Theorem rhmdvdsr 14455

Description: A ring homomorphism preserves the divisibility relation. (Contributed by Thierry Arnoux, 22-Oct-2017.)

Hypotheses

Ref	Expression
rhmdvdsr.x	⊢ 𝑋 = (Base‘𝑅)
rhmdvdsr.m	⊢ ∥ = (∥r‘𝑅)
rhmdvdsr.n	⊢ / = (∥r‘𝑆)

Assertion

Ref	Expression
rhmdvdsr	⊢ (((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ 𝐴 ∥ 𝐵) → (𝐹‘𝐴) / (𝐹‘𝐵))

Proof of Theorem rhmdvdsr

Dummy variables 𝑦 𝑐 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpl1 1031	. . 3 ⊢ (((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ 𝐴 ∥ 𝐵) → 𝐹 ∈ (𝑅 RingHom 𝑆))
2		simpl2 1032	. . 3 ⊢ (((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ 𝐴 ∥ 𝐵) → 𝐴 ∈ 𝑋)
3		rhmdvdsr.x	. . . . 5 ⊢ 𝑋 = (Base‘𝑅)
4		eqid 2238	. . . . 5 ⊢ (Base‘𝑆) = (Base‘𝑆)
5	3, 4	rhmf 14443	. . . 4 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → 𝐹:𝑋⟶(Base‘𝑆))
6	5	ffvelcdmda 5834	. . 3 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐴 ∈ 𝑋) → (𝐹‘𝐴) ∈ (Base‘𝑆))
7	1, 2, 6	syl2anc 415	. 2 ⊢ (((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ 𝐴 ∥ 𝐵) → (𝐹‘𝐴) ∈ (Base‘𝑆))
8		simpll1 1067	. . . . . 6 ⊢ ((((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ 𝐴 ∥ 𝐵) ∧ 𝑐 ∈ 𝑋) → 𝐹 ∈ (𝑅 RingHom 𝑆))
9		simpr 110	. . . . . 6 ⊢ ((((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ 𝐴 ∥ 𝐵) ∧ 𝑐 ∈ 𝑋) → 𝑐 ∈ 𝑋)
10	5	ffvelcdmda 5834	. . . . . 6 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝑐 ∈ 𝑋) → (𝐹‘𝑐) ∈ (Base‘𝑆))
11	8, 9, 10	syl2anc 415	. . . . 5 ⊢ ((((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ 𝐴 ∥ 𝐵) ∧ 𝑐 ∈ 𝑋) → (𝐹‘𝑐) ∈ (Base‘𝑆))
12	11	ralrimiva 2623	. . . 4 ⊢ (((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ 𝐴 ∥ 𝐵) → ∀𝑐 ∈ 𝑋 (𝐹‘𝑐) ∈ (Base‘𝑆))
13	2	adantr 276	. . . . . . 7 ⊢ ((((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ 𝐴 ∥ 𝐵) ∧ 𝑐 ∈ 𝑋) → 𝐴 ∈ 𝑋)
14		eqid 2238	. . . . . . . 8 ⊢ (.r‘𝑅) = (.r‘𝑅)
15		eqid 2238	. . . . . . . 8 ⊢ (.r‘𝑆) = (.r‘𝑆)
16	3, 14, 15	rhmmul 14444	. . . . . . 7 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝑐 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋) → (𝐹‘(𝑐(.r‘𝑅)𝐴)) = ((𝐹‘𝑐)(.r‘𝑆)(𝐹‘𝐴)))
17	8, 9, 13, 16	syl3anc 1278	. . . . . 6 ⊢ ((((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ 𝐴 ∥ 𝐵) ∧ 𝑐 ∈ 𝑋) → (𝐹‘(𝑐(.r‘𝑅)𝐴)) = ((𝐹‘𝑐)(.r‘𝑆)(𝐹‘𝐴)))
