Theorem domnprodeq0 33536

Description: A product over a domain is zero exactly when one of the factors is zero. Generalization of domneq0 20789 for any number of factors. See also domnprodn0 33535. (Contributed by Thierry Arnoux, 15-Feb-2026.)

Hypotheses

Ref	Expression
domnprodeq0.m	⊢ 𝑀 = (mulGrp‘𝑅)
domnprodeq0.b	⊢ 𝐵 = (Base‘𝑅)
domnprodeq0.1	⊢ 0 = (0g‘𝑅)
domnprodeq0.r	⊢ (𝜑 → 𝑅 ∈ IDomn)
domnprodeq0.2	⊢ (𝜑 → 𝐴 ∈ Fin)
domnprodeq0.f	⊢ (𝜑 → 𝐹:𝐴⟶𝐵)

Assertion

Ref	Expression
domnprodeq0	⊢ (𝜑 → ((𝑀 Σg 𝐹) = 0 ↔ 0 ∈ ran 𝐹))

Proof of Theorem domnprodeq0

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

Step	Hyp	Ref	Expression
1		mpteq1 5201	. . . . . . 7 ⊢ (𝑎 = ∅ → (𝑘 ∈ 𝑎 ↦ (𝐹‘𝑘)) = (𝑘 ∈ ∅ ↦ (𝐹‘𝑘)))
2		mpt0 6675	. . . . . . 7 ⊢ (𝑘 ∈ ∅ ↦ (𝐹‘𝑘)) = ∅
3	1, 2	eqtrdi 2820	. . . . . 6 ⊢ (𝑎 = ∅ → (𝑘 ∈ 𝑎 ↦ (𝐹‘𝑘)) = ∅)
4	3	oveq2d 7424	. . . . 5 ⊢ (𝑎 = ∅ → (𝑀 Σg (𝑘 ∈ 𝑎 ↦ (𝐹‘𝑘))) = (𝑀 Σg ∅))
5	4	eqeq1d 2771	. . . 4 ⊢ (𝑎 = ∅ → ((𝑀 Σg (𝑘 ∈ 𝑎 ↦ (𝐹‘𝑘))) = 0 ↔ (𝑀 Σg ∅) = 0 ))
6	3	rneqd 5926	. . . . 5 ⊢ (𝑎 = ∅ → ran (𝑘 ∈ 𝑎 ↦ (𝐹‘𝑘)) = ran ∅)
7	6	eleq2d 2855	. . . 4 ⊢ (𝑎 = ∅ → ( 0 ∈ ran (𝑘 ∈ 𝑎 ↦ (𝐹‘𝑘)) ↔ 0 ∈ ran ∅))
8	5, 7	bibi12d 348	. . 3 ⊢ (𝑎 = ∅ → (((𝑀 Σg (𝑘 ∈ 𝑎 ↦ (𝐹‘𝑘))) = 0 ↔ 0 ∈ ran (𝑘 ∈ 𝑎 ↦ (𝐹‘𝑘))) ↔ ((𝑀 Σg ∅) = 0 ↔ 0 ∈ ran ∅)))
9		mpteq1 5201	. . . . . 6 ⊢ (𝑎 = 𝑏 → (𝑘 ∈ 𝑎 ↦ (𝐹‘𝑘)) = (𝑘 ∈ 𝑏 ↦ (𝐹‘𝑘)))
10	9	oveq2d 7424	. . . . 5 ⊢ (𝑎 = 𝑏 → (𝑀 Σg (𝑘 ∈ 𝑎 ↦ (𝐹‘𝑘))) = (𝑀 Σg (𝑘 ∈ 𝑏 ↦ (𝐹‘𝑘))))
11	10	eqeq1d 2771	. . . 4 ⊢ (𝑎 = 𝑏 → ((𝑀 Σg (𝑘 ∈ 𝑎 ↦ (𝐹‘𝑘))) = 0 ↔ (𝑀 Σg (𝑘 ∈ 𝑏 ↦ (𝐹‘𝑘))) = 0 ))
12	9	rneqd 5926	. . . . 5 ⊢ (𝑎 = 𝑏 → ran (𝑘 ∈ 𝑎 ↦ (𝐹‘𝑘)) = ran (𝑘 ∈ 𝑏 ↦ (𝐹‘𝑘)))
13	12	eleq2d 2855	. . . 4 ⊢ (𝑎 = 𝑏 → ( 0 ∈ ran (𝑘 ∈ 𝑎 ↦ (𝐹‘𝑘)) ↔ 0 ∈ ran (𝑘 ∈ 𝑏 ↦ (𝐹‘𝑘))))
