sseqfv2 - Mathbox for Thierry Arnoux

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Theorem sseqfv2 33381

Description: Value of the strong sequence builder function. (Contributed by Thierry Arnoux, 21-Apr-2019.)

**Hypotheses**
Ref	Expression
sseqval.1	⊢ (𝜑 → 𝑆 ∈ V)
sseqval.2	⊢ (𝜑 → 𝑀 ∈ Word 𝑆)
sseqval.3	⊢ 𝑊 = (Word 𝑆 ∩ (^◡♯ “ (ℤ_≥‘(♯‘𝑀))))
sseqval.4	⊢ (𝜑 → 𝐹:𝑊⟶𝑆)
sseqfv2.4	⊢ (𝜑 → 𝑁 ∈ (ℤ_≥‘(♯‘𝑀)))

**Assertion**
Ref	Expression
sseqfv2	⊢ (𝜑 → ((𝑀seq_str𝐹)‘𝑁) = (lastS‘(seq(♯‘𝑀)((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥 ++ ⟨“(𝐹‘𝑥)”⟩)), (ℕ₀ × {(𝑀 ++ ⟨“(𝐹‘𝑀)”⟩)}))‘𝑁)))

Distinct variable groups: 𝑥,𝑦,𝐹 𝑥,𝑀,𝑦 𝜑,𝑥,𝑦 𝑥,𝑊,𝑦 Allowed substitution hints: 𝑆(𝑥,𝑦) 𝑁(𝑥,𝑦)