18	17	ralrimiva 2623	. . . . 5 ⊢ (((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ 𝐴 ∥ 𝐵) → ∀𝑐 ∈ 𝑋 (𝐹‘(𝑐(.r‘𝑅)𝐴)) = ((𝐹‘𝑐)(.r‘𝑆)(𝐹‘𝐴)))
19		simpr 110	. . . . . 6 ⊢ (((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ 𝐴 ∥ 𝐵) → 𝐴 ∥ 𝐵)
20	3	a1i 9	. . . . . . 7 ⊢ (((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ 𝐴 ∥ 𝐵) → 𝑋 = (Base‘𝑅))
21		rhmdvdsr.m	. . . . . . . 8 ⊢ ∥ = (∥r‘𝑅)
22	21	a1i 9	. . . . . . 7 ⊢ (((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ 𝐴 ∥ 𝐵) → ∥ = (∥r‘𝑅))
23		rhmrcl1 14435	. . . . . . . . . 10 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → 𝑅 ∈ Ring)
24	23	3ad2ant1 1049	. . . . . . . . 9 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → 𝑅 ∈ Ring)
25	24	adantr 276	. . . . . . . 8 ⊢ (((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ 𝐴 ∥ 𝐵) → 𝑅 ∈ Ring)
26		ringsrg 14325	. . . . . . . 8 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ SRing)
27	25, 26	syl 14	. . . . . . 7 ⊢ (((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ 𝐴 ∥ 𝐵) → 𝑅 ∈ SRing)
28		eqidd 2239	. . . . . . 7 ⊢ (((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ 𝐴 ∥ 𝐵) → (.r‘𝑅) = (.r‘𝑅))
29	20, 22, 27, 28, 2	dvdsr2d 14375	. . . . . 6 ⊢ (((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ 𝐴 ∥ 𝐵) → (𝐴 ∥ 𝐵 ↔ ∃𝑐 ∈ 𝑋 (𝑐(.r‘𝑅)𝐴) = 𝐵))
30	19, 29	mpbid 147	. . . . 5 ⊢ (((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ 𝐴 ∥ 𝐵) → ∃𝑐 ∈ 𝑋 (𝑐(.r‘𝑅)𝐴) = 𝐵)
31		r19.29 2688	. . . . . 6 ⊢ ((∀𝑐 ∈ 𝑋 (𝐹‘(𝑐(.r‘𝑅)𝐴)) = ((𝐹‘𝑐)(.r‘𝑆)(𝐹‘𝐴)) ∧ ∃𝑐 ∈ 𝑋 (𝑐(.r‘𝑅)𝐴) = 𝐵) → ∃𝑐 ∈ 𝑋 ((𝐹‘(𝑐(.r‘𝑅)𝐴)) = ((𝐹‘𝑐)(.r‘𝑆)(𝐹‘𝐴)) ∧ (𝑐(.r‘𝑅)𝐴) = 𝐵))
32		simpl 109	. . . . . . . 8 ⊢ (((𝐹‘(𝑐(.r‘𝑅)𝐴)) = ((𝐹‘𝑐)(.r‘𝑆)(𝐹‘𝐴)) ∧ (𝑐(.r‘𝑅)𝐴) = 𝐵) → (𝐹‘(𝑐(.r‘𝑅)𝐴)) = ((𝐹‘𝑐)(.r‘𝑆)(𝐹‘𝐴)))
33		simpr 110	. . . . . . . . 9 ⊢ (((𝐹‘(𝑐(.r‘𝑅)𝐴)) = ((𝐹‘𝑐)(.r‘𝑆)(𝐹‘𝐴)) ∧ (𝑐(.r‘𝑅)𝐴) = 𝐵) → (𝑐(.r‘𝑅)𝐴) = 𝐵)
34	33	fveq2d 5694	. . . . . . . 8 ⊢ (((𝐹‘(𝑐(.r‘𝑅)𝐴)) = ((𝐹‘𝑐)(.r‘𝑆)(𝐹‘𝐴)) ∧ (𝑐(.r‘𝑅)𝐴) = 𝐵) → (𝐹‘(𝑐(.r‘𝑅)𝐴)) = (𝐹‘𝐵))
35	32, 34	eqtr3d 2273	. . . . . . 7 ⊢ (((𝐹‘(𝑐(.r‘𝑅)𝐴)) = ((𝐹‘𝑐)(.r‘𝑆)(𝐹‘𝐴)) ∧ (𝑐(.r‘𝑅)𝐴) = 𝐵) → ((𝐹‘𝑐)(.r‘𝑆)(𝐹‘𝐴)) = (𝐹‘𝐵))
36	35	reximi 2647	. . . . . 6 ⊢ (∃𝑐 ∈ 𝑋 ((𝐹‘(𝑐(.r‘𝑅)𝐴)) = ((𝐹‘𝑐)(.r‘𝑆)(𝐹‘𝐴)) ∧ (𝑐(.r‘𝑅)𝐴) = 𝐵) → ∃𝑐 ∈ 𝑋 ((𝐹‘𝑐)(.r‘𝑆)(𝐹‘𝐴)) = (𝐹‘𝐵))
