mndlactf1o - Mathbox for Thierry Arnoux

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Theorem mndlactf1o 33018

Description: An element 𝑋 of a monoid 𝐸 is invertible iff its left-translation 𝐹 is bijective. See also grplactf1o 19075. Remark in chapter I. of [BourbakiAlg1] p. 17. (Contributed by Thierry Arnoux, 3-Aug-2025.)

**Hypotheses**
Ref	Expression
mndlactf1o.b	⊢ 𝐵 = (Base‘𝐸)
mndlactf1o.z	⊢ 0 = (0_g‘𝐸)
mndlactf1o.p	⊢ + = (+_g‘𝐸)
mndlactf1o.f	⊢ 𝐹 = (𝑎 ∈ 𝐵 ↦ (𝑋 + 𝑎))
mndlactf1o.e	⊢ (𝜑 → 𝐸 ∈ Mnd)
mndlactf1o.x	⊢ (𝜑 → 𝑋 ∈ 𝐵)

**Assertion**
Ref	Expression
mndlactf1o	⊢ (𝜑 → (𝐹:𝐵–1-1-onto→𝐵 ↔ ∃𝑦 ∈ 𝐵 ((𝑋 + 𝑦) = 0 ∧ (𝑦 + 𝑋) = 0 )))

Distinct variable groups: + ,𝑎,𝑦 0 ,𝑎,𝑦 𝐵,𝑎,𝑦 𝐹,𝑎,𝑦 𝑋,𝑎,𝑦 𝜑,𝑎,𝑦 Allowed substitution hints: 𝐸(𝑦,𝑎)

Proof of Theorem mndlactf1o Dummy variables 𝑢 𝑣 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		oveq2 7439	. . . . . . 7 ⊢ (𝑦 = 𝑢 → (𝑋 + 𝑦) = (𝑋 + 𝑢))
2	1	eqeq1d 2737	. . . . . 6 ⊢ (𝑦 = 𝑢 → ((𝑋 + 𝑦) = 0 ↔ (𝑋 + 𝑢) = 0 ))
3		oveq1 7438	. . . . . . 7 ⊢ (𝑦 = 𝑢 → (𝑦 + 𝑋) = (𝑢 + 𝑋))
4	3	eqeq1d 2737	. . . . . 6 ⊢ (𝑦 = 𝑢 → ((𝑦 + 𝑋) = 0 ↔ (𝑢 + 𝑋) = 0 ))
5	2, 4	anbi12d 632	. . . . 5 ⊢ (𝑦 = 𝑢 → (((𝑋 + 𝑦) = 0 ∧ (𝑦 + 𝑋) = 0 ) ↔ ((𝑋 + 𝑢) = 0 ∧ (𝑢 + 𝑋) = 0 )))
6		simplr 769	. . . . 5 ⊢ ((((((𝜑 ∧ 𝐹:𝐵–1-1-onto→𝐵) ∧ 𝑣 ∈ 𝐵) ∧ (𝑣 + 𝑋) = 0 ) ∧ 𝑢 ∈ 𝐵) ∧ (𝑋 + 𝑢) = 0 ) → 𝑢 ∈ 𝐵)
7		simpr 484	. . . . . 6 ⊢ ((((((𝜑 ∧ 𝐹:𝐵–1-1-onto→𝐵) ∧ 𝑣 ∈ 𝐵) ∧ (𝑣 + 𝑋) = 0 ) ∧ 𝑢 ∈ 𝐵) ∧ (𝑋 + 𝑢) = 0 ) → (𝑋 + 𝑢) = 0 )
8		mndlactf1o.b	. . . . . . . . 9 ⊢ 𝐵 = (Base‘𝐸)
9		mndlactf1o.z	. . . . . . . . 9 ⊢ 0 = (0_g‘𝐸)
10		mndlactf1o.p	. . . . . . . . 9 ⊢ + = (+_g‘𝐸)
11		mndlactf1o.e	. . . . . . . . . 10 ⊢ (𝜑 → 𝐸 ∈ Mnd)
12	11	ad5antr 734	. . . . . . . . 9 ⊢ ((((((𝜑 ∧ 𝐹:𝐵–1-1-onto→𝐵) ∧ 𝑣 ∈ 𝐵) ∧ (𝑣 + 𝑋) = 0 ) ∧ 𝑢 ∈ 𝐵) ∧ (𝑋 + 𝑢) = 0 ) → 𝐸 ∈ Mnd)
