Theorem isotr 5476

Description: Composition (transitive) law for isomorphism. Proposition 6.30(3) of [TakeutiZaring] p. 33. (Contributed by NM, 27-Apr-2004.) (Proof shortened by Mario Carneiro, 5-Dec-2016.)

Assertion

Ref	Expression
isotr	⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐺 Isom 𝑆, 𝑇 (𝐵, 𝐶)) → (𝐺 ∘ 𝐻) Isom 𝑅, 𝑇 (𝐴, 𝐶))

Proof of Theorem isotr

Dummy variables 𝑥 𝑦 𝑧 𝑤 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpl 107	. . . 4 ⊢ ((𝐺:𝐵–1-1-onto→𝐶 ∧ ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐵 (𝑧𝑆𝑤 ↔ (𝐺‘𝑧)𝑇(𝐺‘𝑤))) → 𝐺:𝐵–1-1-onto→𝐶)
2		simpl 107	. . . 4 ⊢ ((𝐻:𝐴–1-1-onto→𝐵 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ↔ (𝐻‘𝑥)𝑆(𝐻‘𝑦))) → 𝐻:𝐴–1-1-onto→𝐵)
3		f1oco 5169	. . . 4 ⊢ ((𝐺:𝐵–1-1-onto→𝐶 ∧ 𝐻:𝐴–1-1-onto→𝐵) → (𝐺 ∘ 𝐻):𝐴–1-1-onto→𝐶)
4	1, 2, 3	syl2anr 284	. . 3 ⊢ (((𝐻:𝐴–1-1-onto→𝐵 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ↔ (𝐻‘𝑥)𝑆(𝐻‘𝑦))) ∧ (𝐺:𝐵–1-1-onto→𝐶 ∧ ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐵 (𝑧𝑆𝑤 ↔ (𝐺‘𝑧)𝑇(𝐺‘𝑤)))) → (𝐺 ∘ 𝐻):𝐴–1-1-onto→𝐶)
5		f1of 5146	. . . . . . . . . . . 12 ⊢ (𝐻:𝐴–1-1-onto→𝐵 → 𝐻:𝐴⟶𝐵)
6	5	ad2antrr 471	. . . . . . . . . . 11 ⊢ (((𝐻:𝐴–1-1-onto→𝐵 ∧ (𝐺:𝐵–1-1-onto→𝐶 ∧ ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐵 (𝑧𝑆𝑤 ↔ (𝐺‘𝑧)𝑇(𝐺‘𝑤)))) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → 𝐻:𝐴⟶𝐵)
7		simprl 497	. . . . . . . . . . 11 ⊢ (((𝐻:𝐴–1-1-onto→𝐵 ∧ (𝐺:𝐵–1-1-onto→𝐶 ∧ ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐵 (𝑧𝑆𝑤 ↔ (𝐺‘𝑧)𝑇(𝐺‘𝑤)))) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → 𝑥 ∈ 𝐴)
8	6, 7	ffvelrnd 5324	. . . . . . . . . 10 ⊢ (((𝐻:𝐴–1-1-onto→𝐵 ∧ (𝐺:𝐵–1-1-onto→𝐶 ∧ ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐵 (𝑧𝑆𝑤 ↔ (𝐺‘𝑧)𝑇(𝐺‘𝑤)))) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → (𝐻‘𝑥) ∈ 𝐵)
9		simprr 498	. . . . . . . . . . 11 ⊢ (((𝐻:𝐴–1-1-onto→𝐵 ∧ (𝐺:𝐵–1-1-onto→𝐶 ∧ ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐵 (𝑧𝑆𝑤 ↔ (𝐺‘𝑧)𝑇(𝐺‘𝑤)))) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → 𝑦 ∈ 𝐴)
