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Theorem gsumval3lem2 18130

Description: Lemma 2 for gsumval3 18131. (Contributed by AV, 31-May-2019.)

**Hypotheses**
Ref	Expression
gsumval3.b	⊢ 𝐵 = (Base‘𝐺)
gsumval3.0	⊢ 0 = (0_g‘𝐺)
gsumval3.p	⊢ + = (+_g‘𝐺)
gsumval3.z	⊢ 𝑍 = (Cntz‘𝐺)
gsumval3.g	⊢ (𝜑 → 𝐺 ∈ Mnd)
gsumval3.a	⊢ (𝜑 → 𝐴 ∈ 𝑉)
gsumval3.f	⊢ (𝜑 → 𝐹:𝐴⟶𝐵)
gsumval3.c	⊢ (𝜑 → ran 𝐹 ⊆ (𝑍‘ran 𝐹))
gsumval3.m	⊢ (𝜑 → 𝑀 ∈ ℕ)
gsumval3.h	⊢ (𝜑 → 𝐻:(1...𝑀)–1-1→𝐴)
gsumval3.n	⊢ (𝜑 → (𝐹 supp 0 ) ⊆ ran 𝐻)
gsumval3.w	⊢ 𝑊 = ((𝐹 ∘ 𝐻) supp 0 )

**Assertion**
Ref	Expression
gsumval3lem2	⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ (¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(#‘𝑊)), 𝑊))) → (𝐺 Σ_g 𝐹) = (seq1( + , (𝐹 ∘ (𝐻 ∘ 𝑓)))‘(#‘𝑊)))

Distinct variable groups: + ,𝑓 𝐴,𝑓 𝜑,𝑓 𝑓,𝐺 𝑓,𝑀 𝐵,𝑓 𝑓,𝐹 𝑓,𝐻 𝑓,𝑊 Allowed substitution hints: 𝑉(𝑓) 0 (𝑓) 𝑍(𝑓)

Proof of Theorem gsumval3lem2 Dummy variables 𝑔 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		gsumval3.h	. . . . . . 7 ⊢ (𝜑 → 𝐻:(1...𝑀)–1-1→𝐴)
2		f1f 6014	. . . . . . 7 ⊢ (𝐻:(1...𝑀)–1-1→𝐴 → 𝐻:(1...𝑀)⟶𝐴)
3	1, 2	syl 17	. . . . . 6 ⊢ (𝜑 → 𝐻:(1...𝑀)⟶𝐴)
4		fzfid 12634	. . . . . 6 ⊢ (𝜑 → (1...𝑀) ∈ Fin)
5		gsumval3.a	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
6		fex2 7014	. . . . . 6 ⊢ ((𝐻:(1...𝑀)⟶𝐴 ∧ (1...𝑀) ∈ Fin ∧ 𝐴 ∈ 𝑉) → 𝐻 ∈ V)
7	3, 4, 5, 6	syl3anc 1318	. . . . 5 ⊢ (𝜑 → 𝐻 ∈ V)
8		vex 3176	. . . . 5 ⊢ 𝑓 ∈ V
9		coexg 7010	. . . . 5 ⊢ ((𝐻 ∈ V ∧ 𝑓 ∈ V) → (𝐻 ∘ 𝑓) ∈ V)
10	7, 8, 9	sylancl 693	. . . 4 ⊢ (𝜑 → (𝐻 ∘ 𝑓) ∈ V)
11	10	ad2antrr 758	. . 3 ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ (¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(#‘𝑊)), 𝑊))) → (𝐻 ∘ 𝑓) ∈ V)
12		gsumval3.b	. . . . 5 ⊢ 𝐵 = (Base‘𝐺)
13		gsumval3.0	. . . . 5 ⊢ 0 = (0_g‘𝐺)
14		gsumval3.p	. . . . 5 ⊢ + = (+_g‘𝐺)
15		gsumval3.z	. . . . 5 ⊢ 𝑍 = (Cntz‘𝐺)
16		gsumval3.g	. . . . 5 ⊢ (𝜑 → 𝐺 ∈ Mnd)
17		gsumval3.f	. . . . 5 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)
18		gsumval3.c	. . . . 5 ⊢ (𝜑 → ran 𝐹 ⊆ (𝑍‘ran 𝐹))
19		gsumval3.m	. . . . 5 ⊢ (𝜑 → 𝑀 ∈ ℕ)
20		gsumval3.n	. . . . 5 ⊢ (𝜑 → (𝐹 supp 0 ) ⊆ ran 𝐻)
21		gsumval3.w	. . . . 5 ⊢ 𝑊 = ((𝐹 ∘ 𝐻) supp 0 )
