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Mirrors > Home > MPE Home > Th. List > recidd | Structured version Visualization version GIF version |
Description: Multiplication of a number and its reciprocal. (Contributed by Mario Carneiro, 27-May-2016.) |
Ref | Expression |
---|---|
div1d.1 | ⊢ (𝜑 → 𝐴 ∈ ℂ) |
reccld.2 | ⊢ (𝜑 → 𝐴 ≠ 0) |
Ref | Expression |
---|---|
recidd | ⊢ (𝜑 → (𝐴 · (1 / 𝐴)) = 1) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | div1d.1 | . 2 ⊢ (𝜑 → 𝐴 ∈ ℂ) | |
2 | reccld.2 | . 2 ⊢ (𝜑 → 𝐴 ≠ 0) | |
3 | recid 11403 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (𝐴 · (1 / 𝐴)) = 1) | |
4 | 1, 2, 3 | syl2anc 587 | 1 ⊢ (𝜑 → (𝐴 · (1 / 𝐴)) = 1) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2114 ≠ wne 2935 (class class class)co 7183 ℂcc 10626 0cc0 10628 1c1 10629 · cmul 10633 / cdiv 11388 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2711 ax-sep 5177 ax-nul 5184 ax-pow 5242 ax-pr 5306 ax-un 7492 ax-resscn 10685 ax-1cn 10686 ax-icn 10687 ax-addcl 10688 ax-addrcl 10689 ax-mulcl 10690 ax-mulrcl 10691 ax-mulcom 10692 ax-addass 10693 ax-mulass 10694 ax-distr 10695 ax-i2m1 10696 ax-1ne0 10697 ax-1rid 10698 ax-rnegex 10699 ax-rrecex 10700 ax-cnre 10701 ax-pre-lttri 10702 ax-pre-lttrn 10703 ax-pre-ltadd 10704 ax-pre-mulgt0 10705 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2541 df-eu 2571 df-clab 2718 df-cleq 2731 df-clel 2812 df-nfc 2882 df-ne 2936 df-nel 3040 df-ral 3059 df-rex 3060 df-reu 3061 df-rmo 3062 df-rab 3063 df-v 3402 df-sbc 3686 df-csb 3801 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4222 df-if 4425 df-pw 4500 df-sn 4527 df-pr 4529 df-op 4533 df-uni 4807 df-br 5041 df-opab 5103 df-mpt 5121 df-id 5439 df-po 5452 df-so 5453 df-xp 5541 df-rel 5542 df-cnv 5543 df-co 5544 df-dm 5545 df-rn 5546 df-res 5547 df-ima 5548 df-iota 6308 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-riota 7140 df-ov 7186 df-oprab 7187 df-mpo 7188 df-er 8333 df-en 8569 df-dom 8570 df-sdom 8571 df-pnf 10768 df-mnf 10769 df-xr 10770 df-ltxr 10771 df-le 10772 df-sub 10963 df-neg 10964 df-div 11389 |
This theorem is referenced by: xlemul1 12779 exprec 13575 harmonic 15320 nmoi2 23496 logtayl 25416 cxprec 25442 isosctrlem2 25570 cxplim 25722 amgmlem 25740 lgamgulmlem2 25780 logfaclbnd 25971 dchrabs 26009 lgseisenlem1 26124 lgseisenlem2 26125 lgsquadlem1 26129 ipasslem4 28782 strlem1 30198 knoppndvlem7 34354 dvtan 35483 irrapxlem2 40258 pellexlem2 40265 pell1234qrreccl 40289 pell14qrdich 40304 reclt0d 42505 stirlinglem3 43200 dirkercncflem2 43228 amgmlemALT 46008 young2d 46010 |
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