Theorem axpre-mulgt0 7572

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 7612. (Contributed by NM, 13-May-1996.) (New usage is discouraged.)

Assertion

Ref	Expression
axpre-mulgt0	⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((0 <ℝ 𝐴 ∧ 0 <ℝ 𝐵) → 0 <ℝ (𝐴 · 𝐵)))

Proof of Theorem axpre-mulgt0

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elreal 7516	. 2 ⊢ (𝐴 ∈ ℝ ↔ ∃𝑥 ∈ R ⟨𝑥, 0R⟩ = 𝐴)
2		elreal 7516	. 2 ⊢ (𝐵 ∈ ℝ ↔ ∃𝑦 ∈ R ⟨𝑦, 0R⟩ = 𝐵)
3		breq2 3879	. . . 4 ⊢ (⟨𝑥, 0R⟩ = 𝐴 → (0 <ℝ ⟨𝑥, 0R⟩ ↔ 0 <ℝ 𝐴))
4	3	anbi1d 456	. . 3 ⊢ (⟨𝑥, 0R⟩ = 𝐴 → ((0 <ℝ ⟨𝑥, 0R⟩ ∧ 0 <ℝ ⟨𝑦, 0R⟩) ↔ (0 <ℝ 𝐴 ∧ 0 <ℝ ⟨𝑦, 0R⟩)))
5		oveq1 5713	. . . 4 ⊢ (⟨𝑥, 0R⟩ = 𝐴 → (⟨𝑥, 0R⟩ · ⟨𝑦, 0R⟩) = (𝐴 · ⟨𝑦, 0R⟩))
6	5	breq2d 3887	. . 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 3879	. . . 4 ⊢ (⟨𝑦, 0R⟩ = 𝐵 → (0 <ℝ ⟨𝑦, 0R⟩ ↔ 0 <ℝ 𝐵))
9	8	anbi2d 455	. . 3 ⊢ (⟨𝑦, 0R⟩ = 𝐵 → ((0 <ℝ 𝐴 ∧ 0 <ℝ ⟨𝑦, 0R⟩) ↔ (0 <ℝ 𝐴 ∧ 0 <ℝ 𝐵)))
10		oveq2 5714	. . . 4 ⊢ (⟨𝑦, 0R⟩ = 𝐵 → (𝐴 · ⟨𝑦, 0R⟩) = (𝐴 · 𝐵))
11	10	breq2d 3887	. . 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 7507	. . . . . 6 ⊢ 0 = ⟨0R, 0R⟩
14	13	breq1i 3882	. . . . 5 ⊢ (0 <ℝ ⟨𝑥, 0R⟩ ↔ ⟨0R, 0R⟩ <ℝ ⟨𝑥, 0R⟩)
15		ltresr 7526	. . . . 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 3882	. . . . 5 ⊢ (0 <ℝ ⟨𝑦, 0R⟩ ↔ ⟨0R, 0R⟩ <ℝ ⟨𝑦, 0R⟩)
18		ltresr 7526	. . . . 5 ⊢ (⟨0R, 0R⟩ <ℝ ⟨𝑦, 0R⟩ ↔ 0R <R 𝑦)
19	17, 18	bitri 183	. . . 4 ⊢ (0 <ℝ ⟨𝑦, 0R⟩ ↔ 0R <R 𝑦)
20		mulgt0sr 7473	. . . 4 ⊢ ((0R <R 𝑥 ∧ 0R <R 𝑦) → 0R <R (𝑥 ·R 𝑦))
21	16, 19, 20	syl2anb 287	. . 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 7525	. . . . 5 ⊢ ((𝑥 ∈ R ∧ 𝑦 ∈ R) → (⟨𝑥, 0R⟩ · ⟨𝑦, 0R⟩) = ⟨(𝑥 ·R 𝑦), 0R⟩)
24	22, 23	breq12d 3888	. . . 4 ⊢ ((𝑥 ∈ R ∧ 𝑦 ∈ R) → (0 <ℝ (⟨𝑥, 0R⟩ · ⟨𝑦, 0R⟩) ↔ ⟨0R, 0R⟩ <ℝ ⟨(𝑥 ·R 𝑦), 0R⟩))
25		ltresr 7526	. . . 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 2674	1 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((0 <ℝ 𝐴 ∧ 0 <ℝ 𝐵) → 0 <ℝ (𝐴 · 𝐵)))

Syntax hints:   → wi 4   ∧ wa 103   = wceq 1299   ∈ wcel 1448  ⟨cop 3477   class class class wbr 3875  (class class class)co 5706  Rcnr 7006  0_Rc0r 7007   ·_R cmr 7011   <_R cltr 7012  ℝcr 7499  0cc0 7500   <_ℝ cltrr 7504   · cmul 7505

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 584  ax-in2 585  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-13 1459  ax-14 1460  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082  ax-coll 3983  ax-sep 3986  ax-nul 3994  ax-pow 4038  ax-pr 4069  ax-un 4293  ax-setind 4390  ax-iinf 4440

This theorem depends on definitions:  df-bi 116  df-dc 787  df-3or 931  df-3an 932  df-tru 1302  df-fal 1305  df-nf 1405  df-sb 1704  df-eu 1963  df-mo 1964  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ne 2268  df-ral 2380  df-rex 2381  df-reu 2382  df-rab 2384  df-v 2643  df-sbc 2863  df-csb 2956  df-dif 3023  df-un 3025  df-in 3027  df-ss 3034  df-nul 3311  df-pw 3459  df-sn 3480  df-pr 3481  df-op 3483  df-uni 3684  df-int 3719  df-iun 3762  df-br 3876  df-opab 3930  df-mpt 3931  df-tr 3967  df-eprel 4149  df-id 4153  df-po 4156  df-iso 4157  df-iord 4226  df-on 4228  df-suc 4231  df-iom 4443  df-xp 4483  df-rel 4484  df-cnv 4485  df-co 4486  df-dm 4487  df-rn 4488  df-res 4489  df-ima 4490  df-iota 5024  df-fun 5061  df-fn 5062  df-f 5063  df-f1 5064  df-fo 5065  df-f1o 5066  df-fv 5067  df-ov 5709  df-oprab 5710  df-mpo 5711  df-1st 5969  df-2nd 5970  df-recs 6132  df-irdg 6197  df-1o 6243  df-2o 6244  df-oadd 6247  df-omul 6248  df-er 6359  df-ec 6361  df-qs 6365  df-ni 7013  df-pli 7014  df-mi 7015  df-lti 7016  df-plpq 7053  df-mpq 7054  df-enq 7056  df-nqqs 7057  df-plqqs 7058  df-mqqs 7059  df-1nqqs 7060  df-rq 7061  df-ltnqqs 7062  df-enq0 7133  df-nq0 7134  df-0nq0 7135  df-plq0 7136  df-mq0 7137  df-inp 7175  df-i1p 7176  df-iplp 7177  df-imp 7178  df-iltp 7179  df-enr 7422  df-nr 7423  df-plr 7424  df-mr 7425  df-ltr 7426  df-0r 7427  df-m1r 7429  df-c 7506  df-0 7507  df-r 7510  df-mul 7512  df-lt 7513

This theorem is referenced by: (None)