efgsp1 - Metamath Proof Explorer

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Theorem efgsp1 17973

Description: If 𝐹 is an extension sequence and 𝐴 is an extension of the last element of 𝐹, then 𝐹 + ⟨“𝐴”⟩ is an extension sequence. (Contributed by Mario Carneiro, 1-Oct-2015.)

**Hypotheses**
Ref	Expression
efgval.w	⊢ 𝑊 = ( I ‘Word (𝐼 × 2_𝑜))
efgval.r	⊢ ∼ = ( ~_FG ‘𝐼)
efgval2.m	⊢ 𝑀 = (𝑦 ∈ 𝐼, 𝑧 ∈ 2_𝑜 ↦ ⟨𝑦, (1_𝑜 ∖ 𝑧)⟩)
efgval2.t	⊢ 𝑇 = (𝑣 ∈ 𝑊 ↦ (𝑛 ∈ (0...(#‘𝑣)), 𝑤 ∈ (𝐼 × 2_𝑜) ↦ (𝑣 splice ⟨𝑛, 𝑛, ⟨“𝑤(𝑀‘𝑤)”⟩⟩)))
efgred.d	⊢ 𝐷 = (𝑊 ∖ ∪ 𝑥 ∈ 𝑊 ran (𝑇‘𝑥))
efgred.s	⊢ 𝑆 = (𝑚 ∈ {𝑡 ∈ (Word 𝑊 ∖ {∅}) ∣ ((𝑡‘0) ∈ 𝐷 ∧ ∀𝑘 ∈ (1..^(#‘𝑡))(𝑡‘𝑘) ∈ ran (𝑇‘(𝑡‘(𝑘 − 1))))} ↦ (𝑚‘((#‘𝑚) − 1)))

**Assertion**
Ref	Expression
efgsp1	⊢ ((𝐹 ∈ dom 𝑆 ∧ 𝐴 ∈ ran (𝑇‘(𝑆‘𝐹))) → (𝐹 ++ ⟨“𝐴”⟩) ∈ dom 𝑆)

Distinct variable groups: 𝑦,𝑧 𝑡,𝑛,𝑣,𝑤,𝑦,𝑧,𝑚,𝑥 𝑚,𝑀 𝑥,𝑛,𝑀,𝑡,𝑣,𝑤 𝑘,𝑚,𝑡,𝑥,𝑇 𝑘,𝑛,𝑣,𝑤,𝑦,𝑧,𝑊,𝑚,𝑡,𝑥 ∼ ,𝑚,𝑡,𝑥,𝑦,𝑧 𝑚,𝐼,𝑛,𝑡,𝑣,𝑤,𝑥,𝑦,𝑧 𝐷,𝑚,𝑡 Allowed substitution hints: 𝐴(𝑥,𝑦,𝑧,𝑤,𝑣,𝑡,𝑘,𝑚,𝑛) 𝐷(𝑥,𝑦,𝑧,𝑤,𝑣,𝑘,𝑛) ∼ (𝑤,𝑣,𝑘,𝑛) 𝑆(𝑥,𝑦,𝑧,𝑤,𝑣,𝑡,𝑘,𝑚,𝑛) 𝑇(𝑦,𝑧,𝑤,𝑣,𝑛) 𝐹(𝑥,𝑦,𝑧,𝑤,𝑣,𝑡,𝑘,𝑚,𝑛) 𝐼(𝑘) 𝑀(𝑦,𝑧,𝑘)

Proof of Theorem efgsp1 Dummy variables 𝑎 𝑖 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		efgval.w	. . . . . . . 8 ⊢ 𝑊 = ( I ‘Word (𝐼 × 2_𝑜))
2		efgval.r	. . . . . . . 8 ⊢ ∼ = ( ~_FG ‘𝐼)
3		efgval2.m	. . . . . . . 8 ⊢ 𝑀 = (𝑦 ∈ 𝐼, 𝑧 ∈ 2_𝑜 ↦ ⟨𝑦, (1_𝑜 ∖ 𝑧)⟩)
4		efgval2.t	. . . . . . . 8 ⊢ 𝑇 = (𝑣 ∈ 𝑊 ↦ (𝑛 ∈ (0...(#‘𝑣)), 𝑤 ∈ (𝐼 × 2_𝑜) ↦ (𝑣 splice ⟨𝑛, 𝑛, ⟨“𝑤(𝑀‘𝑤)”⟩⟩)))
5		efgred.d	. . . . . . . 8 ⊢ 𝐷 = (𝑊 ∖ ∪ 𝑥 ∈ 𝑊 ran (𝑇‘𝑥))
6		efgred.s	. . . . . . . 8 ⊢ 𝑆 = (𝑚 ∈ {𝑡 ∈ (Word 𝑊 ∖ {∅}) ∣ ((𝑡‘0) ∈ 𝐷 ∧ ∀𝑘 ∈ (1..^(#‘𝑡))(𝑡‘𝑘) ∈ ran (𝑇‘(𝑡‘(𝑘 − 1))))} ↦ (𝑚‘((#‘𝑚) − 1)))
7	1, 2, 3, 4, 5, 6	efgsdm 17966	. . . . . . 7 ⊢ (𝐹 ∈ dom 𝑆 ↔ (𝐹 ∈ (Word 𝑊 ∖ {∅}) ∧ (𝐹‘0) ∈ 𝐷 ∧ ∀𝑖 ∈ (1..^(#‘𝐹))(𝐹‘𝑖) ∈ ran (𝑇‘(𝐹‘(𝑖 − 1)))))
8	7	simp1bi 1069	. . . . . 6 ⊢ (𝐹 ∈ dom 𝑆 → 𝐹 ∈ (Word 𝑊 ∖ {∅}))
9	8	adantr 480	. . . . 5 ⊢ ((𝐹 ∈ dom 𝑆 ∧ 𝐴 ∈ ran (𝑇‘(𝑆‘𝐹))) → 𝐹 ∈ (Word 𝑊 ∖ {∅}))
10	9	eldifad 3552	. . . 4 ⊢ ((𝐹 ∈ dom 𝑆 ∧ 𝐴 ∈ ran (𝑇‘(𝑆‘𝐹))) → 𝐹 ∈ Word 𝑊)
