Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > sumeq2dv | GIF version |
Description: Equality deduction for sum. (Contributed by NM, 3-Jan-2006.) (Revised by Mario Carneiro, 31-Jan-2014.) |
Ref | Expression |
---|---|
sumeq2dv.1 | ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 = 𝐶) |
Ref | Expression |
---|---|
sumeq2dv | ⊢ (𝜑 → Σ𝑘 ∈ 𝐴 𝐵 = Σ𝑘 ∈ 𝐴 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sumeq2dv.1 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 = 𝐶) | |
2 | 1 | ralrimiva 2505 | . 2 ⊢ (𝜑 → ∀𝑘 ∈ 𝐴 𝐵 = 𝐶) |
3 | 2 | sumeq2d 11141 | 1 ⊢ (𝜑 → Σ𝑘 ∈ 𝐴 𝐵 = Σ𝑘 ∈ 𝐴 𝐶) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 = wceq 1331 ∈ wcel 1480 Σcsu 11127 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 ax-cnex 7716 ax-resscn 7717 ax-1cn 7718 ax-1re 7719 ax-icn 7720 ax-addcl 7721 ax-addrcl 7722 ax-mulcl 7723 ax-addcom 7725 ax-addass 7727 ax-distr 7729 ax-i2m1 7730 ax-0lt1 7731 ax-0id 7733 ax-rnegex 7734 ax-cnre 7736 ax-pre-ltirr 7737 ax-pre-ltwlin 7738 ax-pre-lttrn 7739 ax-pre-ltadd 7741 |
This theorem depends on definitions: df-bi 116 df-dc 820 df-3or 963 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-nel 2404 df-ral 2421 df-rex 2422 df-reu 2423 df-rab 2425 df-v 2688 df-sbc 2910 df-csb 3004 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-if 3475 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-int 3772 df-br 3930 df-opab 3990 df-mpt 3991 df-id 4215 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-f1 5128 df-f1o 5130 df-fv 5131 df-riota 5730 df-ov 5777 df-oprab 5778 df-mpo 5779 df-recs 6202 df-frec 6288 df-pnf 7807 df-mnf 7808 df-xr 7809 df-ltxr 7810 df-le 7811 df-sub 7940 df-neg 7941 df-inn 8726 df-n0 8983 df-z 9060 df-uz 9332 df-fz 9796 df-seqfrec 10224 df-sumdc 11128 |
This theorem is referenced by: sumeq2sdv 11144 2sumeq2dv 11145 sumeq12dv 11146 sumeq12rdv 11147 sumfct 11148 fsumf1o 11164 fisumss 11166 fsumsplit 11181 isummulc1 11201 isumdivapc 11202 isumge0 11204 sumsplitdc 11206 fsum2dlemstep 11208 fsumshftm 11219 fisum0diag2 11221 fsummulc1 11223 fsumdivapc 11224 fsumneg 11225 fsumsub 11226 fsum2mul 11227 telfsumo2 11241 fsumparts 11244 hashiun 11252 hash2iun 11253 hash2iun1dif1 11254 binomlem 11257 binom1p 11259 isum1p 11266 arisum 11272 trireciplem 11274 geosergap 11280 geo2sum 11288 mertenslemi1 11309 mertenslem2 11310 mertensabs 11311 efval2 11376 efaddlem 11385 cvgcmp2nlemabs 13232 |
Copyright terms: Public domain | W3C validator |