efgsp1 - Metamath Proof Explorer

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Theorem efgsp1 18150

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 18143	. . . . . . 7 ⊢ (𝐹 ∈ dom 𝑆 ↔ (𝐹 ∈ (Word 𝑊 ∖ {∅}) ∧ (𝐹‘0) ∈ 𝐷 ∧ ∀𝑖 ∈ (1..^(#‘𝐹))(𝐹‘𝑖) ∈ ran (𝑇‘(𝐹‘(𝑖 − 1)))))
8	7	simp1bi 1076	. . . . . 6 ⊢ (𝐹 ∈ dom 𝑆 → 𝐹 ∈ (Word 𝑊 ∖ {∅}))
9	8	adantr 481	. . . . 5 ⊢ ((𝐹 ∈ dom 𝑆 ∧ 𝐴 ∈ ran (𝑇‘(𝑆‘𝐹))) → 𝐹 ∈ (Word 𝑊 ∖ {∅}))
10	9	eldifad 3586	. . . 4 ⊢ ((𝐹 ∈ dom 𝑆 ∧ 𝐴 ∈ ran (𝑇‘(𝑆‘𝐹))) → 𝐹 ∈ Word 𝑊)
11	1, 2, 3, 4, 5, 6	efgsf 18142	. . . . . . . . . . . 12 ⊢ 𝑆:{𝑡 ∈ (Word 𝑊 ∖ {∅}) ∣ ((𝑡‘0) ∈ 𝐷 ∧ ∀𝑘 ∈ (1..^(#‘𝑡))(𝑡‘𝑘) ∈ ran (𝑇‘(𝑡‘(𝑘 − 1))))}⟶𝑊
12	11	fdmi 6052	. . . . . . . . . . . . 13 ⊢ dom 𝑆 = {𝑡 ∈ (Word 𝑊 ∖ {∅}) ∣ ((𝑡‘0) ∈ 𝐷 ∧ ∀𝑘 ∈ (1..^(#‘𝑡))(𝑡‘𝑘) ∈ ran (𝑇‘(𝑡‘(𝑘 − 1))))}
13	12	feq2i 6037	. . . . . . . . . . . 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 221	. . . . . . . . . . 11 ⊢ 𝑆:dom 𝑆⟶𝑊
15	14	ffvelrni 6358	. . . . . . . . . 10 ⊢ (𝐹 ∈ dom 𝑆 → (𝑆‘𝐹) ∈ 𝑊)
16	15	adantr 481	. . . . . . . . 9 ⊢ ((𝐹 ∈ dom 𝑆 ∧ 𝐴 ∈ ran (𝑇‘(𝑆‘𝐹))) → (𝑆‘𝐹) ∈ 𝑊)
17	1, 2, 3, 4	efgtf 18135	. . . . . . . . 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 479	. . . . . . 7 ⊢ ((𝐹 ∈ dom 𝑆 ∧ 𝐴 ∈ ran (𝑇‘(𝑆‘𝐹))) → (𝑇‘(𝑆‘𝐹)):((0...(#‘(𝑆‘𝐹))) × (𝐼 × 2_𝑜))⟶𝑊)
20		frn 6053	. . . . . . 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 477	. . . . . 6 ⊢ ((𝐹 ∈ dom 𝑆 ∧ 𝐴 ∈ ran (𝑇‘(𝑆‘𝐹))) → 𝐴 ∈ ran (𝑇‘(𝑆‘𝐹)))
23	21, 22	sseldd 3604	. . . . 5 ⊢ ((𝐹 ∈ dom 𝑆 ∧ 𝐴 ∈ ran (𝑇‘(𝑆‘𝐹))) → 𝐴 ∈ 𝑊)
24	23	s1cld 13383	. . . 4 ⊢ ((𝐹 ∈ dom 𝑆 ∧ 𝐴 ∈ ran (𝑇‘(𝑆‘𝐹))) → ⟨“𝐴”⟩ ∈ Word 𝑊)
25		ccatcl 13359	. . . 4 ⊢ ((𝐹 ∈ Word 𝑊 ∧ ⟨“𝐴”⟩ ∈ Word 𝑊) → (𝐹 ++ ⟨“𝐴”⟩) ∈ Word 𝑊)
