![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > pnpncand | Structured version Visualization version GIF version |
Description: Addition/subtraction cancellation law. (Contributed by Scott Fenton, 14-Dec-2017.) |
Ref | Expression |
---|---|
pnpncand.1 | ⊢ (𝜑 → 𝐴 ∈ ℂ) |
pnpncand.2 | ⊢ (𝜑 → 𝐵 ∈ ℂ) |
pnpncand.3 | ⊢ (𝜑 → 𝐶 ∈ ℂ) |
Ref | Expression |
---|---|
pnpncand | ⊢ (𝜑 → ((𝐴 + (𝐵 − 𝐶)) + (𝐶 − 𝐵)) = 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pnpncand.1 | . . . 4 ⊢ (𝜑 → 𝐴 ∈ ℂ) | |
2 | pnpncand.2 | . . . . 5 ⊢ (𝜑 → 𝐵 ∈ ℂ) | |
3 | pnpncand.3 | . . . . 5 ⊢ (𝜑 → 𝐶 ∈ ℂ) | |
4 | 2, 3 | subcld 11553 | . . . 4 ⊢ (𝜑 → (𝐵 − 𝐶) ∈ ℂ) |
5 | 1, 4 | addcld 11215 | . . 3 ⊢ (𝜑 → (𝐴 + (𝐵 − 𝐶)) ∈ ℂ) |
6 | 5, 2, 3 | subsub2d 11582 | . 2 ⊢ (𝜑 → ((𝐴 + (𝐵 − 𝐶)) − (𝐵 − 𝐶)) = ((𝐴 + (𝐵 − 𝐶)) + (𝐶 − 𝐵))) |
7 | 1, 4 | pncand 11554 | . 2 ⊢ (𝜑 → ((𝐴 + (𝐵 − 𝐶)) − (𝐵 − 𝐶)) = 𝐴) |
8 | 6, 7 | eqtr3d 2773 | 1 ⊢ (𝜑 → ((𝐴 + (𝐵 − 𝐶)) + (𝐶 − 𝐵)) = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1541 ∈ wcel 2106 (class class class)co 7393 ℂcc 11090 + caddc 11095 − cmin 11426 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2702 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7708 ax-resscn 11149 ax-1cn 11150 ax-icn 11151 ax-addcl 11152 ax-addrcl 11153 ax-mulcl 11154 ax-mulrcl 11155 ax-mulcom 11156 ax-addass 11157 ax-mulass 11158 ax-distr 11159 ax-i2m1 11160 ax-1ne0 11161 ax-1rid 11162 ax-rnegex 11163 ax-rrecex 11164 ax-cnre 11165 ax-pre-lttri 11166 ax-pre-lttrn 11167 ax-pre-ltadd 11168 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4319 df-if 4523 df-pw 4598 df-sn 4623 df-pr 4625 df-op 4629 df-uni 4902 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-po 5581 df-so 5582 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-iota 6484 df-fun 6534 df-fn 6535 df-f 6536 df-f1 6537 df-fo 6538 df-f1o 6539 df-fv 6540 df-riota 7349 df-ov 7396 df-oprab 7397 df-mpo 7398 df-er 8686 df-en 8923 df-dom 8924 df-sdom 8925 df-pnf 11232 df-mnf 11233 df-ltxr 11235 df-sub 11428 |
This theorem is referenced by: fprodser 15875 |
Copyright terms: Public domain | W3C validator |