13		mndlactf1o.x	. . . . . . . . . 10 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
14	13	ad5antr 734	. . . . . . . . 9 ⊢ ((((((𝜑 ∧ 𝐹:𝐵–1-1-onto→𝐵) ∧ 𝑣 ∈ 𝐵) ∧ (𝑣 + 𝑋) = 0 ) ∧ 𝑢 ∈ 𝐵) ∧ (𝑋 + 𝑢) = 0 ) → 𝑋 ∈ 𝐵)
15		simp-4r 784	. . . . . . . . 9 ⊢ ((((((𝜑 ∧ 𝐹:𝐵–1-1-onto→𝐵) ∧ 𝑣 ∈ 𝐵) ∧ (𝑣 + 𝑋) = 0 ) ∧ 𝑢 ∈ 𝐵) ∧ (𝑋 + 𝑢) = 0 ) → 𝑣 ∈ 𝐵)
16		simpllr 776	. . . . . . . . 9 ⊢ ((((((𝜑 ∧ 𝐹:𝐵–1-1-onto→𝐵) ∧ 𝑣 ∈ 𝐵) ∧ (𝑣 + 𝑋) = 0 ) ∧ 𝑢 ∈ 𝐵) ∧ (𝑋 + 𝑢) = 0 ) → (𝑣 + 𝑋) = 0 )
17	8, 9, 10, 12, 14, 15, 6, 16, 7	mndlrinv 33012	. . . . . . . 8 ⊢ ((((((𝜑 ∧ 𝐹:𝐵–1-1-onto→𝐵) ∧ 𝑣 ∈ 𝐵) ∧ (𝑣 + 𝑋) = 0 ) ∧ 𝑢 ∈ 𝐵) ∧ (𝑋 + 𝑢) = 0 ) → 𝑣 = 𝑢)
18	17	oveq1d 7446	. . . . . . 7 ⊢ ((((((𝜑 ∧ 𝐹:𝐵–1-1-onto→𝐵) ∧ 𝑣 ∈ 𝐵) ∧ (𝑣 + 𝑋) = 0 ) ∧ 𝑢 ∈ 𝐵) ∧ (𝑋 + 𝑢) = 0 ) → (𝑣 + 𝑋) = (𝑢 + 𝑋))
19	18, 16	eqtr3d 2777	. . . . . 6 ⊢ ((((((𝜑 ∧ 𝐹:𝐵–1-1-onto→𝐵) ∧ 𝑣 ∈ 𝐵) ∧ (𝑣 + 𝑋) = 0 ) ∧ 𝑢 ∈ 𝐵) ∧ (𝑋 + 𝑢) = 0 ) → (𝑢 + 𝑋) = 0 )
20	7, 19	jca 511	. . . . 5 ⊢ ((((((𝜑 ∧ 𝐹:𝐵–1-1-onto→𝐵) ∧ 𝑣 ∈ 𝐵) ∧ (𝑣 + 𝑋) = 0 ) ∧ 𝑢 ∈ 𝐵) ∧ (𝑋 + 𝑢) = 0 ) → ((𝑋 + 𝑢) = 0 ∧ (𝑢 + 𝑋) = 0 ))
21	5, 6, 20	rspcedvdw 3625	. . . 4 ⊢ ((((((𝜑 ∧ 𝐹:𝐵–1-1-onto→𝐵) ∧ 𝑣 ∈ 𝐵) ∧ (𝑣 + 𝑋) = 0 ) ∧ 𝑢 ∈ 𝐵) ∧ (𝑋 + 𝑢) = 0 ) → ∃𝑦 ∈ 𝐵 ((𝑋 + 𝑦) = 0 ∧ (𝑦 + 𝑋) = 0 ))
22		f1ofo 6856	. . . . . . 7 ⊢ (𝐹:𝐵–1-1-onto→𝐵 → 𝐹:𝐵–onto→𝐵)
23	22	adantl 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝐹:𝐵–1-1-onto→𝐵) → 𝐹:𝐵–onto→𝐵)
24		mndlactf1o.f	. . . . . . . 8 ⊢ 𝐹 = (𝑎 ∈ 𝐵 ↦ (𝑋 + 𝑎))
25	8, 9, 10, 24, 11, 13	mndlactfo 33015	. . . . . . 7 ⊢ (𝜑 → (𝐹:𝐵–onto→𝐵 ↔ ∃𝑢 ∈ 𝐵 (𝑋 + 𝑢) = 0 ))
26	25	biimpa 476	. . . . . 6 ⊢ ((𝜑 ∧ 𝐹:𝐵–onto→𝐵) → ∃𝑢 ∈ 𝐵 (𝑋 + 𝑢) = 0 )
27	23, 26	syldan 591	. . . . 5 ⊢ ((𝜑 ∧ 𝐹:𝐵–1-1-onto→𝐵) → ∃𝑢 ∈ 𝐵 (𝑋 + 𝑢) = 0 )
28	27	ad2antrr 726	. . . 4 ⊢ ((((𝜑 ∧ 𝐹:𝐵–1-1-onto→𝐵) ∧ 𝑣 ∈ 𝐵) ∧ (𝑣 + 𝑋) = 0 ) → ∃𝑢 ∈ 𝐵 (𝑋 + 𝑢) = 0 )