22	12, 13, 14, 15, 16, 5, 17, 18, 19, 1, 20, 21	gsumval3lem1 18129	. . . 4 ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ (¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(#‘𝑊)), 𝑊))) → (𝐻 ∘ 𝑓):(1...(#‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))
23		resexg 5362	. . . . . . . . 9 ⊢ (𝐻 ∈ V → (𝐻 ↾ 𝑊) ∈ V)
24	7, 23	syl 17	. . . . . . . 8 ⊢ (𝜑 → (𝐻 ↾ 𝑊) ∈ V)
25	24	ad2antrr 758	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ (¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(#‘𝑊)), 𝑊))) → (𝐻 ↾ 𝑊) ∈ V)
26	1	ad2antrr 758	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ (¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(#‘𝑊)), 𝑊))) → 𝐻:(1...𝑀)–1-1→𝐴)
27		suppssdm 7195	. . . . . . . . . . . 12 ⊢ ((𝐹 ∘ 𝐻) supp 0 ) ⊆ dom (𝐹 ∘ 𝐻)
28	21, 27	eqsstri 3598	. . . . . . . . . . 11 ⊢ 𝑊 ⊆ dom (𝐹 ∘ 𝐻)
29		fco 5971	. . . . . . . . . . . . 13 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐻:(1...𝑀)⟶𝐴) → (𝐹 ∘ 𝐻):(1...𝑀)⟶𝐵)
30	17, 3, 29	syl2anc 691	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐹 ∘ 𝐻):(1...𝑀)⟶𝐵)
31		fdm 5964	. . . . . . . . . . . 12 ⊢ ((𝐹 ∘ 𝐻):(1...𝑀)⟶𝐵 → dom (𝐹 ∘ 𝐻) = (1...𝑀))
32	30, 31	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → dom (𝐹 ∘ 𝐻) = (1...𝑀))
33	28, 32	syl5sseq 3616	. . . . . . . . . 10 ⊢ (𝜑 → 𝑊 ⊆ (1...𝑀))
34	33	ad2antrr 758	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ (¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(#‘𝑊)), 𝑊))) → 𝑊 ⊆ (1...𝑀))
35		f1ores 6064	. . . . . . . . 9 ⊢ ((𝐻:(1...𝑀)–1-1→𝐴 ∧ 𝑊 ⊆ (1...𝑀)) → (𝐻 ↾ 𝑊):𝑊–1-1-onto→(𝐻 “ 𝑊))
36	26, 34, 35	syl2anc 691	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ (¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(#‘𝑊)), 𝑊))) → (𝐻 ↾ 𝑊):𝑊–1-1-onto→(𝐻 “ 𝑊))
37	21	imaeq2i 5383	. . . . . . . . . . 11 ⊢ (𝐻 “ 𝑊) = (𝐻 “ ((𝐹 ∘ 𝐻) supp 0 ))
38		fex 6394	. . . . . . . . . . . . . . 15 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐴 ∈ 𝑉) → 𝐹 ∈ V)
39	17, 5, 38	syl2anc 691	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐹 ∈ V)
40		ovex 6577	. . . . . . . . . . . . . . 15 ⊢ (1...𝑀) ∈ V
41		fex 6394	. . . . . . . . . . . . . . 15 ⊢ ((𝐻:(1...𝑀)⟶𝐴 ∧ (1...𝑀) ∈ V) → 𝐻 ∈ V)
42	3, 40, 41	sylancl 693	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐻 ∈ V)
43	39, 42	jca 553	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐹 ∈ V ∧ 𝐻 ∈ V))