11	1, 2, 3, 4, 5, 6	efgsf 17965	. . . . . . . . . . . 12 ⊢ 𝑆:{𝑡 ∈ (Word 𝑊 ∖ {∅}) ∣ ((𝑡‘0) ∈ 𝐷 ∧ ∀𝑘 ∈ (1..^(#‘𝑡))(𝑡‘𝑘) ∈ ran (𝑇‘(𝑡‘(𝑘 − 1))))}⟶𝑊
12	11	fdmi 5965	. . . . . . . . . . . . 13 ⊢ dom 𝑆 = {𝑡 ∈ (Word 𝑊 ∖ {∅}) ∣ ((𝑡‘0) ∈ 𝐷 ∧ ∀𝑘 ∈ (1..^(#‘𝑡))(𝑡‘𝑘) ∈ ran (𝑇‘(𝑡‘(𝑘 − 1))))}
13	12	feq2i 5950	. . . . . . . . . . . 12 ⊢ (𝑆:dom 𝑆⟶𝑊 ↔ 𝑆:{𝑡 ∈ (Word 𝑊 ∖ {∅}) ∣ ((𝑡‘0) ∈ 𝐷 ∧ ∀𝑘 ∈ (1..^(#‘𝑡))(𝑡‘𝑘) ∈ ran (𝑇‘(𝑡‘(𝑘 − 1))))}⟶𝑊)
14	11, 13	mpbir 220	. . . . . . . . . . 11 ⊢ 𝑆:dom 𝑆⟶𝑊
15	14	ffvelrni 6266	. . . . . . . . . 10 ⊢ (𝐹 ∈ dom 𝑆 → (𝑆‘𝐹) ∈ 𝑊)
16	15	adantr 480	. . . . . . . . 9 ⊢ ((𝐹 ∈ dom 𝑆 ∧ 𝐴 ∈ ran (𝑇‘(𝑆‘𝐹))) → (𝑆‘𝐹) ∈ 𝑊)
17	1, 2, 3, 4	efgtf 17958	. . . . . . . . 9 ⊢ ((𝑆‘𝐹) ∈ 𝑊 → ((𝑇‘(𝑆‘𝐹)) = (𝑎 ∈ (0...(#‘(𝑆‘𝐹))), 𝑖 ∈ (𝐼 × 2_𝑜) ↦ ((𝑆‘𝐹) splice ⟨𝑎, 𝑎, ⟨“𝑖(𝑀‘𝑖)”⟩⟩)) ∧ (𝑇‘(𝑆‘𝐹)):((0...(#‘(𝑆‘𝐹))) × (𝐼 × 2_𝑜))⟶𝑊))
18	16, 17	syl 17	. . . . . . . 8 ⊢ ((𝐹 ∈ dom 𝑆 ∧ 𝐴 ∈ ran (𝑇‘(𝑆‘𝐹))) → ((𝑇‘(𝑆‘𝐹)) = (𝑎 ∈ (0...(#‘(𝑆‘𝐹))), 𝑖 ∈ (𝐼 × 2_𝑜) ↦ ((𝑆‘𝐹) splice ⟨𝑎, 𝑎, ⟨“𝑖(𝑀‘𝑖)”⟩⟩)) ∧ (𝑇‘(𝑆‘𝐹)):((0...(#‘(𝑆‘𝐹))) × (𝐼 × 2_𝑜))⟶𝑊))
19	18	simprd 478	. . . . . . 7 ⊢ ((𝐹 ∈ dom 𝑆 ∧ 𝐴 ∈ ran (𝑇‘(𝑆‘𝐹))) → (𝑇‘(𝑆‘𝐹)):((0...(#‘(𝑆‘𝐹))) × (𝐼 × 2_𝑜))⟶𝑊)
20		frn 5966	. . . . . . 7 ⊢ ((𝑇‘(𝑆‘𝐹)):((0...(#‘(𝑆‘𝐹))) × (𝐼 × 2_𝑜))⟶𝑊 → ran (𝑇‘(𝑆‘𝐹)) ⊆ 𝑊)
21	19, 20	syl 17	. . . . . 6 ⊢ ((𝐹 ∈ dom 𝑆 ∧ 𝐴 ∈ ran (𝑇‘(𝑆‘𝐹))) → ran (𝑇‘(𝑆‘𝐹)) ⊆ 𝑊)
22		simpr 476	. . . . . 6 ⊢ ((𝐹 ∈ dom 𝑆 ∧ 𝐴 ∈ ran (𝑇‘(𝑆‘𝐹))) → 𝐴 ∈ ran (𝑇‘(𝑆‘𝐹)))
23	21, 22	sseldd 3569	. . . . 5 ⊢ ((𝐹 ∈ dom 𝑆 ∧ 𝐴 ∈ ran (𝑇‘(𝑆‘𝐹))) → 𝐴 ∈ 𝑊)
24	23	s1cld 13236	. . . 4 ⊢ ((𝐹 ∈ dom 𝑆 ∧ 𝐴 ∈ ran (𝑇‘(𝑆‘𝐹))) → ⟨“𝐴”⟩ ∈ Word 𝑊)
25		ccatcl 13212	. . . 4 ⊢ ((𝐹 ∈ Word 𝑊 ∧ ⟨“𝐴”⟩ ∈ Word 𝑊) → (𝐹 ++ ⟨“𝐴”⟩) ∈ Word 𝑊)
26	10, 24, 25	syl2anc 691	. . 3 ⊢ ((𝐹 ∈ dom 𝑆 ∧ 𝐴 ∈ ran (𝑇‘(𝑆‘𝐹))) → (𝐹 ++ ⟨“𝐴”⟩) ∈ Word 𝑊)
27		ccatlen 13213	. . . . . . 7 ⊢ ((𝐹 ∈ Word 𝑊 ∧ ⟨“𝐴”⟩ ∈ Word 𝑊) → (#‘(𝐹 ++ ⟨“𝐴”⟩)) = ((#‘𝐹) + (#‘⟨“𝐴”⟩)))
28	10, 24, 27	syl2anc 691	. . . . . 6 ⊢ ((𝐹 ∈ dom 𝑆 ∧ 𝐴 ∈ ran (𝑇‘(𝑆‘𝐹))) → (#‘(𝐹 ++ ⟨“𝐴”⟩)) = ((#‘𝐹) + (#‘⟨“𝐴”⟩)))