Proof of Theorem sseqfv2 Dummy variables 𝑎 𝑏 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		sseqval.1	. . . 4 ⊢ (𝜑 → 𝑆 ∈ V)
2		sseqval.2	. . . 4 ⊢ (𝜑 → 𝑀 ∈ Word 𝑆)
3		sseqval.3	. . . 4 ⊢ 𝑊 = (Word 𝑆 ∩ (^◡♯ “ (ℤ_≥‘(♯‘𝑀))))
4		sseqval.4	. . . 4 ⊢ (𝜑 → 𝐹:𝑊⟶𝑆)
5	1, 2, 3, 4	sseqval 33375	. . 3 ⊢ (𝜑 → (𝑀seq_str𝐹) = (𝑀 ∪ (lastS ∘ seq(♯‘𝑀)((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥 ++ ⟨“(𝐹‘𝑥)”⟩)), (ℕ₀ × {(𝑀 ++ ⟨“(𝐹‘𝑀)”⟩)})))))
6	5	fveq1d 6890	. 2 ⊢ (𝜑 → ((𝑀seq_str𝐹)‘𝑁) = ((𝑀 ∪ (lastS ∘ seq(♯‘𝑀)((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥 ++ ⟨“(𝐹‘𝑥)”⟩)), (ℕ₀ × {(𝑀 ++ ⟨“(𝐹‘𝑀)”⟩)}))))‘𝑁))
7		wrdfn 14474	. . . 4 ⊢ (𝑀 ∈ Word 𝑆 → 𝑀 Fn (0..^(♯‘𝑀)))
8	2, 7	syl 17	. . 3 ⊢ (𝜑 → 𝑀 Fn (0..^(♯‘𝑀)))
9		fvex 6901	. . . . . 6 ⊢ (𝑥‘((♯‘𝑥) − 1)) ∈ V
10		df-lsw 14509	. . . . . 6 ⊢ lastS = (𝑥 ∈ V ↦ (𝑥‘((♯‘𝑥) − 1)))
11	9, 10	fnmpti 6690	. . . . 5 ⊢ lastS Fn V
12	11	a1i 11	. . . 4 ⊢ (𝜑 → lastS Fn V)
13		lencl 14479	. . . . . . 7 ⊢ (𝑀 ∈ Word 𝑆 → (♯‘𝑀) ∈ ℕ₀)
14	2, 13	syl 17	. . . . . 6 ⊢ (𝜑 → (♯‘𝑀) ∈ ℕ₀)
15	14	nn0zd 12580	. . . . 5 ⊢ (𝜑 → (♯‘𝑀) ∈ ℤ)
16		seqfn 13974	. . . . 5 ⊢ ((♯‘𝑀) ∈ ℤ → seq(♯‘𝑀)((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥 ++ ⟨“(𝐹‘𝑥)”⟩)), (ℕ₀ × {(𝑀 ++ ⟨“(𝐹‘𝑀)”⟩)})) Fn (ℤ_≥‘(♯‘𝑀)))
17	15, 16	syl 17	. . . 4 ⊢ (𝜑 → seq(♯‘𝑀)((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥 ++ ⟨“(𝐹‘𝑥)”⟩)), (ℕ₀ × {(𝑀 ++ ⟨“(𝐹‘𝑀)”⟩)})) Fn (ℤ_≥‘(♯‘𝑀)))
18		ssv 4005	. . . . 5 ⊢ ran seq(♯‘𝑀)((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥 ++ ⟨“(𝐹‘𝑥)”⟩)), (ℕ₀ × {(𝑀 ++ ⟨“(𝐹‘𝑀)”⟩)})) ⊆ V
19	18	a1i 11	. . . 4 ⊢ (𝜑 → ran seq(♯‘𝑀)((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥 ++ ⟨“(𝐹‘𝑥)”⟩)), (ℕ₀ × {(𝑀 ++ ⟨“(𝐹‘𝑀)”⟩)})) ⊆ V)
20		fnco 6664	. . . 4 ⊢ ((lastS Fn V ∧ seq(♯‘𝑀)((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥 ++ ⟨“(𝐹‘𝑥)”⟩)), (ℕ₀ × {(𝑀 ++ ⟨“(𝐹‘𝑀)”⟩)})) Fn (ℤ_≥‘(♯‘𝑀)) ∧ ran seq(♯‘𝑀)((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥 ++ ⟨“(𝐹‘𝑥)”⟩)), (ℕ₀ × {(𝑀 ++ ⟨“(𝐹‘𝑀)”⟩)})) ⊆ V) → (lastS ∘ seq(♯‘𝑀)((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥 ++ ⟨“(𝐹‘𝑥)”⟩)), (ℕ₀ × {(𝑀 ++ ⟨“(𝐹‘𝑀)”⟩)}))) Fn (ℤ_≥‘(♯‘𝑀)))
21	12, 17, 19, 20	syl3anc 1371	. . 3 ⊢ (𝜑 → (lastS ∘ seq(♯‘𝑀)((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥 ++ ⟨“(𝐹‘𝑥)”⟩)), (ℕ₀ × {(𝑀 ++ ⟨“(𝐹‘𝑀)”⟩)}))) Fn (ℤ_≥‘(♯‘𝑀)))
22		fzouzdisj 13664	. . . 4 ⊢ ((0..^(♯‘𝑀)) ∩ (ℤ_≥‘(♯‘𝑀))) = ∅
23	22	a1i 11	. . 3 ⊢ (𝜑 → ((0..^(♯‘𝑀)) ∩ (ℤ_≥‘(♯‘𝑀))) = ∅)
24		sseqfv2.4	. . 3 ⊢ (𝜑 → 𝑁 ∈ (ℤ_≥‘(♯‘𝑀)))
25		fvun2 6980	. . 3 ⊢ ((𝑀 Fn (0..^(♯‘𝑀)) ∧ (lastS ∘ seq(♯‘𝑀)((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥 ++ ⟨“(𝐹‘𝑥)”⟩)), (ℕ₀ × {(𝑀 ++ ⟨“(𝐹‘𝑀)”⟩)}))) Fn (ℤ_≥‘(♯‘𝑀)) ∧ (((0..^(♯‘𝑀)) ∩ (ℤ_≥‘(♯‘𝑀))) = ∅ ∧ 𝑁 ∈ (ℤ_≥‘(♯‘𝑀)))) → ((𝑀 ∪ (lastS ∘ seq(♯‘𝑀)((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥 ++ ⟨“(𝐹‘𝑥)”⟩)), (ℕ₀ × {(𝑀 ++ ⟨“(𝐹‘𝑀)”⟩)}))))‘𝑁) = ((lastS ∘ seq(♯‘𝑀)((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥 ++ ⟨“(𝐹‘𝑥)”⟩)), (ℕ₀ × {(𝑀 ++ ⟨“(𝐹‘𝑀)”⟩)})))‘𝑁))
26	8, 21, 23, 24, 25	syl112anc 1374	. 2 ⊢ (𝜑 → ((𝑀 ∪ (lastS ∘ seq(♯‘𝑀)((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥 ++ ⟨“(𝐹‘𝑥)”⟩)), (ℕ₀ × {(𝑀 ++ ⟨“(𝐹‘𝑀)”⟩)}))))‘𝑁) = ((lastS ∘ seq(♯‘𝑀)((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥 ++ ⟨“(𝐹‘𝑥)”⟩)), (ℕ₀ × {(𝑀 ++ ⟨“(𝐹‘𝑀)”⟩)})))‘𝑁))