44		f1fun 6016	. . . . . . . . . . . . . . 15 ⊢ (𝐻:(1...𝑀)–1-1→𝐴 → Fun 𝐻)
45	1, 44	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → Fun 𝐻)
46	45, 20	jca 553	. . . . . . . . . . . . 13 ⊢ (𝜑 → (Fun 𝐻 ∧ (𝐹 supp 0 ) ⊆ ran 𝐻))
47		imacosupp 7222	. . . . . . . . . . . . 13 ⊢ ((𝐹 ∈ V ∧ 𝐻 ∈ V) → ((Fun 𝐻 ∧ (𝐹 supp 0 ) ⊆ ran 𝐻) → (𝐻 “ ((𝐹 ∘ 𝐻) supp 0 )) = (𝐹 supp 0 )))
48	43, 46, 47	sylc 63	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐻 “ ((𝐹 ∘ 𝐻) supp 0 )) = (𝐹 supp 0 ))
49	48	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑊 ≠ ∅) → (𝐻 “ ((𝐹 ∘ 𝐻) supp 0 )) = (𝐹 supp 0 ))
50	37, 49	syl5eq 2656	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑊 ≠ ∅) → (𝐻 “ 𝑊) = (𝐹 supp 0 ))
51	50	adantr 480	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ (¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(#‘𝑊)), 𝑊))) → (𝐻 “ 𝑊) = (𝐹 supp 0 ))
52		f1oeq3 6042	. . . . . . . . 9 ⊢ ((𝐻 “ 𝑊) = (𝐹 supp 0 ) → ((𝐻 ↾ 𝑊):𝑊–1-1-onto→(𝐻 “ 𝑊) ↔ (𝐻 ↾ 𝑊):𝑊–1-1-onto→(𝐹 supp 0 )))
53	51, 52	syl 17	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ (¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(#‘𝑊)), 𝑊))) → ((𝐻 ↾ 𝑊):𝑊–1-1-onto→(𝐻 “ 𝑊) ↔ (𝐻 ↾ 𝑊):𝑊–1-1-onto→(𝐹 supp 0 )))
54	36, 53	mpbid 221	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ (¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(#‘𝑊)), 𝑊))) → (𝐻 ↾ 𝑊):𝑊–1-1-onto→(𝐹 supp 0 ))
55		f1oen3g 7857	. . . . . . 7 ⊢ (((𝐻 ↾ 𝑊) ∈ V ∧ (𝐻 ↾ 𝑊):𝑊–1-1-onto→(𝐹 supp 0 )) → 𝑊 ≈ (𝐹 supp 0 ))
56	25, 54, 55	syl2anc 691	. . . . . 6 ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ (¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(#‘𝑊)), 𝑊))) → 𝑊 ≈ (𝐹 supp 0 ))
57		fzfi 12633	. . . . . . . . 9 ⊢ (1...𝑀) ∈ Fin
58		ssfi 8065	. . . . . . . . 9 ⊢ (((1...𝑀) ∈ Fin ∧ 𝑊 ⊆ (1...𝑀)) → 𝑊 ∈ Fin)
59	57, 33, 58	sylancr 694	. . . . . . . 8 ⊢ (𝜑 → 𝑊 ∈ Fin)
60	59	ad2antrr 758	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ (¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(#‘𝑊)), 𝑊))) → 𝑊 ∈ Fin)
61		f1f1orn 6061	. . . . . . . . . . . . 13 ⊢ (𝐻:(1...𝑀)–1-1→𝐴 → 𝐻:(1...𝑀)–1-1-onto→ran 𝐻)
62	1, 61	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐻:(1...𝑀)–1-1-onto→ran 𝐻)
