Theorem axpre-mulext 6577

Description: Strong extensionality of multiplication (expressed in terms of <_ℝ). Axiom for real and complex numbers, derived from set theory. This construction-dependent theorem should not be referenced directly; instead, use ax-pre-mulext 6605.

(Contributed by Jim Kingdon, 18-Feb-2020.) (New usage is discouraged.)

Assertion

Ref	Expression
axpre-mulext	⊢ ((A ∈ ℝ ∧ B ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((A · 𝐶) <ℝ (B · 𝐶) → (A <ℝ B ∨ B <ℝ A)))

Proof of Theorem axpre-mulext

Dummy variables x y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elreal 6536	. 2 ⊢ (A ∈ ℝ ↔ ∃x ∈ R ⟨x, 0R⟩ = A)
2		elreal 6536	. 2 ⊢ (B ∈ ℝ ↔ ∃y ∈ R ⟨y, 0R⟩ = B)
3		elreal 6536	. 2 ⊢ (𝐶 ∈ ℝ ↔ ∃z ∈ R ⟨z, 0R⟩ = 𝐶)
4		oveq1 5431	. . . 4 ⊢ (⟨x, 0R⟩ = A → (⟨x, 0R⟩ · ⟨z, 0R⟩) = (A · ⟨z, 0R⟩))
5	4	breq1d 3737	. . 3 ⊢ (⟨x, 0R⟩ = A → ((⟨x, 0R⟩ · ⟨z, 0R⟩) <ℝ (⟨y, 0R⟩ · ⟨z, 0R⟩) ↔ (A · ⟨z, 0R⟩) <ℝ (⟨y, 0R⟩ · ⟨z, 0R⟩)))
6		breq1 3730	. . . 4 ⊢ (⟨x, 0R⟩ = A → (⟨x, 0R⟩ <ℝ ⟨y, 0R⟩ ↔ A <ℝ ⟨y, 0R⟩))
7		breq2 3731	. . . 4 ⊢ (⟨x, 0R⟩ = A → (⟨y, 0R⟩ <ℝ ⟨x, 0R⟩ ↔ ⟨y, 0R⟩ <ℝ A))
8	6, 7	orbi12d 691	. . 3 ⊢ (⟨x, 0R⟩ = A → ((⟨x, 0R⟩ <ℝ ⟨y, 0R⟩ ∨ ⟨y, 0R⟩ <ℝ ⟨x, 0R⟩) ↔ (A <ℝ ⟨y, 0R⟩ ∨ ⟨y, 0R⟩ <ℝ A)))
9	5, 8	imbi12d 223	. 2 ⊢ (⟨x, 0R⟩ = A → (((⟨x, 0R⟩ · ⟨z, 0R⟩) <ℝ (⟨y, 0R⟩ · ⟨z, 0R⟩) → (⟨x, 0R⟩ <ℝ ⟨y, 0R⟩ ∨ ⟨y, 0R⟩ <ℝ ⟨x, 0R⟩)) ↔ ((A · ⟨z, 0R⟩) <ℝ (⟨y, 0R⟩ · ⟨z, 0R⟩) → (A <ℝ ⟨y, 0R⟩ ∨ ⟨y, 0R⟩ <ℝ A))))
10		oveq1 5431	. . . 4 ⊢ (⟨y, 0R⟩ = B → (⟨y, 0R⟩ · ⟨z, 0R⟩) = (B · ⟨z, 0R⟩))
11	10	breq2d 3739	. . 3 ⊢ (⟨y, 0R⟩ = B → ((A · ⟨z, 0R⟩) <ℝ (⟨y, 0R⟩ · ⟨z, 0R⟩) ↔ (A · ⟨z, 0R⟩) <ℝ (B · ⟨z, 0R⟩)))
12		breq2 3731	. . . 4 ⊢ (⟨y, 0R⟩ = B → (A <ℝ ⟨y, 0R⟩ ↔ A <ℝ B))
13		breq1 3730	. . . 4 ⊢ (⟨y, 0R⟩ = B → (⟨y, 0R⟩ <ℝ A ↔ B <ℝ A))
14	12, 13	orbi12d 691	. . 3 ⊢ (⟨y, 0R⟩ = B → ((A <ℝ ⟨y, 0R⟩ ∨ ⟨y, 0R⟩ <ℝ A) ↔ (A <ℝ B ∨ B <ℝ A)))
