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Theorem rnghmsubcsetclem2 20535

Description: Lemma 2 for rnghmsubcsetc 20536. (Contributed by AV, 9-Mar-2020.)

**Hypotheses**
Ref	Expression
rnghmsubcsetc.c	⊢ 𝐶 = (ExtStrCat‘𝑈)
rnghmsubcsetc.u	⊢ (𝜑 → 𝑈 ∈ 𝑉)
rnghmsubcsetc.b	⊢ (𝜑 → 𝐵 = (Rng ∩ 𝑈))
rnghmsubcsetc.h	⊢ (𝜑 → 𝐻 = ( RngHom ↾ (𝐵 × 𝐵)))

**Assertion**
Ref	Expression
rnghmsubcsetclem2	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ∀𝑓 ∈ (𝑥𝐻𝑦)∀𝑔 ∈ (𝑦𝐻𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥𝐻𝑧))

Distinct variable groups: 𝐵,𝑓,𝑔,𝑥,𝑦,𝑧 𝐶,𝑓,𝑔,𝑥,𝑦,𝑧 𝑓,𝐻,𝑔,𝑥,𝑦,𝑧 𝑥,𝑈,𝑦 𝜑,𝑓,𝑔,𝑥,𝑦,𝑧 Allowed substitution hints: 𝑈(𝑧,𝑓,𝑔) 𝑉(𝑥,𝑦,𝑧,𝑓,𝑔)