14	11, 13	bibi12d 348	. . 3 ⊢ (𝑎 = 𝑏 → (((𝑀 Σg (𝑘 ∈ 𝑎 ↦ (𝐹‘𝑘))) = 0 ↔ 0 ∈ ran (𝑘 ∈ 𝑎 ↦ (𝐹‘𝑘))) ↔ ((𝑀 Σg (𝑘 ∈ 𝑏 ↦ (𝐹‘𝑘))) = 0 ↔ 0 ∈ ran (𝑘 ∈ 𝑏 ↦ (𝐹‘𝑘)))))
15		mpteq1 5201	. . . . . 6 ⊢ (𝑎 = (𝑏 ∪ {𝑙}) → (𝑘 ∈ 𝑎 ↦ (𝐹‘𝑘)) = (𝑘 ∈ (𝑏 ∪ {𝑙}) ↦ (𝐹‘𝑘)))
16	15	oveq2d 7424	. . . . 5 ⊢ (𝑎 = (𝑏 ∪ {𝑙}) → (𝑀 Σg (𝑘 ∈ 𝑎 ↦ (𝐹‘𝑘))) = (𝑀 Σg (𝑘 ∈ (𝑏 ∪ {𝑙}) ↦ (𝐹‘𝑘))))
17	16	eqeq1d 2771	. . . 4 ⊢ (𝑎 = (𝑏 ∪ {𝑙}) → ((𝑀 Σg (𝑘 ∈ 𝑎 ↦ (𝐹‘𝑘))) = 0 ↔ (𝑀 Σg (𝑘 ∈ (𝑏 ∪ {𝑙}) ↦ (𝐹‘𝑘))) = 0 ))
18	15	rneqd 5926	. . . . 5 ⊢ (𝑎 = (𝑏 ∪ {𝑙}) → ran (𝑘 ∈ 𝑎 ↦ (𝐹‘𝑘)) = ran (𝑘 ∈ (𝑏 ∪ {𝑙}) ↦ (𝐹‘𝑘)))
19	18	eleq2d 2855	. . . 4 ⊢ (𝑎 = (𝑏 ∪ {𝑙}) → ( 0 ∈ ran (𝑘 ∈ 𝑎 ↦ (𝐹‘𝑘)) ↔ 0 ∈ ran (𝑘 ∈ (𝑏 ∪ {𝑙}) ↦ (𝐹‘𝑘))))
20	17, 19	bibi12d 348	. . 3 ⊢ (𝑎 = (𝑏 ∪ {𝑙}) → (((𝑀 Σg (𝑘 ∈ 𝑎 ↦ (𝐹‘𝑘))) = 0 ↔ 0 ∈ ran (𝑘 ∈ 𝑎 ↦ (𝐹‘𝑘))) ↔ ((𝑀 Σg (𝑘 ∈ (𝑏 ∪ {𝑙}) ↦ (𝐹‘𝑘))) = 0 ↔ 0 ∈ ran (𝑘 ∈ (𝑏 ∪ {𝑙}) ↦ (𝐹‘𝑘)))))
21		mpteq1 5201	. . . . . 6 ⊢ (𝑎 = 𝐴 → (𝑘 ∈ 𝑎 ↦ (𝐹‘𝑘)) = (𝑘 ∈ 𝐴 ↦ (𝐹‘𝑘)))
22	21	oveq2d 7424	. . . . 5 ⊢ (𝑎 = 𝐴 → (𝑀 Σg (𝑘 ∈ 𝑎 ↦ (𝐹‘𝑘))) = (𝑀 Σg (𝑘 ∈ 𝐴 ↦ (𝐹‘𝑘))))
23	22	eqeq1d 2771	. . . 4 ⊢ (𝑎 = 𝐴 → ((𝑀 Σg (𝑘 ∈ 𝑎 ↦ (𝐹‘𝑘))) = 0 ↔ (𝑀 Σg (𝑘 ∈ 𝐴 ↦ (𝐹‘𝑘))) = 0 ))
24	21	rneqd 5926	. . . . 5 ⊢ (𝑎 = 𝐴 → ran (𝑘 ∈ 𝑎 ↦ (𝐹‘𝑘)) = ran (𝑘 ∈ 𝐴 ↦ (𝐹‘𝑘)))
25	24	eleq2d 2855	. . . 4 ⊢ (𝑎 = 𝐴 → ( 0 ∈ ran (𝑘 ∈ 𝑎 ↦ (𝐹‘𝑘)) ↔ 0 ∈ ran (𝑘 ∈ 𝐴 ↦ (𝐹‘𝑘))))
26	23, 25	bibi12d 348	. . 3 ⊢ (𝑎 = 𝐴 → (((𝑀 Σg (𝑘 ∈ 𝑎 ↦ (𝐹‘𝑘))) = 0 ↔ 0 ∈ ran (𝑘 ∈ 𝑎 ↦ (𝐹‘𝑘))) ↔ ((𝑀 Σg (𝑘 ∈ 𝐴 ↦ (𝐹‘𝑘))) = 0 ↔ 0 ∈ ran (𝑘 ∈ 𝐴 ↦ (𝐹‘𝑘)))))
27		domnprodeq0.m	. . . . . . . . 9 ⊢ 𝑀 = (mulGrp‘𝑅)
28		eqid 2769	. . . . . . . . 9 ⊢ (1r‘𝑅) = (1r‘𝑅)
29	27, 28	ringidval 20261	. . . . . . . 8 ⊢ (1r‘𝑅) = (0g‘𝑀)
