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Mirrors > Home > MPE Home > Th. List > subsdird | Structured version Visualization version GIF version |
Description: Distribution of surreal multiplication over subtraction. (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 |
---|---|
subsdird | โข (๐ โ ((๐ด -s ๐ต) ยทs ๐ถ) = ((๐ด ยทs ๐ถ) -s (๐ต ยทs ๐ถ))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | addsdid.3 | . . 3 โข (๐ โ ๐ถ โ No ) | |
2 | addsdid.1 | . . 3 โข (๐ โ ๐ด โ No ) | |
3 | addsdid.2 | . . 3 โข (๐ โ ๐ต โ No ) | |
4 | 1, 2, 3 | subsdid 27580 | . 2 โข (๐ โ (๐ถ ยทs (๐ด -s ๐ต)) = ((๐ถ ยทs ๐ด) -s (๐ถ ยทs ๐ต))) |
5 | 2, 3 | subscld 27502 | . . 3 โข (๐ โ (๐ด -s ๐ต) โ No ) |
6 | 5, 1 | mulscomd 27563 | . 2 โข (๐ โ ((๐ด -s ๐ต) ยทs ๐ถ) = (๐ถ ยทs (๐ด -s ๐ต))) |
7 | 2, 1 | mulscomd 27563 | . . 3 โข (๐ โ (๐ด ยทs ๐ถ) = (๐ถ ยทs ๐ด)) |
8 | 3, 1 | mulscomd 27563 | . . 3 โข (๐ โ (๐ต ยทs ๐ถ) = (๐ถ ยทs ๐ต)) |
9 | 7, 8 | oveq12d 7414 | . 2 โข (๐ โ ((๐ด ยทs ๐ถ) -s (๐ต ยทs ๐ถ)) = ((๐ถ ยทs ๐ด) -s (๐ถ ยทs ๐ต))) |
10 | 4, 6, 9 | 3eqtr4d 2783 | 1 โข (๐ โ ((๐ด -s ๐ต) ยทs ๐ถ) = ((๐ด ยทs ๐ถ) -s (๐ต ยทs ๐ถ))) |
Colors of variables: wff setvar class |
Syntax hints: โ wi 4 = wceq 1542 โ wcel 2107 (class class class)co 7396 No csur 27110 -s csubs 27462 ยทs cmuls 27529 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5281 ax-sep 5295 ax-nul 5302 ax-pow 5359 ax-pr 5423 ax-un 7712 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-pss 3965 df-nul 4321 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-tp 4629 df-op 4631 df-ot 4633 df-uni 4905 df-int 4947 df-iun 4995 df-br 5145 df-opab 5207 df-mpt 5228 df-tr 5262 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-se 5628 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6292 df-ord 6359 df-on 6360 df-suc 6362 df-iota 6487 df-fun 6537 df-fn 6538 df-f 6539 df-f1 6540 df-fo 6541 df-f1o 6542 df-fv 6543 df-riota 7352 df-ov 7399 df-oprab 7400 df-mpo 7401 df-1st 7962 df-2nd 7963 df-frecs 8253 df-wrecs 8284 df-recs 8358 df-1o 8453 df-2o 8454 df-nadd 8653 df-no 27113 df-slt 27114 df-bday 27115 df-sle 27215 df-sslt 27250 df-scut 27252 df-0s 27292 df-made 27309 df-old 27310 df-left 27312 df-right 27313 df-norec 27389 df-norec2 27400 df-adds 27411 df-negs 27463 df-subs 27464 df-muls 27530 |
This theorem is referenced by: mulsasslem3 27587 precsexlem11 27630 |
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