26	10, 24, 25	syl2anc 693	. . 3 ⊢ ((𝐹 ∈ dom 𝑆 ∧ 𝐴 ∈ ran (𝑇‘(𝑆‘𝐹))) → (𝐹 ++ ⟨“𝐴”⟩) ∈ Word 𝑊)
27		ccatlen 13360	. . . . . . 7 ⊢ ((𝐹 ∈ Word 𝑊 ∧ ⟨“𝐴”⟩ ∈ Word 𝑊) → (#‘(𝐹 ++ ⟨“𝐴”⟩)) = ((#‘𝐹) + (#‘⟨“𝐴”⟩)))
28	10, 24, 27	syl2anc 693	. . . . . 6 ⊢ ((𝐹 ∈ dom 𝑆 ∧ 𝐴 ∈ ran (𝑇‘(𝑆‘𝐹))) → (#‘(𝐹 ++ ⟨“𝐴”⟩)) = ((#‘𝐹) + (#‘⟨“𝐴”⟩)))
29		s1len 13385	. . . . . . 7 ⊢ (#‘⟨“𝐴”⟩) = 1
30	29	oveq2i 6661	. . . . . 6 ⊢ ((#‘𝐹) + (#‘⟨“𝐴”⟩)) = ((#‘𝐹) + 1)
31	28, 30	syl6eq 2672	. . . . 5 ⊢ ((𝐹 ∈ dom 𝑆 ∧ 𝐴 ∈ ran (𝑇‘(𝑆‘𝐹))) → (#‘(𝐹 ++ ⟨“𝐴”⟩)) = ((#‘𝐹) + 1))
32		lencl 13324	. . . . . 6 ⊢ (𝐹 ∈ Word 𝑊 → (#‘𝐹) ∈ ℕ₀)
33		nn0p1nn 11332	. . . . . 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 2701	. . . 4 ⊢ ((𝐹 ∈ dom 𝑆 ∧ 𝐴 ∈ ran (𝑇‘(𝑆‘𝐹))) → (#‘(𝐹 ++ ⟨“𝐴”⟩)) ∈ ℕ)
36		wrdfin 13323	. . . . 5 ⊢ ((𝐹 ++ ⟨“𝐴”⟩) ∈ Word 𝑊 → (𝐹 ++ ⟨“𝐴”⟩) ∈ Fin)
37		hashnncl 13157	. . . . 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 222	. . 3 ⊢ ((𝐹 ∈ dom 𝑆 ∧ 𝐴 ∈ ran (𝑇‘(𝑆‘𝐹))) → (𝐹 ++ ⟨“𝐴”⟩) ≠ ∅)
40		eldifsn 4317	. . 3 ⊢ ((𝐹 ++ ⟨“𝐴”⟩) ∈ (Word 𝑊 ∖ {∅}) ↔ ((𝐹 ++ ⟨“𝐴”⟩) ∈ Word 𝑊 ∧ (𝐹 ++ ⟨“𝐴”⟩) ≠ ∅))
41	26, 39, 40	sylanbrc 698	. 2 ⊢ ((𝐹 ∈ dom 𝑆 ∧ 𝐴 ∈ ran (𝑇‘(𝑆‘𝐹))) → (𝐹 ++ ⟨“𝐴”⟩) ∈ (Word 𝑊 ∖ {∅}))
42		eldifsni 4320	. . . . . . 7 ⊢ (𝐹 ∈ (Word 𝑊 ∖ {∅}) → 𝐹 ≠ ∅)
43	9, 42	syl 17	. . . . . 6 ⊢ ((𝐹 ∈ dom 𝑆 ∧ 𝐴 ∈ ran (𝑇‘(𝑆‘𝐹))) → 𝐹 ≠ ∅)
44		wrdfin 13323	. . . . . . 7 ⊢ (𝐹 ∈ Word 𝑊 → 𝐹 ∈ Fin)
45		hashnncl 13157	. . . . . . 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 247	. . . . 5 ⊢ ((𝐹 ∈ dom 𝑆 ∧ 𝐴 ∈ ran (𝑇‘(𝑆‘𝐹))) → (#‘𝐹) ∈ ℕ)
48		lbfzo0 12507	. . . . 5 ⊢ (0 ∈ (0..^(#‘𝐹)) ↔ (#‘𝐹) ∈ ℕ)
49	47, 48	sylibr 224	. . . 4 ⊢ ((𝐹 ∈ dom 𝑆 ∧ 𝐴 ∈ ran (𝑇‘(𝑆‘𝐹))) → 0 ∈ (0..^(#‘𝐹)))