29		s1len 13238	. . . . . . 7 ⊢ (#‘⟨“𝐴”⟩) = 1
30	29	oveq2i 6560	. . . . . 6 ⊢ ((#‘𝐹) + (#‘⟨“𝐴”⟩)) = ((#‘𝐹) + 1)
31	28, 30	syl6eq 2660	. . . . 5 ⊢ ((𝐹 ∈ dom 𝑆 ∧ 𝐴 ∈ ran (𝑇‘(𝑆‘𝐹))) → (#‘(𝐹 ++ ⟨“𝐴”⟩)) = ((#‘𝐹) + 1))
32		lencl 13179	. . . . . 6 ⊢ (𝐹 ∈ Word 𝑊 → (#‘𝐹) ∈ ℕ₀)
33		nn0p1nn 11209	. . . . . 6 ⊢ ((#‘𝐹) ∈ ℕ₀ → ((#‘𝐹) + 1) ∈ ℕ)
34	10, 32, 33	3syl 18	. . . . 5 ⊢ ((𝐹 ∈ dom 𝑆 ∧ 𝐴 ∈ ran (𝑇‘(𝑆‘𝐹))) → ((#‘𝐹) + 1) ∈ ℕ)
35	31, 34	eqeltrd 2688	. . . 4 ⊢ ((𝐹 ∈ dom 𝑆 ∧ 𝐴 ∈ ran (𝑇‘(𝑆‘𝐹))) → (#‘(𝐹 ++ ⟨“𝐴”⟩)) ∈ ℕ)
36		wrdfin 13178	. . . . 5 ⊢ ((𝐹 ++ ⟨“𝐴”⟩) ∈ Word 𝑊 → (𝐹 ++ ⟨“𝐴”⟩) ∈ Fin)
37		hashnncl 13018	. . . . 5 ⊢ ((𝐹 ++ ⟨“𝐴”⟩) ∈ Fin → ((#‘(𝐹 ++ ⟨“𝐴”⟩)) ∈ ℕ ↔ (𝐹 ++ ⟨“𝐴”⟩) ≠ ∅))
38	26, 36, 37	3syl 18	. . . 4 ⊢ ((𝐹 ∈ dom 𝑆 ∧ 𝐴 ∈ ran (𝑇‘(𝑆‘𝐹))) → ((#‘(𝐹 ++ ⟨“𝐴”⟩)) ∈ ℕ ↔ (𝐹 ++ ⟨“𝐴”⟩) ≠ ∅))
39	35, 38	mpbid 221	. . 3 ⊢ ((𝐹 ∈ dom 𝑆 ∧ 𝐴 ∈ ran (𝑇‘(𝑆‘𝐹))) → (𝐹 ++ ⟨“𝐴”⟩) ≠ ∅)
40		eldifsn 4260	. . 3 ⊢ ((𝐹 ++ ⟨“𝐴”⟩) ∈ (Word 𝑊 ∖ {∅}) ↔ ((𝐹 ++ ⟨“𝐴”⟩) ∈ Word 𝑊 ∧ (𝐹 ++ ⟨“𝐴”⟩) ≠ ∅))
41	26, 39, 40	sylanbrc 695	. 2 ⊢ ((𝐹 ∈ dom 𝑆 ∧ 𝐴 ∈ ran (𝑇‘(𝑆‘𝐹))) → (𝐹 ++ ⟨“𝐴”⟩) ∈ (Word 𝑊 ∖ {∅}))
42		eldifsni 4261	. . . . . . 7 ⊢ (𝐹 ∈ (Word 𝑊 ∖ {∅}) → 𝐹 ≠ ∅)
43	9, 42	syl 17	. . . . . 6 ⊢ ((𝐹 ∈ dom 𝑆 ∧ 𝐴 ∈ ran (𝑇‘(𝑆‘𝐹))) → 𝐹 ≠ ∅)
44		wrdfin 13178	. . . . . . 7 ⊢ (𝐹 ∈ Word 𝑊 → 𝐹 ∈ Fin)
45		hashnncl 13018	. . . . . . 7 ⊢ (𝐹 ∈ Fin → ((#‘𝐹) ∈ ℕ ↔ 𝐹 ≠ ∅))
46	10, 44, 45	3syl 18	. . . . . 6 ⊢ ((𝐹 ∈ dom 𝑆 ∧ 𝐴 ∈ ran (𝑇‘(𝑆‘𝐹))) → ((#‘𝐹) ∈ ℕ ↔ 𝐹 ≠ ∅))
47	43, 46	mpbird 246	. . . . 5 ⊢ ((𝐹 ∈ dom 𝑆 ∧ 𝐴 ∈ ran (𝑇‘(𝑆‘𝐹))) → (#‘𝐹) ∈ ℕ)
48		lbfzo0 12375	. . . . 5 ⊢ (0 ∈ (0..^(#‘𝐹)) ↔ (#‘𝐹) ∈ ℕ)
49	47, 48	sylibr 223	. . . 4 ⊢ ((𝐹 ∈ dom 𝑆 ∧ 𝐴 ∈ ran (𝑇‘(𝑆‘𝐹))) → 0 ∈ (0..^(#‘𝐹)))
50		ccatval1 13214	. . . 4 ⊢ ((𝐹 ∈ Word 𝑊 ∧ ⟨“𝐴”⟩ ∈ Word 𝑊 ∧ 0 ∈ (0..^(#‘𝐹))) → ((𝐹 ++ ⟨“𝐴”⟩)‘0) = (𝐹‘0))
51	10, 24, 49, 50	syl3anc 1318	. . 3 ⊢ ((𝐹 ∈ dom 𝑆 ∧ 𝐴 ∈ ran (𝑇‘(𝑆‘𝐹))) → ((𝐹 ++ ⟨“𝐴”⟩)‘0) = (𝐹‘0))
52	7	simp2bi 1070	. . . 4 ⊢ (𝐹 ∈ dom 𝑆 → (𝐹‘0) ∈ 𝐷)
53	52	adantr 480	. . 3 ⊢ ((𝐹 ∈ dom 𝑆 ∧ 𝐴 ∈ ran (𝑇‘(𝑆‘𝐹))) → (𝐹‘0) ∈ 𝐷)
