Theorem rhmsscmap2 46870

Description: The unital ring homomorphisms between unital rings (in a universe) are a subcategory subset of the mappings between base sets of unital rings (in the same universe). (Contributed by AV, 6-Mar-2020.)

Hypotheses

Ref	Expression
rhmsscmap.u	⊢ (𝜑 → 𝑈 ∈ 𝑉)
rhmsscmap.r	⊢ (𝜑 → 𝑅 = (Ring ∩ 𝑈))

Assertion

Ref	Expression
rhmsscmap2	⊢ (𝜑 → ( RingHom ↾ (𝑅 × 𝑅)) ⊆cat (𝑥 ∈ 𝑅, 𝑦 ∈ 𝑅 ↦ ((Base‘𝑦) ↑m (Base‘𝑥))))

Distinct variable group:   𝑥,𝑅,𝑦

Allowed substitution hints:   𝜑(𝑥,𝑦)   𝑈(𝑥,𝑦)   𝑉(𝑥,𝑦)

Proof of Theorem rhmsscmap2

Dummy variables 𝑎 𝑏 ℎ are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ssidd 4004	. 2 ⊢ (𝜑 → 𝑅 ⊆ 𝑅)
2		eqid 2732	. . . . . . 7 ⊢ (Base‘𝑎) = (Base‘𝑎)
3		eqid 2732	. . . . . . 7 ⊢ (Base‘𝑏) = (Base‘𝑏)
4	2, 3	rhmf 20255	. . . . . 6 ⊢ (ℎ ∈ (𝑎 RingHom 𝑏) → ℎ:(Base‘𝑎)⟶(Base‘𝑏))
5		simpr 485	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑎 ∈ 𝑅 ∧ 𝑏 ∈ 𝑅)) ∧ ℎ:(Base‘𝑎)⟶(Base‘𝑏)) → ℎ:(Base‘𝑎)⟶(Base‘𝑏))
6		fvex 6901	. . . . . . . . . 10 ⊢ (Base‘𝑏) ∈ V
7		fvex 6901	. . . . . . . . . 10 ⊢ (Base‘𝑎) ∈ V
8	6, 7	pm3.2i 471	. . . . . . . . 9 ⊢ ((Base‘𝑏) ∈ V ∧ (Base‘𝑎) ∈ V)
9		elmapg 8829	. . . . . . . . 9 ⊢ (((Base‘𝑏) ∈ V ∧ (Base‘𝑎) ∈ V) → (ℎ ∈ ((Base‘𝑏) ↑m (Base‘𝑎)) ↔ ℎ:(Base‘𝑎)⟶(Base‘𝑏)))
10	8, 9	mp1i 13	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑎 ∈ 𝑅 ∧ 𝑏 ∈ 𝑅)) ∧ ℎ:(Base‘𝑎)⟶(Base‘𝑏)) → (ℎ ∈ ((Base‘𝑏) ↑m (Base‘𝑎)) ↔ ℎ:(Base‘𝑎)⟶(Base‘𝑏)))
11	5, 10	mpbird 256	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑎 ∈ 𝑅 ∧ 𝑏 ∈ 𝑅)) ∧ ℎ:(Base‘𝑎)⟶(Base‘𝑏)) → ℎ ∈ ((Base‘𝑏) ↑m (Base‘𝑎)))
12	11	ex 413	. . . . . 6 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑅 ∧ 𝑏 ∈ 𝑅)) → (ℎ:(Base‘𝑎)⟶(Base‘𝑏) → ℎ ∈ ((Base‘𝑏) ↑m (Base‘𝑎))))
13	4, 12	syl5 34	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑅 ∧ 𝑏 ∈ 𝑅)) → (ℎ ∈ (𝑎 RingHom 𝑏) → ℎ ∈ ((Base‘𝑏) ↑m (Base‘𝑎))))
14	13	ssrdv 3987	. . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑅 ∧ 𝑏 ∈ 𝑅)) → (𝑎 RingHom 𝑏) ⊆ ((Base‘𝑏) ↑m (Base‘𝑎)))
15		ovres 7569	. . . . 5 ⊢ ((𝑎 ∈ 𝑅 ∧ 𝑏 ∈ 𝑅) → (𝑎( RingHom ↾ (𝑅 × 𝑅))𝑏) = (𝑎 RingHom 𝑏))
16	15	adantl 482	. . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑅 ∧ 𝑏 ∈ 𝑅)) → (𝑎( RingHom ↾ (𝑅 × 𝑅))𝑏) = (𝑎 RingHom 𝑏))
17		eqidd 2733	. . . . . 6 ⊢ ((𝑎 ∈ 𝑅 ∧ 𝑏 ∈ 𝑅) → (𝑥 ∈ 𝑅, 𝑦 ∈ 𝑅 ↦ ((Base‘𝑦) ↑m (Base‘𝑥))) = (𝑥 ∈ 𝑅, 𝑦 ∈ 𝑅 ↦ ((Base‘𝑦) ↑m (Base‘𝑥))))
18		fveq2 6888	. . . . . . . 8 ⊢ (𝑦 = 𝑏 → (Base‘𝑦) = (Base‘𝑏))
19		fveq2 6888	. . . . . . . 8 ⊢ (𝑥 = 𝑎 → (Base‘𝑥) = (Base‘𝑎))
20	18, 19	oveqan12rd 7425	. . . . . . 7 ⊢ ((𝑥 = 𝑎 ∧ 𝑦 = 𝑏) → ((Base‘𝑦) ↑m (Base‘𝑥)) = ((Base‘𝑏) ↑m (Base‘𝑎)))
21	20	adantl 482	. . . . . 6 ⊢ (((𝑎 ∈ 𝑅 ∧ 𝑏 ∈ 𝑅) ∧ (𝑥 = 𝑎 ∧ 𝑦 = 𝑏)) → ((Base‘𝑦) ↑m (Base‘𝑥)) = ((Base‘𝑏) ↑m (Base‘𝑎)))
