Theorem nnmord 6026

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 6025	. . . . . 6 ⊢ (((B ∈ 𝜔 ∧ 𝐶 ∈ 𝜔) ∧ ∅ ∈ 𝐶) → (A ∈ B → (𝐶 ·𝑜 A) ∈ (𝐶 ·𝑜 B)))
2	1	ex 108	. . . . 5 ⊢ ((B ∈ 𝜔 ∧ 𝐶 ∈ 𝜔) → (∅ ∈ 𝐶 → (A ∈ B → (𝐶 ·𝑜 A) ∈ (𝐶 ·𝑜 B))))
3	2	com23 72	. . . 4 ⊢ ((B ∈ 𝜔 ∧ 𝐶 ∈ 𝜔) → (A ∈ B → (∅ ∈ 𝐶 → (𝐶 ·𝑜 A) ∈ (𝐶 ·𝑜 B))))
4	3	impd 242	. . 3 ⊢ ((B ∈ 𝜔 ∧ 𝐶 ∈ 𝜔) → ((A ∈ B ∧ ∅ ∈ 𝐶) → (𝐶 ·𝑜 A) ∈ (𝐶 ·𝑜 B)))
5	4	3adant1 921	. 2 ⊢ ((A ∈ 𝜔 ∧ B ∈ 𝜔 ∧ 𝐶 ∈ 𝜔) → ((A ∈ B ∧ ∅ ∈ 𝐶) → (𝐶 ·𝑜 A) ∈ (𝐶 ·𝑜 B)))
6		ne0i 3224	. . . . . . . 8 ⊢ ((𝐶 ·𝑜 A) ∈ (𝐶 ·𝑜 B) → (𝐶 ·𝑜 B) ≠ ∅)
7		nnm0r 5997	. . . . . . . . . 10 ⊢ (B ∈ 𝜔 → (∅ ·𝑜 B) = ∅)
8		oveq1 5462	. . . . . . . . . . 11 ⊢ (𝐶 = ∅ → (𝐶 ·𝑜 B) = (∅ ·𝑜 B))
9	8	eqeq1d 2045	. . . . . . . . . 10 ⊢ (𝐶 = ∅ → ((𝐶 ·𝑜 B) = ∅ ↔ (∅ ·𝑜 B) = ∅))
10	7, 9	syl5ibrcom 146	. . . . . . . . 9 ⊢ (B ∈ 𝜔 → (𝐶 = ∅ → (𝐶 ·𝑜 B) = ∅))
11	10	necon3d 2243	. . . . . . . 8 ⊢ (B ∈ 𝜔 → ((𝐶 ·𝑜 B) ≠ ∅ → 𝐶 ≠ ∅))
12	6, 11	syl5 28	. . . . . . 7 ⊢ (B ∈ 𝜔 → ((𝐶 ·𝑜 A) ∈ (𝐶 ·𝑜 B) → 𝐶 ≠ ∅))
13	12	adantr 261	. . . . . 6 ⊢ ((B ∈ 𝜔 ∧ 𝐶 ∈ 𝜔) → ((𝐶 ·𝑜 A) ∈ (𝐶 ·𝑜 B) → 𝐶 ≠ ∅))
14		nn0eln0 4284	. . . . . . 7 ⊢ (𝐶 ∈ 𝜔 → (∅ ∈ 𝐶 ↔ 𝐶 ≠ ∅))
15	14	adantl 262	. . . . . 6 ⊢ ((B ∈ 𝜔 ∧ 𝐶 ∈ 𝜔) → (∅ ∈ 𝐶 ↔ 𝐶 ≠ ∅))
16	13, 15	sylibrd 158	. . . . 5 ⊢ ((B ∈ 𝜔 ∧ 𝐶 ∈ 𝜔) → ((𝐶 ·𝑜 A) ∈ (𝐶 ·𝑜 B) → ∅ ∈ 𝐶))
17	16	3adant1 921	. . . 4 ⊢ ((A ∈ 𝜔 ∧ B ∈ 𝜔 ∧ 𝐶 ∈ 𝜔) → ((𝐶 ·𝑜 A) ∈ (𝐶 ·𝑜 B) → ∅ ∈ 𝐶))
18		oveq2 5463	. . . . . . . . . 10 ⊢ (A = B → (𝐶 ·𝑜 A) = (𝐶 ·𝑜 B))
19	18	a1i 9	. . . . . . . . 9 ⊢ (((A ∈ 𝜔 ∧ B ∈ 𝜔 ∧ 𝐶 ∈ 𝜔) ∧ ∅ ∈ 𝐶) → (A = B → (𝐶 ·𝑜 A) = (𝐶 ·𝑜 B)))
20		nnmordi 6025	. . . . . . . . . 10 ⊢ (((A ∈ 𝜔 ∧ 𝐶 ∈ 𝜔) ∧ ∅ ∈ 𝐶) → (B ∈ A → (𝐶 ·𝑜 B) ∈ (𝐶 ·𝑜 A)))
