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Mirrors > Home > MPE Home > Th. List > divsassd | Structured version Visualization version GIF version |
Description: An associative law for surreal division. (Contributed by Scott Fenton, 16-Mar-2025.) |
Ref | Expression |
---|---|
divsassd.1 | ⊢ (𝜑 → 𝐴 ∈ No ) |
divsassd.2 | ⊢ (𝜑 → 𝐵 ∈ No ) |
divsassd.3 | ⊢ (𝜑 → 𝐶 ∈ No ) |
divsassd.4 | ⊢ (𝜑 → 𝐶 ≠ 0s ) |
Ref | Expression |
---|---|
divsassd | ⊢ (𝜑 → ((𝐴 ·s 𝐵) /su 𝐶) = (𝐴 ·s (𝐵 /su 𝐶))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | divsassd.1 | . 2 ⊢ (𝜑 → 𝐴 ∈ No ) | |
2 | divsassd.2 | . 2 ⊢ (𝜑 → 𝐵 ∈ No ) | |
3 | divsassd.3 | . 2 ⊢ (𝜑 → 𝐶 ∈ No ) | |
4 | divsassd.4 | . 2 ⊢ (𝜑 → 𝐶 ≠ 0s ) | |
5 | 3, 4 | recsexd 28253 | . 2 ⊢ (𝜑 → ∃𝑥 ∈ No (𝐶 ·s 𝑥) = 1s ) |
6 | 1, 2, 3, 4, 5 | divsasswd 28237 | 1 ⊢ (𝜑 → ((𝐴 ·s 𝐵) /su 𝐶) = (𝐴 ·s (𝐵 /su 𝐶))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2103 ≠ wne 2942 (class class class)co 7445 No csur 27693 0s c0s 27876 ·s cmuls 28141 /su cdivs 28222 |
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 2105 ax-9 2113 ax-10 2136 ax-11 2153 ax-12 2173 ax-ext 2705 ax-rep 5306 ax-sep 5320 ax-nul 5327 ax-pow 5386 ax-pr 5450 ax-un 7766 ax-dc 10511 |
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 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2890 df-ne 2943 df-ral 3064 df-rex 3073 df-rmo 3383 df-reu 3384 df-rab 3439 df-v 3484 df-sbc 3799 df-csb 3916 df-dif 3973 df-un 3975 df-in 3977 df-ss 3987 df-pss 3990 df-nul 4348 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 4973 df-iun 5021 df-br 5170 df-opab 5232 df-mpt 5253 df-tr 5287 df-id 5597 df-eprel 5603 df-po 5611 df-so 5612 df-fr 5654 df-se 5655 df-we 5656 df-xp 5705 df-rel 5706 df-cnv 5707 df-co 5708 df-dm 5709 df-rn 5710 df-res 5711 df-ima 5712 df-pred 6331 df-ord 6397 df-on 6398 df-lim 6399 df-suc 6400 df-iota 6524 df-fun 6574 df-fn 6575 df-f 6576 df-f1 6577 df-fo 6578 df-f1o 6579 df-fv 6580 df-riota 7401 df-ov 7448 df-oprab 7449 df-mpo 7450 df-om 7900 df-1st 8026 df-2nd 8027 df-frecs 8318 df-wrecs 8349 df-recs 8423 df-rdg 8462 df-1o 8518 df-2o 8519 df-oadd 8522 df-nadd 8718 df-no 27696 df-slt 27697 df-bday 27698 df-sle 27799 df-sslt 27835 df-scut 27837 df-0s 27878 df-1s 27879 df-made 27895 df-old 27896 df-left 27898 df-right 27899 df-norec 27980 df-norec2 27991 df-adds 28002 df-negs 28062 df-subs 28063 df-muls 28142 df-divs 28223 |
This theorem is referenced by: zs12bday 28433 |
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