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| Mirrors > Home > MPE Home > Th. List > negsdi | Structured version Visualization version GIF version | ||
| Description: Distribution of surreal negative over addition. (Contributed by Scott Fenton, 5-Feb-2025.) |
| Ref | Expression |
|---|---|
| negsdi | ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → ( -us ‘(𝐴 +s 𝐵)) = (( -us ‘𝐴) +s ( -us ‘𝐵))) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | addscl 28014 | . . . 4 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (𝐴 +s 𝐵) ∈ No ) | |
| 2 | 1 | negsidd 28074 | . . 3 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → ((𝐴 +s 𝐵) +s ( -us ‘(𝐴 +s 𝐵))) = 0s ) |
| 3 | negsid 28073 | . . . . 5 ⊢ (𝐴 ∈ No → (𝐴 +s ( -us ‘𝐴)) = 0s ) | |
| 4 | negsid 28073 | . . . . 5 ⊢ (𝐵 ∈ No → (𝐵 +s ( -us ‘𝐵)) = 0s ) | |
| 5 | 3, 4 | oveqan12d 7450 | . . . 4 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → ((𝐴 +s ( -us ‘𝐴)) +s (𝐵 +s ( -us ‘𝐵))) = ( 0s +s 0s )) |
| 6 | 0sno 27871 | . . . . 5 ⊢ 0s ∈ No | |
| 7 | addslid 28001 | . . . . 5 ⊢ ( 0s ∈ No → ( 0s +s 0s ) = 0s ) | |
| 8 | 6, 7 | ax-mp 5 | . . . 4 ⊢ ( 0s +s 0s ) = 0s |
| 9 | 5, 8 | eqtr2di 2794 | . . 3 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → 0s = ((𝐴 +s ( -us ‘𝐴)) +s (𝐵 +s ( -us ‘𝐵)))) |
| 10 | simpl 482 | . . . 4 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → 𝐴 ∈ No ) | |
| 11 | 10 | negscld 28069 | . . . 4 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → ( -us ‘𝐴) ∈ No ) |
| 12 | simpr 484 | . . . 4 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → 𝐵 ∈ No ) | |
| 13 | 12 | negscld 28069 | . . . 4 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → ( -us ‘𝐵) ∈ No ) |
| 14 | 10, 11, 12, 13 | adds4d 28042 | . . 3 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → ((𝐴 +s ( -us ‘𝐴)) +s (𝐵 +s ( -us ‘𝐵))) = ((𝐴 +s 𝐵) +s (( -us ‘𝐴) +s ( -us ‘𝐵)))) |
| 15 | 2, 9, 14 | 3eqtrd 2781 | . 2 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → ((𝐴 +s 𝐵) +s ( -us ‘(𝐴 +s 𝐵))) = ((𝐴 +s 𝐵) +s (( -us ‘𝐴) +s ( -us ‘𝐵)))) |
| 16 | 1 | negscld 28069 | . . 3 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → ( -us ‘(𝐴 +s 𝐵)) ∈ No ) |
| 17 | negscl 28068 | . . . 4 ⊢ (𝐴 ∈ No → ( -us ‘𝐴) ∈ No ) | |
| 18 | negscl 28068 | . . . 4 ⊢ (𝐵 ∈ No → ( -us ‘𝐵) ∈ No ) | |
| 19 | addscl 28014 | . . . 4 ⊢ ((( -us ‘𝐴) ∈ No ∧ ( -us ‘𝐵) ∈ No ) → (( -us ‘𝐴) +s ( -us ‘𝐵)) ∈ No ) | |
| 20 | 17, 18, 19 | syl2an 596 | . . 3 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (( -us ‘𝐴) +s ( -us ‘𝐵)) ∈ No ) |
| 21 | 16, 20, 1 | addscan1d 28033 | . 2 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (((𝐴 +s 𝐵) +s ( -us ‘(𝐴 +s 𝐵))) = ((𝐴 +s 𝐵) +s (( -us ‘𝐴) +s ( -us ‘𝐵))) ↔ ( -us ‘(𝐴 +s 𝐵)) = (( -us ‘𝐴) +s ( -us ‘𝐵)))) |
| 22 | 15, 21 | mpbid 232 | 1 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → ( -us ‘(𝐴 +s 𝐵)) = (( -us ‘𝐴) +s ( -us ‘𝐵))) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ‘cfv 6561 (class class class)co 7431 No csur 27684 0s c0s 27867 +s cadds 27992 -us cnegs 28051 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-tp 4631 df-op 4633 df-ot 4635 df-uni 4908 df-int 4947 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-se 5638 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-1o 8506 df-2o 8507 df-nadd 8704 df-no 27687 df-slt 27688 df-bday 27689 df-sle 27790 df-sslt 27826 df-scut 27828 df-0s 27869 df-made 27886 df-old 27887 df-left 27889 df-right 27890 df-norec 27971 df-norec2 27982 df-adds 27993 df-negs 28053 |
| This theorem is referenced by: negsubsdi2d 28110 subsubs4d 28124 zscut 28393 renegscl 28430 readdscl 28431 |
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