**Proof of Theorem rnghmsubcsetclem2**
Step	Hyp	Ref	Expression
1		simpl 482	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝜑)
2	1	adantr 480	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → 𝜑)
3	2	adantr 480	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) ∧ (𝑓 ∈ (𝑥𝐻𝑦) ∧ 𝑔 ∈ (𝑦𝐻𝑧))) → 𝜑)
4		simpr 484	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵))
5	4	adantr 480	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) ∧ (𝑓 ∈ (𝑥𝐻𝑦) ∧ 𝑔 ∈ (𝑦𝐻𝑧))) → (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵))
6		simpr 484	. . . . . . 7 ⊢ ((𝑓 ∈ (𝑥𝐻𝑦) ∧ 𝑔 ∈ (𝑦𝐻𝑧)) → 𝑔 ∈ (𝑦𝐻𝑧))
7	6	adantl 481	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) ∧ (𝑓 ∈ (𝑥𝐻𝑦) ∧ 𝑔 ∈ (𝑦𝐻𝑧))) → 𝑔 ∈ (𝑦𝐻𝑧))
8		rnghmsubcsetc.h	. . . . . . 7 ⊢ (𝜑 → 𝐻 = ( RngHom ↾ (𝐵 × 𝐵)))
9	8	rnghmresel 20523	. . . . . 6 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ 𝑔 ∈ (𝑦𝐻𝑧)) → 𝑔 ∈ (𝑦 RngHom 𝑧))
10	3, 5, 7, 9	syl3anc 1373	. . . . 5 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) ∧ (𝑓 ∈ (𝑥𝐻𝑦) ∧ 𝑔 ∈ (𝑦𝐻𝑧))) → 𝑔 ∈ (𝑦 RngHom 𝑧))
11		simpr 484	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ 𝐵)
12		simpl 482	. . . . . . . 8 ⊢ ((𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) → 𝑦 ∈ 𝐵)
13	11, 12	anim12i 613	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵))
14	13	adantr 480	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) ∧ (𝑓 ∈ (𝑥𝐻𝑦) ∧ 𝑔 ∈ (𝑦𝐻𝑧))) → (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵))
15		simprl 770	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) ∧ (𝑓 ∈ (𝑥𝐻𝑦) ∧ 𝑔 ∈ (𝑦𝐻𝑧))) → 𝑓 ∈ (𝑥𝐻𝑦))
16	8	rnghmresel 20523	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ 𝑓 ∈ (𝑥𝐻𝑦)) → 𝑓 ∈ (𝑥 RngHom 𝑦))
17	3, 14, 15, 16	syl3anc 1373	. . . . 5 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) ∧ (𝑓 ∈ (𝑥𝐻𝑦) ∧ 𝑔 ∈ (𝑦𝐻𝑧))) → 𝑓 ∈ (𝑥 RngHom 𝑦))
18		rnghmco 20360	. . . . 5 ⊢ ((𝑔 ∈ (𝑦 RngHom 𝑧) ∧ 𝑓 ∈ (𝑥 RngHom 𝑦)) → (𝑔 ∘ 𝑓) ∈ (𝑥 RngHom 𝑧))
19	10, 17, 18	syl2anc 584	. . . 4 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) ∧ (𝑓 ∈ (𝑥𝐻𝑦) ∧ 𝑔 ∈ (𝑦𝐻𝑧))) → (𝑔 ∘ 𝑓) ∈ (𝑥 RngHom 𝑧))
20		rnghmsubcsetc.c	. . . . 5 ⊢ 𝐶 = (ExtStrCat‘𝑈)
21		rnghmsubcsetc.u	. . . . . 6 ⊢ (𝜑 → 𝑈 ∈ 𝑉)
22	21	ad3antrrr 730	. . . . 5 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) ∧ (𝑓 ∈ (𝑥𝐻𝑦) ∧ 𝑔 ∈ (𝑦𝐻𝑧))) → 𝑈 ∈ 𝑉)
23		eqid 2729	. . . . 5 ⊢ (comp‘𝐶) = (comp‘𝐶)
24		rnghmsubcsetc.b	. . . . . . . . . 10 ⊢ (𝜑 → 𝐵 = (Rng ∩ 𝑈))
25	24	eleq2d 2814	. . . . . . . . 9 ⊢ (𝜑 → (𝑥 ∈ 𝐵 ↔ 𝑥 ∈ (Rng ∩ 𝑈)))
26		elinel2 4155	. . . . . . . . 9 ⊢ (𝑥 ∈ (Rng ∩ 𝑈) → 𝑥 ∈ 𝑈)
27	25, 26	biimtrdi 253	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ 𝐵 → 𝑥 ∈ 𝑈))
28	27	imp 406	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ 𝑈)
29	28	adantr 480	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → 𝑥 ∈ 𝑈)
30	29	adantr 480	. . . . 5 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) ∧ (𝑓 ∈ (𝑥𝐻𝑦) ∧ 𝑔 ∈ (𝑦𝐻𝑧))) → 𝑥 ∈ 𝑈)
