Theorem islmim 19828

Description: An isomorphism of left modules is a bijective homomorphism. (Contributed by Stefan O'Rear, 21-Jan-2015.)

Hypotheses

Ref	Expression
islmim.b	⊢ 𝐵 = (Base‘𝑅)
islmim.c	⊢ 𝐶 = (Base‘𝑆)

Assertion

Ref	Expression
islmim	⊢ (𝐹 ∈ (𝑅 LMIso 𝑆) ↔ (𝐹 ∈ (𝑅 LMHom 𝑆) ∧ 𝐹:𝐵–1-1-onto→𝐶))

Proof of Theorem islmim

Dummy variables 𝑎 𝑏 𝑐 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-lmim 19789	. . 3 ⊢ LMIso = (𝑎 ∈ LMod, 𝑏 ∈ LMod ↦ {𝑐 ∈ (𝑎 LMHom 𝑏) ∣ 𝑐:(Base‘𝑎)–1-1-onto→(Base‘𝑏)})
2		ovex 7183	. . . 4 ⊢ (𝑎 LMHom 𝑏) ∈ V
3	2	rabex 5228	. . 3 ⊢ {𝑐 ∈ (𝑎 LMHom 𝑏) ∣ 𝑐:(Base‘𝑎)–1-1-onto→(Base‘𝑏)} ∈ V
4		oveq12 7159	. . . 4 ⊢ ((𝑎 = 𝑅 ∧ 𝑏 = 𝑆) → (𝑎 LMHom 𝑏) = (𝑅 LMHom 𝑆))
5		fveq2 6665	. . . . . 6 ⊢ (𝑎 = 𝑅 → (Base‘𝑎) = (Base‘𝑅))
6		islmim.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝑅)
7	5, 6	syl6eqr 2874	. . . . 5 ⊢ (𝑎 = 𝑅 → (Base‘𝑎) = 𝐵)
8		fveq2 6665	. . . . . 6 ⊢ (𝑏 = 𝑆 → (Base‘𝑏) = (Base‘𝑆))
9		islmim.c	. . . . . 6 ⊢ 𝐶 = (Base‘𝑆)
10	8, 9	syl6eqr 2874	. . . . 5 ⊢ (𝑏 = 𝑆 → (Base‘𝑏) = 𝐶)
11		f1oeq23 6602	. . . . 5 ⊢ (((Base‘𝑎) = 𝐵 ∧ (Base‘𝑏) = 𝐶) → (𝑐:(Base‘𝑎)–1-1-onto→(Base‘𝑏) ↔ 𝑐:𝐵–1-1-onto→𝐶))
12	7, 10, 11	syl2an 597	. . . 4 ⊢ ((𝑎 = 𝑅 ∧ 𝑏 = 𝑆) → (𝑐:(Base‘𝑎)–1-1-onto→(Base‘𝑏) ↔ 𝑐:𝐵–1-1-onto→𝐶))
13	4, 12	rabeqbidv 3486	. . 3 ⊢ ((𝑎 = 𝑅 ∧ 𝑏 = 𝑆) → {𝑐 ∈ (𝑎 LMHom 𝑏) ∣ 𝑐:(Base‘𝑎)–1-1-onto→(Base‘𝑏)} = {𝑐 ∈ (𝑅 LMHom 𝑆) ∣ 𝑐:𝐵–1-1-onto→𝐶})
14	1, 3, 13	elovmpo 7384	. 2 ⊢ (𝐹 ∈ (𝑅 LMIso 𝑆) ↔ (𝑅 ∈ LMod ∧ 𝑆 ∈ LMod ∧ 𝐹 ∈ {𝑐 ∈ (𝑅 LMHom 𝑆) ∣ 𝑐:𝐵–1-1-onto→𝐶}))
15		df-3an 1085	. 2 ⊢ ((𝑅 ∈ LMod ∧ 𝑆 ∈ LMod ∧ 𝐹 ∈ {𝑐 ∈ (𝑅 LMHom 𝑆) ∣ 𝑐:𝐵–1-1-onto→𝐶}) ↔ ((𝑅 ∈ LMod ∧ 𝑆 ∈ LMod) ∧ 𝐹 ∈ {𝑐 ∈ (𝑅 LMHom 𝑆) ∣ 𝑐:𝐵–1-1-onto→𝐶}))
16		f1oeq1 6599	. . . . 5 ⊢ (𝑐 = 𝐹 → (𝑐:𝐵–1-1-onto→𝐶 ↔ 𝐹:𝐵–1-1-onto→𝐶))