30	29	gsum0 18738	. . . . . . 7 ⊢ (𝑀 Σg ∅) = (1r‘𝑅)
31	30	a1i 11	. . . . . 6 ⊢ (𝜑 → (𝑀 Σg ∅) = (1r‘𝑅))
32		domnprodeq0.r	. . . . . . . 8 ⊢ (𝜑 → 𝑅 ∈ IDomn)
33	32	idomdomd 20806	. . . . . . 7 ⊢ (𝜑 → 𝑅 ∈ Domn)
34		domnnzr 20787	. . . . . . 7 ⊢ (𝑅 ∈ Domn → 𝑅 ∈ NzRing)
35		domnprodeq0.1	. . . . . . . 8 ⊢ 0 = (0g‘𝑅)
36	28, 35	nzrnz 20594	. . . . . . 7 ⊢ (𝑅 ∈ NzRing → (1r‘𝑅) ≠ 0 )
37	33, 34, 36	3syl 19	. . . . . 6 ⊢ (𝜑 → (1r‘𝑅) ≠ 0 )
38	31, 37	eqnetrd 3031	. . . . 5 ⊢ (𝜑 → (𝑀 Σg ∅) ≠ 0 )
39	38	neneqd 2969	. . . 4 ⊢ (𝜑 → ¬ (𝑀 Σg ∅) = 0 )
40		noel 4299	. . . . . 6 ⊢ ¬ 0 ∈ ∅
41		rn0 5914	. . . . . . 7 ⊢ ran ∅ = ∅
42	41	eleq2i 2861	. . . . . 6 ⊢ ( 0 ∈ ran ∅ ↔ 0 ∈ ∅)
43	40, 42	mtbir 326	. . . . 5 ⊢ ¬ 0 ∈ ran ∅
44	43	a1i 11	. . . 4 ⊢ (𝜑 → ¬ 0 ∈ ran ∅)
45	39, 44	2falsed 379	. . 3 ⊢ (𝜑 → ((𝑀 Σg ∅) = 0 ↔ 0 ∈ ran ∅))
46		simpr 489	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑏 ⊆ 𝐴) ∧ 𝑙 ∈ (𝐴 ∖ 𝑏)) ∧ ((𝑀 Σg (𝑘 ∈ 𝑏 ↦ (𝐹‘𝑘))) = 0 ↔ 0 ∈ ran (𝑘 ∈ 𝑏 ↦ (𝐹‘𝑘)))) → ((𝑀 Σg (𝑘 ∈ 𝑏 ↦ (𝐹‘𝑘))) = 0 ↔ 0 ∈ ran (𝑘 ∈ 𝑏 ↦ (𝐹‘𝑘))))
47	46	orbi1d 929	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑏 ⊆ 𝐴) ∧ 𝑙 ∈ (𝐴 ∖ 𝑏)) ∧ ((𝑀 Σg (𝑘 ∈ 𝑏 ↦ (𝐹‘𝑘))) = 0 ↔ 0 ∈ ran (𝑘 ∈ 𝑏 ↦ (𝐹‘𝑘)))) → (((𝑀 Σg (𝑘 ∈ 𝑏 ↦ (𝐹‘𝑘))) = 0 ∨ (𝐹‘𝑙) = 0 ) ↔ ( 0 ∈ ran (𝑘 ∈ 𝑏 ↦ (𝐹‘𝑘)) ∨ (𝐹‘𝑙) = 0 )))
48		domnprodeq0.b	. . . . . . . . . . 11 ⊢ 𝐵 = (Base‘𝑅)
49	27, 48	mgpbas 20217	. . . . . . . . . 10 ⊢ 𝐵 = (Base‘𝑀)
50		eqid 2769	. . . . . . . . . . 11 ⊢ (.r‘𝑅) = (.r‘𝑅)
51	27, 50	mgpplusg 20216	. . . . . . . . . 10 ⊢ (.r‘𝑅) = (+g‘𝑀)
52	32	idomcringd 20807	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑅 ∈ CRing)
53	27	crngmgp 20319	. . . . . . . . . . . 12 ⊢ (𝑅 ∈ CRing → 𝑀 ∈ CMnd)
