Theorem rhmsscmap 44365

Description: The unital ring homomorphisms between unital rings (in a universe) are a subcategory subset of the mappings between base sets of extensible structures (in the same universe). (Contributed by AV, 9-Mar-2020.)

Hypotheses

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

Assertion

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

Distinct variable groups:   𝑥,𝑅,𝑦   𝑥,𝑈,𝑦   𝜑,𝑥,𝑦

Allowed substitution hints:   𝑉(𝑥,𝑦)

Proof of Theorem rhmsscmap

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

Step	Hyp	Ref	Expression
1		rhmsscmap.r	. . 3 ⊢ (𝜑 → 𝑅 = (Ring ∩ 𝑈))
2		inss2 4199	. . 3 ⊢ (Ring ∩ 𝑈) ⊆ 𝑈
3	1, 2	eqsstrdi 4014	. 2 ⊢ (𝜑 → 𝑅 ⊆ 𝑈)
4		eqid 2820	. . . . . . 7 ⊢ (Base‘𝑎) = (Base‘𝑎)
5		eqid 2820	. . . . . . 7 ⊢ (Base‘𝑏) = (Base‘𝑏)
6	4, 5	rhmf 19473	. . . . . 6 ⊢ (ℎ ∈ (𝑎 RingHom 𝑏) → ℎ:(Base‘𝑎)⟶(Base‘𝑏))
7		simpr 487	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑎 ∈ 𝑅 ∧ 𝑏 ∈ 𝑅)) ∧ ℎ:(Base‘𝑎)⟶(Base‘𝑏)) → ℎ:(Base‘𝑎)⟶(Base‘𝑏))
8		fvex 6676	. . . . . . . . . 10 ⊢ (Base‘𝑏) ∈ V
9		fvex 6676	. . . . . . . . . 10 ⊢ (Base‘𝑎) ∈ V
10	8, 9	pm3.2i 473	. . . . . . . . 9 ⊢ ((Base‘𝑏) ∈ V ∧ (Base‘𝑎) ∈ V)
11		elmapg 8412	. . . . . . . . 9 ⊢ (((Base‘𝑏) ∈ V ∧ (Base‘𝑎) ∈ V) → (ℎ ∈ ((Base‘𝑏) ↑m (Base‘𝑎)) ↔ ℎ:(Base‘𝑎)⟶(Base‘𝑏)))
12	10, 11	mp1i 13	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑎 ∈ 𝑅 ∧ 𝑏 ∈ 𝑅)) ∧ ℎ:(Base‘𝑎)⟶(Base‘𝑏)) → (ℎ ∈ ((Base‘𝑏) ↑m (Base‘𝑎)) ↔ ℎ:(Base‘𝑎)⟶(Base‘𝑏)))
13	7, 12	mpbird 259	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑎 ∈ 𝑅 ∧ 𝑏 ∈ 𝑅)) ∧ ℎ:(Base‘𝑎)⟶(Base‘𝑏)) → ℎ ∈ ((Base‘𝑏) ↑m (Base‘𝑎)))
14	13	ex 415	. . . . . 6 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑅 ∧ 𝑏 ∈ 𝑅)) → (ℎ:(Base‘𝑎)⟶(Base‘𝑏) → ℎ ∈ ((Base‘𝑏) ↑m (Base‘𝑎))))
15	6, 14	syl5 34	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑅 ∧ 𝑏 ∈ 𝑅)) → (ℎ ∈ (𝑎 RingHom 𝑏) → ℎ ∈ ((Base‘𝑏) ↑m (Base‘𝑎))))
16	15	ssrdv 3966	. . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑅 ∧ 𝑏 ∈ 𝑅)) → (𝑎 RingHom 𝑏) ⊆ ((Base‘𝑏) ↑m (Base‘𝑎)))
17		ovres 7307	. . . . 5 ⊢ ((𝑎 ∈ 𝑅 ∧ 𝑏 ∈ 𝑅) → (𝑎( RingHom ↾ (𝑅 × 𝑅))𝑏) = (𝑎 RingHom 𝑏))
18	17	adantl 484	. . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑅 ∧ 𝑏 ∈ 𝑅)) → (𝑎( RingHom ↾ (𝑅 × 𝑅))𝑏) = (𝑎 RingHom 𝑏))
19		eqidd 2821	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑅 ∧ 𝑏 ∈ 𝑅)) → (𝑥 ∈ 𝑈, 𝑦 ∈ 𝑈 ↦ ((Base‘𝑦) ↑m (Base‘𝑥))) = (𝑥 ∈ 𝑈, 𝑦 ∈ 𝑈 ↦ ((Base‘𝑦) ↑m (Base‘𝑥))))
20		fveq2 6663	. . . . . . 7 ⊢ (𝑦 = 𝑏 → (Base‘𝑦) = (Base‘𝑏))
21		fveq2 6663	. . . . . . 7 ⊢ (𝑥 = 𝑎 → (Base‘𝑥) = (Base‘𝑎))
22	20, 21	oveqan12rd 7169	. . . . . 6 ⊢ ((𝑥 = 𝑎 ∧ 𝑦 = 𝑏) → ((Base‘𝑦) ↑m (Base‘𝑥)) = ((Base‘𝑏) ↑m (Base‘𝑎)))
23	22	adantl 484	. . . . 5 ⊢ (((𝜑 ∧ (𝑎 ∈ 𝑅 ∧ 𝑏 ∈ 𝑅)) ∧ (𝑥 = 𝑎 ∧ 𝑦 = 𝑏)) → ((Base‘𝑦) ↑m (Base‘𝑥)) = ((Base‘𝑏) ↑m (Base‘𝑎)))