63		f1oen3g 7857	. . . . . . . . . . . 12 ⊢ ((𝐻 ∈ V ∧ 𝐻:(1...𝑀)–1-1-onto→ran 𝐻) → (1...𝑀) ≈ ran 𝐻)
64	7, 62, 63	syl2anc 691	. . . . . . . . . . 11 ⊢ (𝜑 → (1...𝑀) ≈ ran 𝐻)
65		enfi 8061	. . . . . . . . . . 11 ⊢ ((1...𝑀) ≈ ran 𝐻 → ((1...𝑀) ∈ Fin ↔ ran 𝐻 ∈ Fin))
66	64, 65	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → ((1...𝑀) ∈ Fin ↔ ran 𝐻 ∈ Fin))
67	57, 66	mpbii 222	. . . . . . . . 9 ⊢ (𝜑 → ran 𝐻 ∈ Fin)
68		ssfi 8065	. . . . . . . . 9 ⊢ ((ran 𝐻 ∈ Fin ∧ (𝐹 supp 0 ) ⊆ ran 𝐻) → (𝐹 supp 0 ) ∈ Fin)
69	67, 20, 68	syl2anc 691	. . . . . . . 8 ⊢ (𝜑 → (𝐹 supp 0 ) ∈ Fin)
70	69	ad2antrr 758	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ (¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(#‘𝑊)), 𝑊))) → (𝐹 supp 0 ) ∈ Fin)
71		hashen 12997	. . . . . . 7 ⊢ ((𝑊 ∈ Fin ∧ (𝐹 supp 0 ) ∈ Fin) → ((#‘𝑊) = (#‘(𝐹 supp 0 )) ↔ 𝑊 ≈ (𝐹 supp 0 )))
72	60, 70, 71	syl2anc 691	. . . . . 6 ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ (¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(#‘𝑊)), 𝑊))) → ((#‘𝑊) = (#‘(𝐹 supp 0 )) ↔ 𝑊 ≈ (𝐹 supp 0 )))
73	56, 72	mpbird 246	. . . . 5 ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ (¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(#‘𝑊)), 𝑊))) → (#‘𝑊) = (#‘(𝐹 supp 0 )))
74	73	fveq2d 6107	. . . 4 ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ (¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(#‘𝑊)), 𝑊))) → (seq1( + , (𝐹 ∘ (𝐻 ∘ 𝑓)))‘(#‘𝑊)) = (seq1( + , (𝐹 ∘ (𝐻 ∘ 𝑓)))‘(#‘(𝐹 supp 0 ))))
75	22, 74	jca 553	. . 3 ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ (¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(#‘𝑊)), 𝑊))) → ((𝐻 ∘ 𝑓):(1...(#‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ) ∧ (seq1( + , (𝐹 ∘ (𝐻 ∘ 𝑓)))‘(#‘𝑊)) = (seq1( + , (𝐹 ∘ (𝐻 ∘ 𝑓)))‘(#‘(𝐹 supp 0 )))))
76		f1oeq1 6040	. . . . 5 ⊢ (𝑔 = (𝐻 ∘ 𝑓) → (𝑔:(1...(#‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ) ↔ (𝐻 ∘ 𝑓):(1...(#‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 )))
77		coeq2 5202	. . . . . . . 8 ⊢ (𝑔 = (𝐻 ∘ 𝑓) → (𝐹 ∘ 𝑔) = (𝐹 ∘ (𝐻 ∘ 𝑓)))
78	77	seqeq3d 12671	. . . . . . 7 ⊢ (𝑔 = (𝐻 ∘ 𝑓) → seq1( + , (𝐹 ∘ 𝑔)) = seq1( + , (𝐹 ∘ (𝐻 ∘ 𝑓))))
79	78	fveq1d 6105	. . . . . 6 ⊢ (𝑔 = (𝐻 ∘ 𝑓) → (seq1( + , (𝐹 ∘ 𝑔))‘(#‘(𝐹 supp 0 ))) = (seq1( + , (𝐹 ∘ (𝐻 ∘ 𝑓)))‘(#‘(𝐹 supp 0 ))))