31	24	eleq2d 2814	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑦 ∈ 𝐵 ↔ 𝑦 ∈ (Rng ∩ 𝑈)))
32		elinel2 4155	. . . . . . . . . . 11 ⊢ (𝑦 ∈ (Rng ∩ 𝑈) → 𝑦 ∈ 𝑈)
33	31, 32	biimtrdi 253	. . . . . . . . . 10 ⊢ (𝜑 → (𝑦 ∈ 𝐵 → 𝑦 ∈ 𝑈))
34	33	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝑦 ∈ 𝐵 → 𝑦 ∈ 𝑈))
35	34	com12 32	. . . . . . . 8 ⊢ (𝑦 ∈ 𝐵 → ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝑦 ∈ 𝑈))
36	35	adantr 480	. . . . . . 7 ⊢ ((𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) → ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝑦 ∈ 𝑈))
37	36	impcom 407	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → 𝑦 ∈ 𝑈)
38	37	adantr 480	. . . . 5 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) ∧ (𝑓 ∈ (𝑥𝐻𝑦) ∧ 𝑔 ∈ (𝑦𝐻𝑧))) → 𝑦 ∈ 𝑈)
39	24	eleq2d 2814	. . . . . . . . . 10 ⊢ (𝜑 → (𝑧 ∈ 𝐵 ↔ 𝑧 ∈ (Rng ∩ 𝑈)))
40		elinel2 4155	. . . . . . . . . 10 ⊢ (𝑧 ∈ (Rng ∩ 𝑈) → 𝑧 ∈ 𝑈)
41	39, 40	biimtrdi 253	. . . . . . . . 9 ⊢ (𝜑 → (𝑧 ∈ 𝐵 → 𝑧 ∈ 𝑈))
42	41	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝑧 ∈ 𝐵 → 𝑧 ∈ 𝑈))
43	42	adantld 490	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ((𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) → 𝑧 ∈ 𝑈))
44	43	imp 406	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → 𝑧 ∈ 𝑈)
45	44	adantr 480	. . . . 5 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) ∧ (𝑓 ∈ (𝑥𝐻𝑦) ∧ 𝑔 ∈ (𝑦𝐻𝑧))) → 𝑧 ∈ 𝑈)
46		eqid 2729	. . . . 5 ⊢ (Base‘𝑥) = (Base‘𝑥)
47		eqid 2729	. . . . 5 ⊢ (Base‘𝑦) = (Base‘𝑦)
48		eqid 2729	. . . . 5 ⊢ (Base‘𝑧) = (Base‘𝑧)
49		simprl 770	. . . . . . . . . . . . . . 15 ⊢ ((𝑦 ∈ 𝐵 ∧ (𝜑 ∧ 𝑥 ∈ 𝐵)) → 𝜑)
50	49	adantr 480	. . . . . . . . . . . . . 14 ⊢ (((𝑦 ∈ 𝐵 ∧ (𝜑 ∧ 𝑥 ∈ 𝐵)) ∧ 𝑓 ∈ (𝑥𝐻𝑦)) → 𝜑)
51	11	anim1i 615	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) → (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵))
52	51	ancoms 458	. . . . . . . . . . . . . . 15 ⊢ ((𝑦 ∈ 𝐵 ∧ (𝜑 ∧ 𝑥 ∈ 𝐵)) → (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵))
53	52	adantr 480	. . . . . . . . . . . . . 14 ⊢ (((𝑦 ∈ 𝐵 ∧ (𝜑 ∧ 𝑥 ∈ 𝐵)) ∧ 𝑓 ∈ (𝑥𝐻𝑦)) → (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵))
54		simpr 484	. . . . . . . . . . . . . 14 ⊢ (((𝑦 ∈ 𝐵 ∧ (𝜑 ∧ 𝑥 ∈ 𝐵)) ∧ 𝑓 ∈ (𝑥𝐻𝑦)) → 𝑓 ∈ (𝑥𝐻𝑦))
55	50, 53, 54, 16	syl3anc 1373	. . . . . . . . . . . . 13 ⊢ (((𝑦 ∈ 𝐵 ∧ (𝜑 ∧ 𝑥 ∈ 𝐵)) ∧ 𝑓 ∈ (𝑥𝐻𝑦)) → 𝑓 ∈ (𝑥 RngHom 𝑦))
56	46, 47	rnghmf 20351	. . . . . . . . . . . . 13 ⊢ (𝑓 ∈ (𝑥 RngHom 𝑦) → 𝑓:(Base‘𝑥)⟶(Base‘𝑦))
57	55, 56	syl 17	. . . . . . . . . . . 12 ⊢ (((𝑦 ∈ 𝐵 ∧ (𝜑 ∧ 𝑥 ∈ 𝐵)) ∧ 𝑓 ∈ (𝑥𝐻𝑦)) → 𝑓:(Base‘𝑥)⟶(Base‘𝑦))
58	57	ex 412	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ 𝐵 ∧ (𝜑 ∧ 𝑥 ∈ 𝐵)) → (𝑓 ∈ (𝑥𝐻𝑦) → 𝑓:(Base‘𝑥)⟶(Base‘𝑦)))
59	58	ex 412	. . . . . . . . . 10 ⊢ (𝑦 ∈ 𝐵 → ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝑓 ∈ (𝑥𝐻𝑦) → 𝑓:(Base‘𝑥)⟶(Base‘𝑦))))
60	59	adantr 480	. . . . . . . . 9 ⊢ ((𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) → ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝑓 ∈ (𝑥𝐻𝑦) → 𝑓:(Base‘𝑥)⟶(Base‘𝑦))))