29	21, 28	r19.29a 3160	. . 3 ⊢ ((((𝜑 ∧ 𝐹:𝐵–1-1-onto→𝐵) ∧ 𝑣 ∈ 𝐵) ∧ (𝑣 + 𝑋) = 0 ) → ∃𝑦 ∈ 𝐵 ((𝑋 + 𝑦) = 0 ∧ (𝑦 + 𝑋) = 0 ))
30		oveq1 7438	. . . . 5 ⊢ (𝑣 = (^◡𝐹‘ 0 ) → (𝑣 + 𝑋) = ((^◡𝐹‘ 0 ) + 𝑋))
31	30	eqeq1d 2737	. . . 4 ⊢ (𝑣 = (^◡𝐹‘ 0 ) → ((𝑣 + 𝑋) = 0 ↔ ((^◡𝐹‘ 0 ) + 𝑋) = 0 ))
32		f1ocnv 6861	. . . . . . 7 ⊢ (𝐹:𝐵–1-1-onto→𝐵 → ^◡𝐹:𝐵–1-1-onto→𝐵)
33		f1of 6849	. . . . . . 7 ⊢ (^◡𝐹:𝐵–1-1-onto→𝐵 → ^◡𝐹:𝐵⟶𝐵)
34	32, 33	syl 17	. . . . . 6 ⊢ (𝐹:𝐵–1-1-onto→𝐵 → ^◡𝐹:𝐵⟶𝐵)
35	34	adantl 481	. . . . 5 ⊢ ((𝜑 ∧ 𝐹:𝐵–1-1-onto→𝐵) → ^◡𝐹:𝐵⟶𝐵)
36	8, 9	mndidcl 18775	. . . . . . 7 ⊢ (𝐸 ∈ Mnd → 0 ∈ 𝐵)
37	11, 36	syl 17	. . . . . 6 ⊢ (𝜑 → 0 ∈ 𝐵)
38	37	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝐹:𝐵–1-1-onto→𝐵) → 0 ∈ 𝐵)
39	35, 38	ffvelcdmd 7105	. . . 4 ⊢ ((𝜑 ∧ 𝐹:𝐵–1-1-onto→𝐵) → (^◡𝐹‘ 0 ) ∈ 𝐵)
40		f1of1 6848	. . . . . 6 ⊢ (𝐹:𝐵–1-1-onto→𝐵 → 𝐹:𝐵–1-1→𝐵)
41	40	adantl 481	. . . . 5 ⊢ ((𝜑 ∧ 𝐹:𝐵–1-1-onto→𝐵) → 𝐹:𝐵–1-1→𝐵)
42	11	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐹:𝐵–1-1-onto→𝐵) → 𝐸 ∈ Mnd)
43	13	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐹:𝐵–1-1-onto→𝐵) → 𝑋 ∈ 𝐵)
44	8, 10, 42, 39, 43	mndcld 33010	. . . . . 6 ⊢ ((𝜑 ∧ 𝐹:𝐵–1-1-onto→𝐵) → ((^◡𝐹‘ 0 ) + 𝑋) ∈ 𝐵)
45	44, 38	jca 511	. . . . 5 ⊢ ((𝜑 ∧ 𝐹:𝐵–1-1-onto→𝐵) → (((^◡𝐹‘ 0 ) + 𝑋) ∈ 𝐵 ∧ 0 ∈ 𝐵))
46	8, 10, 9	mndrid 18781	. . . . . . 7 ⊢ ((𝐸 ∈ Mnd ∧ 𝑋 ∈ 𝐵) → (𝑋 + 0 ) = 𝑋)
47	42, 43, 46	syl2anc 584	. . . . . 6 ⊢ ((𝜑 ∧ 𝐹:𝐵–1-1-onto→𝐵) → (𝑋 + 0 ) = 𝑋)
48		oveq2 7439	. . . . . . 7 ⊢ (𝑎 = 0 → (𝑋 + 𝑎) = (𝑋 + 0 ))
49		ovexd 7466	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐹:𝐵–1-1-onto→𝐵) → (𝑋 + 0 ) ∈ V)
50	24, 48, 38, 49	fvmptd3 7039	. . . . . 6 ⊢ ((𝜑 ∧ 𝐹:𝐵–1-1-onto→𝐵) → (𝐹‘ 0 ) = (𝑋 + 0 ))
51		oveq2 7439	. . . . . . . 8 ⊢ (𝑎 = ((^◡𝐹‘ 0 ) + 𝑋) → (𝑋 + 𝑎) = (𝑋 + ((^◡𝐹‘ 0 ) + 𝑋)))