50		ccatval1 13361	. . . 4 ⊢ ((𝐹 ∈ Word 𝑊 ∧ ⟨“𝐴”⟩ ∈ Word 𝑊 ∧ 0 ∈ (0..^(#‘𝐹))) → ((𝐹 ++ ⟨“𝐴”⟩)‘0) = (𝐹‘0))
51	10, 24, 49, 50	syl3anc 1326	. . 3 ⊢ ((𝐹 ∈ dom 𝑆 ∧ 𝐴 ∈ ran (𝑇‘(𝑆‘𝐹))) → ((𝐹 ++ ⟨“𝐴”⟩)‘0) = (𝐹‘0))
52	7	simp2bi 1077	. . . 4 ⊢ (𝐹 ∈ dom 𝑆 → (𝐹‘0) ∈ 𝐷)
53	52	adantr 481	. . 3 ⊢ ((𝐹 ∈ dom 𝑆 ∧ 𝐴 ∈ ran (𝑇‘(𝑆‘𝐹))) → (𝐹‘0) ∈ 𝐷)
54	51, 53	eqeltrd 2701	. 2 ⊢ ((𝐹 ∈ dom 𝑆 ∧ 𝐴 ∈ ran (𝑇‘(𝑆‘𝐹))) → ((𝐹 ++ ⟨“𝐴”⟩)‘0) ∈ 𝐷)
55	7	simp3bi 1078	. . . . . 6 ⊢ (𝐹 ∈ dom 𝑆 → ∀𝑖 ∈ (1..^(#‘𝐹))(𝐹‘𝑖) ∈ ran (𝑇‘(𝐹‘(𝑖 − 1))))
56	55	adantr 481	. . . . 5 ⊢ ((𝐹 ∈ dom 𝑆 ∧ 𝐴 ∈ ran (𝑇‘(𝑆‘𝐹))) → ∀𝑖 ∈ (1..^(#‘𝐹))(𝐹‘𝑖) ∈ ran (𝑇‘(𝐹‘(𝑖 − 1))))
57		fzo0ss1 12498	. . . . . . . . . . 11 ⊢ (1..^(#‘𝐹)) ⊆ (0..^(#‘𝐹))
58	57	sseli 3599	. . . . . . . . . 10 ⊢ (𝑖 ∈ (1..^(#‘𝐹)) → 𝑖 ∈ (0..^(#‘𝐹)))
59		ccatval1 13361	. . . . . . . . . 10 ⊢ ((𝐹 ∈ Word 𝑊 ∧ ⟨“𝐴”⟩ ∈ Word 𝑊 ∧ 𝑖 ∈ (0..^(#‘𝐹))) → ((𝐹 ++ ⟨“𝐴”⟩)‘𝑖) = (𝐹‘𝑖))
60	58, 59	syl3an3 1361	. . . . . . . . 9 ⊢ ((𝐹 ∈ Word 𝑊 ∧ ⟨“𝐴”⟩ ∈ Word 𝑊 ∧ 𝑖 ∈ (1..^(#‘𝐹))) → ((𝐹 ++ ⟨“𝐴”⟩)‘𝑖) = (𝐹‘𝑖))
61		elfzoel2 12469	. . . . . . . . . . . . . . . 16 ⊢ (𝑖 ∈ (1..^(#‘𝐹)) → (#‘𝐹) ∈ ℤ)
62		peano2zm 11420	. . . . . . . . . . . . . . . 16 ⊢ ((#‘𝐹) ∈ ℤ → ((#‘𝐹) − 1) ∈ ℤ)
63	61, 62	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝑖 ∈ (1..^(#‘𝐹)) → ((#‘𝐹) − 1) ∈ ℤ)
64	61	zred 11482	. . . . . . . . . . . . . . . 16 ⊢ (𝑖 ∈ (1..^(#‘𝐹)) → (#‘𝐹) ∈ ℝ)
65	64	lem1d 10957	. . . . . . . . . . . . . . 15 ⊢ (𝑖 ∈ (1..^(#‘𝐹)) → ((#‘𝐹) − 1) ≤ (#‘𝐹))
66		eluz2 11693	. . . . . . . . . . . . . . 15 ⊢ ((#‘𝐹) ∈ (ℤ_≥‘((#‘𝐹) − 1)) ↔ (((#‘𝐹) − 1) ∈ ℤ ∧ (#‘𝐹) ∈ ℤ ∧ ((#‘𝐹) − 1) ≤ (#‘𝐹)))
67	63, 61, 65, 66	syl3anbrc 1246	. . . . . . . . . . . . . 14 ⊢ (𝑖 ∈ (1..^(#‘𝐹)) → (#‘𝐹) ∈ (ℤ_≥‘((#‘𝐹) − 1)))
