Theorem nnmord 7599

Description: Ordering property of multiplication. Proposition 8.19 of [TakeutiZaring] p. 63, limited to natural numbers. (Contributed by NM, 22-Jan-1996.) (Revised by Mario Carneiro, 15-Nov-2014.)

Assertion

Ref	Expression
nnmord	⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) → ((𝐴 ∈ 𝐵 ∧ ∅ ∈ 𝐶) ↔ (𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 𝐵)))

Proof of Theorem nnmord

Step	Hyp	Ref	Expression
1		nnmordi 7598	. . . . . 6 ⊢ (((𝐵 ∈ ω ∧ 𝐶 ∈ ω) ∧ ∅ ∈ 𝐶) → (𝐴 ∈ 𝐵 → (𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 𝐵)))
2	1	ex 449	. . . . 5 ⊢ ((𝐵 ∈ ω ∧ 𝐶 ∈ ω) → (∅ ∈ 𝐶 → (𝐴 ∈ 𝐵 → (𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 𝐵))))
3	2	com23 84	. . . 4 ⊢ ((𝐵 ∈ ω ∧ 𝐶 ∈ ω) → (𝐴 ∈ 𝐵 → (∅ ∈ 𝐶 → (𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 𝐵))))
4	3	impd 446	. . 3 ⊢ ((𝐵 ∈ ω ∧ 𝐶 ∈ ω) → ((𝐴 ∈ 𝐵 ∧ ∅ ∈ 𝐶) → (𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 𝐵)))
5	4	3adant1 1072	. 2 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) → ((𝐴 ∈ 𝐵 ∧ ∅ ∈ 𝐶) → (𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 𝐵)))
6		ne0i 3880	. . . . . . . 8 ⊢ ((𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 𝐵) → (𝐶 ·𝑜 𝐵) ≠ ∅)
7		nnm0r 7577	. . . . . . . . . 10 ⊢ (𝐵 ∈ ω → (∅ ·𝑜 𝐵) = ∅)
8		oveq1 6556	. . . . . . . . . . 11 ⊢ (𝐶 = ∅ → (𝐶 ·𝑜 𝐵) = (∅ ·𝑜 𝐵))
9	8	eqeq1d 2612	. . . . . . . . . 10 ⊢ (𝐶 = ∅ → ((𝐶 ·𝑜 𝐵) = ∅ ↔ (∅ ·𝑜 𝐵) = ∅))
10	7, 9	syl5ibrcom 236	. . . . . . . . 9 ⊢ (𝐵 ∈ ω → (𝐶 = ∅ → (𝐶 ·𝑜 𝐵) = ∅))
11	10	necon3d 2803	. . . . . . . 8 ⊢ (𝐵 ∈ ω → ((𝐶 ·𝑜 𝐵) ≠ ∅ → 𝐶 ≠ ∅))
12	6, 11	syl5 33	. . . . . . 7 ⊢ (𝐵 ∈ ω → ((𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 𝐵) → 𝐶 ≠ ∅))
13	12	adantr 480	. . . . . 6 ⊢ ((𝐵 ∈ ω ∧ 𝐶 ∈ ω) → ((𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 𝐵) → 𝐶 ≠ ∅))
14		nnord 6965	. . . . . . . 8 ⊢ (𝐶 ∈ ω → Ord 𝐶)
15		ord0eln0 5696	. . . . . . . 8 ⊢ (Ord 𝐶 → (∅ ∈ 𝐶 ↔ 𝐶 ≠ ∅))
16	14, 15	syl 17	. . . . . . 7 ⊢ (𝐶 ∈ ω → (∅ ∈ 𝐶 ↔ 𝐶 ≠ ∅))
17	16	adantl 481	. . . . . 6 ⊢ ((𝐵 ∈ ω ∧ 𝐶 ∈ ω) → (∅ ∈ 𝐶 ↔ 𝐶 ≠ ∅))
18	13, 17	sylibrd 248	. . . . 5 ⊢ ((𝐵 ∈ ω ∧ 𝐶 ∈ ω) → ((𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 𝐵) → ∅ ∈ 𝐶))
19	18	3adant1 1072	. . . 4 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) → ((𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 𝐵) → ∅ ∈ 𝐶))
20		oveq2 6557	. . . . . . . . . 10 ⊢ (𝐴 = 𝐵 → (𝐶 ·𝑜 𝐴) = (𝐶 ·𝑜 𝐵))
21	20	a1i 11	. . . . . . . . 9 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) ∧ ∅ ∈ 𝐶) → (𝐴 = 𝐵 → (𝐶 ·𝑜 𝐴) = (𝐶 ·𝑜 𝐵)))
22		nnmordi 7598	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ω ∧ 𝐶 ∈ ω) ∧ ∅ ∈ 𝐶) → (𝐵 ∈ 𝐴 → (𝐶 ·𝑜 𝐵) ∈ (𝐶 ·𝑜 𝐴)))
23	22	3adantl2 1211	. . . . . . . . 9 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) ∧ ∅ ∈ 𝐶) → (𝐵 ∈ 𝐴 → (𝐶 ·𝑜 𝐵) ∈ (𝐶 ·𝑜 𝐴)))
24	21, 23	orim12d 879	. . . . . . . 8 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) ∧ ∅ ∈ 𝐶) → ((𝐴 = 𝐵 ∨ 𝐵 ∈ 𝐴) → ((𝐶 ·𝑜 𝐴) = (𝐶 ·𝑜 𝐵) ∨ (𝐶 ·𝑜 𝐵) ∈ (𝐶 ·𝑜 𝐴))))
25	24	con3d 147	. . . . . . 7 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) ∧ ∅ ∈ 𝐶) → (¬ ((𝐶 ·𝑜 𝐴) = (𝐶 ·𝑜 𝐵) ∨ (𝐶 ·𝑜 𝐵) ∈ (𝐶 ·𝑜 𝐴)) → ¬ (𝐴 = 𝐵 ∨ 𝐵 ∈ 𝐴)))
26		simpl3 1059	. . . . . . . . 9 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) ∧ ∅ ∈ 𝐶) → 𝐶 ∈ ω)