54	52, 53	syl 18	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑀 ∈ CMnd)
55	54	ad2antrr 738	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑏 ⊆ 𝐴) ∧ 𝑙 ∈ (𝐴 ∖ 𝑏)) → 𝑀 ∈ CMnd)
56		domnprodeq0.2	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐴 ∈ Fin)
57	56	ad2antrr 738	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑏 ⊆ 𝐴) ∧ 𝑙 ∈ (𝐴 ∖ 𝑏)) → 𝐴 ∈ Fin)
58		simplr 780	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑏 ⊆ 𝐴) ∧ 𝑙 ∈ (𝐴 ∖ 𝑏)) → 𝑏 ⊆ 𝐴)
59	57, 58	ssfid 9225	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑏 ⊆ 𝐴) ∧ 𝑙 ∈ (𝐴 ∖ 𝑏)) → 𝑏 ∈ Fin)
60		domnprodeq0.f	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)
61	60	ad3antrrr 742	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑏 ⊆ 𝐴) ∧ 𝑙 ∈ (𝐴 ∖ 𝑏)) ∧ 𝑘 ∈ 𝑏) → 𝐹:𝐴⟶𝐵)
62	58	sselda 3945	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑏 ⊆ 𝐴) ∧ 𝑙 ∈ (𝐴 ∖ 𝑏)) ∧ 𝑘 ∈ 𝑏) → 𝑘 ∈ 𝐴)
63	61, 62	ffvelcdmd 7078	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑏 ⊆ 𝐴) ∧ 𝑙 ∈ (𝐴 ∖ 𝑏)) ∧ 𝑘 ∈ 𝑏) → (𝐹‘𝑘) ∈ 𝐵)
64		simpr 489	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑏 ⊆ 𝐴) ∧ 𝑙 ∈ (𝐴 ∖ 𝑏)) → 𝑙 ∈ (𝐴 ∖ 𝑏))
65	64	eldifbd 3926	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑏 ⊆ 𝐴) ∧ 𝑙 ∈ (𝐴 ∖ 𝑏)) → ¬ 𝑙 ∈ 𝑏)
66	60	ad2antrr 738	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑏 ⊆ 𝐴) ∧ 𝑙 ∈ (𝐴 ∖ 𝑏)) → 𝐹:𝐴⟶𝐵)
67	64	eldifad 3925	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑏 ⊆ 𝐴) ∧ 𝑙 ∈ (𝐴 ∖ 𝑏)) → 𝑙 ∈ 𝐴)
68	66, 67	ffvelcdmd 7078	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑏 ⊆ 𝐴) ∧ 𝑙 ∈ (𝐴 ∖ 𝑏)) → (𝐹‘𝑙) ∈ 𝐵)
69		fveq2 6879	. . . . . . . . . 10 ⊢ (𝑘 = 𝑙 → (𝐹‘𝑘) = (𝐹‘𝑙))