21	20	3adantl2 1060	. . . . . . . . 9 ⊢ (((A ∈ 𝜔 ∧ B ∈ 𝜔 ∧ 𝐶 ∈ 𝜔) ∧ ∅ ∈ 𝐶) → (B ∈ A → (𝐶 ·𝑜 B) ∈ (𝐶 ·𝑜 A)))
22	19, 21	orim12d 699	. . . . . . . 8 ⊢ (((A ∈ 𝜔 ∧ B ∈ 𝜔 ∧ 𝐶 ∈ 𝜔) ∧ ∅ ∈ 𝐶) → ((A = B ∨ B ∈ A) → ((𝐶 ·𝑜 A) = (𝐶 ·𝑜 B) ∨ (𝐶 ·𝑜 B) ∈ (𝐶 ·𝑜 A))))
23	22	con3d 560	. . . . . . 7 ⊢ (((A ∈ 𝜔 ∧ B ∈ 𝜔 ∧ 𝐶 ∈ 𝜔) ∧ ∅ ∈ 𝐶) → (¬ ((𝐶 ·𝑜 A) = (𝐶 ·𝑜 B) ∨ (𝐶 ·𝑜 B) ∈ (𝐶 ·𝑜 A)) → ¬ (A = B ∨ B ∈ A)))
24		simpl3 908	. . . . . . . . 9 ⊢ (((A ∈ 𝜔 ∧ B ∈ 𝜔 ∧ 𝐶 ∈ 𝜔) ∧ ∅ ∈ 𝐶) → 𝐶 ∈ 𝜔)
25		simpl1 906	. . . . . . . . 9 ⊢ (((A ∈ 𝜔 ∧ B ∈ 𝜔 ∧ 𝐶 ∈ 𝜔) ∧ ∅ ∈ 𝐶) → A ∈ 𝜔)
26		nnmcl 5999	. . . . . . . . 9 ⊢ ((𝐶 ∈ 𝜔 ∧ A ∈ 𝜔) → (𝐶 ·𝑜 A) ∈ 𝜔)
27	24, 25, 26	syl2anc 391	. . . . . . . 8 ⊢ (((A ∈ 𝜔 ∧ B ∈ 𝜔 ∧ 𝐶 ∈ 𝜔) ∧ ∅ ∈ 𝐶) → (𝐶 ·𝑜 A) ∈ 𝜔)
28		simpl2 907	. . . . . . . . 9 ⊢ (((A ∈ 𝜔 ∧ B ∈ 𝜔 ∧ 𝐶 ∈ 𝜔) ∧ ∅ ∈ 𝐶) → B ∈ 𝜔)
29		nnmcl 5999	. . . . . . . . 9 ⊢ ((𝐶 ∈ 𝜔 ∧ B ∈ 𝜔) → (𝐶 ·𝑜 B) ∈ 𝜔)
30	24, 28, 29	syl2anc 391	. . . . . . . 8 ⊢ (((A ∈ 𝜔 ∧ B ∈ 𝜔 ∧ 𝐶 ∈ 𝜔) ∧ ∅ ∈ 𝐶) → (𝐶 ·𝑜 B) ∈ 𝜔)
31		nntri2 6012	. . . . . . . 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))))
32	27, 30, 31	syl2anc 391	. . . . . . 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))))
33		nntri2 6012	. . . . . . . 8 ⊢ ((A ∈ 𝜔 ∧ B ∈ 𝜔) → (A ∈ B ↔ ¬ (A = B ∨ B ∈ A)))
34	25, 28, 33	syl2anc 391	. . . . . . 7 ⊢ (((A ∈ 𝜔 ∧ B ∈ 𝜔 ∧ 𝐶 ∈ 𝜔) ∧ ∅ ∈ 𝐶) → (A ∈ B ↔ ¬ (A = B ∨ B ∈ A)))
35	23, 32, 34	3imtr4d 192	. . . . . 6 ⊢ (((A ∈ 𝜔 ∧ B ∈ 𝜔 ∧ 𝐶 ∈ 𝜔) ∧ ∅ ∈ 𝐶) → ((𝐶 ·𝑜 A) ∈ (𝐶 ·𝑜 B) → A ∈ B))
36	35	ex 108	. . . . 5 ⊢ ((A ∈ 𝜔 ∧ B ∈ 𝜔 ∧ 𝐶 ∈ 𝜔) → (∅ ∈ 𝐶 → ((𝐶 ·𝑜 A) ∈ (𝐶 ·𝑜 B) → A ∈ B)))
