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| Mirrors > Home > MPE Home > Th. List > slemul1ad | Structured version Visualization version GIF version | ||
| Description: Multiplication of both sides of surreal less-than or equal by a non-negative number. (Contributed by Scott Fenton, 17-Apr-2025.) | 
| Ref | Expression | 
|---|---|
| slemul1ad.1 | ⊢ (𝜑 → 𝐴 ∈ No ) | 
| slemul1ad.2 | ⊢ (𝜑 → 𝐵 ∈ No ) | 
| slemul1ad.3 | ⊢ (𝜑 → 𝐶 ∈ No ) | 
| slemul1ad.4 | ⊢ (𝜑 → 0s ≤s 𝐶) | 
| slemul1ad.5 | ⊢ (𝜑 → 𝐴 ≤s 𝐵) | 
| Ref | Expression | 
|---|---|
| slemul1ad | ⊢ (𝜑 → (𝐴 ·s 𝐶) ≤s (𝐵 ·s 𝐶)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | slemul1ad.5 | . . . 4 ⊢ (𝜑 → 𝐴 ≤s 𝐵) | |
| 2 | 1 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ 0s <s 𝐶) → 𝐴 ≤s 𝐵) | 
| 3 | slemul1ad.1 | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ No ) | |
| 4 | 3 | adantr 480 | . . . 4 ⊢ ((𝜑 ∧ 0s <s 𝐶) → 𝐴 ∈ No ) | 
| 5 | slemul1ad.2 | . . . . 5 ⊢ (𝜑 → 𝐵 ∈ No ) | |
| 6 | 5 | adantr 480 | . . . 4 ⊢ ((𝜑 ∧ 0s <s 𝐶) → 𝐵 ∈ No ) | 
| 7 | slemul1ad.3 | . . . . 5 ⊢ (𝜑 → 𝐶 ∈ No ) | |
| 8 | 7 | adantr 480 | . . . 4 ⊢ ((𝜑 ∧ 0s <s 𝐶) → 𝐶 ∈ No ) | 
| 9 | simpr 484 | . . . 4 ⊢ ((𝜑 ∧ 0s <s 𝐶) → 0s <s 𝐶) | |
| 10 | 4, 6, 8, 9 | slemul1d 28201 | . . 3 ⊢ ((𝜑 ∧ 0s <s 𝐶) → (𝐴 ≤s 𝐵 ↔ (𝐴 ·s 𝐶) ≤s (𝐵 ·s 𝐶))) | 
| 11 | 2, 10 | mpbid 232 | . 2 ⊢ ((𝜑 ∧ 0s <s 𝐶) → (𝐴 ·s 𝐶) ≤s (𝐵 ·s 𝐶)) | 
| 12 | 0sno 27871 | . . . . . 6 ⊢ 0s ∈ No | |
| 13 | slerflex 27808 | . . . . . 6 ⊢ ( 0s ∈ No → 0s ≤s 0s ) | |
| 14 | 12, 13 | mp1i 13 | . . . . 5 ⊢ (𝜑 → 0s ≤s 0s ) | 
| 15 | muls01 28138 | . . . . . 6 ⊢ (𝐴 ∈ No → (𝐴 ·s 0s ) = 0s ) | |
| 16 | 3, 15 | syl 17 | . . . . 5 ⊢ (𝜑 → (𝐴 ·s 0s ) = 0s ) | 
| 17 | muls01 28138 | . . . . . 6 ⊢ (𝐵 ∈ No → (𝐵 ·s 0s ) = 0s ) | |
| 18 | 5, 17 | syl 17 | . . . . 5 ⊢ (𝜑 → (𝐵 ·s 0s ) = 0s ) | 
| 19 | 14, 16, 18 | 3brtr4d 5175 | . . . 4 ⊢ (𝜑 → (𝐴 ·s 0s ) ≤s (𝐵 ·s 0s )) | 
| 20 | oveq2 7439 | . . . . 5 ⊢ ( 0s = 𝐶 → (𝐴 ·s 0s ) = (𝐴 ·s 𝐶)) | |
| 21 | oveq2 7439 | . . . . 5 ⊢ ( 0s = 𝐶 → (𝐵 ·s 0s ) = (𝐵 ·s 𝐶)) | |
| 22 | 20, 21 | breq12d 5156 | . . . 4 ⊢ ( 0s = 𝐶 → ((𝐴 ·s 0s ) ≤s (𝐵 ·s 0s ) ↔ (𝐴 ·s 𝐶) ≤s (𝐵 ·s 𝐶))) | 
| 23 | 19, 22 | syl5ibcom 245 | . . 3 ⊢ (𝜑 → ( 0s = 𝐶 → (𝐴 ·s 𝐶) ≤s (𝐵 ·s 𝐶))) | 
| 24 | 23 | imp 406 | . 2 ⊢ ((𝜑 ∧ 0s = 𝐶) → (𝐴 ·s 𝐶) ≤s (𝐵 ·s 𝐶)) | 
| 25 | slemul1ad.4 | . . 3 ⊢ (𝜑 → 0s ≤s 𝐶) | |
| 26 | sleloe 27799 | . . . 4 ⊢ (( 0s ∈ No ∧ 𝐶 ∈ No ) → ( 0s ≤s 𝐶 ↔ ( 0s <s 𝐶 ∨ 0s = 𝐶))) | |
| 27 | 12, 7, 26 | sylancr 587 | . . 3 ⊢ (𝜑 → ( 0s ≤s 𝐶 ↔ ( 0s <s 𝐶 ∨ 0s = 𝐶))) | 
| 28 | 25, 27 | mpbid 232 | . 2 ⊢ (𝜑 → ( 0s <s 𝐶 ∨ 0s = 𝐶)) | 
| 29 | 11, 24, 28 | mpjaodan 961 | 1 ⊢ (𝜑 → (𝐴 ·s 𝐶) ≤s (𝐵 ·s 𝐶)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∨ wo 848 = wceq 1540 ∈ wcel 2108 class class class wbr 5143 (class class class)co 7431 No csur 27684 <s cslt 27685 ≤s csle 27789 0s c0s 27867 ·s cmuls 28132 | 
| 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 df-muls 28133 | 
| This theorem is referenced by: sltmul12ad 28209 | 
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