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Mirrors > Home > MPE Home > Th. List > divsmulwd | Structured version Visualization version GIF version |
Description: Relationship between surreal division and multiplication. Weak version that does not assume reciprocals. (Contributed by Scott Fenton, 12-Mar-2025.) |
Ref | Expression |
---|---|
divsmulwd.1 | ⊢ (𝜑 → 𝐴 ∈ No ) |
divsmulwd.2 | ⊢ (𝜑 → 𝐵 ∈ No ) |
divsmulwd.3 | ⊢ (𝜑 → 𝐶 ∈ No ) |
divsmulwd.4 | ⊢ (𝜑 → 𝐶 ≠ 0s ) |
divsmulwd.5 | ⊢ (𝜑 → ∃𝑥 ∈ No (𝐶 ·s 𝑥) = 1s ) |
Ref | Expression |
---|---|
divsmulwd | ⊢ (𝜑 → ((𝐴 /su 𝐶) = 𝐵 ↔ (𝐶 ·s 𝐵) = 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | divsmulwd.1 | . 2 ⊢ (𝜑 → 𝐴 ∈ No ) | |
2 | divsmulwd.2 | . 2 ⊢ (𝜑 → 𝐵 ∈ No ) | |
3 | divsmulwd.3 | . . 3 ⊢ (𝜑 → 𝐶 ∈ No ) | |
4 | divsmulwd.4 | . . 3 ⊢ (𝜑 → 𝐶 ≠ 0s ) | |
5 | 3, 4 | jca 511 | . 2 ⊢ (𝜑 → (𝐶 ∈ No ∧ 𝐶 ≠ 0s )) |
6 | divsmulwd.5 | . 2 ⊢ (𝜑 → ∃𝑥 ∈ No (𝐶 ·s 𝑥) = 1s ) | |
7 | divsmulw 28086 | . 2 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ (𝐶 ∈ No ∧ 𝐶 ≠ 0s )) ∧ ∃𝑥 ∈ No (𝐶 ·s 𝑥) = 1s ) → ((𝐴 /su 𝐶) = 𝐵 ↔ (𝐶 ·s 𝐵) = 𝐴)) | |
8 | 1, 2, 5, 6, 7 | syl31anc 1371 | 1 ⊢ (𝜑 → ((𝐴 /su 𝐶) = 𝐵 ↔ (𝐶 ·s 𝐵) = 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1534 ∈ wcel 2099 ≠ wne 2936 ∃wrex 3066 (class class class)co 7415 No csur 27567 0s c0s 27749 1s c1s 27750 ·s cmuls 28000 /su cdivs 28081 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5280 ax-sep 5294 ax-nul 5301 ax-pow 5360 ax-pr 5424 ax-un 7735 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2937 df-ral 3058 df-rex 3067 df-rmo 3372 df-reu 3373 df-rab 3429 df-v 3472 df-sbc 3776 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-pss 3964 df-nul 4320 df-if 4526 df-pw 4601 df-sn 4626 df-pr 4628 df-tp 4630 df-op 4632 df-ot 4634 df-uni 4905 df-int 4946 df-iun 4994 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5571 df-eprel 5577 df-po 5585 df-so 5586 df-fr 5628 df-se 5629 df-we 5630 df-xp 5679 df-rel 5680 df-cnv 5681 df-co 5682 df-dm 5683 df-rn 5684 df-res 5685 df-ima 5686 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 7371 df-ov 7418 df-oprab 7419 df-mpo 7420 df-1st 7988 df-2nd 7989 df-frecs 8281 df-wrecs 8312 df-recs 8386 df-1o 8481 df-2o 8482 df-nadd 8681 df-no 27570 df-slt 27571 df-bday 27572 df-sle 27672 df-sslt 27708 df-scut 27710 df-0s 27751 df-1s 27752 df-made 27768 df-old 27769 df-left 27771 df-right 27772 df-norec 27849 df-norec2 27860 df-adds 27871 df-negs 27928 df-subs 27929 df-muls 28001 df-divs 28082 |
This theorem is referenced by: divscan2wd 28090 divsmuld 28114 |
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