 5682 df-dm 5683 df-rn 5684 df-res 5685 df-ima 5686 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-er 8724 df-en 8965 df-dom 8966 df-sdom 8967 df-pnf 11289 df-mnf 11290 df-xr 11291 df-ltxr 11292 df-le 11293 df-sub 11485 df-neg 11486 |
This theorem is referenced by: qredeu 16652 gexexlem 19844 ssscongptld 26845 heron 26861 dcubic 26869 mpodvdsmulf1o 27217 dvdsmulf1o 27219 dchrsum2 27292 sumdchr2 27294 2sqlem8 27450 ax5seg 28867 ipasslem4 30762 oddpwdc 34199 rxp11d 42073 pell1234qrreccl 42546 pell14qrdich 42561 sumnnodd 45285 cevathlem1 46522 |
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