54	51, 53	eqeltrd 2688	. 2 ⊢ ((𝐹 ∈ dom 𝑆 ∧ 𝐴 ∈ ran (𝑇‘(𝑆‘𝐹))) → ((𝐹 ++ ⟨“𝐴”⟩)‘0) ∈ 𝐷)
55	7	simp3bi 1071	. . . . . 6 ⊢ (𝐹 ∈ dom 𝑆 → ∀𝑖 ∈ (1..^(#‘𝐹))(𝐹‘𝑖) ∈ ran (𝑇‘(𝐹‘(𝑖 − 1))))
56	55	adantr 480	. . . . 5 ⊢ ((𝐹 ∈ dom 𝑆 ∧ 𝐴 ∈ ran (𝑇‘(𝑆‘𝐹))) → ∀𝑖 ∈ (1..^(#‘𝐹))(𝐹‘𝑖) ∈ ran (𝑇‘(𝐹‘(𝑖 − 1))))
57		fzo0ss1 12367	. . . . . . . . . . 11 ⊢ (1..^(#‘𝐹)) ⊆ (0..^(#‘𝐹))
58	57	sseli 3564	. . . . . . . . . 10 ⊢ (𝑖 ∈ (1..^(#‘𝐹)) → 𝑖 ∈ (0..^(#‘𝐹)))
59		ccatval1 13214	. . . . . . . . . 10 ⊢ ((𝐹 ∈ Word 𝑊 ∧ ⟨“𝐴”⟩ ∈ Word 𝑊 ∧ 𝑖 ∈ (0..^(#‘𝐹))) → ((𝐹 ++ ⟨“𝐴”⟩)‘𝑖) = (𝐹‘𝑖))
60	58, 59	syl3an3 1353	. . . . . . . . 9 ⊢ ((𝐹 ∈ Word 𝑊 ∧ ⟨“𝐴”⟩ ∈ Word 𝑊 ∧ 𝑖 ∈ (1..^(#‘𝐹))) → ((𝐹 ++ ⟨“𝐴”⟩)‘𝑖) = (𝐹‘𝑖))
61		elfzoel2 12338	. . . . . . . . . . . . . . . 16 ⊢ (𝑖 ∈ (1..^(#‘𝐹)) → (#‘𝐹) ∈ ℤ)
62		peano2zm 11297	. . . . . . . . . . . . . . . 16 ⊢ ((#‘𝐹) ∈ ℤ → ((#‘𝐹) − 1) ∈ ℤ)
63	61, 62	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝑖 ∈ (1..^(#‘𝐹)) → ((#‘𝐹) − 1) ∈ ℤ)
64	61	zred 11358	. . . . . . . . . . . . . . . 16 ⊢ (𝑖 ∈ (1..^(#‘𝐹)) → (#‘𝐹) ∈ ℝ)
65	64	lem1d 10836	. . . . . . . . . . . . . . 15 ⊢ (𝑖 ∈ (1..^(#‘𝐹)) → ((#‘𝐹) − 1) ≤ (#‘𝐹))
66		eluz2 11569	. . . . . . . . . . . . . . 15 ⊢ ((#‘𝐹) ∈ (ℤ_≥‘((#‘𝐹) − 1)) ↔ (((#‘𝐹) − 1) ∈ ℤ ∧ (#‘𝐹) ∈ ℤ ∧ ((#‘𝐹) − 1) ≤ (#‘𝐹)))
67	63, 61, 65, 66	syl3anbrc 1239	. . . . . . . . . . . . . 14 ⊢ (𝑖 ∈ (1..^(#‘𝐹)) → (#‘𝐹) ∈ (ℤ_≥‘((#‘𝐹) − 1)))
68		fzoss2 12365	. . . . . . . . . . . . . 14 ⊢ ((#‘𝐹) ∈ (ℤ_≥‘((#‘𝐹) − 1)) → (0..^((#‘𝐹) − 1)) ⊆ (0..^(#‘𝐹)))
69	67, 68	syl 17	. . . . . . . . . . . . 13 ⊢ (𝑖 ∈ (1..^(#‘𝐹)) → (0..^((#‘𝐹) − 1)) ⊆ (0..^(#‘𝐹)))
70		elfzoelz 12339	. . . . . . . . . . . . . . 15 ⊢ (𝑖 ∈ (1..^(#‘𝐹)) → 𝑖 ∈ ℤ)
71		elfzom1b 12433	. . . . . . . . . . . . . . 15 ⊢ ((𝑖 ∈ ℤ ∧ (#‘𝐹) ∈ ℤ) → (𝑖 ∈ (1..^(#‘𝐹)) ↔ (𝑖 − 1) ∈ (0..^((#‘𝐹) − 1))))
72	70, 61, 71	syl2anc 691	. . . . . . . . . . . . . 14 ⊢ (𝑖 ∈ (1..^(#‘𝐹)) → (𝑖 ∈ (1..^(#‘𝐹)) ↔ (𝑖 − 1) ∈ (0..^((#‘𝐹) − 1))))