17	16	elrab 3680	. . . 4 ⊢ (𝐹 ∈ {𝑐 ∈ (𝑅 LMHom 𝑆) ∣ 𝑐:𝐵–1-1-onto→𝐶} ↔ (𝐹 ∈ (𝑅 LMHom 𝑆) ∧ 𝐹:𝐵–1-1-onto→𝐶))
18	17	anbi2i 624	. . 3 ⊢ (((𝑅 ∈ LMod ∧ 𝑆 ∈ LMod) ∧ 𝐹 ∈ {𝑐 ∈ (𝑅 LMHom 𝑆) ∣ 𝑐:𝐵–1-1-onto→𝐶}) ↔ ((𝑅 ∈ LMod ∧ 𝑆 ∈ LMod) ∧ (𝐹 ∈ (𝑅 LMHom 𝑆) ∧ 𝐹:𝐵–1-1-onto→𝐶)))
19		lmhmlmod1 19799	. . . . . 6 ⊢ (𝐹 ∈ (𝑅 LMHom 𝑆) → 𝑅 ∈ LMod)
20		lmhmlmod2 19798	. . . . . 6 ⊢ (𝐹 ∈ (𝑅 LMHom 𝑆) → 𝑆 ∈ LMod)
21	19, 20	jca 514	. . . . 5 ⊢ (𝐹 ∈ (𝑅 LMHom 𝑆) → (𝑅 ∈ LMod ∧ 𝑆 ∈ LMod))
22	21	adantr 483	. . . 4 ⊢ ((𝐹 ∈ (𝑅 LMHom 𝑆) ∧ 𝐹:𝐵–1-1-onto→𝐶) → (𝑅 ∈ LMod ∧ 𝑆 ∈ LMod))
23	22	pm4.71ri 563	. . 3 ⊢ ((𝐹 ∈ (𝑅 LMHom 𝑆) ∧ 𝐹:𝐵–1-1-onto→𝐶) ↔ ((𝑅 ∈ LMod ∧ 𝑆 ∈ LMod) ∧ (𝐹 ∈ (𝑅 LMHom 𝑆) ∧ 𝐹:𝐵–1-1-onto→𝐶)))
24	18, 23	bitr4i 280	. 2 ⊢ (((𝑅 ∈ LMod ∧ 𝑆 ∈ LMod) ∧ 𝐹 ∈ {𝑐 ∈ (𝑅 LMHom 𝑆) ∣ 𝑐:𝐵–1-1-onto→𝐶}) ↔ (𝐹 ∈ (𝑅 LMHom 𝑆) ∧ 𝐹:𝐵–1-1-onto→𝐶))
25	14, 15, 24	3bitri 299	1 ⊢ (𝐹 ∈ (𝑅 LMIso 𝑆) ↔ (𝐹 ∈ (𝑅 LMHom 𝑆) ∧ 𝐹:𝐵–1-1-onto→𝐶))

Syntax hints:   ↔ wb 208   ∧ wa 398   ∧ w3a 1083   = wceq 1533   ∈ wcel 2110  {crab 3142  –1-1-onto→wf1o 6349  ‘cfv 6350  (class class class)co 7150  Basecbs 16477  LModclmod 19628   LMHom clmhm 19785   LMIso clmim 19786

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2156  ax-12 2172  ax-ext 2793  ax-sep 5196  ax-nul 5203  ax-pow 5259  ax-pr 5322

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3497  df-sbc 3773  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4833  df-br 5060  df-opab 5122  df-id 5455  df-xp 5556  df-rel 5557  df-cnv 5558  df-co 5559  df-dm 5560  df-rn 5561  df-iota 6309  df-fun 6352  df-fn 6353  df-f 6354  df-f1 6355  df-fo 6356  df-f1o 6357  df-fv 6358  df-ov 7153  df-oprab 7154  df-mpo 7155  df-lmhm 19788  df-lmim 19789

This theorem is referenced by:  lmimf1o  19829  lmimlmhm  19830  islmim2  19832  indlcim  20978  lmimco  20982  dimkerim  31018  frlmsnic  39142  pwssplit4  39682