Theorem isotr 7270

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 482	. . . 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 482	. . . 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 6786	. . . 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 597	. . 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 6763	. . . . . . . . . . . 12 ⊢ (𝐻:𝐴–1-1-onto→𝐵 → 𝐻:𝐴⟶𝐵)
6	5	ad2antrr 726	. . . . . . . . . . 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 770	. . . . . . . . . . 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	ffvelcdmd 7018	. . . . . . . . . 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 772	. . . . . . . . . . 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	ffvelcdmd 7018	. . . . . . . . . 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 777	. . . . . . . . . 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 5094	. . . . . . . . . . . 12 ⊢ (𝑧 = (𝐻‘𝑥) → (𝑧𝑆𝑤 ↔ (𝐻‘𝑥)𝑆𝑤))
13		fveq2 6822	. . . . . . . . . . . . 13 ⊢ (𝑧 = (𝐻‘𝑥) → (𝐺‘𝑧) = (𝐺‘(𝐻‘𝑥)))
14	13	breq1d 5101	. . . . . . . . . . . 12 ⊢ (𝑧 = (𝐻‘𝑥) → ((𝐺‘𝑧)𝑇(𝐺‘𝑤) ↔ (𝐺‘(𝐻‘𝑥))𝑇(𝐺‘𝑤)))
15	12, 14	bibi12d 345	. . . . . . . . . . 11 ⊢ (𝑧 = (𝐻‘𝑥) → ((𝑧𝑆𝑤 ↔ (𝐺‘𝑧)𝑇(𝐺‘𝑤)) ↔ ((𝐻‘𝑥)𝑆𝑤 ↔ (𝐺‘(𝐻‘𝑥))𝑇(𝐺‘𝑤))))
16		breq2 5095	. . . . . . . . . . . 12 ⊢ (𝑤 = (𝐻‘𝑦) → ((𝐻‘𝑥)𝑆𝑤 ↔ (𝐻‘𝑥)𝑆(𝐻‘𝑦)))
17		fveq2 6822	. . . . . . . . . . . . 13 ⊢ (𝑤 = (𝐻‘𝑦) → (𝐺‘𝑤) = (𝐺‘(𝐻‘𝑦)))
18	17	breq2d 5103	. . . . . . . . . . . 12 ⊢ (𝑤 = (𝐻‘𝑦) → ((𝐺‘(𝐻‘𝑥))𝑇(𝐺‘𝑤) ↔ (𝐺‘(𝐻‘𝑥))𝑇(𝐺‘(𝐻‘𝑦))))
19	16, 18	bibi12d 345	. . . . . . . . . . 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 3589	. . . . . . . . . 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 837	. . . . . . . . 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 6921	. . . . . . . . . . 11 ⊢ ((𝐻:𝐴⟶𝐵 ∧ 𝑥 ∈ 𝐴) → ((𝐺 ∘ 𝐻)‘𝑥) = (𝐺‘(𝐻‘𝑥)))
23	6, 7, 22	syl2anc 584	. . . . . . . . . 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 6921	. . . . . . . . . . 11 ⊢ ((𝐻:𝐴⟶𝐵 ∧ 𝑦 ∈ 𝐴) → ((𝐺 ∘ 𝐻)‘𝑦) = (𝐺‘(𝐻‘𝑦)))
25	6, 9, 24	syl2anc 584	. . . . . . . . . 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 5104	. . . . . . . . 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 282	. . . . . . . 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 342	. . . . . . 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 3194	. . . . . 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 229	. . . . 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 451	. . . 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 406	. . 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 511	. 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 6490	. . 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 6490	. . 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 628	. 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 6490	. 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 292	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 206   ∧ wa 395   = wceq 1541   ∈ wcel 2111  ∀wral 3047   class class class wbr 5091   ∘ ccom 5620  ⟶wf 6477  –1-1-onto→wf1o 6480  ‘cfv 6481   Isom wiso 6482

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5234  ax-nul 5244  ax-pr 5370

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4476  df-sn 4577  df-pr 4579  df-op 4583  df-uni 4860  df-br 5092  df-opab 5154  df-id 5511  df-xp 5622  df-rel 5623  df-cnv 5624  df-co 5625  df-dm 5626  df-rn 5627  df-res 5628  df-ima 5629  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488  df-fv 6489  df-isom 6490

This theorem is referenced by:  weisoeq  7289  oieu  9425  fz1isolem  14365  erdsze2lem2  35236  fzisoeu  45340