![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > pncan3oi | Structured version Visualization version GIF version |
Description: Subtraction and addition of equals. Almost but not exactly the same as pncan3i 11537 and pncan 11466, this order happens often when applying "operations to both sides" so create a theorem specifically for it. A deduction version of this is available as pncand 11572. (Contributed by David A. Wheeler, 11-Oct-2018.) |
Ref | Expression |
---|---|
pncan3oi.1 | ⊢ 𝐴 ∈ ℂ |
pncan3oi.2 | ⊢ 𝐵 ∈ ℂ |
Ref | Expression |
---|---|
pncan3oi | ⊢ ((𝐴 + 𝐵) − 𝐵) = 𝐴 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pncan3oi.1 | . 2 ⊢ 𝐴 ∈ ℂ | |
2 | pncan3oi.2 | . 2 ⊢ 𝐵 ∈ ℂ | |
3 | pncan 11466 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 + 𝐵) − 𝐵) = 𝐴) | |
4 | 1, 2, 3 | mp2an 691 | 1 ⊢ ((𝐴 + 𝐵) − 𝐵) = 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1542 ∈ wcel 2107 (class class class)co 7409 ℂcc 11108 + caddc 11113 − cmin 11444 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 ax-resscn 11167 ax-1cn 11168 ax-icn 11169 ax-addcl 11170 ax-addrcl 11171 ax-mulcl 11172 ax-mulrcl 11173 ax-mulcom 11174 ax-addass 11175 ax-mulass 11176 ax-distr 11177 ax-i2m1 11178 ax-1ne0 11179 ax-1rid 11180 ax-rnegex 11181 ax-rrecex 11182 ax-cnre 11183 ax-pre-lttri 11184 ax-pre-lttrn 11185 ax-pre-ltadd 11186 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5575 df-po 5589 df-so 5590 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-riota 7365 df-ov 7412 df-oprab 7413 df-mpo 7414 df-er 8703 df-en 8940 df-dom 8941 df-sdom 8942 df-pnf 11250 df-mnf 11251 df-ltxr 11253 df-sub 11446 |
This theorem is referenced by: mvrraddi 11477 mvlladdi 11478 climcndslem1 15795 3dvds 16274 2503prm 17073 ovolicc2lem4 25037 eff1o 26058 basellem8 26592 bposlem6 26792 bposlem8 26794 lnfn0i 31295 lmatfvlem 32795 quad3 34655 poimirlem16 36504 poimirlem17 36505 poimirlem19 36507 poimirlem20 36508 fdc 36613 heiborlem6 36684 areaquad 41965 inductionexd 42906 stoweidlem34 44750 fouriersw 44947 mvlraddi 47817 mvrladdi 47818 |
Copyright terms: Public domain | W3C validator |