10	6, 9	ffvelrnd 5324	. . . . . . . . . 10 ⊢ (((𝐻:𝐴–1-1-onto→𝐵 ∧ (𝐺:𝐵–1-1-onto→𝐶 ∧ ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐵 (𝑧𝑆𝑤 ↔ (𝐺‘𝑧)𝑇(𝐺‘𝑤)))) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → (𝐻‘𝑦) ∈ 𝐵)
11		simplrr 502	. . . . . . . . . 10 ⊢ (((𝐻:𝐴–1-1-onto→𝐵 ∧ (𝐺:𝐵–1-1-onto→𝐶 ∧ ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐵 (𝑧𝑆𝑤 ↔ (𝐺‘𝑧)𝑇(𝐺‘𝑤)))) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐵 (𝑧𝑆𝑤 ↔ (𝐺‘𝑧)𝑇(𝐺‘𝑤)))
12		breq1 3788	. . . . . . . . . . . 12 ⊢ (𝑧 = (𝐻‘𝑥) → (𝑧𝑆𝑤 ↔ (𝐻‘𝑥)𝑆𝑤))
13		fveq2 5198	. . . . . . . . . . . . 13 ⊢ (𝑧 = (𝐻‘𝑥) → (𝐺‘𝑧) = (𝐺‘(𝐻‘𝑥)))
14	13	breq1d 3795	. . . . . . . . . . . 12 ⊢ (𝑧 = (𝐻‘𝑥) → ((𝐺‘𝑧)𝑇(𝐺‘𝑤) ↔ (𝐺‘(𝐻‘𝑥))𝑇(𝐺‘𝑤)))
15	12, 14	bibi12d 233	. . . . . . . . . . 11 ⊢ (𝑧 = (𝐻‘𝑥) → ((𝑧𝑆𝑤 ↔ (𝐺‘𝑧)𝑇(𝐺‘𝑤)) ↔ ((𝐻‘𝑥)𝑆𝑤 ↔ (𝐺‘(𝐻‘𝑥))𝑇(𝐺‘𝑤))))
16		breq2 3789	. . . . . . . . . . . 12 ⊢ (𝑤 = (𝐻‘𝑦) → ((𝐻‘𝑥)𝑆𝑤 ↔ (𝐻‘𝑥)𝑆(𝐻‘𝑦)))
17		fveq2 5198	. . . . . . . . . . . . 13 ⊢ (𝑤 = (𝐻‘𝑦) → (𝐺‘𝑤) = (𝐺‘(𝐻‘𝑦)))
18	17	breq2d 3797	. . . . . . . . . . . 12 ⊢ (𝑤 = (𝐻‘𝑦) → ((𝐺‘(𝐻‘𝑥))𝑇(𝐺‘𝑤) ↔ (𝐺‘(𝐻‘𝑥))𝑇(𝐺‘(𝐻‘𝑦))))
19	16, 18	bibi12d 233	. . . . . . . . . . 11 ⊢ (𝑤 = (𝐻‘𝑦) → (((𝐻‘𝑥)𝑆𝑤 ↔ (𝐺‘(𝐻‘𝑥))𝑇(𝐺‘𝑤)) ↔ ((𝐻‘𝑥)𝑆(𝐻‘𝑦) ↔ (𝐺‘(𝐻‘𝑥))𝑇(𝐺‘(𝐻‘𝑦)))))
20	15, 19	rspc2va 2714	. . . . . . . . . 10 ⊢ ((((𝐻‘𝑥) ∈ 𝐵 ∧ (𝐻‘𝑦) ∈ 𝐵) ∧ ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐵 (𝑧𝑆𝑤 ↔ (𝐺‘𝑧)𝑇(𝐺‘𝑤))) → ((𝐻‘𝑥)𝑆(𝐻‘𝑦) ↔ (𝐺‘(𝐻‘𝑥))𝑇(𝐺‘(𝐻‘𝑦))))
21	8, 10, 11, 20	syl21anc 1168	. . . . . . . . 9 ⊢ (((𝐻:𝐴–1-1-onto→𝐵 ∧ (𝐺:𝐵–1-1-onto→𝐶 ∧ ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐵 (𝑧𝑆𝑤 ↔ (𝐺‘𝑧)𝑇(𝐺‘𝑤)))) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → ((𝐻‘𝑥)𝑆(𝐻‘𝑦) ↔ (𝐺‘(𝐻‘𝑥))𝑇(𝐺‘(𝐻‘𝑦))))
22		fvco3 5265	. . . . . . . . . . 11 ⊢ ((𝐻:𝐴⟶𝐵 ∧ 𝑥 ∈ 𝐴) → ((𝐺 ∘ 𝐻)‘𝑥) = (𝐺‘(𝐻‘𝑥)))
23	6, 7, 22	syl2anc 403	. . . . . . . . . 10 ⊢ (((𝐻:𝐴–1-1-onto→𝐵 ∧ (𝐺:𝐵–1-1-onto→𝐶 ∧ ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐵 (𝑧𝑆𝑤 ↔ (𝐺‘𝑧)𝑇(𝐺‘𝑤)))) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → ((𝐺 ∘ 𝐻)‘𝑥) = (𝐺‘(𝐻‘𝑥)))
24		fvco3 5265	. . . . . . . . . . 11 ⊢ ((𝐻:𝐴⟶𝐵 ∧ 𝑦 ∈ 𝐴) → ((𝐺 ∘ 𝐻)‘𝑦) = (𝐺‘(𝐻‘𝑦)))
