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Mirrors > Home > MPE Home > Th. List > subeq0ad | Structured version Visualization version GIF version |
Description: The difference of two complex numbers is zero iff they are equal. Deduction form of subeq0 10914. Generalization of subeq0d 11007. (Contributed by David Moews, 28-Feb-2017.) |
Ref | Expression |
---|---|
negidd.1 | ⊢ (𝜑 → 𝐴 ∈ ℂ) |
pncand.2 | ⊢ (𝜑 → 𝐵 ∈ ℂ) |
Ref | Expression |
---|---|
subeq0ad | ⊢ (𝜑 → ((𝐴 − 𝐵) = 0 ↔ 𝐴 = 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | negidd.1 | . 2 ⊢ (𝜑 → 𝐴 ∈ ℂ) | |
2 | pncand.2 | . 2 ⊢ (𝜑 → 𝐵 ∈ ℂ) | |
3 | subeq0 10914 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 − 𝐵) = 0 ↔ 𝐴 = 𝐵)) | |
4 | 1, 2, 3 | syl2anc 586 | 1 ⊢ (𝜑 → ((𝐴 − 𝐵) = 0 ↔ 𝐴 = 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 = wceq 1537 ∈ wcel 2114 (class class class)co 7158 ℂcc 10537 0cc0 10539 − cmin 10872 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-po 5476 df-so 5477 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-er 8291 df-en 8512 df-dom 8513 df-sdom 8514 df-pnf 10679 df-mnf 10680 df-ltxr 10682 df-sub 10874 |
This theorem is referenced by: subne0ad 11010 subeq0bd 11068 muleqadd 11286 mulcan1g 11295 ofsubeq0 11637 nn0n0n1ge2 11965 mod0 13247 modirr 13313 addmodlteq 13317 sqreulem 14721 sqreu 14722 tanaddlem 15521 fldivp1 16235 4sqlem11 16293 4sqlem16 16298 znf1o 20700 cphsqrtcl2 23792 rrxmet 24013 dvcobr 24545 dvcnvlem 24575 cmvth 24590 dvlip 24592 lhop1lem 24612 ftc1lem5 24639 aalioulem2 24924 sineq0 25111 tanarg 25204 affineequiv 25403 quad2 25419 dcubic 25426 eqeelen 26692 colinearalg 26698 axcontlem7 26758 ipasslem9 28617 ip2eqi 28635 hi2eq 28884 lnopeqi 29787 riesz3i 29841 signslema 31834 circlemeth 31913 poimirlem32 34926 broucube 34928 rrnmet 35109 eqrabdioph 39381 pellexlem1 39433 sineq0ALT 41278 digexp 44674 eenglngeehlnmlem2 44732 2itscp 44775 |
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