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Mirrors > Home > MPE Home > Th. List > divdivs1d | Structured version Visualization version GIF version |
Description: Surreal division into a fraction. (Contributed by Scott Fenton, 7-Aug-2025.) |
Ref | Expression |
---|---|
divdivs1d.1 | ⊢ (𝜑 → 𝐴 ∈ No ) |
divdivs1d.2 | ⊢ (𝜑 → 𝐵 ∈ No ) |
divdivs1d.3 | ⊢ (𝜑 → 𝐶 ∈ No ) |
divdivs1d.4 | ⊢ (𝜑 → 𝐵 ≠ 0s ) |
divdivs1d.5 | ⊢ (𝜑 → 𝐶 ≠ 0s ) |
Ref | Expression |
---|---|
divdivs1d | ⊢ (𝜑 → ((𝐴 /su 𝐵) /su 𝐶) = (𝐴 /su (𝐵 ·s 𝐶))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | divdivs1d.2 | . . . . . 6 ⊢ (𝜑 → 𝐵 ∈ No ) | |
2 | divdivs1d.3 | . . . . . 6 ⊢ (𝜑 → 𝐶 ∈ No ) | |
3 | divdivs1d.1 | . . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ No ) | |
4 | 1, 2 | mulscld 28176 | . . . . . . 7 ⊢ (𝜑 → (𝐵 ·s 𝐶) ∈ No ) |
5 | divdivs1d.4 | . . . . . . . 8 ⊢ (𝜑 → 𝐵 ≠ 0s ) | |
6 | divdivs1d.5 | . . . . . . . 8 ⊢ (𝜑 → 𝐶 ≠ 0s ) | |
7 | 1, 2 | mulsne0bd 28227 | . . . . . . . 8 ⊢ (𝜑 → ((𝐵 ·s 𝐶) ≠ 0s ↔ (𝐵 ≠ 0s ∧ 𝐶 ≠ 0s ))) |
8 | 5, 6, 7 | mpbir2and 713 | . . . . . . 7 ⊢ (𝜑 → (𝐵 ·s 𝐶) ≠ 0s ) |
9 | 3, 4, 8 | divscld 28263 | . . . . . 6 ⊢ (𝜑 → (𝐴 /su (𝐵 ·s 𝐶)) ∈ No ) |
10 | 1, 2, 9 | mulsassd 28208 | . . . . 5 ⊢ (𝜑 → ((𝐵 ·s 𝐶) ·s (𝐴 /su (𝐵 ·s 𝐶))) = (𝐵 ·s (𝐶 ·s (𝐴 /su (𝐵 ·s 𝐶))))) |
11 | 3, 4, 8 | divscan2d 28264 | . . . . 5 ⊢ (𝜑 → ((𝐵 ·s 𝐶) ·s (𝐴 /su (𝐵 ·s 𝐶))) = 𝐴) |
12 | 10, 11 | eqtr3d 2777 | . . . 4 ⊢ (𝜑 → (𝐵 ·s (𝐶 ·s (𝐴 /su (𝐵 ·s 𝐶)))) = 𝐴) |
13 | 2, 9 | mulscld 28176 | . . . . 5 ⊢ (𝜑 → (𝐶 ·s (𝐴 /su (𝐵 ·s 𝐶))) ∈ No ) |
14 | 3, 13, 1, 5 | divsmuld 28261 | . . . 4 ⊢ (𝜑 → ((𝐴 /su 𝐵) = (𝐶 ·s (𝐴 /su (𝐵 ·s 𝐶))) ↔ (𝐵 ·s (𝐶 ·s (𝐴 /su (𝐵 ·s 𝐶)))) = 𝐴)) |
15 | 12, 14 | mpbird 257 | . . 3 ⊢ (𝜑 → (𝐴 /su 𝐵) = (𝐶 ·s (𝐴 /su (𝐵 ·s 𝐶)))) |
16 | 15 | eqcomd 2741 | . 2 ⊢ (𝜑 → (𝐶 ·s (𝐴 /su (𝐵 ·s 𝐶))) = (𝐴 /su 𝐵)) |
17 | 3, 1, 5 | divscld 28263 | . . 3 ⊢ (𝜑 → (𝐴 /su 𝐵) ∈ No ) |
18 | 17, 9, 2, 6 | divsmuld 28261 | . 2 ⊢ (𝜑 → (((𝐴 /su 𝐵) /su 𝐶) = (𝐴 /su (𝐵 ·s 𝐶)) ↔ (𝐶 ·s (𝐴 /su (𝐵 ·s 𝐶))) = (𝐴 /su 𝐵))) |
19 | 16, 18 | mpbird 257 | 1 ⊢ (𝜑 → ((𝐴 /su 𝐵) /su 𝐶) = (𝐴 /su (𝐵 ·s 𝐶))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2106 ≠ wne 2938 (class class class)co 7431 No csur 27699 0s c0s 27882 ·s cmuls 28147 /su cdivs 28228 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-dc 10484 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-tp 4636 df-op 4638 df-ot 4640 df-uni 4913 df-int 4952 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-se 5642 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8013 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-2o 8506 df-oadd 8509 df-nadd 8703 df-no 27702 df-slt 27703 df-bday 27704 df-sle 27805 df-sslt 27841 df-scut 27843 df-0s 27884 df-1s 27885 df-made 27901 df-old 27902 df-left 27904 df-right 27905 df-norec 27986 df-norec2 27997 df-adds 28008 df-negs 28068 df-subs 28069 df-muls 28148 df-divs 28229 |
This theorem is referenced by: cutpw2 28432 pw2cut 28435 zs12bday 28439 |
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