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Theorem axpre-mulgt0 10312
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 10438. This construction-dependent theorem should not be referenced directly; instead, use ax-pre-mulgt0 10336. (Contributed by NM, 13-May-1996.) (New usage is discouraged.)
Assertion
Ref Expression
axpre-mulgt0 ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((0 < 𝐴 ∧ 0 < 𝐵) → 0 < (𝐴 · 𝐵)))

Proof of Theorem axpre-mulgt0
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elreal 10275 . 2 (𝐴 ∈ ℝ ↔ ∃𝑥R𝑥, 0R⟩ = 𝐴)
2 elreal 10275 . 2 (𝐵 ∈ ℝ ↔ ∃𝑦R𝑦, 0R⟩ = 𝐵)
3 breq2 4879 . . . 4 (⟨𝑥, 0R⟩ = 𝐴 → (0 <𝑥, 0R⟩ ↔ 0 < 𝐴))
43anbi1d 623 . . 3 (⟨𝑥, 0R⟩ = 𝐴 → ((0 <𝑥, 0R⟩ ∧ 0 <𝑦, 0R⟩) ↔ (0 < 𝐴 ∧ 0 <𝑦, 0R⟩)))
5 oveq1 6917 . . . 4 (⟨𝑥, 0R⟩ = 𝐴 → (⟨𝑥, 0R⟩ · ⟨𝑦, 0R⟩) = (𝐴 · ⟨𝑦, 0R⟩))
65breq2d 4887 . . 3 (⟨𝑥, 0R⟩ = 𝐴 → (0 < (⟨𝑥, 0R⟩ · ⟨𝑦, 0R⟩) ↔ 0 < (𝐴 · ⟨𝑦, 0R⟩)))
74, 6imbi12d 336 . 2 (⟨𝑥, 0R⟩ = 𝐴 → (((0 <𝑥, 0R⟩ ∧ 0 <𝑦, 0R⟩) → 0 < (⟨𝑥, 0R⟩ · ⟨𝑦, 0R⟩)) ↔ ((0 < 𝐴 ∧ 0 <𝑦, 0R⟩) → 0 < (𝐴 · ⟨𝑦, 0R⟩))))
8 breq2 4879 . . . 4 (⟨𝑦, 0R⟩ = 𝐵 → (0 <𝑦, 0R⟩ ↔ 0 < 𝐵))
98anbi2d 622 . . 3 (⟨𝑦, 0R⟩ = 𝐵 → ((0 < 𝐴 ∧ 0 <𝑦, 0R⟩) ↔ (0 < 𝐴 ∧ 0 < 𝐵)))
10 oveq2 6918 . . . 4 (⟨𝑦, 0R⟩ = 𝐵 → (𝐴 · ⟨𝑦, 0R⟩) = (𝐴 · 𝐵))
1110breq2d 4887 . . 3 (⟨𝑦, 0R⟩ = 𝐵 → (0 < (𝐴 · ⟨𝑦, 0R⟩) ↔ 0 < (𝐴 · 𝐵)))
129, 11imbi12d 336 . 2 (⟨𝑦, 0R⟩ = 𝐵 → (((0 < 𝐴 ∧ 0 <𝑦, 0R⟩) → 0 < (𝐴 · ⟨𝑦, 0R⟩)) ↔ ((0 < 𝐴 ∧ 0 < 𝐵) → 0 < (𝐴 · 𝐵))))
13 df-0 10266 . . . . . 6 0 = ⟨0R, 0R
1413breq1i 4882 . . . . 5 (0 <𝑥, 0R⟩ ↔ ⟨0R, 0R⟩ <𝑥, 0R⟩)
15 ltresr 10284 . . . . 5 (⟨0R, 0R⟩ <𝑥, 0R⟩ ↔ 0R <R 𝑥)
1614, 15bitri 267 . . . 4 (0 <𝑥, 0R⟩ ↔ 0R <R 𝑥)
1713breq1i 4882 . . . . 5 (0 <𝑦, 0R⟩ ↔ ⟨0R, 0R⟩ <𝑦, 0R⟩)
18 ltresr 10284 . . . . 5 (⟨0R, 0R⟩ <𝑦, 0R⟩ ↔ 0R <R 𝑦)
1917, 18bitri 267 . . . 4 (0 <𝑦, 0R⟩ ↔ 0R <R 𝑦)
20 mulgt0sr 10249 . . . 4 ((0R <R 𝑥 ∧ 0R <R 𝑦) → 0R <R (𝑥 ·R 𝑦))
2116, 19, 20syl2anb 591 . . 3 ((0 <𝑥, 0R⟩ ∧ 0 <𝑦, 0R⟩) → 0R <R (𝑥 ·R 𝑦))
2213a1i 11 . . . . 5 ((𝑥R𝑦R) → 0 = ⟨0R, 0R⟩)
23 mulresr 10283 . . . . 5 ((𝑥R𝑦R) → (⟨𝑥, 0R⟩ · ⟨𝑦, 0R⟩) = ⟨(𝑥 ·R 𝑦), 0R⟩)
2422, 23breq12d 4888 . . . 4 ((𝑥R𝑦R) → (0 < (⟨𝑥, 0R⟩ · ⟨𝑦, 0R⟩) ↔ ⟨0R, 0R⟩ < ⟨(𝑥 ·R 𝑦), 0R⟩))
25 ltresr 10284 . . . 4 (⟨0R, 0R⟩ < ⟨(𝑥 ·R 𝑦), 0R⟩ ↔ 0R <R (𝑥 ·R 𝑦))
