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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 11295. This construction-dependent theorem should not be referenced directly; instead, use ax-pre-mulgt0 11193. (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 11132 | . 2 ⊢ (𝐴 ∈ ℝ ↔ ∃𝑥 ∈ R 〈𝑥, 0R〉 = 𝐴) | |
2 | elreal 11132 | . 2 ⊢ (𝐵 ∈ ℝ ↔ ∃𝑦 ∈ R 〈𝑦, 0R〉 = 𝐵) | |
3 | breq2 5152 | . . . 4 ⊢ (〈𝑥, 0R〉 = 𝐴 → (0 <ℝ 〈𝑥, 0R〉 ↔ 0 <ℝ 𝐴)) | |
4 | 3 | anbi1d 629 | . . 3 ⊢ (〈𝑥, 0R〉 = 𝐴 → ((0 <ℝ 〈𝑥, 0R〉 ∧ 0 <ℝ 〈𝑦, 0R〉) ↔ (0 <ℝ 𝐴 ∧ 0 <ℝ 〈𝑦, 0R〉))) |
5 | oveq1 7419 | . . . 4 ⊢ (〈𝑥, 0R〉 = 𝐴 → (〈𝑥, 0R〉 · 〈𝑦, 0R〉) = (𝐴 · 〈𝑦, 0R〉)) | |
6 | 5 | breq2d 5160 | . . 3 ⊢ (〈𝑥, 0R〉 = 𝐴 → (0 <ℝ (〈𝑥, 0R〉 · 〈𝑦, 0R〉) ↔ 0 <ℝ (𝐴 · 〈𝑦, 0R〉))) |
7 | 4, 6 | imbi12d 344 | . 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 5152 | . . . 4 ⊢ (〈𝑦, 0R〉 = 𝐵 → (0 <ℝ 〈𝑦, 0R〉 ↔ 0 <ℝ 𝐵)) | |
9 | 8 | anbi2d 628 | . . 3 ⊢ (〈𝑦, 0R〉 = 𝐵 → ((0 <ℝ 𝐴 ∧ 0 <ℝ 〈𝑦, 0R〉) ↔ (0 <ℝ 𝐴 ∧ 0 <ℝ 𝐵))) |
10 | oveq2 7420 | . . . 4 ⊢ (〈𝑦, 0R〉 = 𝐵 → (𝐴 · 〈𝑦, 0R〉) = (𝐴 · 𝐵)) | |
11 | 10 | breq2d 5160 | . . 3 ⊢ (〈𝑦, 0R〉 = 𝐵 → (0 <ℝ (𝐴 · 〈𝑦, 0R〉) ↔ 0 <ℝ (𝐴 · 𝐵))) |
12 | 9, 11 | imbi12d 344 | . 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 11123 | . . . . . 6 ⊢ 0 = 〈0R, 0R〉 | |
14 | 13 | breq1i 5155 | . . . . 5 ⊢ (0 <ℝ 〈𝑥, 0R〉 ↔ 〈0R, 0R〉 <ℝ 〈𝑥, 0R〉) |
15 | ltresr 11141 | . . . . 5 ⊢ (〈0R, 0R〉 <ℝ 〈𝑥, 0R〉 ↔ 0R <R 𝑥) | |
16 | 14, 15 | bitri 275 | . . . 4 ⊢ (0 <ℝ 〈𝑥, 0R〉 ↔ 0R <R 𝑥) |
17 | 13 | breq1i 5155 | . . . . 5 ⊢ (0 <ℝ 〈𝑦, 0R〉 ↔ 〈0R, 0R〉 <ℝ 〈𝑦, 0R〉) |
18 | ltresr 11141 | . . . . 5 ⊢ (〈0R, 0R〉 <ℝ 〈𝑦, 0R〉 ↔ 0R <R 𝑦) | |
19 | 17, 18 | bitri 275 | . . . 4 ⊢ (0 <ℝ 〈𝑦, 0R〉 ↔ 0R <R 𝑦) |
20 | mulgt0sr 11106 | . . . 4 ⊢ ((0R <R 𝑥 ∧ 0R <R 𝑦) → 0R <R (𝑥 ·R 𝑦)) | |
21 | 16, 19, 20 | syl2anb 597 | . . 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 11140 | . . . . 5 ⊢ ((𝑥 ∈ R ∧ 𝑦 ∈ R) → (〈𝑥, 0R〉 · 〈𝑦, 0R〉) = 〈(𝑥 ·R 𝑦), 0R〉) | |
24 | 22, 23 | breq12d 5161 | . . . 4 ⊢ ((𝑥 ∈ R ∧ 𝑦 ∈ R) → (0 <ℝ (〈𝑥, 0R〉 · 〈𝑦, 0R〉) ↔ 〈0R, 0R〉 <ℝ 〈(𝑥 ·R 𝑦), 0R〉)) |
25 | ltresr 11141 | . . . 4 ⊢ (〈0R, 0R〉 <ℝ 〈(𝑥 ·R 𝑦), 0R〉 ↔ 0R <R (𝑥 ·R 𝑦)) | |
26 | 24, 25 | bitrdi 287 | . . 3 ⊢ ((𝑥 ∈ R ∧ 𝑦 ∈ R) → (0 <ℝ (〈𝑥, 0R〉 · 〈𝑦, 0R〉) ↔ 0R <R (𝑥 ·R 𝑦))) |
27 | 21, 26 | imbitrrid 245 | . 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 3516 | 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 395 = wceq 1540 ∈ wcel 2105 〈cop 4634 class class class wbr 5148 (class class class)co 7412 Rcnr 10866 0Rc0r 10867 ·R cmr 10871 <R cltr 10872 ℝcr 11115 0cc0 11116 <ℝ cltrr 11120 · cmul 11121 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 ax-inf2 9642 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7860 df-1st 7979 df-2nd 7980 df-frecs 8272 df-wrecs 8303 df-recs 8377 df-rdg 8416 df-1o 8472 df-oadd 8476 df-omul 8477 df-er 8709 df-ec 8711 df-qs 8715 df-ni 10873 df-pli 10874 df-mi 10875 df-lti 10876 df-plpq 10909 df-mpq 10910 df-ltpq 10911 df-enq 10912 df-nq 10913 df-erq 10914 df-plq 10915 df-mq 10916 df-1nq 10917 df-rq 10918 df-ltnq 10919 df-np 10982 df-1p 10983 df-plp 10984 df-mp 10985 df-ltp 10986 df-enr 11056 df-nr 11057 df-plr 11058 df-mr 11059 df-ltr 11060 df-0r 11061 df-m1r 11063 df-c 11122 df-0 11123 df-r 11126 df-mul 11128 df-lt 11129 |
This theorem is referenced by: (None) |
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