![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
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 7958. (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 7857 | . 2 ⊢ (𝐴 ∈ ℝ ↔ ∃𝑥 ∈ R 〈𝑥, 0R〉 = 𝐴) | |
2 | elreal 7857 | . 2 ⊢ (𝐵 ∈ ℝ ↔ ∃𝑦 ∈ R 〈𝑦, 0R〉 = 𝐵) | |
3 | breq2 4022 | . . . 4 ⊢ (〈𝑥, 0R〉 = 𝐴 → (0 <ℝ 〈𝑥, 0R〉 ↔ 0 <ℝ 𝐴)) | |
4 | 3 | anbi1d 465 | . . 3 ⊢ (〈𝑥, 0R〉 = 𝐴 → ((0 <ℝ 〈𝑥, 0R〉 ∧ 0 <ℝ 〈𝑦, 0R〉) ↔ (0 <ℝ 𝐴 ∧ 0 <ℝ 〈𝑦, 0R〉))) |
5 | oveq1 5903 | . . . 4 ⊢ (〈𝑥, 0R〉 = 𝐴 → (〈𝑥, 0R〉 · 〈𝑦, 0R〉) = (𝐴 · 〈𝑦, 0R〉)) | |
6 | 5 | breq2d 4030 | . . 3 ⊢ (〈𝑥, 0R〉 = 𝐴 → (0 <ℝ (〈𝑥, 0R〉 · 〈𝑦, 0R〉) ↔ 0 <ℝ (𝐴 · 〈𝑦, 0R〉))) |
7 | 4, 6 | imbi12d 234 | . 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 4022 | . . . 4 ⊢ (〈𝑦, 0R〉 = 𝐵 → (0 <ℝ 〈𝑦, 0R〉 ↔ 0 <ℝ 𝐵)) | |
9 | 8 | anbi2d 464 | . . 3 ⊢ (〈𝑦, 0R〉 = 𝐵 → ((0 <ℝ 𝐴 ∧ 0 <ℝ 〈𝑦, 0R〉) ↔ (0 <ℝ 𝐴 ∧ 0 <ℝ 𝐵))) |
10 | oveq2 5904 | . . . 4 ⊢ (〈𝑦, 0R〉 = 𝐵 → (𝐴 · 〈𝑦, 0R〉) = (𝐴 · 𝐵)) | |
11 | 10 | breq2d 4030 | . . 3 ⊢ (〈𝑦, 0R〉 = 𝐵 → (0 <ℝ (𝐴 · 〈𝑦, 0R〉) ↔ 0 <ℝ (𝐴 · 𝐵))) |
12 | 9, 11 | imbi12d 234 | . 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 7848 | . . . . . 6 ⊢ 0 = 〈0R, 0R〉 | |
14 | 13 | breq1i 4025 | . . . . 5 ⊢ (0 <ℝ 〈𝑥, 0R〉 ↔ 〈0R, 0R〉 <ℝ 〈𝑥, 0R〉) |
15 | ltresr 7868 | . . . . 5 ⊢ (〈0R, 0R〉 <ℝ 〈𝑥, 0R〉 ↔ 0R <R 𝑥) | |
16 | 14, 15 | bitri 184 | . . . 4 ⊢ (0 <ℝ 〈𝑥, 0R〉 ↔ 0R <R 𝑥) |
17 | 13 | breq1i 4025 | . . . . 5 ⊢ (0 <ℝ 〈𝑦, 0R〉 ↔ 〈0R, 0R〉 <ℝ 〈𝑦, 0R〉) |
18 | ltresr 7868 | . . . . 5 ⊢ (〈0R, 0R〉 <ℝ 〈𝑦, 0R〉 ↔ 0R <R 𝑦) | |
19 | 17, 18 | bitri 184 | . . . 4 ⊢ (0 <ℝ 〈𝑦, 0R〉 ↔ 0R <R 𝑦) |
20 | mulgt0sr 7807 | . . . 4 ⊢ ((0R <R 𝑥 ∧ 0R <R 𝑦) → 0R <R (𝑥 ·R 𝑦)) | |
21 | 16, 19, 20 | syl2anb 291 | . . 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 7867 | . . . . 5 ⊢ ((𝑥 ∈ R ∧ 𝑦 ∈ R) → (〈𝑥, 0R〉 · 〈𝑦, 0R〉) = 〈(𝑥 ·R 𝑦), 0R〉) | |
24 | 22, 23 | breq12d 4031 | . . . 4 ⊢ ((𝑥 ∈ R ∧ 𝑦 ∈ R) → (0 <ℝ (〈𝑥, 0R〉 · 〈𝑦, 0R〉) ↔ 〈0R, 0R〉 <ℝ 〈(𝑥 ·R 𝑦), 0R〉)) |
25 | ltresr 7868 | . . . 4 ⊢ (〈0R, 0R〉 <ℝ 〈(𝑥 ·R 𝑦), 0R〉 ↔ 0R <R (𝑥 ·R 𝑦)) | |