80	79	eqeq2d 2620	. . . . 5 ⊢ (𝑔 = (𝐻 ∘ 𝑓) → ((seq1( + , (𝐹 ∘ (𝐻 ∘ 𝑓)))‘(#‘𝑊)) = (seq1( + , (𝐹 ∘ 𝑔))‘(#‘(𝐹 supp 0 ))) ↔ (seq1( + , (𝐹 ∘ (𝐻 ∘ 𝑓)))‘(#‘𝑊)) = (seq1( + , (𝐹 ∘ (𝐻 ∘ 𝑓)))‘(#‘(𝐹 supp 0 )))))
81	76, 80	anbi12d 743	. . . 4 ⊢ (𝑔 = (𝐻 ∘ 𝑓) → ((𝑔:(1...(#‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ) ∧ (seq1( + , (𝐹 ∘ (𝐻 ∘ 𝑓)))‘(#‘𝑊)) = (seq1( + , (𝐹 ∘ 𝑔))‘(#‘(𝐹 supp 0 )))) ↔ ((𝐻 ∘ 𝑓):(1...(#‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ) ∧ (seq1( + , (𝐹 ∘ (𝐻 ∘ 𝑓)))‘(#‘𝑊)) = (seq1( + , (𝐹 ∘ (𝐻 ∘ 𝑓)))‘(#‘(𝐹 supp 0 ))))))
82	81	spcegv 3267	. . 3 ⊢ ((𝐻 ∘ 𝑓) ∈ V → (((𝐻 ∘ 𝑓):(1...(#‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ) ∧ (seq1( + , (𝐹 ∘ (𝐻 ∘ 𝑓)))‘(#‘𝑊)) = (seq1( + , (𝐹 ∘ (𝐻 ∘ 𝑓)))‘(#‘(𝐹 supp 0 )))) → ∃𝑔(𝑔:(1...(#‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ) ∧ (seq1( + , (𝐹 ∘ (𝐻 ∘ 𝑓)))‘(#‘𝑊)) = (seq1( + , (𝐹 ∘ 𝑔))‘(#‘(𝐹 supp 0 ))))))
83	11, 75, 82	sylc 63	. 2 ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ (¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(#‘𝑊)), 𝑊))) → ∃𝑔(𝑔:(1...(#‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ) ∧ (seq1( + , (𝐹 ∘ (𝐻 ∘ 𝑓)))‘(#‘𝑊)) = (seq1( + , (𝐹 ∘ 𝑔))‘(#‘(𝐹 supp 0 )))))
84	16	ad2antrr 758	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ (¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(#‘𝑊)), 𝑊))) → 𝐺 ∈ Mnd)
85	5	ad2antrr 758	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ (¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(#‘𝑊)), 𝑊))) → 𝐴 ∈ 𝑉)
86	17	ad2antrr 758	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ (¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(#‘𝑊)), 𝑊))) → 𝐹:𝐴⟶𝐵)
87	18	ad2antrr 758	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ (¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(#‘𝑊)), 𝑊))) → ran 𝐹 ⊆ (𝑍‘ran 𝐹))
88	21	neeq1i 2846	. . . . . . . . . 10 ⊢ (𝑊 ≠ ∅ ↔ ((𝐹 ∘ 𝐻) supp 0 ) ≠ ∅)
89		supp0cosupp0 7221	. . . . . . . . . . . 12 ⊢ ((𝐹 ∈ V ∧ 𝐻 ∈ V) → ((𝐹 supp 0 ) = ∅ → ((𝐹 ∘ 𝐻) supp 0 ) = ∅))
90	89	necon3d 2803	. . . . . . . . . . 11 ⊢ ((𝐹 ∈ V ∧ 𝐻 ∈ V) → (((𝐹 ∘ 𝐻) supp 0 ) ≠ ∅ → (𝐹 supp 0 ) ≠ ∅))
91	39, 42, 90	syl2anc 691	. . . . . . . . . 10 ⊢ (𝜑 → (((𝐹 ∘ 𝐻) supp 0 ) ≠ ∅ → (𝐹 supp 0 ) ≠ ∅))