37	31, 36	syl 14	. . . . 5 ⊢ ((∀𝑐 ∈ 𝑋 (𝐹‘(𝑐(.r‘𝑅)𝐴)) = ((𝐹‘𝑐)(.r‘𝑆)(𝐹‘𝐴)) ∧ ∃𝑐 ∈ 𝑋 (𝑐(.r‘𝑅)𝐴) = 𝐵) → ∃𝑐 ∈ 𝑋 ((𝐹‘𝑐)(.r‘𝑆)(𝐹‘𝐴)) = (𝐹‘𝐵))
38	18, 30, 37	syl2anc 415	. . . 4 ⊢ (((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ 𝐴 ∥ 𝐵) → ∃𝑐 ∈ 𝑋 ((𝐹‘𝑐)(.r‘𝑆)(𝐹‘𝐴)) = (𝐹‘𝐵))
39		r19.29 2688	. . . 4 ⊢ ((∀𝑐 ∈ 𝑋 (𝐹‘𝑐) ∈ (Base‘𝑆) ∧ ∃𝑐 ∈ 𝑋 ((𝐹‘𝑐)(.r‘𝑆)(𝐹‘𝐴)) = (𝐹‘𝐵)) → ∃𝑐 ∈ 𝑋 ((𝐹‘𝑐) ∈ (Base‘𝑆) ∧ ((𝐹‘𝑐)(.r‘𝑆)(𝐹‘𝐴)) = (𝐹‘𝐵)))
40	12, 38, 39	syl2anc 415	. . 3 ⊢ (((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ 𝐴 ∥ 𝐵) → ∃𝑐 ∈ 𝑋 ((𝐹‘𝑐) ∈ (Base‘𝑆) ∧ ((𝐹‘𝑐)(.r‘𝑆)(𝐹‘𝐴)) = (𝐹‘𝐵)))
41		oveq1 6082	. . . . . 6 ⊢ (𝑦 = (𝐹‘𝑐) → (𝑦(.r‘𝑆)(𝐹‘𝐴)) = ((𝐹‘𝑐)(.r‘𝑆)(𝐹‘𝐴)))
42	41	eqeq1d 2247	. . . . 5 ⊢ (𝑦 = (𝐹‘𝑐) → ((𝑦(.r‘𝑆)(𝐹‘𝐴)) = (𝐹‘𝐵) ↔ ((𝐹‘𝑐)(.r‘𝑆)(𝐹‘𝐴)) = (𝐹‘𝐵)))
43	42	rspcev 2929	. . . 4 ⊢ (((𝐹‘𝑐) ∈ (Base‘𝑆) ∧ ((𝐹‘𝑐)(.r‘𝑆)(𝐹‘𝐴)) = (𝐹‘𝐵)) → ∃𝑦 ∈ (Base‘𝑆)(𝑦(.r‘𝑆)(𝐹‘𝐴)) = (𝐹‘𝐵))
44	43	rexlimivw 2664	. . 3 ⊢ (∃𝑐 ∈ 𝑋 ((𝐹‘𝑐) ∈ (Base‘𝑆) ∧ ((𝐹‘𝑐)(.r‘𝑆)(𝐹‘𝐴)) = (𝐹‘𝐵)) → ∃𝑦 ∈ (Base‘𝑆)(𝑦(.r‘𝑆)(𝐹‘𝐴)) = (𝐹‘𝐵))
45	40, 44	syl 14	. 2 ⊢ (((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ 𝐴 ∥ 𝐵) → ∃𝑦 ∈ (Base‘𝑆)(𝑦(.r‘𝑆)(𝐹‘𝐴)) = (𝐹‘𝐵))
46		eqidd 2239	. . 3 ⊢ (((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ 𝐴 ∥ 𝐵) → (Base‘𝑆) = (Base‘𝑆))
47		rhmdvdsr.n	. . . 4 ⊢ / = (∥r‘𝑆)
48	47	a1i 9	. . 3 ⊢ (((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ 𝐴 ∥ 𝐵) → / = (∥r‘𝑆))
49		rhmrcl2 14436	. . . . . 6 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → 𝑆 ∈ Ring)
50	49	3ad2ant1 1049	. . . . 5 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → 𝑆 ∈ Ring)
51	50	adantr 276	. . . 4 ⊢ (((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ 𝐴 ∥ 𝐵) → 𝑆 ∈ Ring)
52		ringsrg 14325	. . . 4 ⊢ (𝑆 ∈ Ring → 𝑆 ∈ SRing)
53	51, 52	syl 14	. . 3 ⊢ (((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ 𝐴 ∥ 𝐵) → 𝑆 ∈ SRing)
54		eqidd 2239	. . 3 ⊢ (((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ 𝐴 ∥ 𝐵) → (.r‘𝑆) = (.r‘𝑆))
55	46, 48, 53, 54	dvdsrd 14374	. 2 ⊢ (((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ 𝐴 ∥ 𝐵) → ((𝐹‘𝐴) / (𝐹‘𝐵) ↔ ((𝐹‘𝐴) ∈ (Base‘𝑆) ∧ ∃𝑦 ∈ (Base‘𝑆)(𝑦(.r‘𝑆)(𝐹‘𝐴)) = (𝐹‘𝐵))))
56	7, 45, 55	mpbir2and 957	1 ⊢ (((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ 𝐴 ∥ 𝐵) → (𝐹‘𝐴) / (𝐹‘𝐵))

