Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > reldmdsmm | Structured version Visualization version GIF version |
Description: The direct sum is a well-behaved binary operator. (Contributed by Stefan O'Rear, 7-Jan-2015.) |
Ref | Expression |
---|---|
reldmdsmm | ⊢ Rel dom ⊕m |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-dsmm 20878 | . 2 ⊢ ⊕m = (𝑠 ∈ V, 𝑟 ∈ V ↦ ((𝑠Xs𝑟) ↾s {𝑓 ∈ X𝑥 ∈ dom 𝑟(Base‘(𝑟‘𝑥)) ∣ {𝑥 ∈ dom 𝑟 ∣ (𝑓‘𝑥) ≠ (0g‘(𝑟‘𝑥))} ∈ Fin})) | |
2 | 1 | reldmmpo 7287 | 1 ⊢ Rel dom ⊕m |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2114 ≠ wne 3018 {crab 3144 Vcvv 3496 dom cdm 5557 Rel wrel 5562 ‘cfv 6357 (class class class)co 7158 Xcixp 8463 Fincfn 8511 Basecbs 16485 ↾s cress 16486 0gc0g 16715 Xscprds 16721 ⊕m cdsmm 20877 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pr 5332 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-rab 3149 df-v 3498 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-br 5069 df-opab 5131 df-xp 5563 df-rel 5564 df-dm 5567 df-oprab 7162 df-mpo 7163 df-dsmm 20878 |
This theorem is referenced by: dsmmval 20880 dsmmval2 20882 |
Copyright terms: Public domain | W3C validator |