Theorem axpre-mulgt0 11081

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 11208. This construction-dependent theorem should not be referenced directly; instead, use ax-pre-mulgt0 11105. (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 11044	. 2 ⊢ (𝐴 ∈ ℝ ↔ ∃𝑥 ∈ R ⟨𝑥, 0R⟩ = 𝐴)
2		elreal 11044	. 2 ⊢ (𝐵 ∈ ℝ ↔ ∃𝑦 ∈ R ⟨𝑦, 0R⟩ = 𝐵)
3		breq2 5099	. . . 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 7360	. . . 4 ⊢ (⟨𝑥, 0R⟩ = 𝐴 → (⟨𝑥, 0R⟩ · ⟨𝑦, 0R⟩) = (𝐴 · ⟨𝑦, 0R⟩))
6	5	breq2d 5107	. . 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 5099	. . . 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 7361	. . . 4 ⊢ (⟨𝑦, 0R⟩ = 𝐵 → (𝐴 · ⟨𝑦, 0R⟩) = (𝐴 · 𝐵))
11	10	breq2d 5107	. . 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 11035	. . . . . 6 ⊢ 0 = ⟨0R, 0R⟩
14	13	breq1i 5102	. . . . 5 ⊢ (0 <ℝ ⟨𝑥, 0R⟩ ↔ ⟨0R, 0R⟩ <ℝ ⟨𝑥, 0R⟩)
15		ltresr 11053	. . . . 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 5102	. . . . 5 ⊢ (0 <ℝ ⟨𝑦, 0R⟩ ↔ ⟨0R, 0R⟩ <ℝ ⟨𝑦, 0R⟩)
18		ltresr 11053	. . . . 5 ⊢ (⟨0R, 0R⟩ <ℝ ⟨𝑦, 0R⟩ ↔ 0R <R 𝑦)
19	17, 18	bitri 275	. . . 4 ⊢ (0 <ℝ ⟨𝑦, 0R⟩ ↔ 0R <R 𝑦)
20		mulgt0sr 11018	. . . 4 ⊢ ((0R <R 𝑥 ∧ 0R <R 𝑦) → 0R <R (𝑥 ·R 𝑦))
21	16, 19, 20	syl2anb 598	. . 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 11052	. . . . 5 ⊢ ((𝑥 ∈ R ∧ 𝑦 ∈ R) → (⟨𝑥, 0R⟩ · ⟨𝑦, 0R⟩) = ⟨(𝑥 ·R 𝑦), 0R⟩)
24	22, 23	breq12d 5108	. . . 4 ⊢ ((𝑥 ∈ R ∧ 𝑦 ∈ R) → (0 <ℝ (⟨𝑥, 0R⟩ · ⟨𝑦, 0R⟩) ↔ ⟨0R, 0R⟩ <ℝ ⟨(𝑥 ·R 𝑦), 0R⟩))
25		ltresr 11053	. . . 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 246	. 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 3481	1 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((0 <ℝ 𝐴 ∧ 0 <ℝ 𝐵) → 0 <ℝ (𝐴 · 𝐵)))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ⟨cop 4585   class class class wbr 5095  (class class class)co 7353  Rcnr 10778  0_Rc0r 10779   ·_R cmr 10783   <_R cltr 10784  ℝcr 11027  0cc0 11028   <_ℝ cltrr 11032   · cmul 11033

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374  ax-un 7675  ax-inf2 9556

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rmo 3345  df-reu 3346  df-rab 3397  df-v 3440  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4862  df-int 4900  df-iun 4946  df-br 5096  df-opab 5158  df-mpt 5177  df-tr 5203  df-id 5518  df-eprel 5523  df-po 5531  df-so 5532  df-fr 5576  df-we 5578  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  df-pred 6253  df-ord 6314  df-on 6315  df-lim 6316  df-suc 6317  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-ov 7356  df-oprab 7357  df-mpo 7358  df-om 7807  df-1st 7931  df-2nd 7932  df-frecs 8221  df-wrecs 8252  df-recs 8301  df-rdg 8339  df-1o 8395  df-oadd 8399  df-omul 8400  df-er 8632  df-ec 8634  df-qs 8638  df-ni 10785  df-pli 10786  df-mi 10787  df-lti 10788  df-plpq 10821  df-mpq 10822  df-ltpq 10823  df-enq 10824  df-nq 10825  df-erq 10826  df-plq 10827  df-mq 10828  df-1nq 10829  df-rq 10830  df-ltnq 10831  df-np 10894  df-1p 10895  df-plp 10896  df-mp 10897  df-ltp 10898  df-enr 10968  df-nr 10969  df-plr 10970  df-mr 10971  df-ltr 10972  df-0r 10973  df-m1r 10975  df-c 11034  df-0 11035  df-r 11038  df-mul 11040  df-lt 11041

This theorem is referenced by: (None)