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Mirrors > Home > MPE Home > Th. List > 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 7411 | . 2 ⊢ (𝑥 = 𝐴 → (𝑥 +s ( -us ‘𝑦)) = (𝐴 +s ( -us ‘𝑦))) | |
2 | fveq2 6884 | . . 3 ⊢ (𝑦 = 𝐵 → ( -us ‘𝑦) = ( -us ‘𝐵)) | |
3 | 2 | oveq2d 7420 | . 2 ⊢ (𝑦 = 𝐵 → (𝐴 +s ( -us ‘𝑦)) = (𝐴 +s ( -us ‘𝐵))) |
4 | df-subs 27886 | . 2 ⊢ -s = (𝑥 ∈ No , 𝑦 ∈ No ↦ (𝑥 +s ( -us ‘𝑦))) | |
5 | ovex 7437 | . 2 ⊢ (𝐴 +s ( -us ‘𝐵)) ∈ V | |
6 | 1, 3, 4, 5 | ovmpo 7563 | 1 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (𝐴 -s 𝐵) = (𝐴 +s ( -us ‘𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1533 ∈ wcel 2098 ‘cfv 6536 (class class class)co 7404 No csur 27524 +s cadds 27827 -us cnegs 27883 -s csubs 27884 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pr 5420 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-rab 3427 df-v 3470 df-sbc 3773 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-br 5142 df-opab 5204 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-iota 6488 df-fun 6538 df-fv 6544 df-ov 7407 df-oprab 7408 df-mpo 7409 df-subs 27886 |
This theorem is referenced by: subsvald 27922 subscl 27923 negsval2 27925 subsid1 27927 subsid 27928 subadds 27929 sltsub1 27935 |
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