70	49, 51, 55, 59, 63, 64, 65, 68, 69	gsumunsn 20026	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑏 ⊆ 𝐴) ∧ 𝑙 ∈ (𝐴 ∖ 𝑏)) → (𝑀 Σg (𝑘 ∈ (𝑏 ∪ {𝑙}) ↦ (𝐹‘𝑘))) = ((𝑀 Σg (𝑘 ∈ 𝑏 ↦ (𝐹‘𝑘)))(.r‘𝑅)(𝐹‘𝑙)))
71	70	eqeq1d 2771	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑏 ⊆ 𝐴) ∧ 𝑙 ∈ (𝐴 ∖ 𝑏)) → ((𝑀 Σg (𝑘 ∈ (𝑏 ∪ {𝑙}) ↦ (𝐹‘𝑘))) = 0 ↔ ((𝑀 Σg (𝑘 ∈ 𝑏 ↦ (𝐹‘𝑘)))(.r‘𝑅)(𝐹‘𝑙)) = 0 ))
72	33	ad2antrr 738	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑏 ⊆ 𝐴) ∧ 𝑙 ∈ (𝐴 ∖ 𝑏)) → 𝑅 ∈ Domn)
73	63	ralrimiva 3163	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑏 ⊆ 𝐴) ∧ 𝑙 ∈ (𝐴 ∖ 𝑏)) → ∀𝑘 ∈ 𝑏 (𝐹‘𝑘) ∈ 𝐵)
74	49, 55, 59, 73	gsummptcl 20033	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑏 ⊆ 𝐴) ∧ 𝑙 ∈ (𝐴 ∖ 𝑏)) → (𝑀 Σg (𝑘 ∈ 𝑏 ↦ (𝐹‘𝑘))) ∈ 𝐵)
75	48, 50, 35	domneq0 20789	. . . . . . . . 9 ⊢ ((𝑅 ∈ Domn ∧ (𝑀 Σg (𝑘 ∈ 𝑏 ↦ (𝐹‘𝑘))) ∈ 𝐵 ∧ (𝐹‘𝑙) ∈ 𝐵) → (((𝑀 Σg (𝑘 ∈ 𝑏 ↦ (𝐹‘𝑘)))(.r‘𝑅)(𝐹‘𝑙)) = 0 ↔ ((𝑀 Σg (𝑘 ∈ 𝑏 ↦ (𝐹‘𝑘))) = 0 ∨ (𝐹‘𝑙) = 0 )))
76	72, 74, 68, 75	syl3anc 1396	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑏 ⊆ 𝐴) ∧ 𝑙 ∈ (𝐴 ∖ 𝑏)) → (((𝑀 Σg (𝑘 ∈ 𝑏 ↦ (𝐹‘𝑘)))(.r‘𝑅)(𝐹‘𝑙)) = 0 ↔ ((𝑀 Σg (𝑘 ∈ 𝑏 ↦ (𝐹‘𝑘))) = 0 ∨ (𝐹‘𝑙) = 0 )))
77	71, 76	bitrd 282	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑏 ⊆ 𝐴) ∧ 𝑙 ∈ (𝐴 ∖ 𝑏)) → ((𝑀 Σg (𝑘 ∈ (𝑏 ∪ {𝑙}) ↦ (𝐹‘𝑘))) = 0 ↔ ((𝑀 Σg (𝑘 ∈ 𝑏 ↦ (𝐹‘𝑘))) = 0 ∨ (𝐹‘𝑙) = 0 )))
78	77	adantr 485	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑏 ⊆ 𝐴) ∧ 𝑙 ∈ (𝐴 ∖ 𝑏)) ∧ ((𝑀 Σg (𝑘 ∈ 𝑏 ↦ (𝐹‘𝑘))) = 0 ↔ 0 ∈ ran (𝑘 ∈ 𝑏 ↦ (𝐹‘𝑘)))) → ((𝑀 Σg (𝑘 ∈ (𝑏 ∪ {𝑙}) ↦ (𝐹‘𝑘))) = 0 ↔ ((𝑀 Σg (𝑘 ∈ 𝑏 ↦ (𝐹‘𝑘))) = 0 ∨ (𝐹‘𝑙) = 0 )))
79		eqid 2769	. . . . . . . . 9 ⊢ (𝑘 ∈ (𝑏 ∪ {𝑙}) ↦ (𝐹‘𝑘)) = (𝑘 ∈ (𝑏 ∪ {𝑙}) ↦ (𝐹‘𝑘))
80		fvex 6892	. . . . . . . . 9 ⊢ (𝐹‘𝑘) ∈ V