27		fnfun 6646	. . . 4 ⊢ (seq(♯‘𝑀)((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥 ++ ⟨“(𝐹‘𝑥)”⟩)), (ℕ₀ × {(𝑀 ++ ⟨“(𝐹‘𝑀)”⟩)})) Fn (ℤ_≥‘(♯‘𝑀)) → Fun seq(♯‘𝑀)((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥 ++ ⟨“(𝐹‘𝑥)”⟩)), (ℕ₀ × {(𝑀 ++ ⟨“(𝐹‘𝑀)”⟩)})))
28	17, 27	syl 17	. . 3 ⊢ (𝜑 → Fun seq(♯‘𝑀)((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥 ++ ⟨“(𝐹‘𝑥)”⟩)), (ℕ₀ × {(𝑀 ++ ⟨“(𝐹‘𝑀)”⟩)})))
29		fvexd 6903	. . . . . 6 ⊢ (𝜑 → ((ℕ₀ × {(𝑀 ++ ⟨“(𝐹‘𝑀)”⟩)})‘(♯‘𝑀)) ∈ V)
30		ovexd 7440	. . . . . 6 ⊢ ((𝜑 ∧ (𝑎 ∈ V ∧ 𝑏 ∈ V)) → (𝑎(𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥 ++ ⟨“(𝐹‘𝑥)”⟩))𝑏) ∈ V)
31		eqid 2732	. . . . . 6 ⊢ (ℤ_≥‘(♯‘𝑀)) = (ℤ_≥‘(♯‘𝑀))
32		fvexd 6903	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ (ℤ_≥‘((♯‘𝑀) + 1))) → ((ℕ₀ × {(𝑀 ++ ⟨“(𝐹‘𝑀)”⟩)})‘𝑎) ∈ V)
33	29, 30, 31, 15, 32	seqf2 13983	. . . . 5 ⊢ (𝜑 → seq(♯‘𝑀)((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥 ++ ⟨“(𝐹‘𝑥)”⟩)), (ℕ₀ × {(𝑀 ++ ⟨“(𝐹‘𝑀)”⟩)})):(ℤ_≥‘(♯‘𝑀))⟶V)
34	33	fdmd 6725	. . . 4 ⊢ (𝜑 → dom seq(♯‘𝑀)((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥 ++ ⟨“(𝐹‘𝑥)”⟩)), (ℕ₀ × {(𝑀 ++ ⟨“(𝐹‘𝑀)”⟩)})) = (ℤ_≥‘(♯‘𝑀)))
35	24, 34	eleqtrrd 2836	. . 3 ⊢ (𝜑 → 𝑁 ∈ dom seq(♯‘𝑀)((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥 ++ ⟨“(𝐹‘𝑥)”⟩)), (ℕ₀ × {(𝑀 ++ ⟨“(𝐹‘𝑀)”⟩)})))
36		fvco 6986	. . 3 ⊢ ((Fun seq(♯‘𝑀)((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥 ++ ⟨“(𝐹‘𝑥)”⟩)), (ℕ₀ × {(𝑀 ++ ⟨“(𝐹‘𝑀)”⟩)})) ∧ 𝑁 ∈ dom seq(♯‘𝑀)((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥 ++ ⟨“(𝐹‘𝑥)”⟩)), (ℕ₀ × {(𝑀 ++ ⟨“(𝐹‘𝑀)”⟩)}))) → ((lastS ∘ seq(♯‘𝑀)((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥 ++ ⟨“(𝐹‘𝑥)”⟩)), (ℕ₀ × {(𝑀 ++ ⟨“(𝐹‘𝑀)”⟩)})))‘𝑁) = (lastS‘(seq(♯‘𝑀)((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥 ++ ⟨“(𝐹‘𝑥)”⟩)), (ℕ₀ × {(𝑀 ++ ⟨“(𝐹‘𝑀)”⟩)}))‘𝑁)))
37	28, 35, 36	syl2anc 584	. 2 ⊢ (𝜑 → ((lastS ∘ seq(♯‘𝑀)((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥 ++ ⟨“(𝐹‘𝑥)”⟩)), (ℕ₀ × {(𝑀 ++ ⟨“(𝐹‘𝑀)”⟩)})))‘𝑁) = (lastS‘(seq(♯‘𝑀)((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥 ++ ⟨“(𝐹‘𝑥)”⟩)), (ℕ₀ × {(𝑀 ++ ⟨“(𝐹‘𝑀)”⟩)}))‘𝑁)))
38	6, 26, 37	3eqtrd 2776	1 ⊢ (𝜑 → ((𝑀seq_str𝐹)‘𝑁) = (lastS‘(seq(♯‘𝑀)((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥 ++ ⟨“(𝐹‘𝑥)”⟩)), (ℕ₀ × {(𝑀 ++ ⟨“(𝐹‘𝑀)”⟩)}))‘𝑁)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 396 = wceq 1541 ∈ wcel 2106 Vcvv 3474 ∪ cun 3945 ∩ cin 3946 ⊆ wss 3947 ∅c0 4321 {csn 4627 × cxp 5673 ^◡ccnv 5674 dom cdm 5675 ran crn 5676 “ cima 5678 ∘ ccom 5679 Fun wfun 6534 Fn wfn 6535 ⟶wf 6536 ‘cfv 6540 (class class class)co 7405 ∈ cmpo 7407 0cc0 11106 1c1 11107 + caddc 11109 − cmin 11440 ℕ₀cn0 12468 ℤcz 12554 ℤ_≥cuz 12818 ..^cfzo 13623 seqcseq 13962 ♯chash 14286 Word cword 14460 lastSclsw 14508 ++ cconcat 14516 ⟨“cs1 14541 seq_strcsseq 33370

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 ax-inf2 9632 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-int 4950 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7852 df-1st 7971 df-2nd 7972 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-1o 8462 df-er 8699 df-map 8818 df-en 8936 df-dom 8937 df-sdom 8938 df-fin 8939 df-card 9930 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-nn 12209 df-n0 12469 df-z 12555 df-uz 12819 df-fz 13481 df-fzo 13624 df-seq 13963 df-hash 14287 df-word 14461 df-lsw 14509 df-s1 14542 df-sseq 33371

This theorem is referenced by: sseqp1 33382

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