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Mirrors > Home > ILE Home > Th. List > axpre-mulgt0 | GIF version |
Description: The product of two positive reals is positive. Axiom for real and complex numbers, derived from set theory. This construction-dependent theorem should not be referenced directly; instead, use ax-pre-mulgt0 7761. (Contributed by NM, 13-May-1996.) (New usage is discouraged.) |
Ref | Expression |
---|---|
axpre-mulgt0 | ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((0 <ℝ 𝐴 ∧ 0 <ℝ 𝐵) → 0 <ℝ (𝐴 · 𝐵))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elreal 7660 | . 2 ⊢ (𝐴 ∈ ℝ ↔ ∃𝑥 ∈ R 〈𝑥, 0R〉 = 𝐴) | |
2 | elreal 7660 | . 2 ⊢ (𝐵 ∈ ℝ ↔ ∃𝑦 ∈ R 〈𝑦, 0R〉 = 𝐵) | |
3 | breq2 3941 | . . . 4 ⊢ (〈𝑥, 0R〉 = 𝐴 → (0 <ℝ 〈𝑥, 0R〉 ↔ 0 <ℝ 𝐴)) | |
4 | 3 | anbi1d 461 | . . 3 ⊢ (〈𝑥, 0R〉 = 𝐴 → ((0 <ℝ 〈𝑥, 0R〉 ∧ 0 <ℝ 〈𝑦, 0R〉) ↔ (0 <ℝ 𝐴 ∧ 0 <ℝ 〈𝑦, 0R〉))) |
5 | oveq1 5789 | . . . 4 ⊢ (〈𝑥, 0R〉 = 𝐴 → (〈𝑥, 0R〉 · 〈𝑦, 0R〉) = (𝐴 · 〈𝑦, 0R〉)) | |
6 | 5 | breq2d 3949 | . . 3 ⊢ (〈𝑥, 0R〉 = 𝐴 → (0 <ℝ (〈𝑥, 0R〉 · 〈𝑦, 0R〉) ↔ 0 <ℝ (𝐴 · 〈𝑦, 0R〉))) |
7 | 4, 6 | imbi12d 233 | . 2 ⊢ (〈𝑥, 0R〉 = 𝐴 → (((0 <ℝ 〈𝑥, 0R〉 ∧ 0 <ℝ 〈𝑦, 0R〉) → 0 <ℝ (〈𝑥, 0R〉 · 〈𝑦, 0R〉)) ↔ ((0 <ℝ 𝐴 ∧ 0 <ℝ 〈𝑦, 0R〉) → 0 <ℝ (𝐴 · 〈𝑦, 0R〉)))) |
8 | breq2 3941 | . . . 4 ⊢ (〈𝑦, 0R〉 = 𝐵 → (0 <ℝ 〈𝑦, 0R〉 ↔ 0 <ℝ 𝐵)) | |
9 | 8 | anbi2d 460 | . . 3 ⊢ (〈𝑦, 0R〉 = 𝐵 → ((0 <ℝ 𝐴 ∧ 0 <ℝ 〈𝑦, 0R〉) ↔ (0 <ℝ 𝐴 ∧ 0 <ℝ 𝐵))) |
10 | oveq2 5790 | . . . 4 ⊢ (〈𝑦, 0R〉 = 𝐵 → (𝐴 · 〈𝑦, 0R〉) = (𝐴 · 𝐵)) | |
11 | 10 | breq2d 3949 | . . 3 ⊢ (〈𝑦, 0R〉 = 𝐵 → (0 <ℝ (𝐴 · 〈𝑦, 0R〉) ↔ 0 <ℝ (𝐴 · 𝐵))) |
12 | 9, 11 | imbi12d 233 | . 2 ⊢ (〈𝑦, 0R〉 = 𝐵 → (((0 <ℝ 𝐴 ∧ 0 <ℝ 〈𝑦, 0R〉) → 0 <ℝ (𝐴 · 〈𝑦, 0R〉)) ↔ ((0 <ℝ 𝐴 ∧ 0 <ℝ 𝐵) → 0 <ℝ (𝐴 · 𝐵)))) |
13 | df-0 7651 | . . . . . 6 ⊢ 0 = 〈0R, 0R〉 | |
14 | 13 | breq1i 3944 | . . . . 5 ⊢ (0 <ℝ 〈𝑥, 0R〉 ↔ 〈0R, 0R〉 <ℝ 〈𝑥, 0R〉) |
15 | ltresr 7671 | . . . . 5 ⊢ (〈0R, 0R〉 <ℝ 〈𝑥, 0R〉 ↔ 0R <R 𝑥) | |
16 | 14, 15 | bitri 183 | . . . 4 ⊢ (0 <ℝ 〈𝑥, 0R〉 ↔ 0R <R 𝑥) |
17 | 13 | breq1i 3944 | . . . . 5 ⊢ (0 <ℝ 〈𝑦, 0R〉 ↔ 〈0R, 0R〉 <ℝ 〈𝑦, 0R〉) |
18 | ltresr 7671 | . . . . 5 ⊢ (〈0R, 0R〉 <ℝ 〈𝑦, 0R〉 ↔ 0R <R 𝑦) | |
19 | 17, 18 | bitri 183 | . . . 4 ⊢ (0 <ℝ 〈𝑦, 0R〉 ↔ 0R <R 𝑦) |
20 | mulgt0sr 7610 | . . . 4 ⊢ ((0R <R 𝑥 ∧ 0R <R 𝑦) → 0R <R (𝑥 ·R 𝑦)) | |
21 | 16, 19, 20 | syl2anb 289 | . . 3 ⊢ ((0 <ℝ 〈𝑥, 0R〉 ∧ 0 <ℝ 〈𝑦, 0R〉) → 0R <R (𝑥 ·R 𝑦)) |
22 | 13 | a1i 9 | . . . . 5 ⊢ ((𝑥 ∈ R ∧ 𝑦 ∈ R) → 0 = 〈0R, 0R〉) |
23 | mulresr 7670 | . . . . 5 ⊢ ((𝑥 ∈ R ∧ 𝑦 ∈ R) → (〈𝑥, 0R〉 · 〈𝑦, 0R〉) = 〈(𝑥 ·R 𝑦), 0R〉) | |
