Theorem axpre-mulext 8108

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 8150.

(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 8048	. 2 ⊢ (𝐴 ∈ ℝ ↔ ∃𝑥 ∈ R ⟨𝑥, 0R⟩ = 𝐴)
2		elreal 8048	. 2 ⊢ (𝐵 ∈ ℝ ↔ ∃𝑦 ∈ R ⟨𝑦, 0R⟩ = 𝐵)
3		elreal 8048	. 2 ⊢ (𝐶 ∈ ℝ ↔ ∃𝑧 ∈ R ⟨𝑧, 0R⟩ = 𝐶)
4		oveq1 6025	. . . 4 ⊢ (⟨𝑥, 0R⟩ = 𝐴 → (⟨𝑥, 0R⟩ · ⟨𝑧, 0R⟩) = (𝐴 · ⟨𝑧, 0R⟩))
5	4	breq1d 4098	. . 3 ⊢ (⟨𝑥, 0R⟩ = 𝐴 → ((⟨𝑥, 0R⟩ · ⟨𝑧, 0R⟩) <ℝ (⟨𝑦, 0R⟩ · ⟨𝑧, 0R⟩) ↔ (𝐴 · ⟨𝑧, 0R⟩) <ℝ (⟨𝑦, 0R⟩ · ⟨𝑧, 0R⟩)))
6		breq1 4091	. . . 4 ⊢ (⟨𝑥, 0R⟩ = 𝐴 → (⟨𝑥, 0R⟩ <ℝ ⟨𝑦, 0R⟩ ↔ 𝐴 <ℝ ⟨𝑦, 0R⟩))
7		breq2 4092	. . . 4 ⊢ (⟨𝑥, 0R⟩ = 𝐴 → (⟨𝑦, 0R⟩ <ℝ ⟨𝑥, 0R⟩ ↔ ⟨𝑦, 0R⟩ <ℝ 𝐴))
8	6, 7	orbi12d 800	. . 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 234	. 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 6025	. . . 4 ⊢ (⟨𝑦, 0R⟩ = 𝐵 → (⟨𝑦, 0R⟩ · ⟨𝑧, 0R⟩) = (𝐵 · ⟨𝑧, 0R⟩))
11	10	breq2d 4100	. . 3 ⊢ (⟨𝑦, 0R⟩ = 𝐵 → ((𝐴 · ⟨𝑧, 0R⟩) <ℝ (⟨𝑦, 0R⟩ · ⟨𝑧, 0R⟩) ↔ (𝐴 · ⟨𝑧, 0R⟩) <ℝ (𝐵 · ⟨𝑧, 0R⟩)))
12		breq2 4092	. . . 4 ⊢ (⟨𝑦, 0R⟩ = 𝐵 → (𝐴 <ℝ ⟨𝑦, 0R⟩ ↔ 𝐴 <ℝ 𝐵))
13		breq1 4091	. . . 4 ⊢ (⟨𝑦, 0R⟩ = 𝐵 → (⟨𝑦, 0R⟩ <ℝ 𝐴 ↔ 𝐵 <ℝ 𝐴))
14	12, 13	orbi12d 800	. . 3 ⊢ (⟨𝑦, 0R⟩ = 𝐵 → ((𝐴 <ℝ ⟨𝑦, 0R⟩ ∨ ⟨𝑦, 0R⟩ <ℝ 𝐴) ↔ (𝐴 <ℝ 𝐵 ∨ 𝐵 <ℝ 𝐴)))
15	11, 14	imbi12d 234	. 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 6026	. . . 4 ⊢ (⟨𝑧, 0R⟩ = 𝐶 → (𝐴 · ⟨𝑧, 0R⟩) = (𝐴 · 𝐶))
17		oveq2 6026	. . . 4 ⊢ (⟨𝑧, 0R⟩ = 𝐶 → (𝐵 · ⟨𝑧, 0R⟩) = (𝐵 · 𝐶))
18	16, 17	breq12d 4101	. . 3 ⊢ (⟨𝑧, 0R⟩ = 𝐶 → ((𝐴 · ⟨𝑧, 0R⟩) <ℝ (𝐵 · ⟨𝑧, 0R⟩) ↔ (𝐴 · 𝐶) <ℝ (𝐵 · 𝐶)))
19	18	imbi1d 231	. 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 8001	. . 3 ⊢ ((𝑥 ∈ R ∧ 𝑦 ∈ R ∧ 𝑧 ∈ R) → ((𝑥 ·R 𝑧) <R (𝑦 ·R 𝑧) → (𝑥 <R 𝑦 ∨ 𝑦 <R 𝑥)))
21		mulresr 8058	. . . . . 6 ⊢ ((𝑥 ∈ R ∧ 𝑧 ∈ R) → (⟨𝑥, 0R⟩ · ⟨𝑧, 0R⟩) = ⟨(𝑥 ·R 𝑧), 0R⟩)
22	21	3adant2 1042	. . . . 5 ⊢ ((𝑥 ∈ R ∧ 𝑦 ∈ R ∧ 𝑧 ∈ R) → (⟨𝑥, 0R⟩ · ⟨𝑧, 0R⟩) = ⟨(𝑥 ·R 𝑧), 0R⟩)
23		mulresr 8058	. . . . . 6 ⊢ ((𝑦 ∈ R ∧ 𝑧 ∈ R) → (⟨𝑦, 0R⟩ · ⟨𝑧, 0R⟩) = ⟨(𝑦 ·R 𝑧), 0R⟩)
24	23	3adant1 1041	. . . . 5 ⊢ ((𝑥 ∈ R ∧ 𝑦 ∈ R ∧ 𝑧 ∈ R) → (⟨𝑦, 0R⟩ · ⟨𝑧, 0R⟩) = ⟨(𝑦 ·R 𝑧), 0R⟩)
25	22, 24	breq12d 4101	. . . 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 8059	. . . 4 ⊢ (⟨(𝑥 ·R 𝑧), 0R⟩ <ℝ ⟨(𝑦 ·R 𝑧), 0R⟩ ↔ (𝑥 ·R 𝑧) <R (𝑦 ·R 𝑧))
27	25, 26	bitrdi 196	. . 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 8059	. . . . 5 ⊢ (⟨𝑥, 0R⟩ <ℝ ⟨𝑦, 0R⟩ ↔ 𝑥 <R 𝑦)
29		ltresr 8059	. . . . 5 ⊢ (⟨𝑦, 0R⟩ <ℝ ⟨𝑥, 0R⟩ ↔ 𝑦 <R 𝑥)
30	28, 29	orbi12i 771	. . . 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 203	. 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 2837	1 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 · 𝐶) <ℝ (𝐵 · 𝐶) → (𝐴 <ℝ 𝐵 ∨ 𝐵 <ℝ 𝐴)))

