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Mirrors > Home > MPE Home > Th. List > mulsgt0 | Structured version Visualization version GIF version |
Description: The product of two positive surreals is positive. Theorem 9 of [Conway] p. 20. (Contributed by Scott Fenton, 6-Mar-2025.) |
Ref | Expression |
---|---|
mulsgt0 | ⊢ (((𝐴 ∈ No ∧ 0s <s 𝐴) ∧ (𝐵 ∈ No ∧ 0s <s 𝐵)) → 0s <s (𝐴 ·s 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0sno 27256 | . . . 4 ⊢ 0s ∈ No | |
2 | 1 | a1i 11 | . . 3 ⊢ (((𝐴 ∈ No ∧ 0s <s 𝐴) ∧ (𝐵 ∈ No ∧ 0s <s 𝐵)) → 0s ∈ No ) |
3 | simpll 765 | . . 3 ⊢ (((𝐴 ∈ No ∧ 0s <s 𝐴) ∧ (𝐵 ∈ No ∧ 0s <s 𝐵)) → 𝐴 ∈ No ) | |
4 | simprl 769 | . . 3 ⊢ (((𝐴 ∈ No ∧ 0s <s 𝐴) ∧ (𝐵 ∈ No ∧ 0s <s 𝐵)) → 𝐵 ∈ No ) | |
5 | simplr 767 | . . 3 ⊢ (((𝐴 ∈ No ∧ 0s <s 𝐴) ∧ (𝐵 ∈ No ∧ 0s <s 𝐵)) → 0s <s 𝐴) | |
6 | simprr 771 | . . 3 ⊢ (((𝐴 ∈ No ∧ 0s <s 𝐴) ∧ (𝐵 ∈ No ∧ 0s <s 𝐵)) → 0s <s 𝐵) | |
7 | 2, 3, 2, 4, 5, 6 | sltmuld 27522 | . 2 ⊢ (((𝐴 ∈ No ∧ 0s <s 𝐴) ∧ (𝐵 ∈ No ∧ 0s <s 𝐵)) → (( 0s ·s 𝐵) -s ( 0s ·s 0s )) <s ((𝐴 ·s 𝐵) -s (𝐴 ·s 0s ))) |
8 | muls02 27526 | . . . . 5 ⊢ (𝐵 ∈ No → ( 0s ·s 𝐵) = 0s ) | |
9 | 4, 8 | syl 17 | . . . 4 ⊢ (((𝐴 ∈ No ∧ 0s <s 𝐴) ∧ (𝐵 ∈ No ∧ 0s <s 𝐵)) → ( 0s ·s 𝐵) = 0s ) |
10 | muls02 27526 | . . . . . 6 ⊢ ( 0s ∈ No → ( 0s ·s 0s ) = 0s ) | |
11 | 1, 10 | ax-mp 5 | . . . . 5 ⊢ ( 0s ·s 0s ) = 0s |
12 | 11 | a1i 11 | . . . 4 ⊢ (((𝐴 ∈ No ∧ 0s <s 𝐴) ∧ (𝐵 ∈ No ∧ 0s <s 𝐵)) → ( 0s ·s 0s ) = 0s ) |
13 | 9, 12 | oveq12d 7412 | . . 3 ⊢ (((𝐴 ∈ No ∧ 0s <s 𝐴) ∧ (𝐵 ∈ No ∧ 0s <s 𝐵)) → (( 0s ·s 𝐵) -s ( 0s ·s 0s )) = ( 0s -s 0s )) |
14 | subsid 27466 | . . . 4 ⊢ ( 0s ∈ No → ( 0s -s 0s ) = 0s ) | |
15 | 1, 14 | ax-mp 5 | . . 3 ⊢ ( 0s -s 0s ) = 0s |
16 | 13, 15 | eqtrdi 2788 | . 2 ⊢ (((𝐴 ∈ No ∧ 0s <s 𝐴) ∧ (𝐵 ∈ No ∧ 0s <s 𝐵)) → (( 0s ·s 𝐵) -s ( 0s ·s 0s )) = 0s ) |
17 | muls01 27497 | . . . . 5 ⊢ (𝐴 ∈ No → (𝐴 ·s 0s ) = 0s ) | |
18 | 3, 17 | syl 17 | . . . 4 ⊢ (((𝐴 ∈ No ∧ 0s <s 𝐴) ∧ (𝐵 ∈ No ∧ 0s <s 𝐵)) → (𝐴 ·s 0s ) = 0s ) |