Syntax hints:   → wi 4   ∧ wa 104   ∧ w3a 1009   = wceq 1402   ∈ wcel 2209  ∀wral 2528  ∃wrex 2529   class class class wbr 4125  ‘cfv 5372  (class class class)co 6075  Basecbs 13330  ._rcmulr 13409  SRingcsrg 14241  Ringcrg 14274  ∥_rcdsr 14365   RingHom crh 14430

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4241  ax-sep 4244  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-cnex 8260  ax-resscn 8261  ax-1cn 8262  ax-1re 8263  ax-icn 8264  ax-addcl 8265  ax-addrcl 8266  ax-mulcl 8267  ax-addcom 8269  ax-addass 8271  ax-i2m1 8274  ax-0lt1 8275  ax-0id 8277  ax-rnegex 8278  ax-pre-ltirr 8281  ax-pre-ltadd 8285

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-iun 4009  df-br 4126  df-opab 4188  df-mpt 4189  df-id 4433  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379  df-fv 5380  df-riota 6028  df-ov 6078  df-oprab 6079  df-mpo 6080  df-1st 6364  df-2nd 6365  df-map 6914  df-pnf 8352  df-mnf 8353  df-ltxr 8355  df-inn 9284  df-2 9342  df-3 9343  df-ndx 13333  df-slot 13334  df-base 13336  df-sets 13337  df-plusg 13421  df-mulr 13422  df-0g 13589  df-mgm 13653  df-sgrp 13694  df-mnd 13707  df-mhm 13743  df-grp 13785  df-minusg 13786  df-ghm 14021  df-cmn 14066  df-abl 14067  df-mgp 14195  df-ur 14238  df-srg 14242  df-ring 14276  df-dvdsr 14368  df-rhm 14432

This theorem is referenced by:  elrhmunit  14457