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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 11146. (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 11146 | . 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 1497 | 1 ⊢ (𝜑 → ((𝐴 − 𝐵) / (𝐶 − 𝐷)) = ((𝐵 − 𝐴) / (𝐷 − 𝐶))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1653 ∈ wcel 2157 ≠ wne 2975 (class class class)co 6882 ℂcc 10226 − cmin 10560 / cdiv 10980 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-8 2159 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2379 ax-ext 2781 ax-sep 4979 ax-nul 4987 ax-pow 5039 ax-pr 5101 ax-un 7187 ax-resscn 10285 ax-1cn 10286 ax-icn 10287 ax-addcl 10288 ax-addrcl 10289 ax-mulcl 10290 ax-mulrcl 10291 ax-mulcom 10292 ax-addass 10293 ax-mulass 10294 ax-distr 10295 ax-i2m1 10296 ax-1ne0 10297 ax-1rid 10298 ax-rnegex 10299 ax-rrecex 10300 ax-cnre 10301 ax-pre-lttri 10302 ax-pre-lttrn 10303 ax-pre-ltadd 10304 ax-pre-mulgt0 10305 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3or 1109 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2593 df-eu 2611 df-clab 2790 df-cleq 2796 df-clel 2799 df-nfc 2934 df-ne 2976 df-nel 3079 df-ral 3098 df-rex 3099 df-reu 3100 df-rmo 3101 df-rab 3102 df-v 3391 df-sbc 3638 df-csb 3733 df-dif 3776 df-un 3778 df-in 3780 df-ss 3787 df-nul 4120 df-if 4282 df-pw 4355 df-sn 4373 df-pr 4375 df-op 4379 df-uni 4633 df-br 4848 df-opab 4910 df-mpt 4927 df-id 5224 df-po 5237 df-so 5238 df-xp 5322 df-rel 5323 df-cnv 5324 df-co 5325 df-dm 5326 df-rn 5327 df-res 5328 df-ima 5329 df-iota 6068 df-fun 6107 df-fn 6108 df-f 6109 df-f1 6110 df-fo 6111 df-f1o 6112 df-fv 6113 df-riota 6843 df-ov 6885 df-oprab 6886 df-mpt2 6887 df-er 7986 df-en 8200 df-dom 8201 df-sdom 8202 df-pnf 10369 df-mnf 10370 df-xr 10371 df-ltxr 10372 df-le 10373 df-sub 10562 df-neg 10563 df-div 10981 |
This theorem is referenced by: pwm1geoser 14942 angpieqvdlem 24911 brbtwn2 26146 knoppndvlem14 33028 bj-bary1 33665 ftc1cnnc 33976 |
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