15	11, 14	imbi12d 223	. 2 ⊢ (⟨y, 0R⟩ = B → (((A · ⟨z, 0R⟩) <ℝ (⟨y, 0R⟩ · ⟨z, 0R⟩) → (A <ℝ ⟨y, 0R⟩ ∨ ⟨y, 0R⟩ <ℝ A)) ↔ ((A · ⟨z, 0R⟩) <ℝ (B · ⟨z, 0R⟩) → (A <ℝ B ∨ B <ℝ A))))
16		oveq2 5432	. . . 4 ⊢ (⟨z, 0R⟩ = 𝐶 → (A · ⟨z, 0R⟩) = (A · 𝐶))
17		oveq2 5432	. . . 4 ⊢ (⟨z, 0R⟩ = 𝐶 → (B · ⟨z, 0R⟩) = (B · 𝐶))
18	16, 17	breq12d 3740	. . 3 ⊢ (⟨z, 0R⟩ = 𝐶 → ((A · ⟨z, 0R⟩) <ℝ (B · ⟨z, 0R⟩) ↔ (A · 𝐶) <ℝ (B · 𝐶)))
19	18	imbi1d 220	. 2 ⊢ (⟨z, 0R⟩ = 𝐶 → (((A · ⟨z, 0R⟩) <ℝ (B · ⟨z, 0R⟩) → (A <ℝ B ∨ B <ℝ A)) ↔ ((A · 𝐶) <ℝ (B · 𝐶) → (A <ℝ B ∨ B <ℝ A))))
20		mulextsr1 6517	. . 3 ⊢ ((x ∈ R ∧ y ∈ R ∧ z ∈ R) → ((x ·R z) <R (y ·R z) → (x <R y ∨ y <R x)))
21		mulresr 6544	. . . . . 6 ⊢ ((x ∈ R ∧ z ∈ R) → (⟨x, 0R⟩ · ⟨z, 0R⟩) = ⟨(x ·R z), 0R⟩)
22	21	3adant2 905	. . . . 5 ⊢ ((x ∈ R ∧ y ∈ R ∧ z ∈ R) → (⟨x, 0R⟩ · ⟨z, 0R⟩) = ⟨(x ·R z), 0R⟩)
23		mulresr 6544	. . . . . 6 ⊢ ((y ∈ R ∧ z ∈ R) → (⟨y, 0R⟩ · ⟨z, 0R⟩) = ⟨(y ·R z), 0R⟩)
24	23	3adant1 904	. . . . 5 ⊢ ((x ∈ R ∧ y ∈ R ∧ z ∈ R) → (⟨y, 0R⟩ · ⟨z, 0R⟩) = ⟨(y ·R z), 0R⟩)
25	22, 24	breq12d 3740	. . . 4 ⊢ ((x ∈ R ∧ y ∈ R ∧ z ∈ R) → ((⟨x, 0R⟩ · ⟨z, 0R⟩) <ℝ (⟨y, 0R⟩ · ⟨z, 0R⟩) ↔ ⟨(x ·R z), 0R⟩ <ℝ ⟨(y ·R z), 0R⟩))
26		ltresr 6545	. . . 4 ⊢ (⟨(x ·R z), 0R⟩ <ℝ ⟨(y ·R z), 0R⟩ ↔ (x ·R z) <R (y ·R z))
27	25, 26	syl6bb 185	. . 3 ⊢ ((x ∈ R ∧ y ∈ R ∧ z ∈ R) → ((⟨x, 0R⟩ · ⟨z, 0R⟩) <ℝ (⟨y, 0R⟩ · ⟨z, 0R⟩) ↔ (x ·R z) <R (y ·R z)))
28		ltresr 6545	. . . . 5 ⊢ (⟨x, 0R⟩ <ℝ ⟨y, 0R⟩ ↔ x <R y)
29		ltresr 6545	. . . . 5 ⊢ (⟨y, 0R⟩ <ℝ ⟨x, 0R⟩ ↔ y <R x)
30	28, 29	orbi12i 665	. . . 4 ⊢ ((⟨x, 0R⟩ <ℝ ⟨y, 0R⟩ ∨ ⟨y, 0R⟩ <ℝ ⟨x, 0R⟩) ↔ (x <R y ∨ y <R x))
31	30	a1i 9	. . 3 ⊢ ((x ∈ R ∧ y ∈ R ∧ z ∈ R) → ((⟨x, 0R⟩ <ℝ ⟨y, 0R⟩ ∨ ⟨y, 0R⟩ <ℝ ⟨x, 0R⟩) ↔ (x <R y ∨ y <R x)))
32	20, 27, 31	3imtr4d 192	. 2 ⊢ ((x ∈ R ∧ y ∈ R ∧ z ∈ R) → ((⟨x, 0R⟩ · ⟨z, 0R⟩) <ℝ (⟨y, 0R⟩ · ⟨z, 0R⟩) → (⟨x, 0R⟩ <ℝ ⟨y, 0R⟩ ∨ ⟨y, 0R⟩ <ℝ ⟨x, 0R⟩)))
33	1, 2, 3, 9, 15, 19, 32	3gencl 2556	1 ⊢ ((A ∈ ℝ ∧ B ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((A · 𝐶) <ℝ (B · 𝐶) → (A <ℝ B ∨ B <ℝ A)))