68		fzoss2 12496	. . . . . . . . . . . . . 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 12470	. . . . . . . . . . . . . . 15 ⊢ (𝑖 ∈ (1..^(#‘𝐹)) → 𝑖 ∈ ℤ)
71		elfzom1b 12567	. . . . . . . . . . . . . . 15 ⊢ ((𝑖 ∈ ℤ ∧ (#‘𝐹) ∈ ℤ) → (𝑖 ∈ (1..^(#‘𝐹)) ↔ (𝑖 − 1) ∈ (0..^((#‘𝐹) − 1))))
72	70, 61, 71	syl2anc 693	. . . . . . . . . . . . . 14 ⊢ (𝑖 ∈ (1..^(#‘𝐹)) → (𝑖 ∈ (1..^(#‘𝐹)) ↔ (𝑖 − 1) ∈ (0..^((#‘𝐹) − 1))))
73	72	ibi 256	. . . . . . . . . . . . 13 ⊢ (𝑖 ∈ (1..^(#‘𝐹)) → (𝑖 − 1) ∈ (0..^((#‘𝐹) − 1)))
74	69, 73	sseldd 3604	. . . . . . . . . . . 12 ⊢ (𝑖 ∈ (1..^(#‘𝐹)) → (𝑖 − 1) ∈ (0..^(#‘𝐹)))
75		ccatval1 13361	. . . . . . . . . . . 12 ⊢ ((𝐹 ∈ Word 𝑊 ∧ ⟨“𝐴”⟩ ∈ Word 𝑊 ∧ (𝑖 − 1) ∈ (0..^(#‘𝐹))) → ((𝐹 ++ ⟨“𝐴”⟩)‘(𝑖 − 1)) = (𝐹‘(𝑖 − 1)))
76	74, 75	syl3an3 1361	. . . . . . . . . . 11 ⊢ ((𝐹 ∈ Word 𝑊 ∧ ⟨“𝐴”⟩ ∈ Word 𝑊 ∧ 𝑖 ∈ (1..^(#‘𝐹))) → ((𝐹 ++ ⟨“𝐴”⟩)‘(𝑖 − 1)) = (𝐹‘(𝑖 − 1)))
77	76	fveq2d 6195	. . . . . . . . . 10 ⊢ ((𝐹 ∈ Word 𝑊 ∧ ⟨“𝐴”⟩ ∈ Word 𝑊 ∧ 𝑖 ∈ (1..^(#‘𝐹))) → (𝑇‘((𝐹 ++ ⟨“𝐴”⟩)‘(𝑖 − 1))) = (𝑇‘(𝐹‘(𝑖 − 1))))
78	77	rneqd 5353	. . . . . . . . 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 2695	. . . . . . . 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 1265	. . . . . . 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 2985	. . . . . 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 693	. . . . 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 247	. . . 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 11353	. . . . . . . . 9 ⊢ ((𝐹 ∈ dom 𝑆 ∧ 𝐴 ∈ ran (𝑇‘(𝑆‘𝐹))) → (#‘𝐹) ∈ ℂ)
86	85	addid2d 10237	. . . . . . . 8 ⊢ ((𝐹 ∈ dom 𝑆 ∧ 𝐴 ∈ ran (𝑇‘(𝑆‘𝐹))) → (0 + (#‘𝐹)) = (#‘𝐹))
