Theorem axpre-mulgt0 11079

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 11207. This construction-dependent theorem should not be referenced directly; instead, use ax-pre-mulgt0 11103. (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 11042	. 2 ⊢ (𝐴 ∈ ℝ ↔ ∃𝑥 ∈ R ⟨𝑥, 0R⟩ = 𝐴)
2		elreal 11042	. 2 ⊢ (𝐵 ∈ ℝ ↔ ∃𝑦 ∈ R ⟨𝑦, 0R⟩ = 𝐵)
3		breq2 5102	. . . 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 7365	. . . 4 ⊢ (⟨𝑥, 0R⟩ = 𝐴 → (⟨𝑥, 0R⟩ · ⟨𝑦, 0R⟩) = (𝐴 · ⟨𝑦, 0R⟩))
6	5	breq2d 5110	. . 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 5102	. . . 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 7366	. . . 4 ⊢ (⟨𝑦, 0R⟩ = 𝐵 → (𝐴 · ⟨𝑦, 0R⟩) = (𝐴 · 𝐵))
11	10	breq2d 5110	. . 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 11033	. . . . . 6 ⊢ 0 = ⟨0R, 0R⟩
14	13	breq1i 5105	. . . . 5 ⊢ (0 <ℝ ⟨𝑥, 0R⟩ ↔ ⟨0R, 0R⟩ <ℝ ⟨𝑥, 0R⟩)
15		ltresr 11051	. . . . 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 5105	. . . . 5 ⊢ (0 <ℝ ⟨𝑦, 0R⟩ ↔ ⟨0R, 0R⟩ <ℝ ⟨𝑦, 0R⟩)
18		ltresr 11051	. . . . 5 ⊢ (⟨0R, 0R⟩ <ℝ ⟨𝑦, 0R⟩ ↔ 0R <R 𝑦)
19	17, 18	bitri 275	. . . 4 ⊢ (0 <ℝ ⟨𝑦, 0R⟩ ↔ 0R <R 𝑦)
20		mulgt0sr 11016	. . . 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 11050	. . . . 5 ⊢ ((𝑥 ∈ R ∧ 𝑦 ∈ R) → (⟨𝑥, 0R⟩ · ⟨𝑦, 0R⟩) = ⟨(𝑥 ·R 𝑦), 0R⟩)
24	22, 23	breq12d 5111	. . . 4 ⊢ ((𝑥 ∈ R ∧ 𝑦 ∈ R) → (0 <ℝ (⟨𝑥, 0R⟩ · ⟨𝑦, 0R⟩) ↔ ⟨0R, 0R⟩ <ℝ ⟨(𝑥 ·R 𝑦), 0R⟩))
25		ltresr 11051	. . . 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 3483	1 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((0 <ℝ 𝐴 ∧ 0 <ℝ 𝐵) → 0 <ℝ (𝐴 · 𝐵)))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2113  ⟨cop 4586   class class class wbr 5098  (class class class)co 7358  Rcnr 10776  0_Rc0r 10777   ·_R cmr 10781   <_R cltr 10782  ℝcr 11025  0cc0 11026   <_ℝ cltrr 11030   · cmul 11031

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 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377  ax-un 7680  ax-inf2 9550

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rmo 3350  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-pss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-int 4903  df-iun 4948  df-br 5099  df-opab 5161  df-mpt 5180  df-tr 5206  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-ov 7361  df-oprab 7362  df-mpo 7363  df-om 7809  df-1st 7933  df-2nd 7934  df-frecs 8223  df-wrecs 8254  df-recs 8303  df-rdg 8341  df-1o 8397  df-oadd 8401  df-omul 8402  df-er 8635  df-ec 8637  df-qs 8641  df-ni 10783  df-pli 10784  df-mi 10785  df-lti 10786  df-plpq 10819  df-mpq 10820  df-ltpq 10821  df-enq 10822  df-nq 10823  df-erq 10824  df-plq 10825  df-mq 10826  df-1nq 10827  df-rq 10828  df-ltnq 10829  df-np 10892  df-1p 10893  df-plp 10894  df-mp 10895  df-ltp 10896  df-enr 10966  df-nr 10967  df-plr 10968  df-mr 10969  df-ltr 10970  df-0r 10971  df-m1r 10973  df-c 11032  df-0 11033  df-r 11036  df-mul 11038  df-lt 11039

This theorem is referenced by: (None)