Theorem islmim 21046

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 21007	. . 3 ⊢ LMIso = (𝑎 ∈ LMod, 𝑏 ∈ LMod ↦ {𝑐 ∈ (𝑎 LMHom 𝑏) ∣ 𝑐:(Base‘𝑎)–1-1-onto→(Base‘𝑏)})
2		ovex 7389	. . . 4 ⊢ (𝑎 LMHom 𝑏) ∈ V
3	2	rabex 5269	. . 3 ⊢ {𝑐 ∈ (𝑎 LMHom 𝑏) ∣ 𝑐:(Base‘𝑎)–1-1-onto→(Base‘𝑏)} ∈ V
4		oveq12 7365	. . . 4 ⊢ ((𝑎 = 𝑅 ∧ 𝑏 = 𝑆) → (𝑎 LMHom 𝑏) = (𝑅 LMHom 𝑆))
5		fveq2 6829	. . . . . 6 ⊢ (𝑎 = 𝑅 → (Base‘𝑎) = (Base‘𝑅))
6		islmim.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝑅)
7	5, 6	eqtr4di 2788	. . . . 5 ⊢ (𝑎 = 𝑅 → (Base‘𝑎) = 𝐵)
8		fveq2 6829	. . . . . 6 ⊢ (𝑏 = 𝑆 → (Base‘𝑏) = (Base‘𝑆))
9		islmim.c	. . . . . 6 ⊢ 𝐶 = (Base‘𝑆)
10	8, 9	eqtr4di 2788	. . . . 5 ⊢ (𝑏 = 𝑆 → (Base‘𝑏) = 𝐶)
11		f1oeq23 6760	. . . . 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 3405	. . 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 7601	. 2 ⊢ (𝐹 ∈ (𝑅 LMIso 𝑆) ↔ (𝑅 ∈ LMod ∧ 𝑆 ∈ LMod ∧ 𝐹 ∈ {𝑐 ∈ (𝑅 LMHom 𝑆) ∣ 𝑐:𝐵–1-1-onto→𝐶}))
15		df-3an 1089	. 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 6757	. . . . 5 ⊢ (𝑐 = 𝐹 → (𝑐:𝐵–1-1-onto→𝐶 ↔ 𝐹:𝐵–1-1-onto→𝐶))
17	16	elrab 3631	. . . 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 21017	. . . . . 6 ⊢ (𝐹 ∈ (𝑅 LMHom 𝑆) → 𝑅 ∈ LMod)
20		lmhmlmod2 21016	. . . . . 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 1087   = wceq 1542   ∈ wcel 2114  {crab 3387  –1-1-onto→wf1o 6486  ‘cfv 6487  (class class class)co 7356  Basecbs 17168  LModclmod 20844   LMHom clmhm 21003   LMIso clmim 21004

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2184  ax-ext 2707  ax-sep 5220  ax-nul 5230  ax-pr 5364

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2538  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2810  df-nfc 2884  df-ne 2931  df-ral 3050  df-rex 3060  df-rab 3388  df-v 3429  df-sbc 3726  df-dif 3888  df-un 3890  df-in 3892  df-ss 3902  df-nul 4264  df-if 4457  df-pw 4533  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4841  df-br 5075  df-opab 5137  df-id 5515  df-xp 5626  df-rel 5627  df-cnv 5628  df-co 5629  df-dm 5630  df-rn 5631  df-iota 6443  df-fun 6489  df-fn 6490  df-f 6491  df-f1 6492  df-fo 6493  df-f1o 6494  df-fv 6495  df-ov 7359  df-oprab 7360  df-mpo 7361  df-lmhm 21006  df-lmim 21007

This theorem is referenced by:  lmimf1o  21047  lmimlmhm  21048  islmim2  21050  indlcim  21809  lmimco  21813  lmhmqusker  33465  dimkerim  33759  frlmsnic  42973  pwssplit4  43505