87	86	fveq2d 6195	. . . . . . 7 ⊢ ((𝐹 ∈ dom 𝑆 ∧ 𝐴 ∈ ran (𝑇‘(𝑆‘𝐹))) → ((𝐹 ++ ⟨“𝐴”⟩)‘(0 + (#‘𝐹))) = ((𝐹 ++ ⟨“𝐴”⟩)‘(#‘𝐹)))
88		1nn 11031	. . . . . . . . . . 11 ⊢ 1 ∈ ℕ
89	29, 88	eqeltri 2697	. . . . . . . . . 10 ⊢ (#‘⟨“𝐴”⟩) ∈ ℕ
90	89	a1i 11	. . . . . . . . 9 ⊢ ((𝐹 ∈ dom 𝑆 ∧ 𝐴 ∈ ran (𝑇‘(𝑆‘𝐹))) → (#‘⟨“𝐴”⟩) ∈ ℕ)
91		lbfzo0 12507	. . . . . . . . 9 ⊢ (0 ∈ (0..^(#‘⟨“𝐴”⟩)) ↔ (#‘⟨“𝐴”⟩) ∈ ℕ)
92	90, 91	sylibr 224	. . . . . . . 8 ⊢ ((𝐹 ∈ dom 𝑆 ∧ 𝐴 ∈ ran (𝑇‘(𝑆‘𝐹))) → 0 ∈ (0..^(#‘⟨“𝐴”⟩)))
93		ccatval3 13363	. . . . . . . 8 ⊢ ((𝐹 ∈ Word 𝑊 ∧ ⟨“𝐴”⟩ ∈ Word 𝑊 ∧ 0 ∈ (0..^(#‘⟨“𝐴”⟩))) → ((𝐹 ++ ⟨“𝐴”⟩)‘(0 + (#‘𝐹))) = (⟨“𝐴”⟩‘0))
94	10, 24, 92, 93	syl3anc 1326	. . . . . . 7 ⊢ ((𝐹 ∈ dom 𝑆 ∧ 𝐴 ∈ ran (𝑇‘(𝑆‘𝐹))) → ((𝐹 ++ ⟨“𝐴”⟩)‘(0 + (#‘𝐹))) = (⟨“𝐴”⟩‘0))
95	87, 94	eqtr3d 2658	. . . . . 6 ⊢ ((𝐹 ∈ dom 𝑆 ∧ 𝐴 ∈ ran (𝑇‘(𝑆‘𝐹))) → ((𝐹 ++ ⟨“𝐴”⟩)‘(#‘𝐹)) = (⟨“𝐴”⟩‘0))
96		s1fv 13390	. . . . . . . 8 ⊢ (𝐴 ∈ ran (𝑇‘(𝑆‘𝐹)) → (⟨“𝐴”⟩‘0) = 𝐴)
97	96	adantl 482	. . . . . . 7 ⊢ ((𝐹 ∈ dom 𝑆 ∧ 𝐴 ∈ ran (𝑇‘(𝑆‘𝐹))) → (⟨“𝐴”⟩‘0) = 𝐴)
98		fzo0end 12560	. . . . . . . . . . . 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 13361	. . . . . . . . . . 11 ⊢ ((𝐹 ∈ Word 𝑊 ∧ ⟨“𝐴”⟩ ∈ Word 𝑊 ∧ ((#‘𝐹) − 1) ∈ (0..^(#‘𝐹))) → ((𝐹 ++ ⟨“𝐴”⟩)‘((#‘𝐹) − 1)) = (𝐹‘((#‘𝐹) − 1)))
101	10, 24, 99, 100	syl3anc 1326	. . . . . . . . . 10 ⊢ ((𝐹 ∈ dom 𝑆 ∧ 𝐴 ∈ ran (𝑇‘(𝑆‘𝐹))) → ((𝐹 ++ ⟨“𝐴”⟩)‘((#‘𝐹) − 1)) = (𝐹‘((#‘𝐹) − 1)))
102	1, 2, 3, 4, 5, 6	efgsval 18144	. . . . . . . . . . 11 ⊢ (𝐹 ∈ dom 𝑆 → (𝑆‘𝐹) = (𝐹‘((#‘𝐹) − 1)))
103	102	adantr 481	. . . . . . . . . 10 ⊢ ((𝐹 ∈ dom 𝑆 ∧ 𝐴 ∈ ran (𝑇‘(𝑆‘𝐹))) → (𝑆‘𝐹) = (𝐹‘((#‘𝐹) − 1)))
104	101, 103	eqtr4d 2659	. . . . . . . . 9 ⊢ ((𝐹 ∈ dom 𝑆 ∧ 𝐴 ∈ ran (𝑇‘(𝑆‘𝐹))) → ((𝐹 ++ ⟨“𝐴”⟩)‘((#‘𝐹) − 1)) = (𝑆‘𝐹))