73	72	ibi 255	. . . . . . . . . . . . 13 ⊢ (𝑖 ∈ (1..^(#‘𝐹)) → (𝑖 − 1) ∈ (0..^((#‘𝐹) − 1)))
74	69, 73	sseldd 3569	. . . . . . . . . . . 12 ⊢ (𝑖 ∈ (1..^(#‘𝐹)) → (𝑖 − 1) ∈ (0..^(#‘𝐹)))
75		ccatval1 13214	. . . . . . . . . . . 12 ⊢ ((𝐹 ∈ Word 𝑊 ∧ ⟨“𝐴”⟩ ∈ Word 𝑊 ∧ (𝑖 − 1) ∈ (0..^(#‘𝐹))) → ((𝐹 ++ ⟨“𝐴”⟩)‘(𝑖 − 1)) = (𝐹‘(𝑖 − 1)))
76	74, 75	syl3an3 1353	. . . . . . . . . . 11 ⊢ ((𝐹 ∈ Word 𝑊 ∧ ⟨“𝐴”⟩ ∈ Word 𝑊 ∧ 𝑖 ∈ (1..^(#‘𝐹))) → ((𝐹 ++ ⟨“𝐴”⟩)‘(𝑖 − 1)) = (𝐹‘(𝑖 − 1)))
77	76	fveq2d 6107	. . . . . . . . . 10 ⊢ ((𝐹 ∈ Word 𝑊 ∧ ⟨“𝐴”⟩ ∈ Word 𝑊 ∧ 𝑖 ∈ (1..^(#‘𝐹))) → (𝑇‘((𝐹 ++ ⟨“𝐴”⟩)‘(𝑖 − 1))) = (𝑇‘(𝐹‘(𝑖 − 1))))
78	77	rneqd 5274	. . . . . . . . 9 ⊢ ((𝐹 ∈ Word 𝑊 ∧ ⟨“𝐴”⟩ ∈ Word 𝑊 ∧ 𝑖 ∈ (1..^(#‘𝐹))) → ran (𝑇‘((𝐹 ++ ⟨“𝐴”⟩)‘(𝑖 − 1))) = ran (𝑇‘(𝐹‘(𝑖 − 1))))
79	60, 78	eleq12d 2682	. . . . . . . 8 ⊢ ((𝐹 ∈ Word 𝑊 ∧ ⟨“𝐴”⟩ ∈ Word 𝑊 ∧ 𝑖 ∈ (1..^(#‘𝐹))) → (((𝐹 ++ ⟨“𝐴”⟩)‘𝑖) ∈ ran (𝑇‘((𝐹 ++ ⟨“𝐴”⟩)‘(𝑖 − 1))) ↔ (𝐹‘𝑖) ∈ ran (𝑇‘(𝐹‘(𝑖 − 1)))))
80	79	3expa 1257	. . . . . . 7 ⊢ (((𝐹 ∈ Word 𝑊 ∧ ⟨“𝐴”⟩ ∈ Word 𝑊) ∧ 𝑖 ∈ (1..^(#‘𝐹))) → (((𝐹 ++ ⟨“𝐴”⟩)‘𝑖) ∈ ran (𝑇‘((𝐹 ++ ⟨“𝐴”⟩)‘(𝑖 − 1))) ↔ (𝐹‘𝑖) ∈ ran (𝑇‘(𝐹‘(𝑖 − 1)))))
81	80	ralbidva 2968	. . . . . 6 ⊢ ((𝐹 ∈ Word 𝑊 ∧ ⟨“𝐴”⟩ ∈ Word 𝑊) → (∀𝑖 ∈ (1..^(#‘𝐹))((𝐹 ++ ⟨“𝐴”⟩)‘𝑖) ∈ ran (𝑇‘((𝐹 ++ ⟨“𝐴”⟩)‘(𝑖 − 1))) ↔ ∀𝑖 ∈ (1..^(#‘𝐹))(𝐹‘𝑖) ∈ ran (𝑇‘(𝐹‘(𝑖 − 1)))))
82	10, 24, 81	syl2anc 691	. . . . 5 ⊢ ((𝐹 ∈ dom 𝑆 ∧ 𝐴 ∈ ran (𝑇‘(𝑆‘𝐹))) → (∀𝑖 ∈ (1..^(#‘𝐹))((𝐹 ++ ⟨“𝐴”⟩)‘𝑖) ∈ ran (𝑇‘((𝐹 ++ ⟨“𝐴”⟩)‘(𝑖 − 1))) ↔ ∀𝑖 ∈ (1..^(#‘𝐹))(𝐹‘𝑖) ∈ ran (𝑇‘(𝐹‘(𝑖 − 1)))))
83	56, 82	mpbird 246	. . . 4 ⊢ ((𝐹 ∈ dom 𝑆 ∧ 𝐴 ∈ ran (𝑇‘(𝑆‘𝐹))) → ∀𝑖 ∈ (1..^(#‘𝐹))((𝐹 ++ ⟨“𝐴”⟩)‘𝑖) ∈ ran (𝑇‘((𝐹 ++ ⟨“𝐴”⟩)‘(𝑖 − 1))))
84	10, 32	syl 17	. . . . . . . . . 10 ⊢ ((𝐹 ∈ dom 𝑆 ∧ 𝐴 ∈ ran (𝑇‘(𝑆‘𝐹))) → (#‘𝐹) ∈ ℕ₀)
85	84	nn0cnd 11230	. . . . . . . . 9 ⊢ ((𝐹 ∈ dom 𝑆 ∧ 𝐴 ∈ ran (𝑇‘(𝑆‘𝐹))) → (#‘𝐹) ∈ ℂ)
86	85	addid2d 10116	. . . . . . . 8 ⊢ ((𝐹 ∈ dom 𝑆 ∧ 𝐴 ∈ ran (𝑇‘(𝑆‘𝐹))) → (0 + (#‘𝐹)) = (#‘𝐹))
87	86	fveq2d 6107	. . . . . . 7 ⊢ ((𝐹 ∈ dom 𝑆 ∧ 𝐴 ∈ ran (𝑇‘(𝑆‘𝐹))) → ((𝐹 ++ ⟨“𝐴”⟩)‘(0 + (#‘𝐹))) = ((𝐹 ++ ⟨“𝐴”⟩)‘(#‘𝐹)))
88		1nn 10908	. . . . . . . . . . 11 ⊢ 1 ∈ ℕ
