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Mirrors > Home > MPE Home > Th. List > mvrladdd | Structured version Visualization version GIF version |
Description: Move RHS left addition to LHS. (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 10842 | . 2 ⊢ (𝜑 → 𝐴 = (𝐶 + 𝐵)) |
5 | 1, 2, 4 | mvrraddd 11040 | 1 ⊢ (𝜑 → (𝐴 − 𝐵) = 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1528 ∈ wcel 2105 (class class class)co 7145 ℂcc 10523 + caddc 10528 − cmin 10858 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-po 5467 df-so 5468 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-er 8278 df-en 8498 df-dom 8499 df-sdom 8500 df-pnf 10665 df-mnf 10666 df-ltxr 10668 df-sub 10860 |
This theorem is referenced by: 2txmxeqx 11765 cvgcmpce 15161 mertens 15230 sin01bnd 15526 cos01bnd 15527 eirrlem 15545 bitsmod 15773 dveflem 24503 mtest 24919 tangtx 25018 efiarg 25117 quart1lem 25360 efiatan2 25422 log2tlbnd 25450 jensenlem2 25492 fsumharmonic 25516 chtublem 25714 bcctr 25778 pcbcctr 25779 bcp1ctr 25782 bposlem9 25795 lgsquadlem1 25883 selberg2lem 26053 logdivbnd 26059 pntrsumo1 26068 pntrsumbnd2 26070 pntrlog2bndlem6 26086 pntpbnd1a 26088 hgt750lemd 31818 bcprod 32867 dnizphlfeqhlf 33712 jm3.1lem1 39492 fzisoeu 41443 supxrgelem 41481 sigarcol 42998 dignn0flhalflem1 44603 1subrec1sub 44620 i2linesd 44808 |
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