27		simpl1 1057	. . . . . . . . 9 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) ∧ ∅ ∈ 𝐶) → 𝐴 ∈ ω)
28		nnmcl 7579	. . . . . . . . 9 ⊢ ((𝐶 ∈ ω ∧ 𝐴 ∈ ω) → (𝐶 ·𝑜 𝐴) ∈ ω)
29	26, 27, 28	syl2anc 691	. . . . . . . 8 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) ∧ ∅ ∈ 𝐶) → (𝐶 ·𝑜 𝐴) ∈ ω)
30		simpl2 1058	. . . . . . . . 9 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) ∧ ∅ ∈ 𝐶) → 𝐵 ∈ ω)
31		nnmcl 7579	. . . . . . . . 9 ⊢ ((𝐶 ∈ ω ∧ 𝐵 ∈ ω) → (𝐶 ·𝑜 𝐵) ∈ ω)
32	26, 30, 31	syl2anc 691	. . . . . . . 8 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) ∧ ∅ ∈ 𝐶) → (𝐶 ·𝑜 𝐵) ∈ ω)
33		nnord 6965	. . . . . . . . 9 ⊢ ((𝐶 ·𝑜 𝐴) ∈ ω → Ord (𝐶 ·𝑜 𝐴))
34		nnord 6965	. . . . . . . . 9 ⊢ ((𝐶 ·𝑜 𝐵) ∈ ω → Ord (𝐶 ·𝑜 𝐵))
35		ordtri2 5675	. . . . . . . . 9 ⊢ ((Ord (𝐶 ·𝑜 𝐴) ∧ Ord (𝐶 ·𝑜 𝐵)) → ((𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 𝐵) ↔ ¬ ((𝐶 ·𝑜 𝐴) = (𝐶 ·𝑜 𝐵) ∨ (𝐶 ·𝑜 𝐵) ∈ (𝐶 ·𝑜 𝐴))))
36	33, 34, 35	syl2an 493	. . . . . . . 8 ⊢ (((𝐶 ·𝑜 𝐴) ∈ ω ∧ (𝐶 ·𝑜 𝐵) ∈ ω) → ((𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 𝐵) ↔ ¬ ((𝐶 ·𝑜 𝐴) = (𝐶 ·𝑜 𝐵) ∨ (𝐶 ·𝑜 𝐵) ∈ (𝐶 ·𝑜 𝐴))))
37	29, 32, 36	syl2anc 691	. . . . . . 7 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) ∧ ∅ ∈ 𝐶) → ((𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 𝐵) ↔ ¬ ((𝐶 ·𝑜 𝐴) = (𝐶 ·𝑜 𝐵) ∨ (𝐶 ·𝑜 𝐵) ∈ (𝐶 ·𝑜 𝐴))))
38		nnord 6965	. . . . . . . . 9 ⊢ (𝐴 ∈ ω → Ord 𝐴)
39		nnord 6965	. . . . . . . . 9 ⊢ (𝐵 ∈ ω → Ord 𝐵)
40		ordtri2 5675	. . . . . . . . 9 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴 ∈ 𝐵 ↔ ¬ (𝐴 = 𝐵 ∨ 𝐵 ∈ 𝐴)))
41	38, 39, 40	syl2an 493	. . . . . . . 8 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ∈ 𝐵 ↔ ¬ (𝐴 = 𝐵 ∨ 𝐵 ∈ 𝐴)))
42	27, 30, 41	syl2anc 691	. . . . . . 7 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) ∧ ∅ ∈ 𝐶) → (𝐴 ∈ 𝐵 ↔ ¬ (𝐴 = 𝐵 ∨ 𝐵 ∈ 𝐴)))
43	25, 37, 42	3imtr4d 282	. . . . . 6 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) ∧ ∅ ∈ 𝐶) → ((𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 𝐵) → 𝐴 ∈ 𝐵))
44	43	ex 449	. . . . 5 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) → (∅ ∈ 𝐶 → ((𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 𝐵) → 𝐴 ∈ 𝐵)))
45	44	com23 84	. . . 4 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) → ((𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 𝐵) → (∅ ∈ 𝐶 → 𝐴 ∈ 𝐵)))
46	19, 45	mpdd 42	. . 3 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) → ((𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 𝐵) → 𝐴 ∈ 𝐵))
47	46, 19	jcad 554	. 2 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) → ((𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 𝐵) → (𝐴 ∈ 𝐵 ∧ ∅ ∈ 𝐶)))
48	5, 47	impbid 201	1 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) → ((𝐴 ∈ 𝐵 ∧ ∅ ∈ 𝐶) ↔ (𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 𝐵)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∨ wo 382   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  ∅c0 3874  Ord word 5639  (class class class)co 6549  ωcom 6957   ·_𝑜 comu 7445

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-oadd 7451  df-omul 7452

This theorem is referenced by:  nnmword  7600  nnneo  7618  ltmpi  9605