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Mirrors > Home > MPE Home > Th. List > addsdid | Structured version Visualization version GIF version |
Description: Distributive law for surreal numbers. Commuted form of part of theorem 7 of [Conway] p. 19. (Contributed by Scott Fenton, 9-Mar-2025.) |
Ref | Expression |
---|---|
addsdid.1 | โข (๐ โ ๐ด โ No ) |
addsdid.2 | โข (๐ โ ๐ต โ No ) |
addsdid.3 | โข (๐ โ ๐ถ โ No ) |
Ref | Expression |
---|---|
addsdid | โข (๐ โ (๐ด ยทs (๐ต +s ๐ถ)) = ((๐ด ยทs ๐ต) +s (๐ด ยทs ๐ถ))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | addsdid.1 | . 2 โข (๐ โ ๐ด โ No ) | |
2 | addsdid.2 | . 2 โข (๐ โ ๐ต โ No ) | |
3 | addsdid.3 | . 2 โข (๐ โ ๐ถ โ No ) | |
4 | addsdi 27969 | . 2 โข ((๐ด โ No โง ๐ต โ No โง ๐ถ โ No ) โ (๐ด ยทs (๐ต +s ๐ถ)) = ((๐ด ยทs ๐ต) +s (๐ด ยทs ๐ถ))) | |
5 | 1, 2, 3, 4 | syl3anc 1370 | 1 โข (๐ โ (๐ด ยทs (๐ต +s ๐ถ)) = ((๐ด ยทs ๐ต) +s (๐ด ยทs ๐ถ))) |
Colors of variables: wff setvar class |
Syntax hints: โ wi 4 = wceq 1540 โ wcel 2105 (class class class)co 7412 No csur 27487 +s cadds 27790 ยทs cmuls 27920 |
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 27490 df-slt 27491 df-bday 27492 df-sle 27592 df-sslt 27628 df-scut 27630 df-0s 27671 df-made 27688 df-old 27689 df-left 27691 df-right 27692 df-norec 27769 df-norec2 27780 df-adds 27791 df-negs 27848 df-subs 27849 df-muls 27921 |
This theorem is referenced by: addsdird 27971 subsdid 27972 mulsasslem3 27979 precsexlem9 28027 n0mulscl 28096 |
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