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Mirrors > Home > ILE Home > Th. List > lt2mulnq | GIF version |
Description: Ordering property of multiplication for positive fractions. (Contributed by Jim Kingdon, 18-Jul-2021.) |
Ref | Expression |
---|---|
lt2mulnq | ⊢ (((𝐴 ∈ Q ∧ 𝐵 ∈ Q) ∧ (𝐶 ∈ Q ∧ 𝐷 ∈ Q)) → ((𝐴 <Q 𝐵 ∧ 𝐶 <Q 𝐷) → (𝐴 ·Q 𝐶) <Q (𝐵 ·Q 𝐷))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ltmnqg 7333 | . . . . . 6 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → (𝐴 <Q 𝐵 ↔ (𝐶 ·Q 𝐴) <Q (𝐶 ·Q 𝐵))) | |
2 | 1 | 3expa 1192 | . . . . 5 ⊢ (((𝐴 ∈ Q ∧ 𝐵 ∈ Q) ∧ 𝐶 ∈ Q) → (𝐴 <Q 𝐵 ↔ (𝐶 ·Q 𝐴) <Q (𝐶 ·Q 𝐵))) |
3 | 2 | adantrr 471 | . . . 4 ⊢ (((𝐴 ∈ Q ∧ 𝐵 ∈ Q) ∧ (𝐶 ∈ Q ∧ 𝐷 ∈ Q)) → (𝐴 <Q 𝐵 ↔ (𝐶 ·Q 𝐴) <Q (𝐶 ·Q 𝐵))) |
4 | mulcomnqg 7315 | . . . . . . 7 ⊢ ((𝐶 ∈ Q ∧ 𝐴 ∈ Q) → (𝐶 ·Q 𝐴) = (𝐴 ·Q 𝐶)) | |
5 | 4 | ancoms 266 | . . . . . 6 ⊢ ((𝐴 ∈ Q ∧ 𝐶 ∈ Q) → (𝐶 ·Q 𝐴) = (𝐴 ·Q 𝐶)) |
6 | 5 | ad2ant2r 501 | . . . . 5 ⊢ (((𝐴 ∈ Q ∧ 𝐵 ∈ Q) ∧ (𝐶 ∈ Q ∧ 𝐷 ∈ Q)) → (𝐶 ·Q 𝐴) = (𝐴 ·Q 𝐶)) |
7 | mulcomnqg 7315 | . . . . . . 7 ⊢ ((𝐶 ∈ Q ∧ 𝐵 ∈ Q) → (𝐶 ·Q 𝐵) = (𝐵 ·Q 𝐶)) | |
8 | 7 | ancoms 266 | . . . . . 6 ⊢ ((𝐵 ∈ Q ∧ 𝐶 ∈ Q) → (𝐶 ·Q 𝐵) = (𝐵 ·Q 𝐶)) |
9 | 8 | ad2ant2lr 502 | . . . . 5 ⊢ (((𝐴 ∈ Q ∧ 𝐵 ∈ Q) ∧ (𝐶 ∈ Q ∧ 𝐷 ∈ Q)) → (𝐶 ·Q 𝐵) = (𝐵 ·Q 𝐶)) |
10 | 6, 9 | breq12d 3989 | . . . 4 ⊢ (((𝐴 ∈ Q ∧ 𝐵 ∈ Q) ∧ (𝐶 ∈ Q ∧ 𝐷 ∈ Q)) → ((𝐶 ·Q 𝐴) <Q (𝐶 ·Q 𝐵) ↔ (𝐴 ·Q 𝐶) <Q (𝐵 ·Q 𝐶))) |
11 | 3, 10 | bitrd 187 | . . 3 ⊢ (((𝐴 ∈ Q ∧ 𝐵 ∈ Q) ∧ (𝐶 ∈ Q ∧ 𝐷 ∈ Q)) → (𝐴 <Q 𝐵 ↔ (𝐴 ·Q 𝐶) <Q (𝐵 ·Q 𝐶))) |
12 | ltmnqg 7333 | . . . . . 6 ⊢ ((𝐶 ∈ Q ∧ 𝐷 ∈ Q ∧ 𝐵 ∈ Q) → (𝐶 <Q 𝐷 ↔ (𝐵 ·Q 𝐶) <Q (𝐵 ·Q 𝐷))) | |
13 | 12 | 3expa 1192 | . . . . 5 ⊢ (((𝐶 ∈ Q ∧ 𝐷 ∈ Q) ∧ 𝐵 ∈ Q) → (𝐶 <Q 𝐷 ↔ (𝐵 ·Q 𝐶) <Q (𝐵 ·Q 𝐷))) |
14 | 13 | ancoms 266 | . . . 4 ⊢ ((𝐵 ∈ Q ∧ (𝐶 ∈ Q ∧ 𝐷 ∈ Q)) → (𝐶 <Q 𝐷 ↔ (𝐵 ·Q 𝐶) <Q (𝐵 ·Q 𝐷))) |
15 | 14 | adantll 468 | . . 3 ⊢ (((𝐴 ∈ Q ∧ 𝐵 ∈ Q) ∧ (𝐶 ∈ Q ∧ 𝐷 ∈ Q)) → (𝐶 <Q 𝐷 ↔ (𝐵 ·Q 𝐶) <Q (𝐵 ·Q 𝐷))) |
