Theorem axpre-mulext 7878

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 7920.

(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 7818	. 2 ⊢ (𝐴 ∈ ℝ ↔ ∃𝑥 ∈ R ⟨𝑥, 0R⟩ = 𝐴)
2		elreal 7818	. 2 ⊢ (𝐵 ∈ ℝ ↔ ∃𝑦 ∈ R ⟨𝑦, 0R⟩ = 𝐵)
3		elreal 7818	. 2 ⊢ (𝐶 ∈ ℝ ↔ ∃𝑧 ∈ R ⟨𝑧, 0R⟩ = 𝐶)
4		oveq1 5876	. . . 4 ⊢ (⟨𝑥, 0R⟩ = 𝐴 → (⟨𝑥, 0R⟩ · ⟨𝑧, 0R⟩) = (𝐴 · ⟨𝑧, 0R⟩))
5	4	breq1d 4010	. . 3 ⊢ (⟨𝑥, 0R⟩ = 𝐴 → ((⟨𝑥, 0R⟩ · ⟨𝑧, 0R⟩) <ℝ (⟨𝑦, 0R⟩ · ⟨𝑧, 0R⟩) ↔ (𝐴 · ⟨𝑧, 0R⟩) <ℝ (⟨𝑦, 0R⟩ · ⟨𝑧, 0R⟩)))
6		breq1 4003	. . . 4 ⊢ (⟨𝑥, 0R⟩ = 𝐴 → (⟨𝑥, 0R⟩ <ℝ ⟨𝑦, 0R⟩ ↔ 𝐴 <ℝ ⟨𝑦, 0R⟩))
7		breq2 4004	. . . 4 ⊢ (⟨𝑥, 0R⟩ = 𝐴 → (⟨𝑦, 0R⟩ <ℝ ⟨𝑥, 0R⟩ ↔ ⟨𝑦, 0R⟩ <ℝ 𝐴))
8	6, 7	orbi12d 793	. . 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 5876	. . . 4 ⊢ (⟨𝑦, 0R⟩ = 𝐵 → (⟨𝑦, 0R⟩ · ⟨𝑧, 0R⟩) = (𝐵 · ⟨𝑧, 0R⟩))
11	10	breq2d 4012	. . 3 ⊢ (⟨𝑦, 0R⟩ = 𝐵 → ((𝐴 · ⟨𝑧, 0R⟩) <ℝ (⟨𝑦, 0R⟩ · ⟨𝑧, 0R⟩) ↔ (𝐴 · ⟨𝑧, 0R⟩) <ℝ (𝐵 · ⟨𝑧, 0R⟩)))
12		breq2 4004	. . . 4 ⊢ (⟨𝑦, 0R⟩ = 𝐵 → (𝐴 <ℝ ⟨𝑦, 0R⟩ ↔ 𝐴 <ℝ 𝐵))
13		breq1 4003	. . . 4 ⊢ (⟨𝑦, 0R⟩ = 𝐵 → (⟨𝑦, 0R⟩ <ℝ 𝐴 ↔ 𝐵 <ℝ 𝐴))
14	12, 13	orbi12d 793	. . 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 5877	. . . 4 ⊢ (⟨𝑧, 0R⟩ = 𝐶 → (𝐴 · ⟨𝑧, 0R⟩) = (𝐴 · 𝐶))
17		oveq2 5877	. . . 4 ⊢ (⟨𝑧, 0R⟩ = 𝐶 → (𝐵 · ⟨𝑧, 0R⟩) = (𝐵 · 𝐶))
18	16, 17	breq12d 4013	. . 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 7771	. . 3 ⊢ ((𝑥 ∈ R ∧ 𝑦 ∈ R ∧ 𝑧 ∈ R) → ((𝑥 ·R 𝑧) <R (𝑦 ·R 𝑧) → (𝑥 <R 𝑦 ∨ 𝑦 <R 𝑥)))
21		mulresr 7828	. . . . . 6 ⊢ ((𝑥 ∈ R ∧ 𝑧 ∈ R) → (⟨𝑥, 0R⟩ · ⟨𝑧, 0R⟩) = ⟨(𝑥 ·R 𝑧), 0R⟩)
22	21	3adant2 1016	. . . . 5 ⊢ ((𝑥 ∈ R ∧ 𝑦 ∈ R ∧ 𝑧 ∈ R) → (⟨𝑥, 0R⟩ · ⟨𝑧, 0R⟩) = ⟨(𝑥 ·R 𝑧), 0R⟩)
23		mulresr 7828	. . . . . 6 ⊢ ((𝑦 ∈ R ∧ 𝑧 ∈ R) → (⟨𝑦, 0R⟩ · ⟨𝑧, 0R⟩) = ⟨(𝑦 ·R 𝑧), 0R⟩)
24	23	3adant1 1015	. . . . 5 ⊢ ((𝑥 ∈ R ∧ 𝑦 ∈ R ∧ 𝑧 ∈ R) → (⟨𝑦, 0R⟩ · ⟨𝑧, 0R⟩) = ⟨(𝑦 ·R 𝑧), 0R⟩)
25	22, 24	breq12d 4013	. . . 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 7829	. . . 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 7829	. . . . 5 ⊢ (⟨𝑥, 0R⟩ <ℝ ⟨𝑦, 0R⟩ ↔ 𝑥 <R 𝑦)
29		ltresr 7829	. . . . 5 ⊢ (⟨𝑦, 0R⟩ <ℝ ⟨𝑥, 0R⟩ ↔ 𝑦 <R 𝑥)
30	28, 29	orbi12i 764	. . . 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 2771	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 708   ∧ w3a 978   = wceq 1353   ∈ wcel 2148  ⟨cop 3594   class class class wbr 4000  (class class class)co 5869  Rcnr 7287  0_Rc0r 7288   ·_R cmr 7292   <_R cltr 7293  ℝcr 7801   <_ℝ cltrr 7806   · cmul 7807

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 4115  ax-sep 4118  ax-nul 4126  ax-pow 4171  ax-pr 4206  ax-un 4430  ax-setind 4533  ax-iinf 4584

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 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-int 3843  df-iun 3886  df-br 4001  df-opab 4062  df-mpt 4063  df-tr 4099  df-eprel 4286  df-id 4290  df-po 4293  df-iso 4294  df-iord 4363  df-on 4365  df-suc 4368  df-iom 4587  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-rn 4634  df-res 4635  df-ima 4636  df-iota 5174  df-fun 5214  df-fn 5215  df-f 5216  df-f1 5217  df-fo 5218  df-f1o 5219  df-fv 5220  df-ov 5872  df-oprab 5873  df-mpo 5874  df-1st 6135  df-2nd 6136  df-recs 6300  df-irdg 6365  df-1o 6411  df-2o 6412  df-oadd 6415  df-omul 6416  df-er 6529  df-ec 6531  df-qs 6535  df-ni 7294  df-pli 7295  df-mi 7296  df-lti 7297  df-plpq 7334  df-mpq 7335  df-enq 7337  df-nqqs 7338  df-plqqs 7339  df-mqqs 7340  df-1nqqs 7341  df-rq 7342  df-ltnqqs 7343  df-enq0 7414  df-nq0 7415  df-0nq0 7416  df-plq0 7417  df-mq0 7418  df-inp 7456  df-i1p 7457  df-iplp 7458  df-imp 7459  df-iltp 7460  df-enr 7716  df-nr 7717  df-plr 7718  df-mr 7719  df-ltr 7720  df-0r 7721  df-m1r 7723  df-c 7808  df-r 7812  df-mul 7814  df-lt 7815

This theorem is referenced by: (None)