Theorem isotr 5496

Description: Composition (transitive) law for isomorphism. Proposition 6.30(3) of [TakeutiZaring] p. 33. (Contributed by set.mm contributors, 27-Apr-2004.)

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 v u are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		f1oco 5309	. . . . 5 ⊢ ((G:B–1-1-onto→C ∧ H:A–1-1-onto→B) → (G ∘ H):A–1-1-onto→C)
2	1	ad2ant2r 727	. . . 4 ⊢ (((G:B–1-1-onto→C ∧ ∀u ∈ B ∀v ∈ B (uSv ↔ (G ‘u)T(G ‘v))) ∧ (H:A–1-1-onto→B ∧ ∀z ∈ A ∀w ∈ A (zRw ↔ (H ‘z)S(H ‘w)))) → (G ∘ H):A–1-1-onto→C)
3	2	ancoms 439	. . 3 ⊢ (((H:A–1-1-onto→B ∧ ∀z ∈ A ∀w ∈ A (zRw ↔ (H ‘z)S(H ‘w))) ∧ (G:B–1-1-onto→C ∧ ∀u ∈ B ∀v ∈ B (uSv ↔ (G ‘u)T(G ‘v)))) → (G ∘ H):A–1-1-onto→C)
4		f1of 5288	. . . . . . . . 9 ⊢ (H:A–1-1-onto→B → H:A–→B)
5		ffvelrn 5416	. . . . . . . . . . 11 ⊢ ((H:A–→B ∧ x ∈ A) → (H ‘x) ∈ B)
6	5	ex 423	. . . . . . . . . 10 ⊢ (H:A–→B → (x ∈ A → (H ‘x) ∈ B))
7		ffvelrn 5416	. . . . . . . . . . 11 ⊢ ((H:A–→B ∧ y ∈ A) → (H ‘y) ∈ B)
8	7	ex 423	. . . . . . . . . 10 ⊢ (H:A–→B → (y ∈ A → (H ‘y) ∈ B))
9	6, 8	anim12d 546	. . . . . . . . 9 ⊢ (H:A–→B → ((x ∈ A ∧ y ∈ A) → ((H ‘x) ∈ B ∧ (H ‘y) ∈ B)))
10	4, 9	syl 15	. . . . . . . 8 ⊢ (H:A–1-1-onto→B → ((x ∈ A ∧ y ∈ A) → ((H ‘x) ∈ B ∧ (H ‘y) ∈ B)))
11	10	adantr 451	. . . . . . 7 ⊢ ((H:A–1-1-onto→B ∧ ∀z ∈ A ∀w ∈ A (zRw ↔ (H ‘z)S(H ‘w))) → ((x ∈ A ∧ y ∈ A) → ((H ‘x) ∈ B ∧ (H ‘y) ∈ B)))
12		breq1 4643	. . . . . . . . . . 11 ⊢ (u = (H ‘x) → (uSv ↔ (H ‘x)Sv))
13		fveq2 5329	. . . . . . . . . . . 12 ⊢ (u = (H ‘x) → (G ‘u) = (G ‘(H ‘x)))
14	13	breq1d 4650	. . . . . . . . . . 11 ⊢ (u = (H ‘x) → ((G ‘u)T(G ‘v) ↔ (G ‘(H ‘x))T(G ‘v)))
15	12, 14	bibi12d 312	. . . . . . . . . 10 ⊢ (u = (H ‘x) → ((uSv ↔ (G ‘u)T(G ‘v)) ↔ ((H ‘x)Sv ↔ (G ‘(H ‘x))T(G ‘v))))
16		breq2 4644	. . . . . . . . . . 11 ⊢ (v = (H ‘y) → ((H ‘x)Sv ↔ (H ‘x)S(H ‘y)))
17		fveq2 5329	. . . . . . . . . . . 12 ⊢ (v = (H ‘y) → (G ‘v) = (G ‘(H ‘y)))