Syntax hints:   → wi 4   ↔ wb 105   ∨ wo 715   ∧ w3a 1004   = wceq 1397   ∈ wcel 2202  ⟨cop 3672   class class class wbr 4088  (class class class)co 6018  Rcnr 7517  0_Rc0r 7518   ·_R cmr 7522   <_R cltr 7523  ℝcr 8031   <_ℝ cltrr 8036   · cmul 8037

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-iinf 4686

This theorem depends on definitions:  df-bi 117  df-dc 842  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-ral 2515  df-rex 2516  df-reu 2517  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-tr 4188  df-eprel 4386  df-id 4390  df-po 4393  df-iso 4394  df-iord 4463  df-on 4465  df-suc 4468  df-iom 4689  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-ov 6021  df-oprab 6022  df-mpo 6023  df-1st 6303  df-2nd 6304  df-recs 6471  df-irdg 6536  df-1o 6582  df-2o 6583  df-oadd 6586  df-omul 6587  df-er 6702  df-ec 6704  df-qs 6708  df-ni 7524  df-pli 7525  df-mi 7526  df-lti 7527  df-plpq 7564  df-mpq 7565  df-enq 7567  df-nqqs 7568  df-plqqs 7569  df-mqqs 7570  df-1nqqs 7571  df-rq 7572  df-ltnqqs 7573  df-enq0 7644  df-nq0 7645  df-0nq0 7646  df-plq0 7647  df-mq0 7648  df-inp 7686  df-i1p 7687  df-iplp 7688  df-imp 7689  df-iltp 7690  df-enr 7946  df-nr 7947  df-plr 7948  df-mr 7949  df-ltr 7950  df-0r 7951  df-m1r 7953  df-c 8038  df-r 8042  df-mul 8044  df-lt 8045

This theorem is referenced by: (None)