92	88, 91	syl5bi 231	. . . . . . . . 9 ⊢ (𝜑 → (𝑊 ≠ ∅ → (𝐹 supp 0 ) ≠ ∅))
93	92	imp 444	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑊 ≠ ∅) → (𝐹 supp 0 ) ≠ ∅)
94	93	adantr 480	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ (¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(#‘𝑊)), 𝑊))) → (𝐹 supp 0 ) ≠ ∅)
95	20	ad2antrr 758	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ (¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(#‘𝑊)), 𝑊))) → (𝐹 supp 0 ) ⊆ ran 𝐻)
96		frn 5966	. . . . . . . . . 10 ⊢ (𝐻:(1...𝑀)⟶𝐴 → ran 𝐻 ⊆ 𝐴)
97	3, 96	syl 17	. . . . . . . . 9 ⊢ (𝜑 → ran 𝐻 ⊆ 𝐴)
98	97	ad2antrr 758	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ (¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(#‘𝑊)), 𝑊))) → ran 𝐻 ⊆ 𝐴)
99	95, 98	sstrd 3578	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ (¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(#‘𝑊)), 𝑊))) → (𝐹 supp 0 ) ⊆ 𝐴)
100	12, 13, 14, 15, 84, 85, 86, 87, 70, 94, 99	gsumval3eu 18128	. . . . . 6 ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ (¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(#‘𝑊)), 𝑊))) → ∃!𝑥∃𝑔(𝑔:(1...(#‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ) ∧ 𝑥 = (seq1( + , (𝐹 ∘ 𝑔))‘(#‘(𝐹 supp 0 )))))
101		iota1 5782	. . . . . 6 ⊢ (∃!𝑥∃𝑔(𝑔:(1...(#‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ) ∧ 𝑥 = (seq1( + , (𝐹 ∘ 𝑔))‘(#‘(𝐹 supp 0 )))) → (∃𝑔(𝑔:(1...(#‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ) ∧ 𝑥 = (seq1( + , (𝐹 ∘ 𝑔))‘(#‘(𝐹 supp 0 )))) ↔ (℩𝑥∃𝑔(𝑔:(1...(#‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ) ∧ 𝑥 = (seq1( + , (𝐹 ∘ 𝑔))‘(#‘(𝐹 supp 0 ))))) = 𝑥))
102	100, 101	syl 17	. . . . 5 ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ (¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(#‘𝑊)), 𝑊))) → (∃𝑔(𝑔:(1...(#‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ) ∧ 𝑥 = (seq1( + , (𝐹 ∘ 𝑔))‘(#‘(𝐹 supp 0 )))) ↔ (℩𝑥∃𝑔(𝑔:(1...(#‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ) ∧ 𝑥 = (seq1( + , (𝐹 ∘ 𝑔))‘(#‘(𝐹 supp 0 ))))) = 𝑥))
103		eqid 2610	. . . . . . 7 ⊢ (𝐹 supp 0 ) = (𝐹 supp 0 )
104		simprl 790	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ (¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(#‘𝑊)), 𝑊))) → ¬ 𝐴 ∈ ran ...)
