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Mirrors > Home > MPE Home > Th. List > divsdird | Structured version Visualization version GIF version |
Description: Distribution of surreal division over addition. (Contributed by Scott Fenton, 13-Aug-2025.) |
Ref | Expression |
---|---|
divsdird.1 | ⊢ (𝜑 → 𝐴 ∈ No ) |
divsdird.2 | ⊢ (𝜑 → 𝐵 ∈ No ) |
divsdird.3 | ⊢ (𝜑 → 𝐶 ∈ No ) |
divsdird.4 | ⊢ (𝜑 → 𝐶 ≠ 0s ) |
Ref | Expression |
---|---|
divsdird | ⊢ (𝜑 → ((𝐴 +s 𝐵) /su 𝐶) = ((𝐴 /su 𝐶) +s (𝐵 /su 𝐶))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | divsdird.1 | . . 3 ⊢ (𝜑 → 𝐴 ∈ No ) | |
2 | divsdird.2 | . . 3 ⊢ (𝜑 → 𝐵 ∈ No ) | |
3 | 1sno 27892 | . . . . 5 ⊢ 1s ∈ No | |
4 | 3 | a1i 11 | . . . 4 ⊢ (𝜑 → 1s ∈ No ) |
5 | divsdird.3 | . . . 4 ⊢ (𝜑 → 𝐶 ∈ No ) | |
6 | divsdird.4 | . . . 4 ⊢ (𝜑 → 𝐶 ≠ 0s ) | |
7 | 4, 5, 6 | divscld 28268 | . . 3 ⊢ (𝜑 → ( 1s /su 𝐶) ∈ No ) |
8 | 1, 2, 7 | addsdird 28203 | . 2 ⊢ (𝜑 → ((𝐴 +s 𝐵) ·s ( 1s /su 𝐶)) = ((𝐴 ·s ( 1s /su 𝐶)) +s (𝐵 ·s ( 1s /su 𝐶)))) |
9 | 1, 2 | addscld 28033 | . . 3 ⊢ (𝜑 → (𝐴 +s 𝐵) ∈ No ) |
10 | 9, 5, 6 | divsrecd 28278 | . 2 ⊢ (𝜑 → ((𝐴 +s 𝐵) /su 𝐶) = ((𝐴 +s 𝐵) ·s ( 1s /su 𝐶))) |
11 | 1, 5, 6 | divsrecd 28278 | . . 3 ⊢ (𝜑 → (𝐴 /su 𝐶) = (𝐴 ·s ( 1s /su 𝐶))) |
12 | 2, 5, 6 | divsrecd 28278 | . . 3 ⊢ (𝜑 → (𝐵 /su 𝐶) = (𝐵 ·s ( 1s /su 𝐶))) |
13 | 11, 12 | oveq12d 7468 | . 2 ⊢ (𝜑 → ((𝐴 /su 𝐶) +s (𝐵 /su 𝐶)) = ((𝐴 ·s ( 1s /su 𝐶)) +s (𝐵 ·s ( 1s /su 𝐶)))) |
14 | 8, 10, 13 | 3eqtr4d 2790 | 1 ⊢ (𝜑 → ((𝐴 +s 𝐵) /su 𝐶) = ((𝐴 /su 𝐶) +s (𝐵 /su 𝐶))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2108 ≠ wne 2946 (class class class)co 7450 No csur 27704 0s c0s 27887 1s c1s 27888 +s cadds 28012 ·s cmuls 28152 /su cdivs 28233 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-rep 5303 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7772 ax-dc 10517 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-ral 3068 df-rex 3077 df-rmo 3388 df-reu 3389 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-pss 3996 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-tp 4653 df-op 4655 df-ot 4657 df-uni 4932 df-int 4971 df-iun 5017 df-br 5167 df-opab 5229 df-mpt 5250 df-tr 5284 df-id 5593 df-eprel 5599 df-po 5607 df-so 5608 df-fr 5652 df-se 5653 df-we 5654 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-pred 6334 df-ord 6400 df-on 6401 df-lim 6402 df-suc 6403 df-iota 6527 df-fun 6577 df-fn 6578 df-f 6579 df-f1 6580 df-fo 6581 df-f1o 6582 df-fv 6583 df-riota 7406 df-ov 7453 df-oprab 7454 df-mpo 7455 df-om 7906 df-1st 8032 df-2nd 8033 df-frecs 8324 df-wrecs 8355 df-recs 8429 df-rdg 8468 df-1o 8524 df-2o 8525 df-oadd 8528 df-nadd 8724 df-no 27707 df-slt 27708 df-bday 27709 df-sle 27810 df-sslt 27846 df-scut 27848 df-0s 27889 df-1s 27890 df-made 27906 df-old 27907 df-left 27909 df-right 27910 df-norec 27991 df-norec2 28002 df-adds 28013 df-negs 28073 df-subs 28074 df-muls 28153 df-divs 28234 |
This theorem is referenced by: addhalfcut 28439 pw2cut 28440 |
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