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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 8096 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 + 𝐵) − 𝐵) = 𝐴) | |
4 | 1, 2, 3 | syl2anc 409 | 1 ⊢ (𝜑 → ((𝐴 + 𝐵) − 𝐵) = 𝐴) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1342 ∈ wcel 2135 (class class class)co 5837 ℂcc 7743 + caddc 7748 − cmin 8061 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-14 2138 ax-ext 2146 ax-sep 4095 ax-pow 4148 ax-pr 4182 ax-setind 4509 ax-resscn 7837 ax-1cn 7838 ax-icn 7840 ax-addcl 7841 ax-addrcl 7842 ax-mulcl 7843 ax-addcom 7845 ax-addass 7847 ax-distr 7849 ax-i2m1 7850 ax-0id 7853 ax-rnegex 7854 ax-cnre 7856 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ne 2335 df-ral 2447 df-rex 2448 df-reu 2449 df-rab 2451 df-v 2724 df-sbc 2948 df-dif 3114 df-un 3116 df-in 3118 df-ss 3125 df-pw 3556 df-sn 3577 df-pr 3578 df-op 3580 df-uni 3785 df-br 3978 df-opab 4039 df-id 4266 df-xp 4605 df-rel 4606 df-cnv 4607 df-co 4608 df-dm 4609 df-iota 5148 df-fun 5185 df-fv 5191 df-riota 5793 df-ov 5840 df-oprab 5841 df-mpo 5842 df-sub 8063 |
This theorem is referenced by: mvlraddd 8254 mvlladdd 8255 mvrraddd 8256 addlsub 8260 pnpncand 8265 pncan1 8267 icoshftf1o 9919 nnsplit 10063 uzsinds 10368 zesq 10563 resqrexlemcalc2 10944 iser3shft 11274 fisumrev2 11374 fprodp1 11528 uzwodc 11956 hashdvds 12139 pythagtriplem4 12186 pythagtriplem6 12188 pythagtriplem7 12189 pythagtriplem12 12193 pythagtriplem14 12195 pcqdiv 12225 ennnfonelemp1 12293 blhalf 12966 trilpolemeq1 13771 trilpolemlt1 13772 nconstwlpolemgt0 13794 |
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