Syntax hints:   → wi 4   ↔ wb 98   ∨ wo 613   ∧ w3a 867   = wceq 1223   ∈ wcel 1366  ⟨cop 3342   class class class wbr 3727  (class class class)co 5424  Rcnr 6143  0_Rc0r 6144   ·_R cmr 6148   <_R cltr 6149  ℝcr 6519   <_ℝ cltrr 6524   · cmul 6525

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 529  ax-in2 530  ax-io 614  ax-5 1309  ax-7 1310  ax-gen 1311  ax-ie1 1355  ax-ie2 1356  ax-8 1368  ax-10 1369  ax-11 1370  ax-i12 1371  ax-bnd 1372  ax-4 1373  ax-13 1377  ax-14 1378  ax-17 1392  ax-i9 1396  ax-ial 1400  ax-i5r 1401  ax-ext 1995  ax-coll 3835  ax-sep 3838  ax-nul 3846  ax-pow 3890  ax-pr 3907  ax-un 4108  ax-setind 4192  ax-iinf 4226

This theorem depends on definitions:  df-bi 110  df-dc 727  df-3or 868  df-3an 869  df-tru 1226  df-fal 1229  df-nf 1323  df-sb 1619  df-eu 1876  df-mo 1877  df-clab 2000  df-cleq 2006  df-clel 2009  df-nfc 2140  df-ne 2179  df-ral 2280  df-rex 2281  df-reu 2282  df-rab 2284  df-v 2528  df-sbc 2733  df-csb 2821  df-dif 2888  df-un 2890  df-in 2892  df-ss 2899  df-nul 3193  df-pw 3325  df-sn 3345  df-pr 3346  df-op 3348  df-uni 3544  df-int 3579  df-iun 3622  df-br 3728  df-opab 3782  df-mpt 3783  df-tr 3818  df-eprel 3989  df-id 3993  df-po 3996  df-iso 3997  df-iord 4041  df-on 4043  df-suc 4046  df-iom 4229  df-xp 4266  df-rel 4267  df-cnv 4268  df-co 4269  df-dm 4270  df-rn 4271  df-res 4272  df-ima 4273  df-iota 4782  df-fun 4819  df-fn 4820  df-f 4821  df-f1 4822  df-fo 4823  df-f1o 4824  df-fv 4825  df-ov 5427  df-oprab 5428  df-mpt2 5429  df-1st 5678  df-2nd 5679  df-recs 5830  df-irdg 5866  df-1o 5904  df-2o 5905  df-oadd 5908  df-omul 5909  df-er 6005  df-ec 6007  df-qs 6011  df-ni 6150  df-pli 6151  df-mi 6152  df-lti 6153  df-plpq 6189  df-mpq 6190  df-enq 6192  df-nqqs 6193  df-plqqs 6194  df-mqqs 6195  df-1nqqs 6196  df-rq 6197  df-ltnqqs 6198  df-enq0 6265  df-nq0 6266  df-0nq0 6267  df-plq0 6268  df-mq0 6269  df-inp 6306  df-i1p 6307  df-iplp 6308  df-imp 6309  df-iltp 6310  df-enr 6464  df-nr 6465  df-plr 6466  df-mr 6467  df-ltr 6468  df-0r 6469  df-m1r 6471  df-c 6526  df-r 6530  df-mul 6532  df-lt 6533

This theorem is referenced by: (None)