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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 28215 | . . 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 27885 | . . . . . 6 ⊢ 0s ∈ No | |
13 | slerflex 27822 | . . . . . 6 ⊢ ( 0s ∈ No → 0s ≤s 0s ) | |
14 | 12, 13 | mp1i 13 | . . . . 5 ⊢ (𝜑 → 0s ≤s 0s ) |
15 | muls01 28152 | . . . . . 6 ⊢ (𝐴 ∈ No → (𝐴 ·s 0s ) = 0s ) | |
16 | 3, 15 | syl 17 | . . . . 5 ⊢ (𝜑 → (𝐴 ·s 0s ) = 0s ) |
17 | muls01 28152 | . . . . . 6 ⊢ (𝐵 ∈ No → (𝐵 ·s 0s ) = 0s ) | |
18 | 5, 17 | syl 17 | . . . . 5 ⊢ (𝜑 → (𝐵 ·s 0s ) = 0s ) |
19 | 14, 16, 18 | 3brtr4d 5179 | . . . 4 ⊢ (𝜑 → (𝐴 ·s 0s ) ≤s (𝐵 ·s 0s )) |
20 | oveq2 7438 | . . . . 5 ⊢ ( 0s = 𝐶 → (𝐴 ·s 0s ) = (𝐴 ·s 𝐶)) | |
21 | oveq2 7438 | . . . . 5 ⊢ ( 0s = 𝐶 → (𝐵 ·s 0s ) = (𝐵 ·s 𝐶)) | |
22 | 20, 21 | breq12d 5160 | . . . 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 27813 | . . . 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 960 | 1 ⊢ (𝜑 → (𝐴 ·s 𝐶) ≤s (𝐵 ·s 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∨ wo 847 = wceq 1536 ∈ wcel 2105 class class class wbr 5147 (class class class)co 7430 No csur 27698 <s cslt 27699 ≤s csle 27803 0s c0s 27881 ·s cmuls 28146 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-rep 5284 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-ral 3059 df-rex 3068 df-rmo 3377 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-pss 3982 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-tp 4635 df-op 4637 df-ot 4639 df-uni 4912 df-int 4951 df-iun 4997 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5640 df-se 5641 df-we 5642 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-pred 6322 df-ord 6388 df-on 6389 df-suc 6391 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-riota 7387 df-ov 7433 df-oprab 7434 df-mpo 7435 df-1st 8012 df-2nd 8013 df-frecs 8304 df-wrecs 8335 df-recs 8409 df-1o 8504 df-2o 8505 df-nadd 8702 df-no 27701 df-slt 27702 df-bday 27703 df-sle 27804 df-sslt 27840 df-scut 27842 df-0s 27883 df-made 27900 df-old 27901 df-left 27903 df-right 27904 df-norec 27985 df-norec2 27996 df-adds 28007 df-negs 28067 df-subs 28068 df-muls 28147 |
This theorem is referenced by: sltmul12ad 28223 |
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