105	12, 13, 14, 15, 84, 85, 86, 87, 70, 94, 103, 104	gsumval3a 18127	. . . . . 6 ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ (¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(#‘𝑊)), 𝑊))) → (𝐺 Σ_g 𝐹) = (℩𝑥∃𝑔(𝑔:(1...(#‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ) ∧ 𝑥 = (seq1( + , (𝐹 ∘ 𝑔))‘(#‘(𝐹 supp 0 ))))))
106	105	eqeq1d 2612	. . . . 5 ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ (¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(#‘𝑊)), 𝑊))) → ((𝐺 Σ_g 𝐹) = 𝑥 ↔ (℩𝑥∃𝑔(𝑔:(1...(#‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ) ∧ 𝑥 = (seq1( + , (𝐹 ∘ 𝑔))‘(#‘(𝐹 supp 0 ))))) = 𝑥))
107	102, 106	bitr4d 270	. . . 4 ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ (¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(#‘𝑊)), 𝑊))) → (∃𝑔(𝑔:(1...(#‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ) ∧ 𝑥 = (seq1( + , (𝐹 ∘ 𝑔))‘(#‘(𝐹 supp 0 )))) ↔ (𝐺 Σ_g 𝐹) = 𝑥))
108	107	alrimiv 1842	. . 3 ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ (¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(#‘𝑊)), 𝑊))) → ∀𝑥(∃𝑔(𝑔:(1...(#‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ) ∧ 𝑥 = (seq1( + , (𝐹 ∘ 𝑔))‘(#‘(𝐹 supp 0 )))) ↔ (𝐺 Σ_g 𝐹) = 𝑥))
109		fvex 6113	. . . 4 ⊢ (seq1( + , (𝐹 ∘ (𝐻 ∘ 𝑓)))‘(#‘𝑊)) ∈ V
110		eqeq1 2614	. . . . . . 7 ⊢ (𝑥 = (seq1( + , (𝐹 ∘ (𝐻 ∘ 𝑓)))‘(#‘𝑊)) → (𝑥 = (seq1( + , (𝐹 ∘ 𝑔))‘(#‘(𝐹 supp 0 ))) ↔ (seq1( + , (𝐹 ∘ (𝐻 ∘ 𝑓)))‘(#‘𝑊)) = (seq1( + , (𝐹 ∘ 𝑔))‘(#‘(𝐹 supp 0 )))))
111	110	anbi2d 736	. . . . . 6 ⊢ (𝑥 = (seq1( + , (𝐹 ∘ (𝐻 ∘ 𝑓)))‘(#‘𝑊)) → ((𝑔:(1...(#‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ) ∧ 𝑥 = (seq1( + , (𝐹 ∘ 𝑔))‘(#‘(𝐹 supp 0 )))) ↔ (𝑔:(1...(#‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ) ∧ (seq1( + , (𝐹 ∘ (𝐻 ∘ 𝑓)))‘(#‘𝑊)) = (seq1( + , (𝐹 ∘ 𝑔))‘(#‘(𝐹 supp 0 ))))))
112	111	exbidv 1837	. . . . 5 ⊢ (𝑥 = (seq1( + , (𝐹 ∘ (𝐻 ∘ 𝑓)))‘(#‘𝑊)) → (∃𝑔(𝑔:(1...(#‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ) ∧ 𝑥 = (seq1( + , (𝐹 ∘ 𝑔))‘(#‘(𝐹 supp 0 )))) ↔ ∃𝑔(𝑔:(1...(#‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ) ∧ (seq1( + , (𝐹 ∘ (𝐻 ∘ 𝑓)))‘(#‘𝑊)) = (seq1( + , (𝐹 ∘ 𝑔))‘(#‘(𝐹 supp 0 ))))))
113		eqeq2 2621	. . . . 5 ⊢ (𝑥 = (seq1( + , (𝐹 ∘ (𝐻 ∘ 𝑓)))‘(#‘𝑊)) → ((𝐺 Σ_g 𝐹) = 𝑥 ↔ (𝐺 Σ_g 𝐹) = (seq1( + , (𝐹 ∘ (𝐻 ∘ 𝑓)))‘(#‘𝑊))))