37	36	com23 72	. . . 4 ⊢ ((A ∈ 𝜔 ∧ B ∈ 𝜔 ∧ 𝐶 ∈ 𝜔) → ((𝐶 ·𝑜 A) ∈ (𝐶 ·𝑜 B) → (∅ ∈ 𝐶 → A ∈ B)))
38	17, 37	mpdd 36	. . 3 ⊢ ((A ∈ 𝜔 ∧ B ∈ 𝜔 ∧ 𝐶 ∈ 𝜔) → ((𝐶 ·𝑜 A) ∈ (𝐶 ·𝑜 B) → A ∈ B))
39	38, 17	jcad 291	. 2 ⊢ ((A ∈ 𝜔 ∧ B ∈ 𝜔 ∧ 𝐶 ∈ 𝜔) → ((𝐶 ·𝑜 A) ∈ (𝐶 ·𝑜 B) → (A ∈ B ∧ ∅ ∈ 𝐶)))
40	5, 39	impbid 120	1 ⊢ ((A ∈ 𝜔 ∧ B ∈ 𝜔 ∧ 𝐶 ∈ 𝜔) → ((A ∈ B ∧ ∅ ∈ 𝐶) ↔ (𝐶 ·𝑜 A) ∈ (𝐶 ·𝑜 B)))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 97   ↔ wb 98   ∨ wo 628   ∧ w3a 884   = wceq 1242   ∈ wcel 1390   ≠ wne 2201  ∅c0 3218  𝜔com 4256  (class class class)co 5455   ·_𝑜 comu 5938

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 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-coll 3863  ax-sep 3866  ax-nul 3874  ax-pow 3918  ax-pr 3935  ax-un 4136  ax-setind 4220  ax-iinf 4254

This theorem depends on definitions:  df-bi 110  df-3or 885  df-3an 886  df-tru 1245  df-fal 1248  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ne 2203  df-ral 2305  df-rex 2306  df-reu 2307  df-rab 2309  df-v 2553  df-sbc 2759  df-csb 2847  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-nul 3219  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-int 3607  df-iun 3650  df-br 3756  df-opab 3810  df-mpt 3811  df-tr 3846  df-id 4021  df-iord 4069  df-on 4071  df-suc 4074  df-iom 4257  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301  df-iota 4810  df-fun 4847  df-fn 4848  df-f 4849  df-f1 4850  df-fo 4851  df-f1o 4852  df-fv 4853  df-ov 5458  df-oprab 5459  df-mpt2 5460  df-1st 5709  df-2nd 5710  df-recs 5861  df-irdg 5897  df-oadd 5944  df-omul 5945

This theorem is referenced by:  nnmword  6027  ltmpig  6323