52		ovexd 7466	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐹:𝐵–1-1-onto→𝐵) → (𝑋 + ((^◡𝐹‘ 0 ) + 𝑋)) ∈ V)
53	24, 51, 44, 52	fvmptd3 7039	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐹:𝐵–1-1-onto→𝐵) → (𝐹‘((^◡𝐹‘ 0 ) + 𝑋)) = (𝑋 + ((^◡𝐹‘ 0 ) + 𝑋)))
54		oveq2 7439	. . . . . . . . . . 11 ⊢ (𝑎 = (^◡𝐹‘ 0 ) → (𝑋 + 𝑎) = (𝑋 + (^◡𝐹‘ 0 )))
55		ovexd 7466	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐹:𝐵–1-1-onto→𝐵) → (𝑋 + (^◡𝐹‘ 0 )) ∈ V)
56	24, 54, 39, 55	fvmptd3 7039	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐹:𝐵–1-1-onto→𝐵) → (𝐹‘(^◡𝐹‘ 0 )) = (𝑋 + (^◡𝐹‘ 0 )))
57		simpr 484	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐹:𝐵–1-1-onto→𝐵) → 𝐹:𝐵–1-1-onto→𝐵)
58		f1ocnvfv2 7297	. . . . . . . . . . 11 ⊢ ((𝐹:𝐵–1-1-onto→𝐵 ∧ 0 ∈ 𝐵) → (𝐹‘(^◡𝐹‘ 0 )) = 0 )
59	57, 38, 58	syl2anc 584	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐹:𝐵–1-1-onto→𝐵) → (𝐹‘(^◡𝐹‘ 0 )) = 0 )
60	56, 59	eqtr3d 2777	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐹:𝐵–1-1-onto→𝐵) → (𝑋 + (^◡𝐹‘ 0 )) = 0 )
61	60	oveq1d 7446	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐹:𝐵–1-1-onto→𝐵) → ((𝑋 + (^◡𝐹‘ 0 )) + 𝑋) = ( 0 + 𝑋))
62	8, 10, 42, 43, 39, 43	mndassd 33011	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐹:𝐵–1-1-onto→𝐵) → ((𝑋 + (^◡𝐹‘ 0 )) + 𝑋) = (𝑋 + ((^◡𝐹‘ 0 ) + 𝑋)))
63	8, 10, 9	mndlid 18780	. . . . . . . . 9 ⊢ ((𝐸 ∈ Mnd ∧ 𝑋 ∈ 𝐵) → ( 0 + 𝑋) = 𝑋)
64	42, 43, 63	syl2anc 584	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐹:𝐵–1-1-onto→𝐵) → ( 0 + 𝑋) = 𝑋)
65	61, 62, 64	3eqtr3d 2783	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐹:𝐵–1-1-onto→𝐵) → (𝑋 + ((^◡𝐹‘ 0 ) + 𝑋)) = 𝑋)
66	53, 65	eqtrd 2775	. . . . . 6 ⊢ ((𝜑 ∧ 𝐹:𝐵–1-1-onto→𝐵) → (𝐹‘((^◡𝐹‘ 0 ) + 𝑋)) = 𝑋)