25	6, 9, 24	syl2anc 403	. . . . . . . . . 10 ⊢ (((𝐻:𝐴–1-1-onto→𝐵 ∧ (𝐺:𝐵–1-1-onto→𝐶 ∧ ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐵 (𝑧𝑆𝑤 ↔ (𝐺‘𝑧)𝑇(𝐺‘𝑤)))) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → ((𝐺 ∘ 𝐻)‘𝑦) = (𝐺‘(𝐻‘𝑦)))
26	23, 25	breq12d 3798	. . . . . . . . 9 ⊢ (((𝐻:𝐴–1-1-onto→𝐵 ∧ (𝐺:𝐵–1-1-onto→𝐶 ∧ ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐵 (𝑧𝑆𝑤 ↔ (𝐺‘𝑧)𝑇(𝐺‘𝑤)))) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → (((𝐺 ∘ 𝐻)‘𝑥)𝑇((𝐺 ∘ 𝐻)‘𝑦) ↔ (𝐺‘(𝐻‘𝑥))𝑇(𝐺‘(𝐻‘𝑦))))
27	21, 26	bitr4d 189	. . . . . . . 8 ⊢ (((𝐻:𝐴–1-1-onto→𝐵 ∧ (𝐺:𝐵–1-1-onto→𝐶 ∧ ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐵 (𝑧𝑆𝑤 ↔ (𝐺‘𝑧)𝑇(𝐺‘𝑤)))) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → ((𝐻‘𝑥)𝑆(𝐻‘𝑦) ↔ ((𝐺 ∘ 𝐻)‘𝑥)𝑇((𝐺 ∘ 𝐻)‘𝑦)))
28	27	bibi2d 230	. . . . . . 7 ⊢ (((𝐻:𝐴–1-1-onto→𝐵 ∧ (𝐺:𝐵–1-1-onto→𝐶 ∧ ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐵 (𝑧𝑆𝑤 ↔ (𝐺‘𝑧)𝑇(𝐺‘𝑤)))) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → ((𝑥𝑅𝑦 ↔ (𝐻‘𝑥)𝑆(𝐻‘𝑦)) ↔ (𝑥𝑅𝑦 ↔ ((𝐺 ∘ 𝐻)‘𝑥)𝑇((𝐺 ∘ 𝐻)‘𝑦))))
29	28	2ralbidva 2388	. . . . . 6 ⊢ ((𝐻:𝐴–1-1-onto→𝐵 ∧ (𝐺:𝐵–1-1-onto→𝐶 ∧ ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐵 (𝑧𝑆𝑤 ↔ (𝐺‘𝑧)𝑇(𝐺‘𝑤)))) → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ↔ (𝐻‘𝑥)𝑆(𝐻‘𝑦)) ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ↔ ((𝐺 ∘ 𝐻)‘𝑥)𝑇((𝐺 ∘ 𝐻)‘𝑦))))
30	29	biimpd 142	. . . . 5 ⊢ ((𝐻:𝐴–1-1-onto→𝐵 ∧ (𝐺:𝐵–1-1-onto→𝐶 ∧ ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐵 (𝑧𝑆𝑤 ↔ (𝐺‘𝑧)𝑇(𝐺‘𝑤)))) → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ↔ (𝐻‘𝑥)𝑆(𝐻‘𝑦)) → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ↔ ((𝐺 ∘ 𝐻)‘𝑥)𝑇((𝐺 ∘ 𝐻)‘𝑦))))
31	30	impancom 256	. . . 4 ⊢ ((𝐻:𝐴–1-1-onto→𝐵 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ↔ (𝐻‘𝑥)𝑆(𝐻‘𝑦))) → ((𝐺:𝐵–1-1-onto→𝐶 ∧ ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐵 (𝑧𝑆𝑤 ↔ (𝐺‘𝑧)𝑇(𝐺‘𝑤))) → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ↔ ((𝐺 ∘ 𝐻)‘𝑥)𝑇((𝐺 ∘ 𝐻)‘𝑦))))
32	31	imp 122	. . 3 ⊢ (((𝐻:𝐴–1-1-onto→𝐵 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ↔ (𝐻‘𝑥)𝑆(𝐻‘𝑦))) ∧ (𝐺:𝐵–1-1-onto→𝐶 ∧ ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐵 (𝑧𝑆𝑤 ↔ (𝐺‘𝑧)𝑇(𝐺‘𝑤)))) → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ↔ ((𝐺 ∘ 𝐻)‘𝑥)𝑇((𝐺 ∘ 𝐻)‘𝑦)))