24 | 22, 23 | breq12d 3950 | . . . 4 ⊢ ((𝑥 ∈ R ∧ 𝑦 ∈ R) → (0 <ℝ (〈𝑥, 0R〉 · 〈𝑦, 0R〉) ↔ 〈0R, 0R〉 <ℝ 〈(𝑥 ·R 𝑦), 0R〉)) |
25 | ltresr 7671 | . . . 4 ⊢ (〈0R, 0R〉 <ℝ 〈(𝑥 ·R 𝑦), 0R〉 ↔ 0R <R (𝑥 ·R 𝑦)) | |
26 | 24, 25 | syl6bb 195 | . . 3 ⊢ ((𝑥 ∈ R ∧ 𝑦 ∈ R) → (0 <ℝ (〈𝑥, 0R〉 · 〈𝑦, 0R〉) ↔ 0R <R (𝑥 ·R 𝑦))) |
27 | 21, 26 | syl5ibr 155 | . 2 ⊢ ((𝑥 ∈ R ∧ 𝑦 ∈ R) → ((0 <ℝ 〈𝑥, 0R〉 ∧ 0 <ℝ 〈𝑦, 0R〉) → 0 <ℝ (〈𝑥, 0R〉 · 〈𝑦, 0R〉))) |
28 | 1, 2, 7, 12, 27 | 2gencl 2722 | 1 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((0 <ℝ 𝐴 ∧ 0 <ℝ 𝐵) → 0 <ℝ (𝐴 · 𝐵))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 = wceq 1332 ∈ wcel 1481 〈cop 3535 class class class wbr 3937 (class class class)co 5782 Rcnr 7129 0Rc0r 7130 ·R cmr 7134 <R cltr 7135 ℝcr 7643 0cc0 7644 <ℝ cltrr 7648 · cmul 7649 |
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 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-13 1492 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-coll 4051 ax-sep 4054 ax-nul 4062 ax-pow 4106 ax-pr 4139 ax-un 4363 ax-setind 4460 ax-iinf 4510 |
This theorem depends on definitions: df-bi 116 df-dc 821 df-3or 964 df-3an 965 df-tru 1335 df-fal 1338 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ne 2310 df-ral 2422 df-rex 2423 df-reu 2424 df-rab 2426 df-v 2691 df-sbc 2914 df-csb 3008 df-dif 3078 df-un 3080 df-in 3082 df-ss 3089 df-nul 3369 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-int 3780 df-iun 3823 df-br 3938 df-opab 3998 df-mpt 3999 df-tr 4035 df-eprel 4219 df-id 4223 df-po 4226 df-iso 4227 df-iord 4296 df-on 4298 df-suc 4301 df-iom 4513 df-xp 4553 df-rel 4554 df-cnv 4555 df-co 4556 df-dm 4557 df-rn 4558 df-res 4559 df-ima 4560 df-iota 5096 df-fun 5133 df-fn 5134 df-f 5135 df-f1 5136 df-fo 5137 df-f1o 5138 df-fv 5139 df-ov 5785 df-oprab 5786 df-mpo 5787 df-1st 6046 df-2nd 6047 df-recs 6210 df-irdg 6275 df-1o 6321 df-2o 6322 df-oadd 6325 df-omul 6326 df-er 6437 df-ec 6439 df-qs 6443 df-ni 7136 df-pli 7137 df-mi 7138 df-lti 7139 df-plpq 7176 df-mpq 7177 df-enq 7179 df-nqqs 7180 df-plqqs 7181 df-mqqs 7182 df-1nqqs 7183 df-rq 7184 df-ltnqqs 7185 df-enq0 7256 df-nq0 7257 df-0nq0 7258 df-plq0 7259 df-mq0 7260 df-inp 7298 df-i1p 7299 df-iplp 7300 df-imp 7301 df-iltp 7302 df-enr 7558 df-nr 7559 df-plr 7560 df-mr 7561 df-ltr 7562 df-0r 7563 df-m1r 7565 df-c 7650 df-0 7651 df-r 7654 df-mul 7656 df-lt 7657 |
This theorem is referenced by: (None) |
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