![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > div2subd | Structured version Visualization version GIF version |
Description: Swap subtrahend and minuend inside the numerator and denominator of a fraction. Deduction form of div2sub 12090. (Contributed by David Moews, 28-Feb-2017.) |
Ref | Expression |
---|---|
div2subd.1 | ⊢ (𝜑 → 𝐴 ∈ ℂ) |
div2subd.2 | ⊢ (𝜑 → 𝐵 ∈ ℂ) |
div2subd.3 | ⊢ (𝜑 → 𝐶 ∈ ℂ) |
div2subd.4 | ⊢ (𝜑 → 𝐷 ∈ ℂ) |
div2subd.5 | ⊢ (𝜑 → 𝐶 ≠ 𝐷) |
Ref | Expression |
---|---|
div2subd | ⊢ (𝜑 → ((𝐴 − 𝐵) / (𝐶 − 𝐷)) = ((𝐵 − 𝐴) / (𝐷 − 𝐶))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | div2subd.1 | . 2 ⊢ (𝜑 → 𝐴 ∈ ℂ) | |
2 | div2subd.2 | . 2 ⊢ (𝜑 → 𝐵 ∈ ℂ) | |
3 | div2subd.3 | . 2 ⊢ (𝜑 → 𝐶 ∈ ℂ) | |
4 | div2subd.4 | . 2 ⊢ (𝜑 → 𝐷 ∈ ℂ) | |
5 | div2subd.5 | . 2 ⊢ (𝜑 → 𝐶 ≠ 𝐷) | |
6 | div2sub 12090 | . 2 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝐶 ∈ ℂ ∧ 𝐷 ∈ ℂ ∧ 𝐶 ≠ 𝐷)) → ((𝐴 − 𝐵) / (𝐶 − 𝐷)) = ((𝐵 − 𝐴) / (𝐷 − 𝐶))) | |
7 | 1, 2, 3, 4, 5, 6 | syl23anc 1374 | 1 ⊢ (𝜑 → ((𝐴 − 𝐵) / (𝐶 − 𝐷)) = ((𝐵 − 𝐴) / (𝐷 − 𝐶))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1534 ∈ wcel 2099 ≠ wne 2930 (class class class)co 7424 ℂcc 11156 − cmin 11494 / cdiv 11921 |
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 5304 ax-nul 5311 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-resscn 11215 ax-1cn 11216 ax-icn 11217 ax-addcl 11218 ax-addrcl 11219 ax-mulcl 11220 ax-mulrcl 11221 ax-mulcom 11222 ax-addass 11223 ax-mulass 11224 ax-distr 11225 ax-i2m1 11226 ax-1ne0 11227 ax-1rid 11228 ax-rnegex 11229 ax-rrecex 11230 ax-cnre 11231 ax-pre-lttri 11232 ax-pre-lttrn 11233 ax-pre-ltadd 11234 ax-pre-mulgt0 11235 |
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-rmo 3364 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4326 df-if 4534 df-pw 4609 df-sn 4634 df-pr 4636 df-op 4640 df-uni 4914 df-br 5154 df-opab 5216 df-mpt 5237 df-id 5580 df-po 5594 df-so 5595 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-iota 6506 df-fun 6556 df-fn 6557 df-f 6558 df-f1 6559 df-fo 6560 df-f1o 6561 df-fv 6562 df-riota 7380 df-ov 7427 df-oprab 7428 df-mpo 7429 df-er 8734 df-en 8975 df-dom 8976 df-sdom 8977 df-pnf 11300 df-mnf 11301 df-xr 11302 df-ltxr 11303 df-le 11304 df-sub 11496 df-neg 11497 df-div 11922 |
This theorem is referenced by: angpieqvdlem 26856 brbtwn2 28839 knoppndvlem14 36228 bj-bary1 37019 ftc1cnnc 37393 |
Copyright terms: Public domain | W3C validator |