33	4, 32	jca 300	. 2 ⊢ (((𝐻:𝐴–1-1-onto→𝐵 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ↔ (𝐻‘𝑥)𝑆(𝐻‘𝑦))) ∧ (𝐺:𝐵–1-1-onto→𝐶 ∧ ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐵 (𝑧𝑆𝑤 ↔ (𝐺‘𝑧)𝑇(𝐺‘𝑤)))) → ((𝐺 ∘ 𝐻):𝐴–1-1-onto→𝐶 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ↔ ((𝐺 ∘ 𝐻)‘𝑥)𝑇((𝐺 ∘ 𝐻)‘𝑦))))
34		df-isom 4931	. . 3 ⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ↔ (𝐻:𝐴–1-1-onto→𝐵 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ↔ (𝐻‘𝑥)𝑆(𝐻‘𝑦))))
35		df-isom 4931	. . 3 ⊢ (𝐺 Isom 𝑆, 𝑇 (𝐵, 𝐶) ↔ (𝐺:𝐵–1-1-onto→𝐶 ∧ ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐵 (𝑧𝑆𝑤 ↔ (𝐺‘𝑧)𝑇(𝐺‘𝑤))))
36	34, 35	anbi12i 447	. 2 ⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐺 Isom 𝑆, 𝑇 (𝐵, 𝐶)) ↔ ((𝐻:𝐴–1-1-onto→𝐵 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ↔ (𝐻‘𝑥)𝑆(𝐻‘𝑦))) ∧ (𝐺:𝐵–1-1-onto→𝐶 ∧ ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐵 (𝑧𝑆𝑤 ↔ (𝐺‘𝑧)𝑇(𝐺‘𝑤)))))
37		df-isom 4931	. 2 ⊢ ((𝐺 ∘ 𝐻) Isom 𝑅, 𝑇 (𝐴, 𝐶) ↔ ((𝐺 ∘ 𝐻):𝐴–1-1-onto→𝐶 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ↔ ((𝐺 ∘ 𝐻)‘𝑥)𝑇((𝐺 ∘ 𝐻)‘𝑦))))
38	33, 36, 37	3imtr4i 199	1 ⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐺 Isom 𝑆, 𝑇 (𝐵, 𝐶)) → (𝐺 ∘ 𝐻) Isom 𝑅, 𝑇 (𝐴, 𝐶))

Syntax hints:   → wi 4   ∧ wa 102   ↔ wb 103   = wceq 1284   ∈ wcel 1433  ∀wral 2348   class class class wbr 3785   ∘ ccom 4367  ⟶wf 4918  –1-1-onto→wf1o 4921  ‘cfv 4922   Isom wiso 4923

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-pow 3948  ax-pr 3964

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rex 2354  df-v 2603  df-sbc 2816  df-un 2977  df-in 2979  df-ss 2986  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-br 3786  df-opab 3840  df-id 4048  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-rn 4374  df-res 4375  df-ima 4376  df-iota 4887  df-fun 4924  df-fn 4925  df-f 4926  df-f1 4927  df-fo 4928  df-f1o 4929  df-fv 4930  df-isom 4931

This theorem is referenced by: (None)