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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 10715. This construction-dependent theorem should not be referenced directly; instead, use ax-pre-mulgt0 10614. (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 10553 | . 2 ⊢ (𝐴 ∈ ℝ ↔ ∃𝑥 ∈ R 〈𝑥, 0R〉 = 𝐴) | |
2 | elreal 10553 | . 2 ⊢ (𝐵 ∈ ℝ ↔ ∃𝑦 ∈ R 〈𝑦, 0R〉 = 𝐵) | |
3 | breq2 5070 | . . . 4 ⊢ (〈𝑥, 0R〉 = 𝐴 → (0 <ℝ 〈𝑥, 0R〉 ↔ 0 <ℝ 𝐴)) | |
4 | 3 | anbi1d 631 | . . 3 ⊢ (〈𝑥, 0R〉 = 𝐴 → ((0 <ℝ 〈𝑥, 0R〉 ∧ 0 <ℝ 〈𝑦, 0R〉) ↔ (0 <ℝ 𝐴 ∧ 0 <ℝ 〈𝑦, 0R〉))) |
5 | oveq1 7163 | . . . 4 ⊢ (〈𝑥, 0R〉 = 𝐴 → (〈𝑥, 0R〉 · 〈𝑦, 0R〉) = (𝐴 · 〈𝑦, 0R〉)) | |
6 | 5 | breq2d 5078 | . . 3 ⊢ (〈𝑥, 0R〉 = 𝐴 → (0 <ℝ (〈𝑥, 0R〉 · 〈𝑦, 0R〉) ↔ 0 <ℝ (𝐴 · 〈𝑦, 0R〉))) |
7 | 4, 6 | imbi12d 347 | . 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 5070 | . . . 4 ⊢ (〈𝑦, 0R〉 = 𝐵 → (0 <ℝ 〈𝑦, 0R〉 ↔ 0 <ℝ 𝐵)) | |
9 | 8 | anbi2d 630 | . . 3 ⊢ (〈𝑦, 0R〉 = 𝐵 → ((0 <ℝ 𝐴 ∧ 0 <ℝ 〈𝑦, 0R〉) ↔ (0 <ℝ 𝐴 ∧ 0 <ℝ 𝐵))) |
10 | oveq2 7164 | . . . 4 ⊢ (〈𝑦, 0R〉 = 𝐵 → (𝐴 · 〈𝑦, 0R〉) = (𝐴 · 𝐵)) | |
11 | 10 | breq2d 5078 | . . 3 ⊢ (〈𝑦, 0R〉 = 𝐵 → (0 <ℝ (𝐴 · 〈𝑦, 0R〉) ↔ 0 <ℝ (𝐴 · 𝐵))) |
12 | 9, 11 | imbi12d 347 | . 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 10544 | . . . . . 6 ⊢ 0 = 〈0R, 0R〉 | |
14 | 13 | breq1i 5073 | . . . . 5 ⊢ (0 <ℝ 〈𝑥, 0R〉 ↔ 〈0R, 0R〉 <ℝ 〈𝑥, 0R〉) |
15 | ltresr 10562 | . . . . 5 ⊢ (〈0R, 0R〉 <ℝ 〈𝑥, 0R〉 ↔ 0R <R 𝑥) | |
16 | 14, 15 | bitri 277 | . . . 4 ⊢ (0 <ℝ 〈𝑥, 0R〉 ↔ 0R <R 𝑥) |
17 | 13 | breq1i 5073 | . . . . 5 ⊢ (0 <ℝ 〈𝑦, 0R〉 ↔ 〈0R, 0R〉 <ℝ 〈𝑦, 0R〉) |
18 | ltresr 10562 | . . . . 5 ⊢ (〈0R, 0R〉 <ℝ 〈𝑦, 0R〉 ↔ 0R <R 𝑦) | |
19 | 17, 18 | bitri 277 | . . . 4 ⊢ (0 <ℝ 〈𝑦, 0R〉 ↔ 0R <R 𝑦) |
20 | mulgt0sr 10527 | . . . 4 ⊢ ((0R <R 𝑥 ∧ 0R <R 𝑦) → 0R <R (𝑥 ·R 𝑦)) | |
21 | 16, 19, 20 | syl2anb 599 | . . 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 10561 | . . . . 5 ⊢ ((𝑥 ∈ R ∧ 𝑦 ∈ R) → (〈𝑥, 0R〉 · 〈𝑦, 0R〉) = 〈(𝑥 ·R 𝑦), 0R〉) | |
24 | 22, 23 | breq12d 5079 | . . . 4 ⊢ ((𝑥 ∈ R ∧ 𝑦 ∈ R) → (0 <ℝ (〈𝑥, 0R〉 · 〈𝑦, 0R〉) ↔ 〈0R, 0R〉 <ℝ 〈(𝑥 ·R 𝑦), 0R〉)) |
25 | ltresr 10562 | . . . 4 ⊢ (〈0R, 0R〉 <ℝ 〈(𝑥 ·R 𝑦), 0R〉 ↔ 0R <R (𝑥 ·R 𝑦)) | |
26 | 24, 25 | syl6bb 289 | . . 3 ⊢ ((𝑥 ∈ R ∧ 𝑦 ∈ R) → (0 <ℝ (〈𝑥, 0R〉 · 〈𝑦, 0R〉) ↔ 0R <R (𝑥 ·R 𝑦))) |
27 | 21, 26 | syl5ibr 248 | . 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 3535 | 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 398 = wceq 1537 ∈ wcel 2114 〈cop 4573 class class class wbr 5066 (class class class)co 7156 Rcnr 10287 0Rc0r 10288 ·R cmr 10292 <R cltr 10293 ℝcr 10536 0cc0 10537 <ℝ cltrr 10541 · cmul 10542 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-inf2 9104 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-1st 7689 df-2nd 7690 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-oadd 8106 df-omul 8107 df-er 8289 df-ec 8291 df-qs 8295 df-ni 10294 df-pli 10295 df-mi 10296 df-lti 10297 df-plpq 10330 df-mpq 10331 df-ltpq 10332 df-enq 10333 df-nq 10334 df-erq 10335 df-plq 10336 df-mq 10337 df-1nq 10338 df-rq 10339 df-ltnq 10340 df-np 10403 df-1p 10404 df-plp 10405 df-mp 10406 df-ltp 10407 df-enr 10477 df-nr 10478 df-plr 10479 df-mr 10480 df-ltr 10481 df-0r 10482 df-m1r 10484 df-c 10543 df-0 10544 df-r 10547 df-mul 10549 df-lt 10550 |
This theorem is referenced by: (None) |
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