89	29, 88	eqeltri 2684	. . . . . . . . . 10 ⊢ (#‘⟨“𝐴”⟩) ∈ ℕ
90	89	a1i 11	. . . . . . . . 9 ⊢ ((𝐹 ∈ dom 𝑆 ∧ 𝐴 ∈ ran (𝑇‘(𝑆‘𝐹))) → (#‘⟨“𝐴”⟩) ∈ ℕ)
91		lbfzo0 12375	. . . . . . . . 9 ⊢ (0 ∈ (0..^(#‘⟨“𝐴”⟩)) ↔ (#‘⟨“𝐴”⟩) ∈ ℕ)
92	90, 91	sylibr 223	. . . . . . . 8 ⊢ ((𝐹 ∈ dom 𝑆 ∧ 𝐴 ∈ ran (𝑇‘(𝑆‘𝐹))) → 0 ∈ (0..^(#‘⟨“𝐴”⟩)))
93		ccatval3 13216	. . . . . . . 8 ⊢ ((𝐹 ∈ Word 𝑊 ∧ ⟨“𝐴”⟩ ∈ Word 𝑊 ∧ 0 ∈ (0..^(#‘⟨“𝐴”⟩))) → ((𝐹 ++ ⟨“𝐴”⟩)‘(0 + (#‘𝐹))) = (⟨“𝐴”⟩‘0))
94	10, 24, 92, 93	syl3anc 1318	. . . . . . 7 ⊢ ((𝐹 ∈ dom 𝑆 ∧ 𝐴 ∈ ran (𝑇‘(𝑆‘𝐹))) → ((𝐹 ++ ⟨“𝐴”⟩)‘(0 + (#‘𝐹))) = (⟨“𝐴”⟩‘0))
95	87, 94	eqtr3d 2646	. . . . . 6 ⊢ ((𝐹 ∈ dom 𝑆 ∧ 𝐴 ∈ ran (𝑇‘(𝑆‘𝐹))) → ((𝐹 ++ ⟨“𝐴”⟩)‘(#‘𝐹)) = (⟨“𝐴”⟩‘0))
96		s1fv 13243	. . . . . . . 8 ⊢ (𝐴 ∈ ran (𝑇‘(𝑆‘𝐹)) → (⟨“𝐴”⟩‘0) = 𝐴)
97	96	adantl 481	. . . . . . 7 ⊢ ((𝐹 ∈ dom 𝑆 ∧ 𝐴 ∈ ran (𝑇‘(𝑆‘𝐹))) → (⟨“𝐴”⟩‘0) = 𝐴)
98		fzo0end 12426	. . . . . . . . . . . 12 ⊢ ((#‘𝐹) ∈ ℕ → ((#‘𝐹) − 1) ∈ (0..^(#‘𝐹)))
99	47, 98	syl 17	. . . . . . . . . . 11 ⊢ ((𝐹 ∈ dom 𝑆 ∧ 𝐴 ∈ ran (𝑇‘(𝑆‘𝐹))) → ((#‘𝐹) − 1) ∈ (0..^(#‘𝐹)))
100		ccatval1 13214	. . . . . . . . . . 11 ⊢ ((𝐹 ∈ Word 𝑊 ∧ ⟨“𝐴”⟩ ∈ Word 𝑊 ∧ ((#‘𝐹) − 1) ∈ (0..^(#‘𝐹))) → ((𝐹 ++ ⟨“𝐴”⟩)‘((#‘𝐹) − 1)) = (𝐹‘((#‘𝐹) − 1)))
101	10, 24, 99, 100	syl3anc 1318	. . . . . . . . . 10 ⊢ ((𝐹 ∈ dom 𝑆 ∧ 𝐴 ∈ ran (𝑇‘(𝑆‘𝐹))) → ((𝐹 ++ ⟨“𝐴”⟩)‘((#‘𝐹) − 1)) = (𝐹‘((#‘𝐹) − 1)))
102	1, 2, 3, 4, 5, 6	efgsval 17967	. . . . . . . . . . 11 ⊢ (𝐹 ∈ dom 𝑆 → (𝑆‘𝐹) = (𝐹‘((#‘𝐹) − 1)))
103	102	adantr 480	. . . . . . . . . 10 ⊢ ((𝐹 ∈ dom 𝑆 ∧ 𝐴 ∈ ran (𝑇‘(𝑆‘𝐹))) → (𝑆‘𝐹) = (𝐹‘((#‘𝐹) − 1)))
104	101, 103	eqtr4d 2647	. . . . . . . . 9 ⊢ ((𝐹 ∈ dom 𝑆 ∧ 𝐴 ∈ ran (𝑇‘(𝑆‘𝐹))) → ((𝐹 ++ ⟨“𝐴”⟩)‘((#‘𝐹) − 1)) = (𝑆‘𝐹))
105	104	fveq2d 6107	. . . . . . . 8 ⊢ ((𝐹 ∈ dom 𝑆 ∧ 𝐴 ∈ ran (𝑇‘(𝑆‘𝐹))) → (𝑇‘((𝐹 ++ ⟨“𝐴”⟩)‘((#‘𝐹) − 1))) = (𝑇‘(𝑆‘𝐹)))
106	105	rneqd 5274	. . . . . . 7 ⊢ ((𝐹 ∈ dom 𝑆 ∧ 𝐴 ∈ ran (𝑇‘(𝑆‘𝐹))) → ran (𝑇‘((𝐹 ++ ⟨“𝐴”⟩)‘((#‘𝐹) − 1))) = ran (𝑇‘(𝑆‘𝐹)))
107	22, 97, 106	3eltr4d 2703	. . . . . 6 ⊢ ((𝐹 ∈ dom 𝑆 ∧ 𝐴 ∈ ran (𝑇‘(𝑆‘𝐹))) → (⟨“𝐴”⟩‘0) ∈ ran (𝑇‘((𝐹 ++ ⟨“𝐴”⟩)‘((#‘𝐹) − 1))))