19 | 18 | oveq2d 7410 | . . 3 ⊢ (((𝐴 ∈ No ∧ 0s <s 𝐴) ∧ (𝐵 ∈ No ∧ 0s <s 𝐵)) → ((𝐴 ·s 𝐵) -s (𝐴 ·s 0s )) = ((𝐴 ·s 𝐵) -s 0s )) |
20 | mulscl 27519 | . . . . 5 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (𝐴 ·s 𝐵) ∈ No ) | |
21 | 20 | ad2ant2r 745 | . . . 4 ⊢ (((𝐴 ∈ No ∧ 0s <s 𝐴) ∧ (𝐵 ∈ No ∧ 0s <s 𝐵)) → (𝐴 ·s 𝐵) ∈ No ) |
22 | subsid1 27465 | . . . 4 ⊢ ((𝐴 ·s 𝐵) ∈ No → ((𝐴 ·s 𝐵) -s 0s ) = (𝐴 ·s 𝐵)) | |
23 | 21, 22 | syl 17 | . . 3 ⊢ (((𝐴 ∈ No ∧ 0s <s 𝐴) ∧ (𝐵 ∈ No ∧ 0s <s 𝐵)) → ((𝐴 ·s 𝐵) -s 0s ) = (𝐴 ·s 𝐵)) |
24 | 19, 23 | eqtrd 2772 | . 2 ⊢ (((𝐴 ∈ No ∧ 0s <s 𝐴) ∧ (𝐵 ∈ No ∧ 0s <s 𝐵)) → ((𝐴 ·s 𝐵) -s (𝐴 ·s 0s )) = (𝐴 ·s 𝐵)) |
25 | 7, 16, 24 | 3brtr3d 5173 | 1 ⊢ (((𝐴 ∈ No ∧ 0s <s 𝐴) ∧ (𝐵 ∈ No ∧ 0s <s 𝐵)) → 0s <s (𝐴 ·s 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1541 ∈ wcel 2106 class class class wbr 5142 (class class class)co 7394 No csur 27072 <s cslt 27073 0s c0s 27252 -s csubs 27424 ·s cmuls 27491 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5357 ax-pr 5421 ax-un 7709 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3775 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-pss 3964 df-nul 4320 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-tp 4628 df-op 4630 df-ot 4632 df-uni 4903 df-int 4945 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5568 df-eprel 5574 df-po 5582 df-so 5583 df-fr 5625 df-se 5626 df-we 5627 df-xp 5676 df-rel 5677 df-cnv 5678 df-co 5679 df-dm 5680 df-rn 5681 df-res 5682 df-ima 5683 df-pred 6290 df-ord 6357 df-on 6358 df-suc 6360 df-iota 6485 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-riota 7350 df-ov 7397 df-oprab 7398 df-mpo 7399 df-1st 7959 df-2nd 7960 df-frecs 8250 df-wrecs 8281 df-recs 8355 df-1o 8450 df-2o 8451 df-nadd 8650 df-no 27075 df-slt 27076 df-bday 27077 df-sle 27177 df-sslt 27212 df-scut 27214 df-0s 27254 df-made 27271 df-old 27272 df-left 27274 df-right 27275 df-norec 27351 df-norec2 27362 df-adds 27373 df-negs 27425 df-subs 27426 df-muls 27492 |
This theorem is referenced by: mulsgt0d 27530 |
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