Theorem axpre-mulgt0 7888

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 7930. (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 7829	. 2 ⊢ (𝐴 ∈ ℝ ↔ ∃𝑥 ∈ R ⟨𝑥, 0R⟩ = 𝐴)
2		elreal 7829	. 2 ⊢ (𝐵 ∈ ℝ ↔ ∃𝑦 ∈ R ⟨𝑦, 0R⟩ = 𝐵)
3		breq2 4009	. . . 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 5884	. . . 4 ⊢ (⟨𝑥, 0R⟩ = 𝐴 → (⟨𝑥, 0R⟩ · ⟨𝑦, 0R⟩) = (𝐴 · ⟨𝑦, 0R⟩))
6	5	breq2d 4017	. . 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 4009	. . . 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 5885	. . . 4 ⊢ (⟨𝑦, 0R⟩ = 𝐵 → (𝐴 · ⟨𝑦, 0R⟩) = (𝐴 · 𝐵))
11	10	breq2d 4017	. . 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 7820	. . . . . 6 ⊢ 0 = ⟨0R, 0R⟩
14	13	breq1i 4012	. . . . 5 ⊢ (0 <ℝ ⟨𝑥, 0R⟩ ↔ ⟨0R, 0R⟩ <ℝ ⟨𝑥, 0R⟩)
15		ltresr 7840	. . . . 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 4012	. . . . 5 ⊢ (0 <ℝ ⟨𝑦, 0R⟩ ↔ ⟨0R, 0R⟩ <ℝ ⟨𝑦, 0R⟩)
18		ltresr 7840	. . . . 5 ⊢ (⟨0R, 0R⟩ <ℝ ⟨𝑦, 0R⟩ ↔ 0R <R 𝑦)
19	17, 18	bitri 184	. . . 4 ⊢ (0 <ℝ ⟨𝑦, 0R⟩ ↔ 0R <R 𝑦)
20		mulgt0sr 7779	. . . 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 7839	. . . . 5 ⊢ ((𝑥 ∈ R ∧ 𝑦 ∈ R) → (⟨𝑥, 0R⟩ · ⟨𝑦, 0R⟩) = ⟨(𝑥 ·R 𝑦), 0R⟩)
24	22, 23	breq12d 4018	. . . 4 ⊢ ((𝑥 ∈ R ∧ 𝑦 ∈ R) → (0 <ℝ (⟨𝑥, 0R⟩ · ⟨𝑦, 0R⟩) ↔ ⟨0R, 0R⟩ <ℝ ⟨(𝑥 ·R 𝑦), 0R⟩))
25		ltresr 7840	. . . 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 2772	1 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((0 <ℝ 𝐴 ∧ 0 <ℝ 𝐵) → 0 <ℝ (𝐴 · 𝐵)))

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1353   ∈ wcel 2148  ⟨cop 3597   class class class wbr 4005  (class class class)co 5877  Rcnr 7298  0_Rc0r 7299   ·_R cmr 7303   <_R cltr 7304  ℝcr 7812  0cc0 7813   <_ℝ cltrr 7817   · cmul 7818

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4120  ax-sep 4123  ax-nul 4131  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538  ax-iinf 4589

This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2741  df-sbc 2965  df-csb 3060  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-nul 3425  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-int 3847  df-iun 3890  df-br 4006  df-opab 4067  df-mpt 4068  df-tr 4104  df-eprel 4291  df-id 4295  df-po 4298  df-iso 4299  df-iord 4368  df-on 4370  df-suc 4373  df-iom 4592  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-f1 5223  df-fo 5224  df-f1o 5225  df-fv 5226  df-ov 5880  df-oprab 5881  df-mpo 5882  df-1st 6143  df-2nd 6144  df-recs 6308  df-irdg 6373  df-1o 6419  df-2o 6420  df-oadd 6423  df-omul 6424  df-er 6537  df-ec 6539  df-qs 6543  df-ni 7305  df-pli 7306  df-mi 7307  df-lti 7308  df-plpq 7345  df-mpq 7346  df-enq 7348  df-nqqs 7349  df-plqqs 7350  df-mqqs 7351  df-1nqqs 7352  df-rq 7353  df-ltnqqs 7354  df-enq0 7425  df-nq0 7426  df-0nq0 7427  df-plq0 7428  df-mq0 7429  df-inp 7467  df-i1p 7468  df-iplp 7469  df-imp 7470  df-iltp 7471  df-enr 7727  df-nr 7728  df-plr 7729  df-mr 7730  df-ltr 7731  df-0r 7732  df-m1r 7734  df-c 7819  df-0 7820  df-r 7823  df-mul 7825  df-lt 7826

This theorem is referenced by: (None)