108	95, 107	eqeltrd 2688	. . . . 5 ⊢ ((𝐹 ∈ dom 𝑆 ∧ 𝐴 ∈ ran (𝑇‘(𝑆‘𝐹))) → ((𝐹 ++ ⟨“𝐴”⟩)‘(#‘𝐹)) ∈ ran (𝑇‘((𝐹 ++ ⟨“𝐴”⟩)‘((#‘𝐹) − 1))))
109		fvex 6113	. . . . . 6 ⊢ (#‘𝐹) ∈ V
110		fveq2 6103	. . . . . . 7 ⊢ (𝑖 = (#‘𝐹) → ((𝐹 ++ ⟨“𝐴”⟩)‘𝑖) = ((𝐹 ++ ⟨“𝐴”⟩)‘(#‘𝐹)))
111		oveq1 6556	. . . . . . . . . 10 ⊢ (𝑖 = (#‘𝐹) → (𝑖 − 1) = ((#‘𝐹) − 1))
112	111	fveq2d 6107	. . . . . . . . 9 ⊢ (𝑖 = (#‘𝐹) → ((𝐹 ++ ⟨“𝐴”⟩)‘(𝑖 − 1)) = ((𝐹 ++ ⟨“𝐴”⟩)‘((#‘𝐹) − 1)))
113	112	fveq2d 6107	. . . . . . . 8 ⊢ (𝑖 = (#‘𝐹) → (𝑇‘((𝐹 ++ ⟨“𝐴”⟩)‘(𝑖 − 1))) = (𝑇‘((𝐹 ++ ⟨“𝐴”⟩)‘((#‘𝐹) − 1))))
114	113	rneqd 5274	. . . . . . 7 ⊢ (𝑖 = (#‘𝐹) → ran (𝑇‘((𝐹 ++ ⟨“𝐴”⟩)‘(𝑖 − 1))) = ran (𝑇‘((𝐹 ++ ⟨“𝐴”⟩)‘((#‘𝐹) − 1))))
115	110, 114	eleq12d 2682	. . . . . 6 ⊢ (𝑖 = (#‘𝐹) → (((𝐹 ++ ⟨“𝐴”⟩)‘𝑖) ∈ ran (𝑇‘((𝐹 ++ ⟨“𝐴”⟩)‘(𝑖 − 1))) ↔ ((𝐹 ++ ⟨“𝐴”⟩)‘(#‘𝐹)) ∈ ran (𝑇‘((𝐹 ++ ⟨“𝐴”⟩)‘((#‘𝐹) − 1)))))
116	109, 115	ralsn 4169	. . . . 5 ⊢ (∀𝑖 ∈ {(#‘𝐹)} ((𝐹 ++ ⟨“𝐴”⟩)‘𝑖) ∈ ran (𝑇‘((𝐹 ++ ⟨“𝐴”⟩)‘(𝑖 − 1))) ↔ ((𝐹 ++ ⟨“𝐴”⟩)‘(#‘𝐹)) ∈ ran (𝑇‘((𝐹 ++ ⟨“𝐴”⟩)‘((#‘𝐹) − 1))))
117	108, 116	sylibr 223	. . . 4 ⊢ ((𝐹 ∈ dom 𝑆 ∧ 𝐴 ∈ ran (𝑇‘(𝑆‘𝐹))) → ∀𝑖 ∈ {(#‘𝐹)} ((𝐹 ++ ⟨“𝐴”⟩)‘𝑖) ∈ ran (𝑇‘((𝐹 ++ ⟨“𝐴”⟩)‘(𝑖 − 1))))
118		ralunb 3756	. . . 4 ⊢ (∀𝑖 ∈ ((1..^(#‘𝐹)) ∪ {(#‘𝐹)})((𝐹 ++ ⟨“𝐴”⟩)‘𝑖) ∈ ran (𝑇‘((𝐹 ++ ⟨“𝐴”⟩)‘(𝑖 − 1))) ↔ (∀𝑖 ∈ (1..^(#‘𝐹))((𝐹 ++ ⟨“𝐴”⟩)‘𝑖) ∈ ran (𝑇‘((𝐹 ++ ⟨“𝐴”⟩)‘(𝑖 − 1))) ∧ ∀𝑖 ∈ {(#‘𝐹)} ((𝐹 ++ ⟨“𝐴”⟩)‘𝑖) ∈ ran (𝑇‘((𝐹 ++ ⟨“𝐴”⟩)‘(𝑖 − 1)))))
119	83, 117, 118	sylanbrc 695	. . 3 ⊢ ((𝐹 ∈ dom 𝑆 ∧ 𝐴 ∈ ran (𝑇‘(𝑆‘𝐹))) → ∀𝑖 ∈ ((1..^(#‘𝐹)) ∪ {(#‘𝐹)})((𝐹 ++ ⟨“𝐴”⟩)‘𝑖) ∈ ran (𝑇‘((𝐹 ++ ⟨“𝐴”⟩)‘(𝑖 − 1))))
120	31	oveq2d 6565	. . . . 5 ⊢ ((𝐹 ∈ dom 𝑆 ∧ 𝐴 ∈ ran (𝑇‘(𝑆‘𝐹))) → (1..^(#‘(𝐹 ++ ⟨“𝐴”⟩))) = (1..^((#‘𝐹) + 1)))
121		nnuz 11599	. . . . . . 7 ⊢ ℕ = (ℤ_≥‘1)
122	47, 121	syl6eleq 2698	. . . . . 6 ⊢ ((𝐹 ∈ dom 𝑆 ∧ 𝐴 ∈ ran (𝑇‘(𝑆‘𝐹))) → (#‘𝐹) ∈ (ℤ_≥‘1))
123		fzosplitsn 12442	. . . . . 6 ⊢ ((#‘𝐹) ∈ (ℤ_≥‘1) → (1..^((#‘𝐹) + 1)) = ((1..^(#‘𝐹)) ∪ {(#‘𝐹)}))
