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Mirrors > Home > ILE Home > Th. List > pncand | GIF version |
Description: Cancellation law for subtraction. (Contributed by Mario Carneiro, 27-May-2016.) |
Ref | Expression |
---|---|
negidd.1 | ⊢ (𝜑 → 𝐴 ∈ ℂ) |
pncand.2 | ⊢ (𝜑 → 𝐵 ∈ ℂ) |
Ref | Expression |
---|---|
pncand | ⊢ (𝜑 → ((𝐴 + 𝐵) − 𝐵) = 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | negidd.1 | . 2 ⊢ (𝜑 → 𝐴 ∈ ℂ) | |
2 | pncand.2 | . 2 ⊢ (𝜑 → 𝐵 ∈ ℂ) | |
3 | pncan 8177 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 + 𝐵) − 𝐵) = 𝐴) | |
4 | 1, 2, 3 | syl2anc 411 | 1 ⊢ (𝜑 → ((𝐴 + 𝐵) − 𝐵) = 𝐴) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1363 ∈ wcel 2158 (class class class)co 5888 ℂcc 7823 + caddc 7828 − cmin 8142 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1457 ax-7 1458 ax-gen 1459 ax-ie1 1503 ax-ie2 1504 ax-8 1514 ax-10 1515 ax-11 1516 ax-i12 1517 ax-bndl 1519 ax-4 1520 ax-17 1536 ax-i9 1540 ax-ial 1544 ax-i5r 1545 ax-14 2161 ax-ext 2169 ax-sep 4133 ax-pow 4186 ax-pr 4221 ax-setind 4548 ax-resscn 7917 ax-1cn 7918 ax-icn 7920 ax-addcl 7921 ax-addrcl 7922 ax-mulcl 7923 ax-addcom 7925 ax-addass 7927 ax-distr 7929 ax-i2m1 7930 ax-0id 7933 ax-rnegex 7934 ax-cnre 7936 |
This theorem depends on definitions: df-bi 117 df-3an 981 df-tru 1366 df-fal 1369 df-nf 1471 df-sb 1773 df-eu 2039 df-mo 2040 df-clab 2174 df-cleq 2180 df-clel 2183 df-nfc 2318 df-ne 2358 df-ral 2470 df-rex 2471 df-reu 2472 df-rab 2474 df-v 2751 df-sbc 2975 df-dif 3143 df-un 3145 df-in 3147 df-ss 3154 df-pw 3589 df-sn 3610 df-pr 3611 df-op 3613 df-uni 3822 df-br 4016 df-opab 4077 df-id 4305 df-xp 4644 df-rel 4645 df-cnv 4646 df-co 4647 df-dm 4648 df-iota 5190 df-fun 5230 df-fv 5236 df-riota 5844 df-ov 5891 df-oprab 5892 df-mpo 5893 df-sub 8144 |
This theorem is referenced by: mvlraddd 8335 mvlladdd 8336 mvrraddd 8337 addlsub 8341 pnpncand 8346 pncan1 8348 icoshftf1o 10005 nnsplit 10151 uzsinds 10456 zesq 10653 resqrexlemcalc2 11038 iser3shft 11368 fisumrev2 11468 fprodp1 11622 uzwodc 12052 hashdvds 12235 pythagtriplem4 12282 pythagtriplem6 12284 pythagtriplem7 12285 pythagtriplem12 12289 pythagtriplem14 12291 pcqdiv 12321 ennnfonelemp1 12421 mulgdirlem 13054 blhalf 14261 trilpolemeq1 15142 trilpolemlt1 15143 nconstwlpolemgt0 15166 |
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