67	47, 50, 66	3eqtr4rd 2786	. . . . 5 ⊢ ((𝜑 ∧ 𝐹:𝐵–1-1-onto→𝐵) → (𝐹‘((^◡𝐹‘ 0 ) + 𝑋)) = (𝐹‘ 0 ))
68		f1fveq 7282	. . . . . 6 ⊢ ((𝐹:𝐵–1-1→𝐵 ∧ (((^◡𝐹‘ 0 ) + 𝑋) ∈ 𝐵 ∧ 0 ∈ 𝐵)) → ((𝐹‘((^◡𝐹‘ 0 ) + 𝑋)) = (𝐹‘ 0 ) ↔ ((^◡𝐹‘ 0 ) + 𝑋) = 0 ))
69	68	biimpa 476	. . . . 5 ⊢ (((𝐹:𝐵–1-1→𝐵 ∧ (((^◡𝐹‘ 0 ) + 𝑋) ∈ 𝐵 ∧ 0 ∈ 𝐵)) ∧ (𝐹‘((^◡𝐹‘ 0 ) + 𝑋)) = (𝐹‘ 0 )) → ((^◡𝐹‘ 0 ) + 𝑋) = 0 )
70	41, 45, 67, 69	syl21anc 838	. . . 4 ⊢ ((𝜑 ∧ 𝐹:𝐵–1-1-onto→𝐵) → ((^◡𝐹‘ 0 ) + 𝑋) = 0 )
71	31, 39, 70	rspcedvdw 3625	. . 3 ⊢ ((𝜑 ∧ 𝐹:𝐵–1-1-onto→𝐵) → ∃𝑣 ∈ 𝐵 (𝑣 + 𝑋) = 0 )
72	29, 71	r19.29a 3160	. 2 ⊢ ((𝜑 ∧ 𝐹:𝐵–1-1-onto→𝐵) → ∃𝑦 ∈ 𝐵 ((𝑋 + 𝑦) = 0 ∧ (𝑦 + 𝑋) = 0 ))
73		oveq1 7438	. . . . . . 7 ⊢ (𝑣 = 𝑦 → (𝑣 + 𝑋) = (𝑦 + 𝑋))
74	73	eqeq1d 2737	. . . . . 6 ⊢ (𝑣 = 𝑦 → ((𝑣 + 𝑋) = 0 ↔ (𝑦 + 𝑋) = 0 ))
75		simplr 769	. . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐵) ∧ ((𝑋 + 𝑦) = 0 ∧ (𝑦 + 𝑋) = 0 )) → 𝑦 ∈ 𝐵)
76		simprr 773	. . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐵) ∧ ((𝑋 + 𝑦) = 0 ∧ (𝑦 + 𝑋) = 0 )) → (𝑦 + 𝑋) = 0 )
77	74, 75, 76	rspcedvdw 3625	. . . . 5 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐵) ∧ ((𝑋 + 𝑦) = 0 ∧ (𝑦 + 𝑋) = 0 )) → ∃𝑣 ∈ 𝐵 (𝑣 + 𝑋) = 0 )
78		oveq2 7439	. . . . . . 7 ⊢ (𝑢 = 𝑦 → (𝑋 + 𝑢) = (𝑋 + 𝑦))
79	78	eqeq1d 2737	. . . . . 6 ⊢ (𝑢 = 𝑦 → ((𝑋 + 𝑢) = 0 ↔ (𝑋 + 𝑦) = 0 ))
80		simprl 771	. . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐵) ∧ ((𝑋 + 𝑦) = 0 ∧ (𝑦 + 𝑋) = 0 )) → (𝑋 + 𝑦) = 0 )
81	79, 75, 80	rspcedvdw 3625	. . . . 5 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐵) ∧ ((𝑋 + 𝑦) = 0 ∧ (𝑦 + 𝑋) = 0 )) → ∃𝑢 ∈ 𝐵 (𝑋 + 𝑢) = 0 )
82	77, 81	jca 511	. . . 4 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐵) ∧ ((𝑋 + 𝑦) = 0 ∧ (𝑦 + 𝑋) = 0 )) → (∃𝑣 ∈ 𝐵 (𝑣 + 𝑋) = 0 ∧ ∃𝑢 ∈ 𝐵 (𝑋 + 𝑢) = 0 ))