26 | 24, 25 | bitrdi 196 | . . 3 ⊢ ((𝑥 ∈ R ∧ 𝑦 ∈ R) → (0 <ℝ (〈𝑥, 0R〉 · 〈𝑦, 0R〉) ↔ 0R <R (𝑥 ·R 𝑦))) |
27 | 21, 26 | imbitrrid 156 | . 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 2785 | 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 104 = wceq 1364 ∈ wcel 2160 〈cop 3610 class class class wbr 4018 (class class class)co 5896 Rcnr 7326 0Rc0r 7327 ·R cmr 7331 <R cltr 7332 ℝcr 7840 0cc0 7841 <ℝ cltrr 7845 · cmul 7846 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2162 ax-14 2163 ax-ext 2171 ax-coll 4133 ax-sep 4136 ax-nul 4144 ax-pow 4192 ax-pr 4227 ax-un 4451 ax-setind 4554 ax-iinf 4605 |
This theorem depends on definitions: df-bi 117 df-dc 836 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ne 2361 df-ral 2473 df-rex 2474 df-reu 2475 df-rab 2477 df-v 2754 df-sbc 2978 df-csb 3073 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-nul 3438 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-int 3860 df-iun 3903 df-br 4019 df-opab 4080 df-mpt 4081 df-tr 4117 df-eprel 4307 df-id 4311 df-po 4314 df-iso 4315 df-iord 4384 df-on 4386 df-suc 4389 df-iom 4608 df-xp 4650 df-rel 4651 df-cnv 4652 df-co 4653 df-dm 4654 df-rn 4655 df-res 4656 df-ima 4657 df-iota 5196 df-fun 5237 df-fn 5238 df-f 5239 df-f1 5240 df-fo 5241 df-f1o 5242 df-fv 5243 df-ov 5899 df-oprab 5900 df-mpo 5901 df-1st 6165 df-2nd 6166 df-recs 6330 df-irdg 6395 df-1o 6441 df-2o 6442 df-oadd 6445 df-omul 6446 df-er 6559 df-ec 6561 df-qs 6565 df-ni 7333 df-pli 7334 df-mi 7335 df-lti 7336 df-plpq 7373 df-mpq 7374 df-enq 7376 df-nqqs 7377 df-plqqs 7378 df-mqqs 7379 df-1nqqs 7380 df-rq 7381 df-ltnqqs 7382 df-enq0 7453 df-nq0 7454 df-0nq0 7455 df-plq0 7456 df-mq0 7457 df-inp 7495 df-i1p 7496 df-iplp 7497 df-imp 7498 df-iltp 7499 df-enr 7755 df-nr 7756 df-plr 7757 df-mr 7758 df-ltr 7759 df-0r 7760 df-m1r 7762 df-c 7847 df-0 7848 df-r 7851 df-mul 7853 df-lt 7854 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |