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Mirrors > Home > MPE Home > Th. List > recid2d | 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 |
---|---|
recid2d | ⊢ (𝜑 → ((1 / 𝐴) · 𝐴) = 1) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | div1d.1 | . 2 ⊢ (𝜑 → 𝐴 ∈ ℂ) | |
2 | reccld.2 | . 2 ⊢ (𝜑 → 𝐴 ≠ 0) | |
3 | recid2 11336 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → ((1 / 𝐴) · 𝐴) = 1) | |
4 | 1, 2, 3 | syl2anc 588 | 1 ⊢ (𝜑 → ((1 / 𝐴) · 𝐴) = 1) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1539 ∈ wcel 2112 ≠ wne 2949 (class class class)co 7143 ℂcc 10558 0cc0 10560 1c1 10561 · cmul 10565 / cdiv 11320 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1912 ax-6 1971 ax-7 2016 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2159 ax-12 2176 ax-ext 2730 ax-sep 5162 ax-nul 5169 ax-pow 5227 ax-pr 5291 ax-un 7452 ax-resscn 10617 ax-1cn 10618 ax-icn 10619 ax-addcl 10620 ax-addrcl 10621 ax-mulcl 10622 ax-mulrcl 10623 ax-mulcom 10624 ax-addass 10625 ax-mulass 10626 ax-distr 10627 ax-i2m1 10628 ax-1ne0 10629 ax-1rid 10630 ax-rnegex 10631 ax-rrecex 10632 ax-cnre 10633 ax-pre-lttri 10634 ax-pre-lttrn 10635 ax-pre-ltadd 10636 ax-pre-mulgt0 10637 |
This theorem depends on definitions: df-bi 210 df-an 401 df-or 846 df-3or 1086 df-3an 1087 df-tru 1542 df-ex 1783 df-nf 1787 df-sb 2071 df-mo 2558 df-eu 2589 df-clab 2737 df-cleq 2751 df-clel 2831 df-nfc 2899 df-ne 2950 df-nel 3054 df-ral 3073 df-rex 3074 df-reu 3075 df-rmo 3076 df-rab 3077 df-v 3409 df-sbc 3694 df-csb 3802 df-dif 3857 df-un 3859 df-in 3861 df-ss 3871 df-nul 4222 df-if 4414 df-pw 4489 df-sn 4516 df-pr 4518 df-op 4522 df-uni 4792 df-br 5026 df-opab 5088 df-mpt 5106 df-id 5423 df-po 5436 df-so 5437 df-xp 5523 df-rel 5524 df-cnv 5525 df-co 5526 df-dm 5527 df-rn 5528 df-res 5529 df-ima 5530 df-iota 6287 df-fun 6330 df-fn 6331 df-f 6332 df-f1 6333 df-fo 6334 df-f1o 6335 df-fv 6336 df-riota 7101 df-ov 7146 df-oprab 7147 df-mpo 7148 df-er 8292 df-en 8521 df-dom 8522 df-sdom 8523 df-pnf 10700 df-mnf 10701 df-xr 10702 df-ltxr 10703 df-le 10704 df-sub 10895 df-neg 10896 df-div 11321 |
This theorem is referenced by: recgt0 11509 cvsmuleqdivd 23820 ncvs1 23843 mbfi1fseqlem4 24403 itg2mulc 24432 advlog 25329 cxproot 25365 rlimcxp 25643 cxp2lim 25646 ftalem5 25746 chpchtsum 25887 dchrinvcl 25921 pntrsumo1 26233 nv1 28542 norm1 29116 branmfn 29972 leopmul 30001 strlem1 30117 mpaaeu 40452 stirlinglem15 43081 dirkercncflem2 43097 eenglngeehlnmlem2 45502 |
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