22		simpl 483	. . . . . 6 ⊢ ((𝑎 ∈ 𝑅 ∧ 𝑏 ∈ 𝑅) → 𝑎 ∈ 𝑅)
23		simpr 485	. . . . . 6 ⊢ ((𝑎 ∈ 𝑅 ∧ 𝑏 ∈ 𝑅) → 𝑏 ∈ 𝑅)
24		ovexd 7440	. . . . . 6 ⊢ ((𝑎 ∈ 𝑅 ∧ 𝑏 ∈ 𝑅) → ((Base‘𝑏) ↑m (Base‘𝑎)) ∈ V)
25	17, 21, 22, 23, 24	ovmpod 7556	. . . . 5 ⊢ ((𝑎 ∈ 𝑅 ∧ 𝑏 ∈ 𝑅) → (𝑎(𝑥 ∈ 𝑅, 𝑦 ∈ 𝑅 ↦ ((Base‘𝑦) ↑m (Base‘𝑥)))𝑏) = ((Base‘𝑏) ↑m (Base‘𝑎)))
26	25	adantl 482	. . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑅 ∧ 𝑏 ∈ 𝑅)) → (𝑎(𝑥 ∈ 𝑅, 𝑦 ∈ 𝑅 ↦ ((Base‘𝑦) ↑m (Base‘𝑥)))𝑏) = ((Base‘𝑏) ↑m (Base‘𝑎)))
27	14, 16, 26	3sstr4d 4028	. . 3 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑅 ∧ 𝑏 ∈ 𝑅)) → (𝑎( RingHom ↾ (𝑅 × 𝑅))𝑏) ⊆ (𝑎(𝑥 ∈ 𝑅, 𝑦 ∈ 𝑅 ↦ ((Base‘𝑦) ↑m (Base‘𝑥)))𝑏))
28	27	ralrimivva 3200	. 2 ⊢ (𝜑 → ∀𝑎 ∈ 𝑅 ∀𝑏 ∈ 𝑅 (𝑎( RingHom ↾ (𝑅 × 𝑅))𝑏) ⊆ (𝑎(𝑥 ∈ 𝑅, 𝑦 ∈ 𝑅 ↦ ((Base‘𝑦) ↑m (Base‘𝑥)))𝑏))
29		rhmfn 46705	. . . . 5 ⊢ RingHom Fn (Ring × Ring)
30	29	a1i 11	. . . 4 ⊢ (𝜑 → RingHom Fn (Ring × Ring))
31		rhmsscmap.r	. . . . . 6 ⊢ (𝜑 → 𝑅 = (Ring ∩ 𝑈))
32		inss1 4227	. . . . . 6 ⊢ (Ring ∩ 𝑈) ⊆ Ring
33	31, 32	eqsstrdi 4035	. . . . 5 ⊢ (𝜑 → 𝑅 ⊆ Ring)
34		xpss12 5690	. . . . 5 ⊢ ((𝑅 ⊆ Ring ∧ 𝑅 ⊆ Ring) → (𝑅 × 𝑅) ⊆ (Ring × Ring))
35	33, 33, 34	syl2anc 584	. . . 4 ⊢ (𝜑 → (𝑅 × 𝑅) ⊆ (Ring × Ring))
36		fnssres 6670	. . . 4 ⊢ (( RingHom Fn (Ring × Ring) ∧ (𝑅 × 𝑅) ⊆ (Ring × Ring)) → ( RingHom ↾ (𝑅 × 𝑅)) Fn (𝑅 × 𝑅))
37	30, 35, 36	syl2anc 584	. . 3 ⊢ (𝜑 → ( RingHom ↾ (𝑅 × 𝑅)) Fn (𝑅 × 𝑅))
38		eqid 2732	. . . . 5 ⊢ (𝑥 ∈ 𝑅, 𝑦 ∈ 𝑅 ↦ ((Base‘𝑦) ↑m (Base‘𝑥))) = (𝑥 ∈ 𝑅, 𝑦 ∈ 𝑅 ↦ ((Base‘𝑦) ↑m (Base‘𝑥)))
39		ovex 7438	. . . . 5 ⊢ ((Base‘𝑦) ↑m (Base‘𝑥)) ∈ V
40	38, 39	fnmpoi 8052	. . . 4 ⊢ (𝑥 ∈ 𝑅, 𝑦 ∈ 𝑅 ↦ ((Base‘𝑦) ↑m (Base‘𝑥))) Fn (𝑅 × 𝑅)
41	40	a1i 11	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝑅, 𝑦 ∈ 𝑅 ↦ ((Base‘𝑦) ↑m (Base‘𝑥))) Fn (𝑅 × 𝑅))
42		incom 4200	. . . . 5 ⊢ (Ring ∩ 𝑈) = (𝑈 ∩ Ring)
43		rhmsscmap.u	. . . . . 6 ⊢ (𝜑 → 𝑈 ∈ 𝑉)
44		inex1g 5318	. . . . . 6 ⊢ (𝑈 ∈ 𝑉 → (𝑈 ∩ Ring) ∈ V)
45	43, 44	syl 17	. . . . 5 ⊢ (𝜑 → (𝑈 ∩ Ring) ∈ V)
46	42, 45	eqeltrid 2837	. . . 4 ⊢ (𝜑 → (Ring ∩ 𝑈) ∈ V)
47	31, 46	eqeltrd 2833	. . 3 ⊢ (𝜑 → 𝑅 ∈ V)
48	37, 41, 47	isssc 17763	. 2 ⊢ (𝜑 → (( RingHom ↾ (𝑅 × 𝑅)) ⊆cat (𝑥 ∈ 𝑅, 𝑦 ∈ 𝑅 ↦ ((Base‘𝑦) ↑m (Base‘𝑥))) ↔ (𝑅 ⊆ 𝑅 ∧ ∀𝑎 ∈ 𝑅 ∀𝑏 ∈ 𝑅 (𝑎( RingHom ↾ (𝑅 × 𝑅))𝑏) ⊆ (𝑎(𝑥 ∈ 𝑅, 𝑦 ∈ 𝑅 ↦ ((Base‘𝑦) ↑m (Base‘𝑥)))𝑏))))
49	1, 28, 48	mpbir2and 711	1 ⊢ (𝜑 → ( RingHom ↾ (𝑅 × 𝑅)) ⊆cat (𝑥 ∈ 𝑅, 𝑦 ∈ 𝑅 ↦ ((Base‘𝑦) ↑m (Base‘𝑥))))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1541   ∈ wcel 2106  ∀wral 3061  Vcvv 3474   ∩ cin 3946   ⊆ wss 3947   class class class wbr 5147   × cxp 5673   ↾ cres 5677   Fn wfn 6535  ⟶wf 6536  ‘cfv 6540  (class class class)co 7405   ∈ cmpo 7407   ↑_m cmap 8816  Basecbs 17140   ⊆_cat cssc 17750  Ringcrg 20049   RingHom crh 20240