18	17	breq2d 4652	. . . . . . . . . . 11 ⊢ (v = (H ‘y) → ((G ‘(H ‘x))T(G ‘v) ↔ (G ‘(H ‘x))T(G ‘(H ‘y))))
19	16, 18	bibi12d 312	. . . . . . . . . 10 ⊢ (v = (H ‘y) → (((H ‘x)Sv ↔ (G ‘(H ‘x))T(G ‘v)) ↔ ((H ‘x)S(H ‘y) ↔ (G ‘(H ‘x))T(G ‘(H ‘y)))))
20	15, 19	rspc2v 2962	. . . . . . . . 9 ⊢ (((H ‘x) ∈ B ∧ (H ‘y) ∈ B) → (∀u ∈ B ∀v ∈ B (uSv ↔ (G ‘u)T(G ‘v)) → ((H ‘x)S(H ‘y) ↔ (G ‘(H ‘x))T(G ‘(H ‘y)))))
21	20	com12 27	. . . . . . . 8 ⊢ (∀u ∈ B ∀v ∈ B (uSv ↔ (G ‘u)T(G ‘v)) → (((H ‘x) ∈ B ∧ (H ‘y) ∈ B) → ((H ‘x)S(H ‘y) ↔ (G ‘(H ‘x))T(G ‘(H ‘y)))))
22	21	adantl 452	. . . . . . 7 ⊢ ((G:B–1-1-onto→C ∧ ∀u ∈ B ∀v ∈ B (uSv ↔ (G ‘u)T(G ‘v))) → (((H ‘x) ∈ B ∧ (H ‘y) ∈ B) → ((H ‘x)S(H ‘y) ↔ (G ‘(H ‘x))T(G ‘(H ‘y)))))
23	11, 22	sylan9 638	. . . . . 6 ⊢ (((H:A–1-1-onto→B ∧ ∀z ∈ A ∀w ∈ A (zRw ↔ (H ‘z)S(H ‘w))) ∧ (G:B–1-1-onto→C ∧ ∀u ∈ B ∀v ∈ B (uSv ↔ (G ‘u)T(G ‘v)))) → ((x ∈ A ∧ y ∈ A) → ((H ‘x)S(H ‘y) ↔ (G ‘(H ‘x))T(G ‘(H ‘y)))))
24	23	imp 418	. . . . 5 ⊢ ((((H:A–1-1-onto→B ∧ ∀z ∈ A ∀w ∈ A (zRw ↔ (H ‘z)S(H ‘w))) ∧ (G:B–1-1-onto→C ∧ ∀u ∈ B ∀v ∈ B (uSv ↔ (G ‘u)T(G ‘v)))) ∧ (x ∈ A ∧ y ∈ A)) → ((H ‘x)S(H ‘y) ↔ (G ‘(H ‘x))T(G ‘(H ‘y))))
25		breq1 4643	. . . . . . . . . 10 ⊢ (z = x → (zRw ↔ xRw))
26		fveq2 5329	. . . . . . . . . . 11 ⊢ (z = x → (H ‘z) = (H ‘x))
27	26	breq1d 4650	. . . . . . . . . 10 ⊢ (z = x → ((H ‘z)S(H ‘w) ↔ (H ‘x)S(H ‘w)))
28	25, 27	bibi12d 312	. . . . . . . . 9 ⊢ (z = x → ((zRw ↔ (H ‘z)S(H ‘w)) ↔ (xRw ↔ (H ‘x)S(H ‘w))))
29		breq2 4644	. . . . . . . . . 10 ⊢ (w = y → (xRw ↔ xRy))
30		fveq2 5329	. . . . . . . . . . 11 ⊢ (w = y → (H ‘w) = (H ‘y))
31	30	breq2d 4652	. . . . . . . . . 10 ⊢ (w = y → ((H ‘x)S(H ‘w) ↔ (H ‘x)S(H ‘y)))
32	29, 31	bibi12d 312	. . . . . . . . 9 ⊢ (w = y → ((xRw ↔ (H ‘x)S(H ‘w)) ↔ (xRy ↔ (H ‘x)S(H ‘y))))
33	28, 32	rspc2v 2962	. . . . . . . 8 ⊢ ((x ∈ A ∧ y ∈ A) → (∀z ∈ A ∀w ∈ A (zRw ↔ (H ‘z)S(H ‘w)) → (xRy ↔ (H ‘x)S(H ‘y))))