16 | 11, 15 | anbi12d 465 | . 2 ⊢ (((𝐴 ∈ Q ∧ 𝐵 ∈ Q) ∧ (𝐶 ∈ Q ∧ 𝐷 ∈ Q)) → ((𝐴 <Q 𝐵 ∧ 𝐶 <Q 𝐷) ↔ ((𝐴 ·Q 𝐶) <Q (𝐵 ·Q 𝐶) ∧ (𝐵 ·Q 𝐶) <Q (𝐵 ·Q 𝐷)))) |
17 | ltsonq 7330 | . . 3 ⊢ <Q Or Q | |
18 | ltrelnq 7297 | . . 3 ⊢ <Q ⊆ (Q × Q) | |
19 | 17, 18 | sotri 4993 | . 2 ⊢ (((𝐴 ·Q 𝐶) <Q (𝐵 ·Q 𝐶) ∧ (𝐵 ·Q 𝐶) <Q (𝐵 ·Q 𝐷)) → (𝐴 ·Q 𝐶) <Q (𝐵 ·Q 𝐷)) |
20 | 16, 19 | syl6bi 162 | 1 ⊢ (((𝐴 ∈ Q ∧ 𝐵 ∈ Q) ∧ (𝐶 ∈ Q ∧ 𝐷 ∈ Q)) → ((𝐴 <Q 𝐵 ∧ 𝐶 <Q 𝐷) → (𝐴 ·Q 𝐶) <Q (𝐵 ·Q 𝐷))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 = wceq 1342 ∈ wcel 2135 class class class wbr 3976 (class class class)co 5836 Qcnq 7212 ·Q cmq 7215 <Q cltq 7217 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-coll 4091 ax-sep 4094 ax-nul 4102 ax-pow 4147 ax-pr 4181 ax-un 4405 ax-setind 4508 ax-iinf 4559 |
This theorem depends on definitions: df-bi 116 df-dc 825 df-3or 968 df-3an 969 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ne 2335 df-ral 2447 df-rex 2448 df-reu 2449 df-rab 2451 df-v 2723 df-sbc 2947 df-csb 3041 df-dif 3113 df-un 3115 df-in 3117 df-ss 3124 df-nul 3405 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-int 3819 df-iun 3862 df-br 3977 df-opab 4038 df-mpt 4039 df-tr 4075 df-eprel 4261 df-id 4265 df-po 4268 df-iso 4269 df-iord 4338 df-on 4340 df-suc 4343 df-iom 4562 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-rn 4609 df-res 4610 df-ima 4611 df-iota 5147 df-fun 5184 df-fn 5185 df-f 5186 df-f1 5187 df-fo 5188 df-f1o 5189 df-fv 5190 df-ov 5839 df-oprab 5840 df-mpo 5841 df-1st 6100 df-2nd 6101 df-recs 6264 df-irdg 6329 df-oadd 6379 df-omul 6380 df-er 6492 df-ec 6494 df-qs 6498 df-ni 7236 df-mi 7238 df-lti 7239 df-mpq 7277 df-enq 7279 df-nqqs 7280 df-mqqs 7282 df-ltnqqs 7285 |
This theorem is referenced by: mulnqprlemrl 7505 mulnqprlemru 7506 |
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