81	79, 80	elrnmpti 5950	. . . . . . . 8 ⊢ ( 0 ∈ ran (𝑘 ∈ (𝑏 ∪ {𝑙}) ↦ (𝐹‘𝑘)) ↔ ∃𝑘 ∈ (𝑏 ∪ {𝑙}) 0 = (𝐹‘𝑘))
82		rexun 4157	. . . . . . . 8 ⊢ (∃𝑘 ∈ (𝑏 ∪ {𝑙}) 0 = (𝐹‘𝑘) ↔ (∃𝑘 ∈ 𝑏 0 = (𝐹‘𝑘) ∨ ∃𝑘 ∈ {𝑙} 0 = (𝐹‘𝑘)))
83		eqid 2769	. . . . . . . . . . 11 ⊢ (𝑘 ∈ 𝑏 ↦ (𝐹‘𝑘)) = (𝑘 ∈ 𝑏 ↦ (𝐹‘𝑘))
84	83, 80	elrnmpti 5950	. . . . . . . . . 10 ⊢ ( 0 ∈ ran (𝑘 ∈ 𝑏 ↦ (𝐹‘𝑘)) ↔ ∃𝑘 ∈ 𝑏 0 = (𝐹‘𝑘))
85	84	bicomi 227	. . . . . . . . 9 ⊢ (∃𝑘 ∈ 𝑏 0 = (𝐹‘𝑘) ↔ 0 ∈ ran (𝑘 ∈ 𝑏 ↦ (𝐹‘𝑘)))
86		vex 3467	. . . . . . . . . 10 ⊢ 𝑙 ∈ V
87	69	eqeq2d 2780	. . . . . . . . . . 11 ⊢ (𝑘 = 𝑙 → ( 0 = (𝐹‘𝑘) ↔ 0 = (𝐹‘𝑙)))
88		eqcom 2776	. . . . . . . . . . 11 ⊢ ( 0 = (𝐹‘𝑙) ↔ (𝐹‘𝑙) = 0 )
89	87, 88	bitrdi 290	. . . . . . . . . 10 ⊢ (𝑘 = 𝑙 → ( 0 = (𝐹‘𝑘) ↔ (𝐹‘𝑙) = 0 ))
90	86, 89	rexsn 4650	. . . . . . . . 9 ⊢ (∃𝑘 ∈ {𝑙} 0 = (𝐹‘𝑘) ↔ (𝐹‘𝑙) = 0 )
91	85, 90	orbi12i 927	. . . . . . . 8 ⊢ ((∃𝑘 ∈ 𝑏 0 = (𝐹‘𝑘) ∨ ∃𝑘 ∈ {𝑙} 0 = (𝐹‘𝑘)) ↔ ( 0 ∈ ran (𝑘 ∈ 𝑏 ↦ (𝐹‘𝑘)) ∨ (𝐹‘𝑙) = 0 ))
92	81, 82, 91	3bitri 300	. . . . . . 7 ⊢ ( 0 ∈ ran (𝑘 ∈ (𝑏 ∪ {𝑙}) ↦ (𝐹‘𝑘)) ↔ ( 0 ∈ ran (𝑘 ∈ 𝑏 ↦ (𝐹‘𝑘)) ∨ (𝐹‘𝑙) = 0 ))
93	92	a1i 11	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑏 ⊆ 𝐴) ∧ 𝑙 ∈ (𝐴 ∖ 𝑏)) ∧ ((𝑀 Σg (𝑘 ∈ 𝑏 ↦ (𝐹‘𝑘))) = 0 ↔ 0 ∈ ran (𝑘 ∈ 𝑏 ↦ (𝐹‘𝑘)))) → ( 0 ∈ ran (𝑘 ∈ (𝑏 ∪ {𝑙}) ↦ (𝐹‘𝑘)) ↔ ( 0 ∈ ran (𝑘 ∈ 𝑏 ↦ (𝐹‘𝑘)) ∨ (𝐹‘𝑙) = 0 )))
94	47, 78, 93	3bitr4d 314	. . . . 5 ⊢ ((((𝜑 ∧ 𝑏 ⊆ 𝐴) ∧ 𝑙 ∈ (𝐴 ∖ 𝑏)) ∧ ((𝑀 Σg (𝑘 ∈ 𝑏 ↦ (𝐹‘𝑘))) = 0 ↔ 0 ∈ ran (𝑘 ∈ 𝑏 ↦ (𝐹‘𝑘)))) → ((𝑀 Σg (𝑘 ∈ (𝑏 ∪ {𝑙}) ↦ (𝐹‘𝑘))) = 0 ↔ 0 ∈ ran (𝑘 ∈ (𝑏 ∪ {𝑙}) ↦ (𝐹‘𝑘))))
95	94	ex 417	. . . 4 ⊢ (((𝜑 ∧ 𝑏 ⊆ 𝐴) ∧ 𝑙 ∈ (𝐴 ∖ 𝑏)) → (((𝑀 Σg (𝑘 ∈ 𝑏 ↦ (𝐹‘𝑘))) = 0 ↔ 0 ∈ ran (𝑘 ∈ 𝑏 ↦ (𝐹‘𝑘))) → ((𝑀 Σg (𝑘 ∈ (𝑏 ∪ {𝑙}) ↦ (𝐹‘𝑘))) = 0 ↔ 0 ∈ ran (𝑘 ∈ (𝑏 ∪ {𝑙}) ↦ (𝐹‘𝑘)))))