83	82	r19.29an 3156	. . 3 ⊢ ((𝜑 ∧ ∃𝑦 ∈ 𝐵 ((𝑋 + 𝑦) = 0 ∧ (𝑦 + 𝑋) = 0 )) → (∃𝑣 ∈ 𝐵 (𝑣 + 𝑋) = 0 ∧ ∃𝑢 ∈ 𝐵 (𝑋 + 𝑢) = 0 ))
84	11	ad2antrr 726	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑣 ∈ 𝐵) ∧ (𝑣 + 𝑋) = 0 ) → 𝐸 ∈ Mnd)
85	13	ad2antrr 726	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑣 ∈ 𝐵) ∧ (𝑣 + 𝑋) = 0 ) → 𝑋 ∈ 𝐵)
86		simplr 769	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑣 ∈ 𝐵) ∧ (𝑣 + 𝑋) = 0 ) → 𝑣 ∈ 𝐵)
87		simpr 484	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑣 ∈ 𝐵) ∧ (𝑣 + 𝑋) = 0 ) → (𝑣 + 𝑋) = 0 )
88	8, 9, 10, 24, 84, 85, 86, 87	mndlactf1 33014	. . . . . 6 ⊢ (((𝜑 ∧ 𝑣 ∈ 𝐵) ∧ (𝑣 + 𝑋) = 0 ) → 𝐹:𝐵–1-1→𝐵)
89	88	r19.29an 3156	. . . . 5 ⊢ ((𝜑 ∧ ∃𝑣 ∈ 𝐵 (𝑣 + 𝑋) = 0 ) → 𝐹:𝐵–1-1→𝐵)
90	25	biimpar 477	. . . . 5 ⊢ ((𝜑 ∧ ∃𝑢 ∈ 𝐵 (𝑋 + 𝑢) = 0 ) → 𝐹:𝐵–onto→𝐵)
91	89, 90	anim12dan 619	. . . 4 ⊢ ((𝜑 ∧ (∃𝑣 ∈ 𝐵 (𝑣 + 𝑋) = 0 ∧ ∃𝑢 ∈ 𝐵 (𝑋 + 𝑢) = 0 )) → (𝐹:𝐵–1-1→𝐵 ∧ 𝐹:𝐵–onto→𝐵))
92		df-f1o 6570	. . . 4 ⊢ (𝐹:𝐵–1-1-onto→𝐵 ↔ (𝐹:𝐵–1-1→𝐵 ∧ 𝐹:𝐵–onto→𝐵))
93	91, 92	sylibr 234	. . 3 ⊢ ((𝜑 ∧ (∃𝑣 ∈ 𝐵 (𝑣 + 𝑋) = 0 ∧ ∃𝑢 ∈ 𝐵 (𝑋 + 𝑢) = 0 )) → 𝐹:𝐵–1-1-onto→𝐵)
94	83, 93	syldan 591	. 2 ⊢ ((𝜑 ∧ ∃𝑦 ∈ 𝐵 ((𝑋 + 𝑦) = 0 ∧ (𝑦 + 𝑋) = 0 )) → 𝐹:𝐵–1-1-onto→𝐵)
95	72, 94	impbida 801	1 ⊢ (𝜑 → (𝐹:𝐵–1-1-onto→𝐵 ↔ ∃𝑦 ∈ 𝐵 ((𝑋 + 𝑦) = 0 ∧ (𝑦 + 𝑋) = 0 )))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1537 ∈ wcel 2106 ∃wrex 3068 Vcvv 3478 ↦ cmpt 5231 ^◡ccnv 5688 ⟶wf 6559 –1-1→wf1 6560 –onto→wfo 6561 –1-1-onto→wf1o 6562 ‘cfv 6563 (class class class)co 7431 Basecbs 17245 +_gcplusg 17298 0_gc0g 17486 Mndcmnd 18760

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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-0g 17488 df-mgm 18666 df-sgrp 18745 df-mnd 18761

This theorem is referenced by: assarrginv 33664

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