24	3	sseld 3959	. . . . . . . 8 ⊢ (𝜑 → (𝑎 ∈ 𝑅 → 𝑎 ∈ 𝑈))
25	24	com12 32	. . . . . . 7 ⊢ (𝑎 ∈ 𝑅 → (𝜑 → 𝑎 ∈ 𝑈))
26	25	adantr 483	. . . . . 6 ⊢ ((𝑎 ∈ 𝑅 ∧ 𝑏 ∈ 𝑅) → (𝜑 → 𝑎 ∈ 𝑈))
27	26	impcom 410	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑅 ∧ 𝑏 ∈ 𝑅)) → 𝑎 ∈ 𝑈)
28	3	sseld 3959	. . . . . . . 8 ⊢ (𝜑 → (𝑏 ∈ 𝑅 → 𝑏 ∈ 𝑈))
29	28	com12 32	. . . . . . 7 ⊢ (𝑏 ∈ 𝑅 → (𝜑 → 𝑏 ∈ 𝑈))
30	29	adantl 484	. . . . . 6 ⊢ ((𝑎 ∈ 𝑅 ∧ 𝑏 ∈ 𝑅) → (𝜑 → 𝑏 ∈ 𝑈))
31	30	impcom 410	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑅 ∧ 𝑏 ∈ 𝑅)) → 𝑏 ∈ 𝑈)
32		ovexd 7184	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑅 ∧ 𝑏 ∈ 𝑅)) → ((Base‘𝑏) ↑m (Base‘𝑎)) ∈ V)
33	19, 23, 27, 31, 32	ovmpod 7295	. . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑅 ∧ 𝑏 ∈ 𝑅)) → (𝑎(𝑥 ∈ 𝑈, 𝑦 ∈ 𝑈 ↦ ((Base‘𝑦) ↑m (Base‘𝑥)))𝑏) = ((Base‘𝑏) ↑m (Base‘𝑎)))
34	16, 18, 33	3sstr4d 4007	. . 3 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑅 ∧ 𝑏 ∈ 𝑅)) → (𝑎( RingHom ↾ (𝑅 × 𝑅))𝑏) ⊆ (𝑎(𝑥 ∈ 𝑈, 𝑦 ∈ 𝑈 ↦ ((Base‘𝑦) ↑m (Base‘𝑥)))𝑏))
35	34	ralrimivva 3190	. 2 ⊢ (𝜑 → ∀𝑎 ∈ 𝑅 ∀𝑏 ∈ 𝑅 (𝑎( RingHom ↾ (𝑅 × 𝑅))𝑏) ⊆ (𝑎(𝑥 ∈ 𝑈, 𝑦 ∈ 𝑈 ↦ ((Base‘𝑦) ↑m (Base‘𝑥)))𝑏))
36		rhmfn 44263	. . . . 5 ⊢ RingHom Fn (Ring × Ring)
37	36	a1i 11	. . . 4 ⊢ (𝜑 → RingHom Fn (Ring × Ring))
38		inss1 4198	. . . . . 6 ⊢ (Ring ∩ 𝑈) ⊆ Ring
39	1, 38	eqsstrdi 4014	. . . . 5 ⊢ (𝜑 → 𝑅 ⊆ Ring)
40		xpss12 5563	. . . . 5 ⊢ ((𝑅 ⊆ Ring ∧ 𝑅 ⊆ Ring) → (𝑅 × 𝑅) ⊆ (Ring × Ring))
41	39, 39, 40	syl2anc 586	. . . 4 ⊢ (𝜑 → (𝑅 × 𝑅) ⊆ (Ring × Ring))
42		fnssres 6463	. . . 4 ⊢ (( RingHom Fn (Ring × Ring) ∧ (𝑅 × 𝑅) ⊆ (Ring × Ring)) → ( RingHom ↾ (𝑅 × 𝑅)) Fn (𝑅 × 𝑅))
43	37, 41, 42	syl2anc 586	. . 3 ⊢ (𝜑 → ( RingHom ↾ (𝑅 × 𝑅)) Fn (𝑅 × 𝑅))
44		eqid 2820	. . . . 5 ⊢ (𝑥 ∈ 𝑈, 𝑦 ∈ 𝑈 ↦ ((Base‘𝑦) ↑m (Base‘𝑥))) = (𝑥 ∈ 𝑈, 𝑦 ∈ 𝑈 ↦ ((Base‘𝑦) ↑m (Base‘𝑥)))
45		ovex 7182	. . . . 5 ⊢ ((Base‘𝑦) ↑m (Base‘𝑥)) ∈ V
46	44, 45	fnmpoi 7761	. . . 4 ⊢ (𝑥 ∈ 𝑈, 𝑦 ∈ 𝑈 ↦ ((Base‘𝑦) ↑m (Base‘𝑥))) Fn (𝑈 × 𝑈)
47	46	a1i 11	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝑈, 𝑦 ∈ 𝑈 ↦ ((Base‘𝑦) ↑m (Base‘𝑥))) Fn (𝑈 × 𝑈))
48		rhmsscmap.u	. . . 4 ⊢ (𝜑 → 𝑈 ∈ 𝑉)
49		elex 3509	. . . 4 ⊢ (𝑈 ∈ 𝑉 → 𝑈 ∈ V)
50	48, 49	syl 17	. . 3 ⊢ (𝜑 → 𝑈 ∈ V)
51	43, 47, 50	isssc 17085	. 2 ⊢ (𝜑 → (( RingHom ↾ (𝑅 × 𝑅)) ⊆cat (𝑥 ∈ 𝑈, 𝑦 ∈ 𝑈 ↦ ((Base‘𝑦) ↑m (Base‘𝑥))) ↔ (𝑅 ⊆ 𝑈 ∧ ∀𝑎 ∈ 𝑅 ∀𝑏 ∈ 𝑅 (𝑎( RingHom ↾ (𝑅 × 𝑅))𝑏) ⊆ (𝑎(𝑥 ∈ 𝑈, 𝑦 ∈ 𝑈 ↦ ((Base‘𝑦) ↑m (Base‘𝑥)))𝑏))))
52	3, 35, 51	mpbir2and 711	1 ⊢ (𝜑 → ( RingHom ↾ (𝑅 × 𝑅)) ⊆cat (𝑥 ∈ 𝑈, 𝑦 ∈ 𝑈 ↦ ((Base‘𝑦) ↑m (Base‘𝑥))))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   = wceq 1536   ∈ wcel 2113  ∀wral 3137  Vcvv 3491   ∩ cin 3928   ⊆ wss 3929   class class class wbr 5059   × cxp 5546   ↾ cres 5550   Fn wfn 6343  ⟶wf 6344  ‘cfv 6348  (class class class)co 7149   ∈ cmpo 7151   ↑_m cmap 8399  Basecbs 16478   ⊆_cat cssc 17072  Ringcrg 19292   RingHom crh 19459

