Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > sumeq1i | GIF version |
Description: Equality inference for sum. (Contributed by NM, 2-Jan-2006.) |
Ref | Expression |
---|---|
sumeq1i.1 | ⊢ 𝐴 = 𝐵 |
Ref | Expression |
---|---|
sumeq1i | ⊢ Σ𝑘 ∈ 𝐴 𝐶 = Σ𝑘 ∈ 𝐵 𝐶 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sumeq1i.1 | . 2 ⊢ 𝐴 = 𝐵 | |
2 | sumeq1 11131 | . 2 ⊢ (𝐴 = 𝐵 → Σ𝑘 ∈ 𝐴 𝐶 = Σ𝑘 ∈ 𝐵 𝐶) | |
3 | 1, 2 | ax-mp 5 | 1 ⊢ Σ𝑘 ∈ 𝐴 𝐶 = Σ𝑘 ∈ 𝐵 𝐶 |
Colors of variables: wff set class |
Syntax hints: = wceq 1331 Σcsu 11129 |
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-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 |
This theorem depends on definitions: df-bi 116 df-dc 820 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-rex 2422 df-v 2688 df-un 3075 df-in 3077 df-ss 3084 df-if 3475 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-br 3930 df-opab 3990 df-mpt 3991 df-cnv 4547 df-dm 4549 df-rn 4550 df-res 4551 df-iota 5088 df-f 5127 df-f1 5128 df-fo 5129 df-f1o 5130 df-fv 5131 df-ov 5777 df-oprab 5778 df-mpo 5779 df-recs 6202 df-frec 6288 df-seqfrec 10226 df-sumdc 11130 |
This theorem is referenced by: sumeq12i 11141 fsump1i 11209 fsum2d 11211 fsumxp 11212 isumnn0nn 11269 arisum 11274 arisum2 11275 geo2sum 11290 efsep 11404 ef4p 11407 dveflem 12865 |
Copyright terms: Public domain | W3C validator |