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Mirrors > Home > MPE Home > Th. List > mvrladdd | Structured version Visualization version GIF version |
Description: Move the left term in a sum on the RHS to the LHS, deduction form. (Contributed by David A. Wheeler, 11-Oct-2018.) |
Ref | Expression |
---|---|
mvrraddd.1 | ⊢ (𝜑 → 𝐵 ∈ ℂ) |
mvrraddd.2 | ⊢ (𝜑 → 𝐶 ∈ ℂ) |
mvrraddd.3 | ⊢ (𝜑 → 𝐴 = (𝐵 + 𝐶)) |
Ref | Expression |
---|---|
mvrladdd | ⊢ (𝜑 → (𝐴 − 𝐵) = 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mvrraddd.2 | . 2 ⊢ (𝜑 → 𝐶 ∈ ℂ) | |
2 | mvrraddd.1 | . 2 ⊢ (𝜑 → 𝐵 ∈ ℂ) | |
3 | mvrraddd.3 | . . 3 ⊢ (𝜑 → 𝐴 = (𝐵 + 𝐶)) | |
4 | 2, 1, 3 | comraddd 11450 | . 2 ⊢ (𝜑 → 𝐴 = (𝐶 + 𝐵)) |
5 | 1, 2, 4 | mvrraddd 11648 | 1 ⊢ (𝜑 → (𝐴 − 𝐵) = 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1534 ∈ wcel 2099 (class class class)co 7414 ℂcc 11128 + caddc 11133 − cmin 11466 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 ax-resscn 11187 ax-1cn 11188 ax-icn 11189 ax-addcl 11190 ax-addrcl 11191 ax-mulcl 11192 ax-mulrcl 11193 ax-mulcom 11194 ax-addass 11195 ax-mulass 11196 ax-distr 11197 ax-i2m1 11198 ax-1ne0 11199 ax-1rid 11200 ax-rnegex 11201 ax-rrecex 11202 ax-cnre 11203 ax-pre-lttri 11204 ax-pre-lttrn 11205 ax-pre-ltadd 11206 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-nel 3042 df-ral 3057 df-rex 3066 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-br 5143 df-opab 5205 df-mpt 5226 df-id 5570 df-po 5584 df-so 5585 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-er 8718 df-en 8956 df-dom 8957 df-sdom 8958 df-pnf 11272 df-mnf 11273 df-ltxr 11275 df-sub 11468 |
This theorem is referenced by: 2txmxeqx 12374 cvgcmpce 15788 mertens 15856 sin01bnd 16153 cos01bnd 16154 eirrlem 16172 bitsmod 16402 dveflem 25898 mtest 26327 tangtx 26427 efiarg 26528 quart1lem 26774 efiatan2 26836 log2tlbnd 26864 jensenlem2 26907 fsumharmonic 26931 chtublem 27131 bcctr 27195 pcbcctr 27196 bcp1ctr 27199 bposlem9 27212 lgsquadlem1 27300 selberg2lem 27470 logdivbnd 27476 pntrsumo1 27485 pntrsumbnd2 27487 pntrlog2bndlem6 27503 pntpbnd1a 27505 hgt750lemd 34216 bcprod 35268 dnizphlfeqhlf 35887 sumcubes 41795 flt4lem5elem 41997 jm3.1lem1 42360 sqrtcval 42994 fzisoeu 44605 supxrgelem 44642 sigarcol 46175 dignn0flhalflem1 47611 1subrec1sub 47701 i2linesd 48135 |
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