61	60	impcom 407	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → (𝑓 ∈ (𝑥𝐻𝑦) → 𝑓:(Base‘𝑥)⟶(Base‘𝑦)))
62	61	com12 32	. . . . . . 7 ⊢ (𝑓 ∈ (𝑥𝐻𝑦) → (((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → 𝑓:(Base‘𝑥)⟶(Base‘𝑦)))
63	62	adantr 480	. . . . . 6 ⊢ ((𝑓 ∈ (𝑥𝐻𝑦) ∧ 𝑔 ∈ (𝑦𝐻𝑧)) → (((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → 𝑓:(Base‘𝑥)⟶(Base‘𝑦)))
64	63	impcom 407	. . . . 5 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) ∧ (𝑓 ∈ (𝑥𝐻𝑦) ∧ 𝑔 ∈ (𝑦𝐻𝑧))) → 𝑓:(Base‘𝑥)⟶(Base‘𝑦))
65	9	3expa 1118	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) ∧ 𝑔 ∈ (𝑦𝐻𝑧)) → 𝑔 ∈ (𝑦 RngHom 𝑧))
66	47, 48	rnghmf 20351	. . . . . . . . . 10 ⊢ (𝑔 ∈ (𝑦 RngHom 𝑧) → 𝑔:(Base‘𝑦)⟶(Base‘𝑧))
67	65, 66	syl 17	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) ∧ 𝑔 ∈ (𝑦𝐻𝑧)) → 𝑔:(Base‘𝑦)⟶(Base‘𝑧))
68	67	ex 412	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → (𝑔 ∈ (𝑦𝐻𝑧) → 𝑔:(Base‘𝑦)⟶(Base‘𝑧)))
69	68	adantlr 715	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → (𝑔 ∈ (𝑦𝐻𝑧) → 𝑔:(Base‘𝑦)⟶(Base‘𝑧)))
70	69	adantld 490	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → ((𝑓 ∈ (𝑥𝐻𝑦) ∧ 𝑔 ∈ (𝑦𝐻𝑧)) → 𝑔:(Base‘𝑦)⟶(Base‘𝑧)))
71	70	imp 406	. . . . 5 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) ∧ (𝑓 ∈ (𝑥𝐻𝑦) ∧ 𝑔 ∈ (𝑦𝐻𝑧))) → 𝑔:(Base‘𝑦)⟶(Base‘𝑧))
72	20, 22, 23, 30, 38, 45, 46, 47, 48, 64, 71	estrcco 18054	. . . 4 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) ∧ (𝑓 ∈ (𝑥𝐻𝑦) ∧ 𝑔 ∈ (𝑦𝐻𝑧))) → (𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) = (𝑔 ∘ 𝑓))
73	8	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝐻 = ( RngHom ↾ (𝐵 × 𝐵)))
74	73	oveqdr 7381	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → (𝑥𝐻𝑧) = (𝑥( RngHom ↾ (𝐵 × 𝐵))𝑧))
75		ovres 7519	. . . . . . 7 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) → (𝑥( RngHom ↾ (𝐵 × 𝐵))𝑧) = (𝑥 RngHom 𝑧))
76	75	ad2ant2l 746	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → (𝑥( RngHom ↾ (𝐵 × 𝐵))𝑧) = (𝑥 RngHom 𝑧))
77	74, 76	eqtrd 2764	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → (𝑥𝐻𝑧) = (𝑥 RngHom 𝑧))
78	77	adantr 480	. . . 4 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) ∧ (𝑓 ∈ (𝑥𝐻𝑦) ∧ 𝑔 ∈ (𝑦𝐻𝑧))) → (𝑥𝐻𝑧) = (𝑥 RngHom 𝑧))
79	19, 72, 78	3eltr4d 2843	. . 3 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) ∧ (𝑓 ∈ (𝑥𝐻𝑦) ∧ 𝑔 ∈ (𝑦𝐻𝑧))) → (𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥𝐻𝑧))
80	79	ralrimivva 3172	. 2 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → ∀𝑓 ∈ (𝑥𝐻𝑦)∀𝑔 ∈ (𝑦𝐻𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥𝐻𝑧))
81	80	ralrimivva 3172	1 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ∀𝑓 ∈ (𝑥𝐻𝑦)∀𝑔 ∈ (𝑦𝐻𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥𝐻𝑧))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2109 ∀wral 3044 ∩ cin 3904 ⟨cop 4585 × cxp 5621 ↾ cres 5625 ∘ ccom 5627 ⟶wf 6482 ‘cfv 6486 (class class class)co 7353 Basecbs 17138 compcco 17191 ExtStrCatcestrc 18046 Rngcrng 20055 RngHom crnghm 20337