105	104	fveq2d 6195	. . . . . . . 8 ⊢ ((𝐹 ∈ dom 𝑆 ∧ 𝐴 ∈ ran (𝑇‘(𝑆‘𝐹))) → (𝑇‘((𝐹 ++ ⟨“𝐴”⟩)‘((#‘𝐹) − 1))) = (𝑇‘(𝑆‘𝐹)))
106	105	rneqd 5353	. . . . . . 7 ⊢ ((𝐹 ∈ dom 𝑆 ∧ 𝐴 ∈ ran (𝑇‘(𝑆‘𝐹))) → ran (𝑇‘((𝐹 ++ ⟨“𝐴”⟩)‘((#‘𝐹) − 1))) = ran (𝑇‘(𝑆‘𝐹)))
107	22, 97, 106	3eltr4d 2716	. . . . . 6 ⊢ ((𝐹 ∈ dom 𝑆 ∧ 𝐴 ∈ ran (𝑇‘(𝑆‘𝐹))) → (⟨“𝐴”⟩‘0) ∈ ran (𝑇‘((𝐹 ++ ⟨“𝐴”⟩)‘((#‘𝐹) − 1))))
108	95, 107	eqeltrd 2701	. . . . 5 ⊢ ((𝐹 ∈ dom 𝑆 ∧ 𝐴 ∈ ran (𝑇‘(𝑆‘𝐹))) → ((𝐹 ++ ⟨“𝐴”⟩)‘(#‘𝐹)) ∈ ran (𝑇‘((𝐹 ++ ⟨“𝐴”⟩)‘((#‘𝐹) − 1))))
109		fvex 6201	. . . . . 6 ⊢ (#‘𝐹) ∈ V
110		fveq2 6191	. . . . . . 7 ⊢ (𝑖 = (#‘𝐹) → ((𝐹 ++ ⟨“𝐴”⟩)‘𝑖) = ((𝐹 ++ ⟨“𝐴”⟩)‘(#‘𝐹)))
111		oveq1 6657	. . . . . . . . . 10 ⊢ (𝑖 = (#‘𝐹) → (𝑖 − 1) = ((#‘𝐹) − 1))
112	111	fveq2d 6195	. . . . . . . . 9 ⊢ (𝑖 = (#‘𝐹) → ((𝐹 ++ ⟨“𝐴”⟩)‘(𝑖 − 1)) = ((𝐹 ++ ⟨“𝐴”⟩)‘((#‘𝐹) − 1)))
113	112	fveq2d 6195	. . . . . . . 8 ⊢ (𝑖 = (#‘𝐹) → (𝑇‘((𝐹 ++ ⟨“𝐴”⟩)‘(𝑖 − 1))) = (𝑇‘((𝐹 ++ ⟨“𝐴”⟩)‘((#‘𝐹) − 1))))
114	113	rneqd 5353	. . . . . . 7 ⊢ (𝑖 = (#‘𝐹) → ran (𝑇‘((𝐹 ++ ⟨“𝐴”⟩)‘(𝑖 − 1))) = ran (𝑇‘((𝐹 ++ ⟨“𝐴”⟩)‘((#‘𝐹) − 1))))
115	110, 114	eleq12d 2695	. . . . . 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 4222	. . . . 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 224	. . . 4 ⊢ ((𝐹 ∈ dom 𝑆 ∧ 𝐴 ∈ ran (𝑇‘(𝑆‘𝐹))) → ∀𝑖 ∈ {(#‘𝐹)} ((𝐹 ++ ⟨“𝐴”⟩)‘𝑖) ∈ ran (𝑇‘((𝐹 ++ ⟨“𝐴”⟩)‘(𝑖 − 1))))
118		ralunb 3794	. . . 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 698	. . 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 6666	. . . . 5 ⊢ ((𝐹 ∈ dom 𝑆 ∧ 𝐴 ∈ ran (𝑇‘(𝑆‘𝐹))) → (1..^(#‘(𝐹 ++ ⟨“𝐴”⟩))) = (1..^((#‘𝐹) + 1)))
121		nnuz 11723	. . . . . . 7 ⊢ ℕ = (ℤ_≥‘1)
122	47, 121	syl6eleq 2711	. . . . . 6 ⊢ ((𝐹 ∈ dom 𝑆 ∧ 𝐴 ∈ ran (𝑇‘(𝑆‘𝐹))) → (#‘𝐹) ∈ (ℤ_≥‘1))
