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| Mirrors > Home > MPE Home > Th. List > subadds | Structured version Visualization version GIF version | ||
| Description: Relationship between addition and subtraction for surreals. (Contributed by Scott Fenton, 3-Feb-2025.) |
| Ref | Expression |
|---|---|
| subadds | ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) → ((𝐴 -s 𝐵) = 𝐶 ↔ (𝐵 +s 𝐶) = 𝐴)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | subsval 28090 | . . . 4 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (𝐴 -s 𝐵) = (𝐴 +s ( -us ‘𝐵))) | |
| 2 | 1 | 3adant3 1133 | . . 3 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) → (𝐴 -s 𝐵) = (𝐴 +s ( -us ‘𝐵))) |
| 3 | 2 | eqeq1d 2739 | . 2 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) → ((𝐴 -s 𝐵) = 𝐶 ↔ (𝐴 +s ( -us ‘𝐵)) = 𝐶)) |
| 4 | simpl 482 | . . . . . . 7 ⊢ ((𝐵 ∈ No ∧ 𝐶 ∈ No ) → 𝐵 ∈ No ) | |
| 5 | simpr 484 | . . . . . . 7 ⊢ ((𝐵 ∈ No ∧ 𝐶 ∈ No ) → 𝐶 ∈ No ) | |
| 6 | negscl 28068 | . . . . . . . 8 ⊢ (𝐵 ∈ No → ( -us ‘𝐵) ∈ No ) | |
| 7 | 6 | adantr 480 | . . . . . . 7 ⊢ ((𝐵 ∈ No ∧ 𝐶 ∈ No ) → ( -us ‘𝐵) ∈ No ) |
| 8 | 4, 5, 7 | adds32d 28040 | . . . . . 6 ⊢ ((𝐵 ∈ No ∧ 𝐶 ∈ No ) → ((𝐵 +s 𝐶) +s ( -us ‘𝐵)) = ((𝐵 +s ( -us ‘𝐵)) +s 𝐶)) |
| 9 | negsid 28073 | . . . . . . . 8 ⊢ (𝐵 ∈ No → (𝐵 +s ( -us ‘𝐵)) = 0s ) | |
| 10 | 9 | adantr 480 | . . . . . . 7 ⊢ ((𝐵 ∈ No ∧ 𝐶 ∈ No ) → (𝐵 +s ( -us ‘𝐵)) = 0s ) |
| 11 | 10 | oveq1d 7446 | . . . . . 6 ⊢ ((𝐵 ∈ No ∧ 𝐶 ∈ No ) → ((𝐵 +s ( -us ‘𝐵)) +s 𝐶) = ( 0s +s 𝐶)) |
| 12 | addslid 28001 | . . . . . . 7 ⊢ (𝐶 ∈ No → ( 0s +s 𝐶) = 𝐶) | |
| 13 | 12 | adantl 481 | . . . . . 6 ⊢ ((𝐵 ∈ No ∧ 𝐶 ∈ No ) → ( 0s +s 𝐶) = 𝐶) |
| 14 | 8, 11, 13 | 3eqtrd 2781 | . . . . 5 ⊢ ((𝐵 ∈ No ∧ 𝐶 ∈ No ) → ((𝐵 +s 𝐶) +s ( -us ‘𝐵)) = 𝐶) |
| 15 | 14 | 3adant1 1131 | . . . 4 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) → ((𝐵 +s 𝐶) +s ( -us ‘𝐵)) = 𝐶) |
| 16 | 15 | eqeq1d 2739 | . . 3 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) → (((𝐵 +s 𝐶) +s ( -us ‘𝐵)) = (𝐴 +s ( -us ‘𝐵)) ↔ 𝐶 = (𝐴 +s ( -us ‘𝐵)))) |
| 17 | eqcom 2744 | . . 3 ⊢ (𝐶 = (𝐴 +s ( -us ‘𝐵)) ↔ (𝐴 +s ( -us ‘𝐵)) = 𝐶) | |
| 18 | 16, 17 | bitrdi 287 | . 2 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) → (((𝐵 +s 𝐶) +s ( -us ‘𝐵)) = (𝐴 +s ( -us ‘𝐵)) ↔ (𝐴 +s ( -us ‘𝐵)) = 𝐶)) |
| 19 | addscl 28014 | . . . 4 ⊢ ((𝐵 ∈ No ∧ 𝐶 ∈ No ) → (𝐵 +s 𝐶) ∈ No ) | |
| 20 | 19 | 3adant1 1131 | . . 3 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) → (𝐵 +s 𝐶) ∈ No ) |
| 21 | simp1 1137 | . . 3 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) → 𝐴 ∈ No ) | |
| 22 | simp2 1138 | . . . 4 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) → 𝐵 ∈ No ) | |
| 23 | 22 | negscld 28069 | . . 3 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) → ( -us ‘𝐵) ∈ No ) |
| 24 | 20, 21, 23 | addscan2d 28032 | . 2 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) → (((𝐵 +s 𝐶) +s ( -us ‘𝐵)) = (𝐴 +s ( -us ‘𝐵)) ↔ (𝐵 +s 𝐶) = 𝐴)) |
| 25 | 3, 18, 24 | 3bitr2d 307 | 1 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) → ((𝐴 -s 𝐵) = 𝐶 ↔ (𝐵 +s 𝐶) = 𝐴)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 ‘cfv 6561 (class class class)co 7431 No csur 27684 0s c0s 27867 +s cadds 27992 -us cnegs 28051 -s csubs 28052 |
| 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 df-subs 28054 |
| This theorem is referenced by: subaddsd 28101 zseo 28406 |
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