124	122, 123	syl 17	. . . . 5 ⊢ ((𝐹 ∈ dom 𝑆 ∧ 𝐴 ∈ ran (𝑇‘(𝑆‘𝐹))) → (1..^((#‘𝐹) + 1)) = ((1..^(#‘𝐹)) ∪ {(#‘𝐹)}))
125	120, 124	eqtrd 2644	. . . 4 ⊢ ((𝐹 ∈ dom 𝑆 ∧ 𝐴 ∈ ran (𝑇‘(𝑆‘𝐹))) → (1..^(#‘(𝐹 ++ ⟨“𝐴”⟩))) = ((1..^(#‘𝐹)) ∪ {(#‘𝐹)}))
126	125	raleqdv 3121	. . 3 ⊢ ((𝐹 ∈ dom 𝑆 ∧ 𝐴 ∈ ran (𝑇‘(𝑆‘𝐹))) → (∀𝑖 ∈ (1..^(#‘(𝐹 ++ ⟨“𝐴”⟩)))((𝐹 ++ ⟨“𝐴”⟩)‘𝑖) ∈ ran (𝑇‘((𝐹 ++ ⟨“𝐴”⟩)‘(𝑖 − 1))) ↔ ∀𝑖 ∈ ((1..^(#‘𝐹)) ∪ {(#‘𝐹)})((𝐹 ++ ⟨“𝐴”⟩)‘𝑖) ∈ ran (𝑇‘((𝐹 ++ ⟨“𝐴”⟩)‘(𝑖 − 1)))))
127	119, 126	mpbird 246	. 2 ⊢ ((𝐹 ∈ dom 𝑆 ∧ 𝐴 ∈ ran (𝑇‘(𝑆‘𝐹))) → ∀𝑖 ∈ (1..^(#‘(𝐹 ++ ⟨“𝐴”⟩)))((𝐹 ++ ⟨“𝐴”⟩)‘𝑖) ∈ ran (𝑇‘((𝐹 ++ ⟨“𝐴”⟩)‘(𝑖 − 1))))
128	1, 2, 3, 4, 5, 6	efgsdm 17966	. 2 ⊢ ((𝐹 ++ ⟨“𝐴”⟩) ∈ dom 𝑆 ↔ ((𝐹 ++ ⟨“𝐴”⟩) ∈ (Word 𝑊 ∖ {∅}) ∧ ((𝐹 ++ ⟨“𝐴”⟩)‘0) ∈ 𝐷 ∧ ∀𝑖 ∈ (1..^(#‘(𝐹 ++ ⟨“𝐴”⟩)))((𝐹 ++ ⟨“𝐴”⟩)‘𝑖) ∈ ran (𝑇‘((𝐹 ++ ⟨“𝐴”⟩)‘(𝑖 − 1)))))
129	41, 54, 127, 128	syl3anbrc 1239	1 ⊢ ((𝐹 ∈ dom 𝑆 ∧ 𝐴 ∈ ran (𝑇‘(𝑆‘𝐹))) → (𝐹 ++ ⟨“𝐴”⟩) ∈ dom 𝑆)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 ∀wral 2896 {crab 2900 ∖ cdif 3537 ∪ cun 3538 ⊆ wss 3540 ∅c0 3874 {csn 4125 ⟨cop 4131 ⟨cotp 4133 ∪ ciun 4455 class class class wbr 4583 ↦ cmpt 4643 I cid 4948 × cxp 5036 dom cdm 5038 ran crn 5039 ⟶wf 5800 ‘cfv 5804 (class class class)co 6549 ↦ cmpt2 6551 1_𝑜c1o 7440 2_𝑜c2o 7441 Fincfn 7841 0cc0 9815 1c1 9816 + caddc 9818 ≤ cle 9954 − cmin 10145 ℕcn 10897 ℕ₀cn0 11169 ℤcz 11254 ℤ_≥cuz 11563 ...cfz 12197 ..^cfzo 12334 #chash 12979 Word cword 13146 ++ cconcat 13148 ⟨“cs1 13149 splice csplice 13151 ⟨“cs2 13437 ~_FG cefg 17942

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-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-ot 4134 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-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-2o 7448 df-oadd 7451 df-er 7629 df-map 7746 df-pm 7747 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-hash 12980 df-word 13154 df-concat 13156 df-s1 13157 df-substr 13158 df-splice 13159 df-s2 13444

This theorem is referenced by: efgsfo 17975 efgredlemd 17980 efgrelexlemb 17986

W3C validator