123		fzosplitsn 12576	. . . . . 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 2656	. . . 4 ⊢ ((𝐹 ∈ dom 𝑆 ∧ 𝐴 ∈ ran (𝑇‘(𝑆‘𝐹))) → (1..^(#‘(𝐹 ++ ⟨“𝐴”⟩))) = ((1..^(#‘𝐹)) ∪ {(#‘𝐹)}))
126	125	raleqdv 3144	. . 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 247	. 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 18143	. 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 1246	1 ⊢ ((𝐹 ∈ dom 𝑆 ∧ 𝐴 ∈ ran (𝑇‘(𝑆‘𝐹))) → (𝐹 ++ ⟨“𝐴”⟩) ∈ dom 𝑆)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 196 ∧ wa 384 ∧ w3a 1037 = wceq 1483 ∈ wcel 1990 ≠ wne 2794 ∀wral 2912 {crab 2916 ∖ cdif 3571 ∪ cun 3572 ⊆ wss 3574 ∅c0 3915 {csn 4177 ⟨cop 4183 ⟨cotp 4185 ∪ ciun 4520 class class class wbr 4653 ↦ cmpt 4729 I cid 5023 × cxp 5112 dom cdm 5114 ran crn 5115 ⟶wf 5884 ‘cfv 5888 (class class class)co 6650 ↦ cmpt2 6652 1_𝑜c1o 7553 2_𝑜c2o 7554 Fincfn 7955 0cc0 9936 1c1 9937 + caddc 9939 ≤ cle 10075 − cmin 10266 ℕcn 11020 ℕ₀cn0 11292 ℤcz 11377 ℤ_≥cuz 11687 ...cfz 12326 ..^cfzo 12465 #chash 13117 Word cword 13291 ++ cconcat 13293 ⟨“cs1 13294 splice csplice 13296 ⟨“cs2 13586 ~_FG cefg 18119

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-ot 4186 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-2o 7561 df-oadd 7564 df-er 7742 df-map 7859 df-pm 7860 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-card 8765 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-n0 11293 df-z 11378 df-uz 11688 df-fz 12327 df-fzo 12466 df-hash 13118 df-word 13299 df-concat 13301 df-s1 13302 df-substr 13303 df-splice 13304 df-s2 13593

This theorem is referenced by: efgsfo 18152 efgredlemd 18157 efgrelexlemb 18163

W3C validator