96	95	anasss 471	. . 3 ⊢ ((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑙 ∈ (𝐴 ∖ 𝑏))) → (((𝑀 Σg (𝑘 ∈ 𝑏 ↦ (𝐹‘𝑘))) = 0 ↔ 0 ∈ ran (𝑘 ∈ 𝑏 ↦ (𝐹‘𝑘))) → ((𝑀 Σg (𝑘 ∈ (𝑏 ∪ {𝑙}) ↦ (𝐹‘𝑘))) = 0 ↔ 0 ∈ ran (𝑘 ∈ (𝑏 ∪ {𝑙}) ↦ (𝐹‘𝑘)))))
97	8, 14, 20, 26, 45, 96, 56	findcard2d 9147	. 2 ⊢ (𝜑 → ((𝑀 Σg (𝑘 ∈ 𝐴 ↦ (𝐹‘𝑘))) = 0 ↔ 0 ∈ ran (𝑘 ∈ 𝐴 ↦ (𝐹‘𝑘))))
98	60	feqmptd 6947	. . . 4 ⊢ (𝜑 → 𝐹 = (𝑘 ∈ 𝐴 ↦ (𝐹‘𝑘)))
99	98	oveq2d 7424	. . 3 ⊢ (𝜑 → (𝑀 Σg 𝐹) = (𝑀 Σg (𝑘 ∈ 𝐴 ↦ (𝐹‘𝑘))))
100	99	eqeq1d 2771	. 2 ⊢ (𝜑 → ((𝑀 Σg 𝐹) = 0 ↔ (𝑀 Σg (𝑘 ∈ 𝐴 ↦ (𝐹‘𝑘))) = 0 ))
101	98	rneqd 5926	. . 3 ⊢ (𝜑 → ran 𝐹 = ran (𝑘 ∈ 𝐴 ↦ (𝐹‘𝑘)))
102	101	eleq2d 2855	. 2 ⊢ (𝜑 → ( 0 ∈ ran 𝐹 ↔ 0 ∈ ran (𝑘 ∈ 𝐴 ↦ (𝐹‘𝑘))))
103	97, 100, 102	3bitr4d 314	1 ⊢ (𝜑 → ((𝑀 Σg 𝐹) = 0 ↔ 0 ∈ ran 𝐹))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 400   ∨ wo 860   = wceq 1567   ∈ wcel 2149   ≠ wne 2964  ∃wrex 3095   ∖ cdif 3910   ∪ cun 3911   ⊆ wss 3913  ∅c0 4294  {csn 4591   ↦ cmpt 5193  ran crn 5660  ⟶wf 6529  ‘cfv 6533  (class class class)co 7408  Fincfn 8939  Basecbs 17265  ._rcmulr 17307  0_gc0g 17488   Σ_g cgsu 17489  CMndccmn 19846  mulGrpcmgp 20212  1_rcur 20259  CRingccrg 20312  NzRingcnzr 20591  Domncdomn 20773  IDomncidom 20774

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-rep 5239  ax-sep 5258  ax-nul 5268  ax-pow 5334  ax-pr 5402  ax-un 7730  ax-cnex 11152  ax-resscn 11153  ax-1cn 11154  ax-icn 11155  ax-addcl 11156  ax-addrcl 11157  ax-mulcl 11158  ax-mulrcl 11159  ax-mulcom 11160  ax-addass 11161  ax-mulass 11162  ax-distr 11163  ax-i2m1 11164  ax-1ne0 11165  ax-1rid 11166  ax-rnegex 11167  ax-rrecex 11168  ax-cnre 11169  ax-pre-lttri 11170  ax-pre-lttrn 11171  ax-pre-ltadd 11172  ax-pre-mulgt0 11173