34	33	impcom 419	. . . . . . 7 ⊢ ((∀z ∈ A ∀w ∈ A (zRw ↔ (H ‘z)S(H ‘w)) ∧ (x ∈ A ∧ y ∈ A)) → (xRy ↔ (H ‘x)S(H ‘y)))
35	34	adantll 694	. . . . . 6 ⊢ (((H:A–1-1-onto→B ∧ ∀z ∈ A ∀w ∈ A (zRw ↔ (H ‘z)S(H ‘w))) ∧ (x ∈ A ∧ y ∈ A)) → (xRy ↔ (H ‘x)S(H ‘y)))
36	35	adantlr 695	. . . . 5 ⊢ ((((H:A–1-1-onto→B ∧ ∀z ∈ A ∀w ∈ A (zRw ↔ (H ‘z)S(H ‘w))) ∧ (G:B–1-1-onto→C ∧ ∀u ∈ B ∀v ∈ B (uSv ↔ (G ‘u)T(G ‘v)))) ∧ (x ∈ A ∧ y ∈ A)) → (xRy ↔ (H ‘x)S(H ‘y)))
37	4	ad2antrr 706	. . . . . 6 ⊢ (((H:A–1-1-onto→B ∧ ∀z ∈ A ∀w ∈ A (zRw ↔ (H ‘z)S(H ‘w))) ∧ (G:B–1-1-onto→C ∧ ∀u ∈ B ∀v ∈ B (uSv ↔ (G ‘u)T(G ‘v)))) → H:A–→B)
38		fvco3 5385	. . . . . . . 8 ⊢ ((H:A–→B ∧ x ∈ A) → ((G ∘ H) ‘x) = (G ‘(H ‘x)))
39		fvco3 5385	. . . . . . . 8 ⊢ ((H:A–→B ∧ y ∈ A) → ((G ∘ H) ‘y) = (G ‘(H ‘y)))
40	38, 39	breqan12d 4655	. . . . . . 7 ⊢ (((H:A–→B ∧ x ∈ A) ∧ (H:A–→B ∧ y ∈ A)) → (((G ∘ H) ‘x)T((G ∘ H) ‘y) ↔ (G ‘(H ‘x))T(G ‘(H ‘y))))
41	40	anandis 803	. . . . . 6 ⊢ ((H:A–→B ∧ (x ∈ A ∧ y ∈ A)) → (((G ∘ H) ‘x)T((G ∘ H) ‘y) ↔ (G ‘(H ‘x))T(G ‘(H ‘y))))
42	37, 41	sylan 457	. . . . 5 ⊢ ((((H:A–1-1-onto→B ∧ ∀z ∈ A ∀w ∈ A (zRw ↔ (H ‘z)S(H ‘w))) ∧ (G:B–1-1-onto→C ∧ ∀u ∈ B ∀v ∈ B (uSv ↔ (G ‘u)T(G ‘v)))) ∧ (x ∈ A ∧ y ∈ A)) → (((G ∘ H) ‘x)T((G ∘ H) ‘y) ↔ (G ‘(H ‘x))T(G ‘(H ‘y))))
43	24, 36, 42	3bitr4d 276	. . . 4 ⊢ ((((H:A–1-1-onto→B ∧ ∀z ∈ A ∀w ∈ A (zRw ↔ (H ‘z)S(H ‘w))) ∧ (G:B–1-1-onto→C ∧ ∀u ∈ B ∀v ∈ B (uSv ↔ (G ‘u)T(G ‘v)))) ∧ (x ∈ A ∧ y ∈ A)) → (xRy ↔ ((G ∘ H) ‘x)T((G ∘ H) ‘y)))
44	43	ralrimivva 2707	. . 3 ⊢ (((H:A–1-1-onto→B ∧ ∀z ∈ A ∀w ∈ A (zRw ↔ (H ‘z)S(H ‘w))) ∧ (G:B–1-1-onto→C ∧ ∀u ∈ B ∀v ∈ B (uSv ↔ (G ‘u)T(G ‘v)))) → ∀x ∈ A ∀y ∈ A (xRy ↔ ((G ∘ H) ‘x)T((G ∘ H) ‘y)))
45	3, 44	jca 518	. 2 ⊢ (((H:A–1-1-onto→B ∧ ∀z ∈ A ∀w ∈ A (zRw ↔ (H ‘z)S(H ‘w))) ∧ (G:B–1-1-onto→C ∧ ∀u ∈ B ∀v ∈ B (uSv ↔ (G ‘u)T(G ‘v)))) → ((G ∘ H):A–1-1-onto→C ∧ ∀x ∈ A ∀y ∈ A (xRy ↔ ((G ∘ H) ‘x)T((G ∘ H) ‘y))))
