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| Mirrors > Home > MPE Home > Th. List > sltsubsubbd | Structured version Visualization version GIF version | ||
| Description: Equivalence for the surreal less-than relationship between differences. (Contributed by Scott Fenton, 6-Feb-2025.) | 
| Ref | Expression | 
|---|---|
| sltsubsubbd.1 | ⊢ (𝜑 → 𝐴 ∈ No ) | 
| sltsubsubbd.2 | ⊢ (𝜑 → 𝐵 ∈ No ) | 
| sltsubsubbd.3 | ⊢ (𝜑 → 𝐶 ∈ No ) | 
| sltsubsubbd.4 | ⊢ (𝜑 → 𝐷 ∈ No ) | 
| Ref | Expression | 
|---|---|
| sltsubsubbd | ⊢ (𝜑 → ((𝐴 -s 𝐶) <s (𝐵 -s 𝐷) ↔ (𝐴 -s 𝐵) <s (𝐶 -s 𝐷))) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | sltsubsubbd.1 | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ No ) | |
| 2 | sltsubsubbd.3 | . . . . 5 ⊢ (𝜑 → 𝐶 ∈ No ) | |
| 3 | npcans 28106 | . . . . 5 ⊢ ((𝐴 ∈ No ∧ 𝐶 ∈ No ) → ((𝐴 -s 𝐶) +s 𝐶) = 𝐴) | |
| 4 | 1, 2, 3 | syl2anc 584 | . . . 4 ⊢ (𝜑 → ((𝐴 -s 𝐶) +s 𝐶) = 𝐴) | 
| 5 | sltsubsubbd.2 | . . . . 5 ⊢ (𝜑 → 𝐵 ∈ No ) | |
| 6 | npcans 28106 | . . . . 5 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → ((𝐴 -s 𝐵) +s 𝐵) = 𝐴) | |
| 7 | 1, 5, 6 | syl2anc 584 | . . . 4 ⊢ (𝜑 → ((𝐴 -s 𝐵) +s 𝐵) = 𝐴) | 
| 8 | 4, 7 | eqtr4d 2779 | . . 3 ⊢ (𝜑 → ((𝐴 -s 𝐶) +s 𝐶) = ((𝐴 -s 𝐵) +s 𝐵)) | 
| 9 | 5, 2 | addscomd 28001 | . . . . 5 ⊢ (𝜑 → (𝐵 +s 𝐶) = (𝐶 +s 𝐵)) | 
| 10 | 9 | oveq1d 7447 | . . . 4 ⊢ (𝜑 → ((𝐵 +s 𝐶) +s ( -us ‘𝐷)) = ((𝐶 +s 𝐵) +s ( -us ‘𝐷))) | 
| 11 | sltsubsubbd.4 | . . . . . . 7 ⊢ (𝜑 → 𝐷 ∈ No ) | |
| 12 | 5, 11 | subsvald 28092 | . . . . . 6 ⊢ (𝜑 → (𝐵 -s 𝐷) = (𝐵 +s ( -us ‘𝐷))) | 
| 13 | 12 | oveq1d 7447 | . . . . 5 ⊢ (𝜑 → ((𝐵 -s 𝐷) +s 𝐶) = ((𝐵 +s ( -us ‘𝐷)) +s 𝐶)) | 
| 14 | 11 | negscld 28070 | . . . . . 6 ⊢ (𝜑 → ( -us ‘𝐷) ∈ No ) | 
| 15 | 5, 14, 2 | adds32d 28041 | . . . . 5 ⊢ (𝜑 → ((𝐵 +s ( -us ‘𝐷)) +s 𝐶) = ((𝐵 +s 𝐶) +s ( -us ‘𝐷))) | 
| 16 | 13, 15 | eqtrd 2776 | . . . 4 ⊢ (𝜑 → ((𝐵 -s 𝐷) +s 𝐶) = ((𝐵 +s 𝐶) +s ( -us ‘𝐷))) | 
| 17 | 2, 11 | subsvald 28092 | . . . . . 6 ⊢ (𝜑 → (𝐶 -s 𝐷) = (𝐶 +s ( -us ‘𝐷))) | 
| 18 | 17 | oveq1d 7447 | . . . . 5 ⊢ (𝜑 → ((𝐶 -s 𝐷) +s 𝐵) = ((𝐶 +s ( -us ‘𝐷)) +s 𝐵)) | 
| 19 | 2, 14, 5 | adds32d 28041 | . . . . 5 ⊢ (𝜑 → ((𝐶 +s ( -us ‘𝐷)) +s 𝐵) = ((𝐶 +s 𝐵) +s ( -us ‘𝐷))) | 
