Theorem isotr 6586

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 473	. . . 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 473	. . . 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 6159	. . . 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 495	. . 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 6137	. . . . . . . . . . . 12 ⊢ (𝐻:𝐴–1-1-onto→𝐵 → 𝐻:𝐴⟶𝐵)
6	5	ad2antrr 762	. . . . . . . . . . 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 794	. . . . . . . . . . 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 6360	. . . . . . . . . 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 796	. . . . . . . . . . 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 6360	. . . . . . . . . 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 801	. . . . . . . . . 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 4656	. . . . . . . . . . . 12 ⊢ (𝑧 = (𝐻‘𝑥) → (𝑧𝑆𝑤 ↔ (𝐻‘𝑥)𝑆𝑤))
13		fveq2 6191	. . . . . . . . . . . . 13 ⊢ (𝑧 = (𝐻‘𝑥) → (𝐺‘𝑧) = (𝐺‘(𝐻‘𝑥)))
14	13	breq1d 4663	. . . . . . . . . . . 12 ⊢ (𝑧 = (𝐻‘𝑥) → ((𝐺‘𝑧)𝑇(𝐺‘𝑤) ↔ (𝐺‘(𝐻‘𝑥))𝑇(𝐺‘𝑤)))
15	12, 14	bibi12d 335	. . . . . . . . . . 11 ⊢ (𝑧 = (𝐻‘𝑥) → ((𝑧𝑆𝑤 ↔ (𝐺‘𝑧)𝑇(𝐺‘𝑤)) ↔ ((𝐻‘𝑥)𝑆𝑤 ↔ (𝐺‘(𝐻‘𝑥))𝑇(𝐺‘𝑤))))
16		breq2 4657	. . . . . . . . . . . 12 ⊢ (𝑤 = (𝐻‘𝑦) → ((𝐻‘𝑥)𝑆𝑤 ↔ (𝐻‘𝑥)𝑆(𝐻‘𝑦)))
17		fveq2 6191	. . . . . . . . . . . . 13 ⊢ (𝑤 = (𝐻‘𝑦) → (𝐺‘𝑤) = (𝐺‘(𝐻‘𝑦)))
18	17	breq2d 4665	. . . . . . . . . . . 12 ⊢ (𝑤 = (𝐻‘𝑦) → ((𝐺‘(𝐻‘𝑥))𝑇(𝐺‘𝑤) ↔ (𝐺‘(𝐻‘𝑥))𝑇(𝐺‘(𝐻‘𝑦))))
19	16, 18	bibi12d 335	. . . . . . . . . . 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 3323	. . . . . . . . . 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 1325	. . . . . . . . 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 6275	. . . . . . . . . . 11 ⊢ ((𝐻:𝐴⟶𝐵 ∧ 𝑥 ∈ 𝐴) → ((𝐺 ∘ 𝐻)‘𝑥) = (𝐺‘(𝐻‘𝑥)))
23	6, 7, 22	syl2anc 693	. . . . . . . . . 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 6275	. . . . . . . . . . 11 ⊢ ((𝐻:𝐴⟶𝐵 ∧ 𝑦 ∈ 𝐴) → ((𝐺 ∘ 𝐻)‘𝑦) = (𝐺‘(𝐻‘𝑦)))
25	6, 9, 24	syl2anc 693	. . . . . . . . . 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 4666	. . . . . . . . 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 271	. . . . . . . 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 332	. . . . . . 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 2988	. . . . . 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 219	. . . . 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 456	. . . 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 445	. . 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 554	. 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 5897	. . 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 5897	. . 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 733	. 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 5897	. 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 281	1 ⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐺 Isom 𝑆, 𝑇 (𝐵, 𝐶)) → (𝐺 ∘ 𝐻) Isom 𝑅, 𝑇 (𝐴, 𝐶))

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   = wceq 1483   ∈ wcel 1990  ∀wral 2912   class class class wbr 4653   ∘ ccom 5118  ⟶wf 5884  –1-1-onto→wf1o 5887  ‘cfv 5888   Isom wiso 5889

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897

This theorem is referenced by:  weisoeq  6605  oieu  8444  fz1isolem  13245  erdsze2lem2  31186  fzisoeu  39514