Theorem islmim 20984

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 20945	. . 3 ⊢ LMIso = (𝑎 ∈ LMod, 𝑏 ∈ LMod ↦ {𝑐 ∈ (𝑎 LMHom 𝑏) ∣ 𝑐:(Base‘𝑎)–1-1-onto→(Base‘𝑏)})
2		ovex 7386	. . . 4 ⊢ (𝑎 LMHom 𝑏) ∈ V
3	2	rabex 5281	. . 3 ⊢ {𝑐 ∈ (𝑎 LMHom 𝑏) ∣ 𝑐:(Base‘𝑎)–1-1-onto→(Base‘𝑏)} ∈ V
4		oveq12 7362	. . . 4 ⊢ ((𝑎 = 𝑅 ∧ 𝑏 = 𝑆) → (𝑎 LMHom 𝑏) = (𝑅 LMHom 𝑆))
5		fveq2 6826	. . . . . 6 ⊢ (𝑎 = 𝑅 → (Base‘𝑎) = (Base‘𝑅))
6		islmim.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝑅)
7	5, 6	eqtr4di 2782	. . . . 5 ⊢ (𝑎 = 𝑅 → (Base‘𝑎) = 𝐵)
8		fveq2 6826	. . . . . 6 ⊢ (𝑏 = 𝑆 → (Base‘𝑏) = (Base‘𝑆))
9		islmim.c	. . . . . 6 ⊢ 𝐶 = (Base‘𝑆)
10	8, 9	eqtr4di 2782	. . . . 5 ⊢ (𝑏 = 𝑆 → (Base‘𝑏) = 𝐶)
11		f1oeq23 6759	. . . . 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 596	. . . 4 ⊢ ((𝑎 = 𝑅 ∧ 𝑏 = 𝑆) → (𝑐:(Base‘𝑎)–1-1-onto→(Base‘𝑏) ↔ 𝑐:𝐵–1-1-onto→𝐶))
13	4, 12	rabeqbidv 3415	. . 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 7598	. 2 ⊢ (𝐹 ∈ (𝑅 LMIso 𝑆) ↔ (𝑅 ∈ LMod ∧ 𝑆 ∈ LMod ∧ 𝐹 ∈ {𝑐 ∈ (𝑅 LMHom 𝑆) ∣ 𝑐:𝐵–1-1-onto→𝐶}))
15		df-3an 1088	. 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 6756	. . . . 5 ⊢ (𝑐 = 𝐹 → (𝑐:𝐵–1-1-onto→𝐶 ↔ 𝐹:𝐵–1-1-onto→𝐶))
17	16	elrab 3650	. . . 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 623	. . 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 20955	. . . . . 6 ⊢ (𝐹 ∈ (𝑅 LMHom 𝑆) → 𝑅 ∈ LMod)
20		lmhmlmod2 20954	. . . . . 6 ⊢ (𝐹 ∈ (𝑅 LMHom 𝑆) → 𝑆 ∈ LMod)
21	19, 20	jca 511	. . . . 5 ⊢ (𝐹 ∈ (𝑅 LMHom 𝑆) → (𝑅 ∈ LMod ∧ 𝑆 ∈ LMod))
22	21	adantr 480	. . . 4 ⊢ ((𝐹 ∈ (𝑅 LMHom 𝑆) ∧ 𝐹:𝐵–1-1-onto→𝐶) → (𝑅 ∈ LMod ∧ 𝑆 ∈ LMod))
23	22	pm4.71ri 560	. . 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 278	. 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 297	1 ⊢ (𝐹 ∈ (𝑅 LMIso 𝑆) ↔ (𝐹 ∈ (𝑅 LMHom 𝑆) ∧ 𝐹:𝐵–1-1-onto→𝐶))

Syntax hints:   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109  {crab 3396  –1-1-onto→wf1o 6485  ‘cfv 6486  (class class class)co 7353  Basecbs 17138  LModclmod 20781   LMHom clmhm 20941   LMIso clmim 20942

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5238  ax-nul 5248  ax-pr 5374

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3397  df-v 3440  df-sbc 3745  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4862  df-br 5096  df-opab 5158  df-id 5518  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-ov 7356  df-oprab 7357  df-mpo 7358  df-lmhm 20944  df-lmim 20945

This theorem is referenced by:  lmimf1o  20985  lmimlmhm  20986  islmim2  20988  indlcim  21765  lmimco  21769  lmhmqusker  33364  dimkerim  33599  frlmsnic  42513  pwssplit4  43062