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Mirrors > Home > MPE Home > Th. List > axpre-mulgt0 | Structured version Visualization version GIF version |
Description: The product of two positive reals is positive. Axiom 21 of 22 for real and complex numbers, derived from ZF set theory. Note: The more general version for extended reals is axmulgt0 9991. This construction-dependent theorem should not be referenced directly; instead, use ax-pre-mulgt0 9892. (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 9831 | . 2 ⊢ (𝐴 ∈ ℝ ↔ ∃𝑥 ∈ R 〈𝑥, 0R〉 = 𝐴) | |
2 | elreal 9831 | . 2 ⊢ (𝐵 ∈ ℝ ↔ ∃𝑦 ∈ R 〈𝑦, 0R〉 = 𝐵) | |
3 | breq2 4587 | . . . 4 ⊢ (〈𝑥, 0R〉 = 𝐴 → (0 <ℝ 〈𝑥, 0R〉 ↔ 0 <ℝ 𝐴)) | |
4 | 3 | anbi1d 737 | . . 3 ⊢ (〈𝑥, 0R〉 = 𝐴 → ((0 <ℝ 〈𝑥, 0R〉 ∧ 0 <ℝ 〈𝑦, 0R〉) ↔ (0 <ℝ 𝐴 ∧ 0 <ℝ 〈𝑦, 0R〉))) |
5 | oveq1 6556 | . . . 4 ⊢ (〈𝑥, 0R〉 = 𝐴 → (〈𝑥, 0R〉 · 〈𝑦, 0R〉) = (𝐴 · 〈𝑦, 0R〉)) | |
6 | 5 | breq2d 4595 | . . 3 ⊢ (〈𝑥, 0R〉 = 𝐴 → (0 <ℝ (〈𝑥, 0R〉 · 〈𝑦, 0R〉) ↔ 0 <ℝ (𝐴 · 〈𝑦, 0R〉))) |
7 | 4, 6 | imbi12d 333 | . 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 4587 | . . . 4 ⊢ (〈𝑦, 0R〉 = 𝐵 → (0 <ℝ 〈𝑦, 0R〉 ↔ 0 <ℝ 𝐵)) | |
9 | 8 | anbi2d 736 | . . 3 ⊢ (〈𝑦, 0R〉 = 𝐵 → ((0 <ℝ 𝐴 ∧ 0 <ℝ 〈𝑦, 0R〉) ↔ (0 <ℝ 𝐴 ∧ 0 <ℝ 𝐵))) |
10 | oveq2 6557 | . . . 4 ⊢ (〈𝑦, 0R〉 = 𝐵 → (𝐴 · 〈𝑦, 0R〉) = (𝐴 · 𝐵)) | |
11 | 10 | breq2d 4595 | . . 3 ⊢ (〈𝑦, 0R〉 = 𝐵 → (0 <ℝ (𝐴 · 〈𝑦, 0R〉) ↔ 0 <ℝ (𝐴 · 𝐵))) |
12 | 9, 11 | imbi12d 333 | . 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 9822 | . . . . . 6 ⊢ 0 = 〈0R, 0R〉 | |
14 | 13 | breq1i 4590 | . . . . 5 ⊢ (0 <ℝ 〈𝑥, 0R〉 ↔ 〈0R, 0R〉 <ℝ 〈𝑥, 0R〉) |
15 | ltresr 9840 | . . . . 5 ⊢ (〈0R, 0R〉 <ℝ 〈𝑥, 0R〉 ↔ 0R <R 𝑥) | |
16 | 14, 15 | bitri 263 | . . . 4 ⊢ (0 <ℝ 〈𝑥, 0R〉 ↔ 0R <R 𝑥) |
17 | 13 | breq1i 4590 | . . . . 5 ⊢ (0 <ℝ 〈𝑦, 0R〉 ↔ 〈0R, 0R〉 <ℝ 〈𝑦, 0R〉) |
18 | ltresr 9840 | . . . . 5 ⊢ (〈0R, 0R〉 <ℝ 〈𝑦, 0R〉 ↔ 0R <R 𝑦) | |
19 | 17, 18 | bitri 263 | . . . 4 ⊢ (0 <ℝ 〈𝑦, 0R〉 ↔ 0R <R 𝑦) |
20 | mulgt0sr 9805 | . . . 4 ⊢ ((0R <R 𝑥 ∧ 0R <R 𝑦) → 0R <R (𝑥 ·R 𝑦)) | |
21 | 16, 19, 20 | syl2anb 495 | . . 3 ⊢ ((0 <ℝ 〈𝑥, 0R〉 ∧ 0 <ℝ 〈𝑦, 0R〉) → 0R <R (𝑥 ·R 𝑦)) |
22 | 13 | a1i 11 | . . . . 5 ⊢ ((𝑥 ∈ R ∧ 𝑦 ∈ R) → 0 = 〈0R, 0R〉) |
23 | mulresr 9839 | . . . . 5 ⊢ ((𝑥 ∈ R ∧ 𝑦 ∈ R) → (〈𝑥, 0R〉 · 〈𝑦, 0R〉) = 〈(𝑥 ·R 𝑦), 0R〉) | |
24 | 22, 23 | breq12d 4596 | . . . 4 ⊢ ((𝑥 ∈ R ∧ 𝑦 ∈ R) → (0 <ℝ (〈𝑥, 0R〉 · 〈𝑦, 0R〉) ↔ 〈0R, 0R〉 <ℝ 〈(𝑥 ·R 𝑦), 0R〉)) |
25 | ltresr 9840 | . . . 4 ⊢ (〈0R, 0R〉 <ℝ 〈(𝑥 ·R 𝑦), 0R〉 ↔ 0R <R (𝑥 ·R 𝑦)) | |
26 | 24, 25 | syl6bb 275 | . . 3 ⊢ ((𝑥 ∈ R ∧ 𝑦 ∈ R) → (0 <ℝ (〈𝑥, 0R〉 · 〈𝑦, 0R〉) ↔ 0R <R (𝑥 ·R 𝑦))) |
27 | 21, 26 | syl5ibr 235 | . 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 3209 | 1 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((0 <ℝ 𝐴 ∧ 0 <ℝ 𝐵) → 0 <ℝ (𝐴 · 𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 〈cop 4131 class class class wbr 4583 (class class class)co 6549 Rcnr 9566 0Rc0r 9567 ·R cmr 9571 <R cltr 9572 ℝcr 9814 0cc0 9815 <ℝ cltrr 9819 · cmul 9820 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-inf2 8421 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-oadd 7451 df-omul 7452 df-er 7629 df-ec 7631 df-qs 7635 df-ni 9573 df-pli 9574 df-mi 9575 df-lti 9576 df-plpq 9609 df-mpq 9610 df-ltpq 9611 df-enq 9612 df-nq 9613 df-erq 9614 df-plq 9615 df-mq 9616 df-1nq 9617 df-rq 9618 df-ltnq 9619 df-np 9682 df-1p 9683 df-plp 9684 df-mp 9685 df-ltp 9686 df-enr 9756 df-nr 9757 df-plr 9758 df-mr 9759 df-ltr 9760 df-0r 9761 df-m1r 9763 df-c 9821 df-0 9822 df-r 9825 df-mul 9827 df-lt 9828 |
This theorem is referenced by: (None) |
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