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 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5221 ax-sep 5238 ax-nul 5248 ax-pow 5307 ax-pr 5374 ax-un 7675 ax-cnex 11084 ax-resscn 11085 ax-1cn 11086 ax-icn 11087 ax-addcl 11088 ax-addrcl 11089 ax-mulcl 11090 ax-mulrcl 11091 ax-mulcom 11092 ax-addass 11093 ax-mulass 11094 ax-distr 11095 ax-i2m1 11096 ax-1ne0 11097 ax-1rid 11098 ax-rnegex 11099 ax-rrecex 11100 ax-cnre 11101 ax-pre-lttri 11102 ax-pre-lttrn 11103 ax-pre-ltadd 11104 ax-pre-mulgt0 11105

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3345 df-reu 3346 df-rab 3397 df-v 3440 df-sbc 3745 df-csb 3854 df-dif 3908 df-un 3910 df-in 3912 df-ss 3922 df-pss 3925 df-nul 4287 df-if 4479 df-pw 4555 df-sn 4580 df-pr 4582 df-tp 4584 df-op 4586 df-uni 4862 df-iun 4946 df-br 5096 df-opab 5158 df-mpt 5177 df-tr 5203 df-id 5518 df-eprel 5523 df-po 5531 df-so 5532 df-fr 5576 df-we 5578 df-xp 5629 df-rel 5630 df-cnv 5631 df-co 5632 df-dm 5633 df-rn 5634 df-res 5635 df-ima 5636 df-pred 6253 df-ord 6314 df-on 6315 df-lim 6316 df-suc 6317 df-iota 6442 df-fun 6488 df-fn 6489 df-f 6490 df-f1 6491 df-fo 6492 df-f1o 6493 df-fv 6494 df-riota 7310 df-ov 7356 df-oprab 7357 df-mpo 7358 df-om 7807 df-1st 7931 df-2nd 7932 df-frecs 8221 df-wrecs 8252 df-recs 8301 df-rdg 8339 df-1o 8395 df-er 8632 df-map 8762 df-en 8880 df-dom 8881 df-sdom 8882 df-fin 8883 df-pnf 11170 df-mnf 11171 df-xr 11172 df-ltxr 11173 df-le 11174 df-sub 11367 df-neg 11368 df-nn 12147 df-2 12209 df-3 12210 df-4 12211 df-5 12212 df-6 12213 df-7 12214 df-8 12215 df-9 12216 df-n0 12403 df-z 12490 df-dec 12610 df-uz 12754 df-fz 13429 df-struct 17076 df-sets 17093 df-slot 17111 df-ndx 17123 df-base 17139 df-plusg 17192 df-hom 17203 df-cco 17204 df-0g 17363 df-estrc 18047 df-mgm 18532 df-mgmhm 18584 df-sgrp 18611 df-mnd 18627 df-mhm 18675 df-grp 18833 df-ghm 19110 df-abl 19680 df-mgp 20044 df-rng 20056 df-rnghm 20339

This theorem is referenced by: rnghmsubcsetc 20536

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