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-nel 3071  df-ral 3086  df-rex 3096  df-rmo 3376  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3933  df-nul 4295  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4874  df-int 4914  df-iun 4959  df-iin 4960  df-br 5111  df-opab 5175  df-mpt 5194  df-tr 5220  df-id 5554  df-eprel 5559  df-po 5567  df-so 5568  df-fr 5612  df-se 5613  df-we 5614  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6299  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6489  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-isom 6542  df-riota 7365  df-ov 7411  df-oprab 7412  df-mpo 7413  df-of 7672  df-om 7859  df-1st 7982  df-2nd 7983  df-supp 8153  df-frecs 8274  df-wrecs 8305  df-recs 8354  df-rdg 8393  df-1o 8449  df-2o 8450  df-er 8690  df-en 8940  df-dom 8941  df-sdom 8942  df-fin 8943  df-fsupp 9318  df-oi 9468  df-card 9921  df-pnf 11241  df-mnf 11242  df-xr 11243  df-ltxr 11244  df-le 11245  df-sub 11439  df-neg 11440  df-nn 12230  df-2 12299  df-n0 12501  df-z 12588  df-uz 12859  df-fz 13532  df-fzo 13679  df-seq 14034  df-hash 14363  df-sets 17220  df-slot 17238  df-ndx 17250  df-base 17266  df-ress 17287  df-plusg 17319  df-0g 17490  df-gsum 17491  df-mre 17634  df-mrc 17635  df-acs 17637  df-mgm 18694  df-sgrp 18773  df-mnd 18789  df-submnd 18838  df-grp 18999  df-minusg 19000  df-mulg 19130  df-cntz 19383  df-cmn 19848  df-abl 19849  df-mgp 20213  df-rng 20227  df-ur 20260  df-ring 20313  df-cring 20314  df-nzr 20592  df-domn 20776  df-idom 20777

This theorem is referenced by:  deg1prod  33814