46		df-iso 4797	. . 3 ⊢ (H Isom R, S (A, B) ↔ (H:A–1-1-onto→B ∧ ∀z ∈ A ∀w ∈ A (zRw ↔ (H ‘z)S(H ‘w))))
47		df-iso 4797	. . 3 ⊢ (G Isom S, T (B, C) ↔ (G:B–1-1-onto→C ∧ ∀u ∈ B ∀v ∈ B (uSv ↔ (G ‘u)T(G ‘v))))
48	46, 47	anbi12i 678	. 2 ⊢ ((H Isom R, S (A, B) ∧ G Isom S, T (B, C)) ↔ ((H:A–1-1-onto→B ∧ ∀z ∈ A ∀w ∈ A (zRw ↔ (H ‘z)S(H ‘w))) ∧ (G:B–1-1-onto→C ∧ ∀u ∈ B ∀v ∈ B (uSv ↔ (G ‘u)T(G ‘v)))))
49		df-iso 4797	. 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))))
50	45, 48, 49	3imtr4i 257	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 176   ∧ wa 358   = wceq 1642   ∈ wcel 1710  ∀wral 2615   class class class wbr 4640   ∘ ccom 4722  –→wf 4778  –1-1-onto→wf1o 4781   ‘cfv 4782   Isom wiso 4783

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4079  ax-xp 4080  ax-cnv 4081  ax-1c 4082  ax-sset 4083  ax-si 4084  ax-ins2 4085  ax-ins3 4086  ax-typlower 4087  ax-sn 4088

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ne 2519  df-ral 2620  df-rex 2621  df-reu 2622  df-rmo 2623  df-rab 2624  df-v 2862  df-sbc 3048  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-symdif 3217  df-ss 3260  df-pss 3262  df-nul 3552  df-if 3664  df-pw 3725  df-sn 3742  df-pr 3743  df-uni 3893  df-int 3928  df-opk 4059  df-1c 4137  df-pw1 4138  df-uni1 4139  df-xpk 4186  df-cnvk 4187  df-ins2k 4188  df-ins3k 4189  df-imak 4190  df-cok 4191  df-p6 4192  df-sik 4193  df-ssetk 4194  df-imagek 4195  df-idk 4196  df-iota 4340  df-0c 4378  df-addc 4379  df-nnc 4380  df-fin 4381  df-lefin 4441  df-ltfin 4442  df-ncfin 4443  df-tfin 4444  df-evenfin 4445  df-oddfin 4446  df-sfin 4447  df-spfin 4448  df-phi 4566  df-op 4567  df-proj1 4568  df-proj2 4569  df-opab 4624  df-br 4641  df-co 4727  df-ima 4728  df-id 4768  df-xp 4785  df-cnv 4786  df-rn 4787  df-dm 4788  df-res 4789  df-fun 4790  df-fn 4791  df-f 4792  df-f1 4793  df-fo 4794  df-f1o 4795  df-fv 4796  df-iso 4797

This theorem is referenced by: (None)