2624, 25syl6bb 279 . . 3 ((𝑥R𝑦R) → (0 < (⟨𝑥, 0R⟩ · ⟨𝑦, 0R⟩) ↔ 0R <R (𝑥 ·R 𝑦)))
2721, 26syl5ibr 238 . 2 ((𝑥R𝑦R) → ((0 <𝑥, 0R⟩ ∧ 0 <𝑦, 0R⟩) → 0 < (⟨𝑥, 0R⟩ · ⟨𝑦, 0R⟩)))
281, 2, 7, 12, 272gencl 3453 1 ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((0 < 𝐴 ∧ 0 < 𝐵) → 0 < (𝐴 · 𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 386   = wceq 1656  wcel 2164  cop 4405   class class class wbr 4875  (class class class)co 6910  Rcnr 10009  0Rc0r 10010   ·R cmr 10014   <R cltr 10015  cr 10258  0cc0 10259   < cltrr 10263   · cmul 10264
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-8 2166  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-sep 5007  ax-nul 5015  ax-pow 5067  ax-pr 5129  ax-un 7214  ax-inf2 8822
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3or 1112  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-ral 3122  df-rex 3123  df-reu 3124  df-rmo 3125  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-pss 3814  df-nul 4147  df-if 4309  df-pw 4382  df-sn 4400  df-pr 4402  df-tp 4404  df-op 4406  df-uni 4661  df-int 4700  df-iun 4744  df-br 4876  df-opab 4938  df-mpt 4955  df-tr 4978  df-id 5252  df-eprel 5257  df-po 5265  df-so 5266  df-fr 5305  df-we 5307  df-xp 5352  df-rel 5353  df-cnv 5354  df-co 5355  df-dm 5356  df-rn 5357  df-res 5358  df-ima 5359  df-pred 5924  df-ord 5970  df-on 5971  df-lim 5972  df-suc 5973  df-iota 6090  df-fun 6129  df-fn 6130  df-f 6131  df-f1 6132  df-fo 6133  df-f1o 6134  df-fv 6135  df-ov 6913  df-oprab 6914  df-mpt2 6915  df-om 7332  df-1st 7433  df-2nd 7434  df-wrecs 7677  df-recs 7739  df-rdg 7777  df-1o 7831  df-oadd 7835  df-omul 7836  df-er 8014  df-ec 8016  df-qs 8020  df-ni 10016  df-pli 10017  df-mi 10018  df-lti 10019  df-plpq 10052  df-mpq 10053  df-ltpq 10054  df-enq 10055  df-nq 10056  df-erq 10057  df-plq 10058  df-mq 10059  df-1nq 10060  df-rq 10061  df-ltnq 10062  df-np 10125  df-1p 10126  df-plp 10127  df-mp 10128  df-ltp 10129  df-enr 10199  df-nr 10200  df-plr 10201  df-mr 10202  df-ltr 10203  df-0r 10204  df-m1r 10206  df-c 10265  df-0 10266  df-r 10269  df-mul 10271  df-lt 10272
This theorem is referenced by: (None)
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