![]() |
Mathbox for Scott Fenton |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > subsval | Structured version Visualization version GIF version |
Description: The value of surreal subtraction. (Contributed by Scott Fenton, 3-Feb-2025.) |
Ref | Expression |
---|---|
subsval | ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (𝐴 -s 𝐵) = (𝐴 +s ( -us ‘𝐵))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oveq1 7358 | . 2 ⊢ (𝑥 = 𝐴 → (𝑥 +s ( -us ‘𝑦)) = (𝐴 +s ( -us ‘𝑦))) | |
2 | fveq2 6839 | . . 3 ⊢ (𝑦 = 𝐵 → ( -us ‘𝑦) = ( -us ‘𝐵)) | |
3 | 2 | oveq2d 7367 | . 2 ⊢ (𝑦 = 𝐵 → (𝐴 +s ( -us ‘𝑦)) = (𝐴 +s ( -us ‘𝐵))) |
4 | df-subs 34309 | . 2 ⊢ -s = (𝑥 ∈ No , 𝑦 ∈ No ↦ (𝑥 +s ( -us ‘𝑦))) | |
5 | ovex 7384 | . 2 ⊢ (𝐴 +s ( -us ‘𝐵)) ∈ V | |
6 | 1, 3, 4, 5 | ovmpo 7509 | 1 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (𝐴 -s 𝐵) = (𝐴 +s ( -us ‘𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ‘cfv 6493 (class class class)co 7351 No csur 26939 +s cadds 34267 -us cnegs 34306 -s csubs 34307 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2708 ax-sep 5254 ax-nul 5261 ax-pr 5382 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2887 df-ne 2942 df-ral 3063 df-rex 3072 df-rab 3406 df-v 3445 df-sbc 3738 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-nul 4281 df-if 4485 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4864 df-br 5104 df-opab 5166 df-id 5529 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-iota 6445 df-fun 6495 df-fv 6501 df-ov 7354 df-oprab 7355 df-mpo 7356 df-subs 34309 |
This theorem is referenced by: subscl 34337 subsid1 34338 subsid 34339 subadds 34340 sltsub1 34344 sltsub2 34345 |
Copyright terms: Public domain | W3C validator |