sseqfv2 - Mathbox for Thierry Arnoux

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Theorem sseqfv2 33393

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 33387	. . 3 ⊢ (𝜑 → (𝑀seq_str𝐹) = (𝑀 ∪ (lastS ∘ seq(♯‘𝑀)((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥 ++ ⟨“(𝐹‘𝑥)”⟩)), (ℕ₀ × {(𝑀 ++ ⟨“(𝐹‘𝑀)”⟩)})))))
6	5	fveq1d 6894	. 2 ⊢ (𝜑 → ((𝑀seq_str𝐹)‘𝑁) = ((𝑀 ∪ (lastS ∘ seq(♯‘𝑀)((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥 ++ ⟨“(𝐹‘𝑥)”⟩)), (ℕ₀ × {(𝑀 ++ ⟨“(𝐹‘𝑀)”⟩)}))))‘𝑁))
7		wrdfn 14478	. . . 4 ⊢ (𝑀 ∈ Word 𝑆 → 𝑀 Fn (0..^(♯‘𝑀)))
8	2, 7	syl 17	. . 3 ⊢ (𝜑 → 𝑀 Fn (0..^(♯‘𝑀)))
9		fvex 6905	. . . . . 6 ⊢ (𝑥‘((♯‘𝑥) − 1)) ∈ V
10		df-lsw 14513	. . . . . 6 ⊢ lastS = (𝑥 ∈ V ↦ (𝑥‘((♯‘𝑥) − 1)))
11	9, 10	fnmpti 6694	. . . . 5 ⊢ lastS Fn V
12	11	a1i 11	. . . 4 ⊢ (𝜑 → lastS Fn V)
13		lencl 14483	. . . . . . 7 ⊢ (𝑀 ∈ Word 𝑆 → (♯‘𝑀) ∈ ℕ₀)
14	2, 13	syl 17	. . . . . 6 ⊢ (𝜑 → (♯‘𝑀) ∈ ℕ₀)
15	14	nn0zd 12584	. . . . 5 ⊢ (𝜑 → (♯‘𝑀) ∈ ℤ)
16		seqfn 13978	. . . . 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 4007	. . . . 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 6668	. . . 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 1372	. . 3 ⊢ (𝜑 → (lastS ∘ seq(♯‘𝑀)((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥 ++ ⟨“(𝐹‘𝑥)”⟩)), (ℕ₀ × {(𝑀 ++ ⟨“(𝐹‘𝑀)”⟩)}))) Fn (ℤ_≥‘(♯‘𝑀)))
22		fzouzdisj 13668	. . . 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 6984	. . 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 1375	. 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 6650	. . . 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 6907	. . . . . 6 ⊢ (𝜑 → ((ℕ₀ × {(𝑀 ++ ⟨“(𝐹‘𝑀)”⟩)})‘(♯‘𝑀)) ∈ V)
30		ovexd 7444	. . . . . 6 ⊢ ((𝜑 ∧ (𝑎 ∈ V ∧ 𝑏 ∈ V)) → (𝑎(𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥 ++ ⟨“(𝐹‘𝑥)”⟩))𝑏) ∈ V)
31		eqid 2733	. . . . . 6 ⊢ (ℤ_≥‘(♯‘𝑀)) = (ℤ_≥‘(♯‘𝑀))
32		fvexd 6907	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ (ℤ_≥‘((♯‘𝑀) + 1))) → ((ℕ₀ × {(𝑀 ++ ⟨“(𝐹‘𝑀)”⟩)})‘𝑎) ∈ V)
33	29, 30, 31, 15, 32	seqf2 13987	. . . . 5 ⊢ (𝜑 → seq(♯‘𝑀)((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥 ++ ⟨“(𝐹‘𝑥)”⟩)), (ℕ₀ × {(𝑀 ++ ⟨“(𝐹‘𝑀)”⟩)})):(ℤ_≥‘(♯‘𝑀))⟶V)
34	33	fdmd 6729	. . . 4 ⊢ (𝜑 → dom seq(♯‘𝑀)((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥 ++ ⟨“(𝐹‘𝑥)”⟩)), (ℕ₀ × {(𝑀 ++ ⟨“(𝐹‘𝑀)”⟩)})) = (ℤ_≥‘(♯‘𝑀)))
35	24, 34	eleqtrrd 2837	. . 3 ⊢ (𝜑 → 𝑁 ∈ dom seq(♯‘𝑀)((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥 ++ ⟨“(𝐹‘𝑥)”⟩)), (ℕ₀ × {(𝑀 ++ ⟨“(𝐹‘𝑀)”⟩)})))
36		fvco 6990	. . 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 585	. 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 2777	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 397 = wceq 1542 ∈ wcel 2107 Vcvv 3475 ∪ cun 3947 ∩ cin 3948 ⊆ wss 3949 ∅c0 4323 {csn 4629 × cxp 5675 ^◡ccnv 5676 dom cdm 5677 ran crn 5678 “ cima 5680 ∘ ccom 5681 Fun wfun 6538 Fn wfn 6539 ⟶wf 6540 ‘cfv 6544 (class class class)co 7409 ∈ cmpo 7411 0cc0 11110 1c1 11111 + caddc 11113 − cmin 11444 ℕ₀cn0 12472 ℤcz 12558 ℤ_≥cuz 12822 ..^cfzo 13627 seqcseq 13966 ♯chash 14290 Word cword 14464 lastSclsw 14512 ++ cconcat 14520 ⟨“cs1 14545 seq_strcsseq 33382

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 ax-inf2 9636 ax-cnex 11166 ax-resscn 11167 ax-1cn 11168 ax-icn 11169 ax-addcl 11170 ax-addrcl 11171 ax-mulcl 11172 ax-mulrcl 11173 ax-mulcom 11174 ax-addass 11175 ax-mulass 11176 ax-distr 11177 ax-i2m1 11178 ax-1ne0 11179 ax-1rid 11180 ax-rnegex 11181 ax-rrecex 11182 ax-cnre 11183 ax-pre-lttri 11184 ax-pre-lttrn 11185 ax-pre-ltadd 11186 ax-pre-mulgt0 11187

This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-int 4952 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-riota 7365 df-ov 7412 df-oprab 7413 df-mpo 7414 df-om 7856 df-1st 7975 df-2nd 7976 df-frecs 8266 df-wrecs 8297 df-recs 8371 df-rdg 8410 df-1o 8466 df-er 8703 df-map 8822 df-en 8940 df-dom 8941 df-sdom 8942 df-fin 8943 df-card 9934 df-pnf 11250 df-mnf 11251 df-xr 11252 df-ltxr 11253 df-le 11254 df-sub 11446 df-neg 11447 df-nn 12213 df-n0 12473 df-z 12559 df-uz 12823 df-fz 13485 df-fzo 13628 df-seq 13967 df-hash 14291 df-word 14465 df-lsw 14513 df-s1 14546 df-sseq 33383

This theorem is referenced by: sseqp1 33394

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