| 20 | 18, 19 | eqtrd 2776 | . . . 4 ⊢ (𝜑 → ((𝐶 -s 𝐷) +s 𝐵) = ((𝐶 +s 𝐵) +s ( -us ‘𝐷))) | 
| 21 | 10, 16, 20 | 3eqtr4d 2786 | . . 3 ⊢ (𝜑 → ((𝐵 -s 𝐷) +s 𝐶) = ((𝐶 -s 𝐷) +s 𝐵)) | 
| 22 | 8, 21 | breq12d 5155 | . 2 ⊢ (𝜑 → (((𝐴 -s 𝐶) +s 𝐶) <s ((𝐵 -s 𝐷) +s 𝐶) ↔ ((𝐴 -s 𝐵) +s 𝐵) <s ((𝐶 -s 𝐷) +s 𝐵))) | 
| 23 | 1, 2 | subscld 28094 | . . 3 ⊢ (𝜑 → (𝐴 -s 𝐶) ∈ No ) | 
| 24 | 5, 11 | subscld 28094 | . . 3 ⊢ (𝜑 → (𝐵 -s 𝐷) ∈ No ) | 
| 25 | 23, 24, 2 | sltadd1d 28032 | . 2 ⊢ (𝜑 → ((𝐴 -s 𝐶) <s (𝐵 -s 𝐷) ↔ ((𝐴 -s 𝐶) +s 𝐶) <s ((𝐵 -s 𝐷) +s 𝐶))) | 
| 26 | 1, 5 | subscld 28094 | . . 3 ⊢ (𝜑 → (𝐴 -s 𝐵) ∈ No ) | 
| 27 | 2, 11 | subscld 28094 | . . 3 ⊢ (𝜑 → (𝐶 -s 𝐷) ∈ No ) | 
| 28 | 26, 27, 5 | sltadd1d 28032 | . 2 ⊢ (𝜑 → ((𝐴 -s 𝐵) <s (𝐶 -s 𝐷) ↔ ((𝐴 -s 𝐵) +s 𝐵) <s ((𝐶 -s 𝐷) +s 𝐵))) | 
| 29 | 22, 25, 28 | 3bitr4d 311 | 1 ⊢ (𝜑 → ((𝐴 -s 𝐶) <s (𝐵 -s 𝐷) ↔ (𝐴 -s 𝐵) <s (𝐶 -s 𝐷))) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 = wceq 1539 ∈ wcel 2107 class class class wbr 5142 ‘cfv 6560 (class class class)co 7432 No csur 27685 <s cslt 27686 +s cadds 27993 -us cnegs 28052 -s csubs 28053 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-rep 5278 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-ral 3061 df-rex 3070 df-rmo 3379 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-tp 4630 df-op 4632 df-ot 4634 df-uni 4907 df-int 4946 df-iun 4992 df-br 5143 df-opab 5205 df-mpt 5225 df-tr 5259 df-id 5577 df-eprel 5583 df-po 5591 df-so 5592 df-fr 5636 df-se 5637 df-we 5638 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-pred 6320 df-ord 6386 df-on 6387 df-suc 6389 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-riota 7389 df-ov 7435 df-oprab 7436 df-mpo 7437 df-1st 8015 df-2nd 8016 df-frecs 8307 df-wrecs 8338 df-recs 8412 df-1o 8507 df-2o 8508 df-nadd 8705 df-no 27688 df-slt 27689 df-bday 27690 df-sle 27791 df-sslt 27827 df-scut 27829 df-0s 27870 df-made 27887 df-old 27888 df-left 27890 df-right 27891 df-norec 27972 df-norec2 27983 df-adds 27994 df-negs 28054 df-subs 28055 | 
| This theorem is referenced by: sltsubsub3bd 28116 slesubsub3bd 28119 mulsproplem6 28148 mulsproplem7 28149 mulsproplem8 28150 | 
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