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7721  ax-cnex 11162  ax-resscn 11163  ax-1cn 11164  ax-icn 11165  ax-addcl 11166  ax-addrcl 11167  ax-mulcl 11168  ax-mulrcl 11169  ax-mulcom 11170  ax-addass 11171  ax-mulass 11172  ax-distr 11173  ax-i2m1 11174  ax-1ne0 11175  ax-1rid 11176  ax-rnegex 11177  ax-rrecex 11178  ax-cnre 11179  ax-pre-lttri 11180  ax-pre-lttrn 11181  ax-pre-ltadd 11182  ax-pre-mulgt0 11183

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6297  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-riota 7361  df-ov 7408  df-oprab 7409  df-mpo 7410  df-om 7852  df-1st 7971  df-2nd 7972  df-frecs 8262  df-wrecs 8293  df-recs 8367  df-rdg 8406  df-er 8699  df-map 8818  df-ixp 8888  df-en 8936  df-dom 8937  df-sdom 8938  df-pnf 11246  df-mnf 11247  df-xr 11248  df-ltxr 11249  df-le 11250  df-sub 11442  df-neg 11443  df-nn 12209  df-2 12271  df-sets 17093  df-slot 17111  df-ndx 17123  df-base 17141  df-plusg 17206  df-0g 17383  df-ssc 17753  df-mhm 18667  df-ghm 19084  df-mgp 19982  df-ur 19999  df-ring 20051  df-rnghom 20243

This theorem is referenced by: (None)