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2792  ax-rep 5183  ax-sep 5196  ax-nul 5203  ax-pow 5259  ax-pr 5323  ax-un 7454  ax-cnex 10586  ax-resscn 10587  ax-1cn 10588  ax-icn 10589  ax-addcl 10590  ax-addrcl 10591  ax-mulcl 10592  ax-mulrcl 10593  ax-mulcom 10594  ax-addass 10595  ax-mulass 10596  ax-distr 10597  ax-i2m1 10598  ax-1ne0 10599  ax-1rid 10600  ax-rnegex 10601  ax-rrecex 10602  ax-cnre 10603  ax-pre-lttri 10604  ax-pre-lttrn 10605  ax-pre-ltadd 10606  ax-pre-mulgt0 10607

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1083  df-3an 1084  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-mo 2621  df-eu 2653  df-clab 2799  df-cleq 2813  df-clel 2892  df-nfc 2962  df-ne 3016  df-nel 3123  df-ral 3142  df-rex 3143  df-reu 3144  df-rab 3146  df-v 3493  df-sbc 3769  df-csb 3877  df-dif 3932  df-un 3934  df-in 3936  df-ss 3945  df-pss 3947  df-nul 4285  df-if 4461  df-pw 4534  df-sn 4561  df-pr 4563  df-tp 4565  df-op 4567  df-uni 4832  df-iun 4914  df-br 5060  df-opab 5122  df-mpt 5140  df-tr 5166  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-riota 7107  df-ov 7152  df-oprab 7153  df-mpo 7154  df-om 7574  df-1st 7682  df-2nd 7683  df-wrecs 7940  df-recs 8001  df-rdg 8039  df-er 8282  df-map 8401  df-ixp 8455  df-en 8503  df-dom 8504  df-sdom 8505  df-pnf 10670  df-mnf 10671  df-xr 10672  df-ltxr 10673  df-le 10674  df-sub 10865  df-neg 10866  df-nn 11632  df-2 11694  df-ndx 16481  df-slot 16482  df-base 16484  df-sets 16485  df-plusg 16573  df-0g 16710  df-ssc 17075  df-mhm 17951  df-ghm 18351  df-mgp 19235  df-ur 19247  df-ring 19294  df-rnghom 19462

This theorem is referenced by:  rhmsubcsetc  44368