Theorem axpre-mulgt0 8090

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 8132. (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 8031	. 2 ⊢ (𝐴 ∈ ℝ ↔ ∃𝑥 ∈ R ⟨𝑥, 0R⟩ = 𝐴)
2		elreal 8031	. 2 ⊢ (𝐵 ∈ ℝ ↔ ∃𝑦 ∈ R ⟨𝑦, 0R⟩ = 𝐵)
3		breq2 4087	. . . 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 6017	. . . 4 ⊢ (⟨𝑥, 0R⟩ = 𝐴 → (⟨𝑥, 0R⟩ · ⟨𝑦, 0R⟩) = (𝐴 · ⟨𝑦, 0R⟩))
6	5	breq2d 4095	. . 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 4087	. . . 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 6018	. . . 4 ⊢ (⟨𝑦, 0R⟩ = 𝐵 → (𝐴 · ⟨𝑦, 0R⟩) = (𝐴 · 𝐵))
11	10	breq2d 4095	. . 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 8022	. . . . . 6 ⊢ 0 = ⟨0R, 0R⟩
14	13	breq1i 4090	. . . . 5 ⊢ (0 <ℝ ⟨𝑥, 0R⟩ ↔ ⟨0R, 0R⟩ <ℝ ⟨𝑥, 0R⟩)
15		ltresr 8042	. . . . 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 4090	. . . . 5 ⊢ (0 <ℝ ⟨𝑦, 0R⟩ ↔ ⟨0R, 0R⟩ <ℝ ⟨𝑦, 0R⟩)
18		ltresr 8042	. . . . 5 ⊢ (⟨0R, 0R⟩ <ℝ ⟨𝑦, 0R⟩ ↔ 0R <R 𝑦)
19	17, 18	bitri 184	. . . 4 ⊢ (0 <ℝ ⟨𝑦, 0R⟩ ↔ 0R <R 𝑦)
20		mulgt0sr 7981	. . . 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 8041	. . . . 5 ⊢ ((𝑥 ∈ R ∧ 𝑦 ∈ R) → (⟨𝑥, 0R⟩ · ⟨𝑦, 0R⟩) = ⟨(𝑥 ·R 𝑦), 0R⟩)
24	22, 23	breq12d 4096	. . . 4 ⊢ ((𝑥 ∈ R ∧ 𝑦 ∈ R) → (0 <ℝ (⟨𝑥, 0R⟩ · ⟨𝑦, 0R⟩) ↔ ⟨0R, 0R⟩ <ℝ ⟨(𝑥 ·R 𝑦), 0R⟩))
25		ltresr 8042	. . . 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 2833	1 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((0 <ℝ 𝐴 ∧ 0 <ℝ 𝐵) → 0 <ℝ (𝐴 · 𝐵)))

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1395   ∈ wcel 2200  ⟨cop 3669   class class class wbr 4083  (class class class)co 6010  Rcnr 7500  0_Rc0r 7501   ·_R cmr 7505   <_R cltr 7506  ℝcr 8014  0cc0 8015   <_ℝ cltrr 8019   · cmul 8020

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4199  ax-sep 4202  ax-nul 4210  ax-pow 4259  ax-pr 4294  ax-un 4525  ax-setind 4630  ax-iinf 4681

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-tr 4183  df-eprel 4381  df-id 4385  df-po 4388  df-iso 4389  df-iord 4458  df-on 4460  df-suc 4463  df-iom 4684  df-xp 4726  df-rel 4727  df-cnv 4728  df-co 4729  df-dm 4730  df-rn 4731  df-res 4732  df-ima 4733  df-iota 5281  df-fun 5323  df-fn 5324  df-f 5325  df-f1 5326  df-fo 5327  df-f1o 5328  df-fv 5329  df-ov 6013  df-oprab 6014  df-mpo 6015  df-1st 6295  df-2nd 6296  df-recs 6462  df-irdg 6527  df-1o 6573  df-2o 6574  df-oadd 6577  df-omul 6578  df-er 6693  df-ec 6695  df-qs 6699  df-ni 7507  df-pli 7508  df-mi 7509  df-lti 7510  df-plpq 7547  df-mpq 7548  df-enq 7550  df-nqqs 7551  df-plqqs 7552  df-mqqs 7553  df-1nqqs 7554  df-rq 7555  df-ltnqqs 7556  df-enq0 7627  df-nq0 7628  df-0nq0 7629  df-plq0 7630  df-mq0 7631  df-inp 7669  df-i1p 7670  df-iplp 7671  df-imp 7672  df-iltp 7673  df-enr 7929  df-nr 7930  df-plr 7931  df-mr 7932  df-ltr 7933  df-0r 7934  df-m1r 7936  df-c 8021  df-0 8022  df-r 8025  df-mul 8027  df-lt 8028

This theorem is referenced by: (None)