![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > subsfn | Structured version Visualization version GIF version |
Description: Surreal subtraction is a function over pairs of surreals. (Contributed by Scott Fenton, 22-Jan-2025.) |
Ref | Expression |
---|---|
subsfn | ⊢ -s Fn ( No × No ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-subs 27464 | . 2 ⊢ -s = (𝑥 ∈ No , 𝑦 ∈ No ↦ (𝑥 +s ( -us ‘𝑦))) | |
2 | ovex 7429 | . 2 ⊢ (𝑥 +s ( -us ‘𝑦)) ∈ V | |
3 | 1, 2 | fnmpoi 8043 | 1 ⊢ -s Fn ( No × No ) |
Colors of variables: wff setvar class |
Syntax hints: × cxp 5670 Fn wfn 6530 ‘cfv 6535 (class class class)co 7396 No csur 27110 +s cadds 27410 -us cnegs 27461 -s csubs 27462 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5295 ax-nul 5302 ax-pr 5423 ax-un 7712 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-nul 4321 df-if 4525 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4905 df-iun 4995 df-br 5145 df-opab 5207 df-mpt 5228 df-id 5570 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 6487 df-fun 6537 df-fn 6538 df-f 6539 df-fv 6543 df-ov 7399 df-oprab 7400 df-mpo 7401 df-1st 7962 df-2nd 7963 df-subs 27464 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |