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| Mirrors > Home > MPE Home > Th. List > ppncand | Structured version Visualization version GIF version | ||
| Description: Cancellation law for mixed addition and subtraction. (Contributed by Mario Carneiro, 27-May-2016.) | 
| Ref | Expression | 
|---|---|
| negidd.1 | ⊢ (𝜑 → 𝐴 ∈ ℂ) | 
| pncand.2 | ⊢ (𝜑 → 𝐵 ∈ ℂ) | 
| subaddd.3 | ⊢ (𝜑 → 𝐶 ∈ ℂ) | 
| Ref | Expression | 
|---|---|
| ppncand | ⊢ (𝜑 → ((𝐴 + 𝐵) + (𝐶 − 𝐵)) = (𝐴 + 𝐶)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | negidd.1 | . 2 ⊢ (𝜑 → 𝐴 ∈ ℂ) | |
| 2 | pncand.2 | . 2 ⊢ (𝜑 → 𝐵 ∈ ℂ) | |
| 3 | subaddd.3 | . 2 ⊢ (𝜑 → 𝐶 ∈ ℂ) | |
| 4 | ppncan 11552 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 + 𝐵) + (𝐶 − 𝐵)) = (𝐴 + 𝐶)) | |
| 5 | 1, 2, 3, 4 | syl3anc 1372 | 1 ⊢ (𝜑 → ((𝐴 + 𝐵) + (𝐶 − 𝐵)) = (𝐴 + 𝐶)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 = wceq 1539 ∈ wcel 2107 (class class class)co 7432 ℂcc 11154 + caddc 11159 − cmin 11493 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 ax-resscn 11213 ax-1cn 11214 ax-icn 11215 ax-addcl 11216 ax-addrcl 11217 ax-mulcl 11218 ax-mulrcl 11219 ax-mulcom 11220 ax-addass 11221 ax-mulass 11222 ax-distr 11223 ax-i2m1 11224 ax-1ne0 11225 ax-1rid 11226 ax-rnegex 11227 ax-rrecex 11228 ax-cnre 11229 ax-pre-lttri 11230 ax-pre-lttrn 11231 ax-pre-ltadd 11232 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-br 5143 df-opab 5205 df-mpt 5225 df-id 5577 df-po 5591 df-so 5592 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-riota 7389 df-ov 7435 df-oprab 7436 df-mpo 7437 df-er 8746 df-en 8987 df-dom 8988 df-sdom 8989 df-pnf 11298 df-mnf 11299 df-ltxr 11301 df-sub 11495 | 
| This theorem is referenced by: cju 12263 remim 15157 absmax 15369 bhmafibid2cn 15504 arisum 15897 binomfallfaclem2 16077 cosadd 16202 addsin 16207 cosmul 16210 addcos 16211 pythagtriplem12 16865 pythagtriplem17 16870 fldivp1 16936 mul4sqlem 16992 vdwlem8 17027 nmparlem 25274 cphipval2 25276 heron 26882 quart1lem 26899 quart1 26900 cosasin 26948 cosatan 26965 2sqblem 27476 dchrisumlem2 27535 cncph 30839 divcnvlin 35734 fwddifnp1 36167 posbezout 42102 sticksstones22 42170 rmxluc 42953 rmyluc 42954 acongeq 43000 sinmulcos 45885 fourierdlem65 46191 itsclc0xyqsolr 48695 | 
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