Theorem isotr 7187

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 6722	. . . 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 596	. . 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 6700	. . . . . . . . . . . 12 ⊢ (𝐻:𝐴–1-1-onto→𝐵 → 𝐻:𝐴⟶𝐵)
6	5	ad2antrr 722	. . . . . . . . . . 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 767	. . . . . . . . . . 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 6944	. . . . . . . . . 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 769	. . . . . . . . . . 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 6944	. . . . . . . . . 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 774	. . . . . . . . . 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 5073	. . . . . . . . . . . 12 ⊢ (𝑧 = (𝐻‘𝑥) → (𝑧𝑆𝑤 ↔ (𝐻‘𝑥)𝑆𝑤))
13		fveq2 6756	. . . . . . . . . . . . 13 ⊢ (𝑧 = (𝐻‘𝑥) → (𝐺‘𝑧) = (𝐺‘(𝐻‘𝑥)))
14	13	breq1d 5080	. . . . . . . . . . . 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 5074	. . . . . . . . . . . 12 ⊢ (𝑤 = (𝐻‘𝑦) → ((𝐻‘𝑥)𝑆𝑤 ↔ (𝐻‘𝑥)𝑆(𝐻‘𝑦)))
17		fveq2 6756	. . . . . . . . . . . . 13 ⊢ (𝑤 = (𝐻‘𝑦) → (𝐺‘𝑤) = (𝐺‘(𝐻‘𝑦)))
18	17	breq2d 5082	. . . . . . . . . . . 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 3563	. . . . . . . . . 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 834	. . . . . . . . 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 6849	. . . . . . . . . . 11 ⊢ ((𝐻:𝐴⟶𝐵 ∧ 𝑥 ∈ 𝐴) → ((𝐺 ∘ 𝐻)‘𝑥) = (𝐺‘(𝐻‘𝑥)))
23	6, 7, 22	syl2anc 583	. . . . . . . . . 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 6849	. . . . . . . . . . 11 ⊢ ((𝐻:𝐴⟶𝐵 ∧ 𝑦 ∈ 𝐴) → ((𝐺 ∘ 𝐻)‘𝑦) = (𝐺‘(𝐻‘𝑦)))
25	6, 9, 24	syl2anc 583	. . . . . . . . . 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 5083	. . . . . . . . 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 281	. . . . . . . 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 3121	. . . . . 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 228	. . . . 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 6427	. . 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 6427	. . 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 626	. 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 6427	. 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 291	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 205   ∧ wa 395   = wceq 1539   ∈ wcel 2108  ∀wral 3063   class class class wbr 5070   ∘ ccom 5584  ⟶wf 6414  –1-1-onto→wf1o 6417  ‘cfv 6418   Isom wiso 6419

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-br 5071  df-opab 5133  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-isom 6427

This theorem is referenced by:  weisoeq  7206  oieu  9228  fz1isolem  14103  erdsze2lem2  33066  fzisoeu  42729