114	112, 113	bibi12d 334	. . . 4 ⊢ (𝑥 = (seq1( + , (𝐹 ∘ (𝐻 ∘ 𝑓)))‘(#‘𝑊)) → ((∃𝑔(𝑔:(1...(#‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ) ∧ 𝑥 = (seq1( + , (𝐹 ∘ 𝑔))‘(#‘(𝐹 supp 0 )))) ↔ (𝐺 Σ_g 𝐹) = 𝑥) ↔ (∃𝑔(𝑔:(1...(#‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ) ∧ (seq1( + , (𝐹 ∘ (𝐻 ∘ 𝑓)))‘(#‘𝑊)) = (seq1( + , (𝐹 ∘ 𝑔))‘(#‘(𝐹 supp 0 )))) ↔ (𝐺 Σ_g 𝐹) = (seq1( + , (𝐹 ∘ (𝐻 ∘ 𝑓)))‘(#‘𝑊)))))
115	109, 114	spcv 3272	. . 3 ⊢ (∀𝑥(∃𝑔(𝑔:(1...(#‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ) ∧ 𝑥 = (seq1( + , (𝐹 ∘ 𝑔))‘(#‘(𝐹 supp 0 )))) ↔ (𝐺 Σ_g 𝐹) = 𝑥) → (∃𝑔(𝑔:(1...(#‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ) ∧ (seq1( + , (𝐹 ∘ (𝐻 ∘ 𝑓)))‘(#‘𝑊)) = (seq1( + , (𝐹 ∘ 𝑔))‘(#‘(𝐹 supp 0 )))) ↔ (𝐺 Σ_g 𝐹) = (seq1( + , (𝐹 ∘ (𝐻 ∘ 𝑓)))‘(#‘𝑊))))
116	108, 115	syl 17	. 2 ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ (¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(#‘𝑊)), 𝑊))) → (∃𝑔(𝑔:(1...(#‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ) ∧ (seq1( + , (𝐹 ∘ (𝐻 ∘ 𝑓)))‘(#‘𝑊)) = (seq1( + , (𝐹 ∘ 𝑔))‘(#‘(𝐹 supp 0 )))) ↔ (𝐺 Σ_g 𝐹) = (seq1( + , (𝐹 ∘ (𝐻 ∘ 𝑓)))‘(#‘𝑊))))
117	83, 116	mpbid 221	1 ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ (¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(#‘𝑊)), 𝑊))) → (𝐺 Σ_g 𝐹) = (seq1( + , (𝐹 ∘ (𝐻 ∘ 𝑓)))‘(#‘𝑊)))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 195 ∧ wa 383 ∀wal 1473 = wceq 1475 ∃wex 1695 ∈ wcel 1977 ∃!weu 2458 ≠ wne 2780 Vcvv 3173 ⊆ wss 3540 ∅c0 3874 class class class wbr 4583 dom cdm 5038 ran crn 5039 ↾ cres 5040 “ cima 5041 ∘ ccom 5042 ℩cio 5766 Fun wfun 5798 ⟶wf 5800 –1-1→wf1 5801 –1-1-onto→wf1o 5803 ‘cfv 5804 Isom wiso 5805 (class class class)co 6549 supp csupp 7182 ≈ cen 7838 Fincfn 7841 1c1 9816 < clt 9953 ℕcn 10897 ...cfz 12197 seqcseq 12663 #chash 12979 Basecbs 15695 +_gcplusg 15768 0_gc0g 15923 Σ_g cgsu 15924 Mndcmnd 17117 Cntzccntz 17571

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892

This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-isom 5813 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-supp 7183 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-oadd 7451 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-card 8648 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-n0 11170 df-z 11255 df-uz 11564 df-fz 12198 df-fzo 12335 df-seq 12664 df-hash 12980 df-0g 15925 df-gsum 15926 df-mgm 17065 df-sgrp 17107 df-mnd 17118 df-cntz 17573

This theorem is referenced by: gsumval3 18131

W3C validator