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Mirrors > Home > MPE Home > Th. List > mulscl | Structured version Visualization version GIF version |
Description: The surreals are closed under multiplication. Theorem 8(i) of [Conway] p. 19. (Contributed by Scott Fenton, 5-Mar-2025.) |
Ref | Expression |
---|---|
mulscl | ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (𝐴 ·s 𝐵) ∈ No ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0sno 27672 | . . . 4 ⊢ 0s ∈ No | |
2 | 1, 1 | pm3.2i 470 | . . 3 ⊢ ( 0s ∈ No ∧ 0s ∈ No ) |
3 | mulsprop 27943 | . . 3 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ ( 0s ∈ No ∧ 0s ∈ No ) ∧ ( 0s ∈ No ∧ 0s ∈ No )) → ((𝐴 ·s 𝐵) ∈ No ∧ (( 0s <s 0s ∧ 0s <s 0s ) → (( 0s ·s 0s ) -s ( 0s ·s 0s )) <s (( 0s ·s 0s ) -s ( 0s ·s 0s ))))) | |
4 | 2, 2, 3 | mp3an23 1452 | . 2 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → ((𝐴 ·s 𝐵) ∈ No ∧ (( 0s <s 0s ∧ 0s <s 0s ) → (( 0s ·s 0s ) -s ( 0s ·s 0s )) <s (( 0s ·s 0s ) -s ( 0s ·s 0s ))))) |
5 | 4 | simpld 494 | 1 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (𝐴 ·s 𝐵) ∈ No ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∈ wcel 2105 class class class wbr 5148 (class class class)co 7412 No csur 27486 <s cslt 27487 0s c0s 27668 -s csubs 27846 ·s cmuls 27919 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-tp 4633 df-op 4635 df-ot 4637 df-uni 4909 df-int 4951 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-se 5632 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-1st 7979 df-2nd 7980 df-frecs 8272 df-wrecs 8303 df-recs 8377 df-1o 8472 df-2o 8473 df-nadd 8671 df-no 27489 df-slt 27490 df-bday 27491 df-sle 27591 df-sslt 27627 df-scut 27629 df-0s 27670 df-made 27687 df-old 27688 df-left 27690 df-right 27691 df-norec 27768 df-norec2 27779 df-adds 27790 df-negs 27847 df-subs 27848 df-muls 27920 |
This theorem is referenced by: mulscld 27948 mulsgt0 27957 absmuls 28051 remulscllem2 28109 remulscl 28110 |
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