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Theorem algr0 16016

Description: The value of the algorithm iterator 𝑅 at 0 is the initial state 𝐴. (Contributed by Paul Chapman, 31-Mar-2011.) (Revised by Mario Carneiro, 28-May-2014.)

**Hypotheses**
Ref	Expression
algrf.1	⊢ 𝑍 = (ℤ_≥‘𝑀)
algrf.2	⊢ 𝑅 = seq𝑀((𝐹 ∘ 1^st ), (𝑍 × {𝐴}))
algrf.3	⊢ (𝜑 → 𝑀 ∈ ℤ)
algrf.4	⊢ (𝜑 → 𝐴 ∈ 𝑆)

**Assertion**
Ref	Expression
algr0	⊢ (𝜑 → (𝑅‘𝑀) = 𝐴)

**Proof of Theorem algr0**
Step	Hyp	Ref	Expression
1		algrf.2	. . 3 ⊢ 𝑅 = seq𝑀((𝐹 ∘ 1^st ), (𝑍 × {𝐴}))
2	1	fveq1i 6678	. 2 ⊢ (𝑅‘𝑀) = (seq𝑀((𝐹 ∘ 1^st ), (𝑍 × {𝐴}))‘𝑀)
3		algrf.3	. . . 4 ⊢ (𝜑 → 𝑀 ∈ ℤ)
4		seq1 13476	. . . 4 ⊢ (𝑀 ∈ ℤ → (seq𝑀((𝐹 ∘ 1^st ), (𝑍 × {𝐴}))‘𝑀) = ((𝑍 × {𝐴})‘𝑀))
5	3, 4	syl 17	. . 3 ⊢ (𝜑 → (seq𝑀((𝐹 ∘ 1^st ), (𝑍 × {𝐴}))‘𝑀) = ((𝑍 × {𝐴})‘𝑀))
6		algrf.4	. . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑆)
7		uzid 12342	. . . . . 6 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ (ℤ_≥‘𝑀))
8	3, 7	syl 17	. . . . 5 ⊢ (𝜑 → 𝑀 ∈ (ℤ_≥‘𝑀))
9		algrf.1	. . . . 5 ⊢ 𝑍 = (ℤ_≥‘𝑀)
10	8, 9	eleqtrrdi 2845	. . . 4 ⊢ (𝜑 → 𝑀 ∈ 𝑍)
11		fvconst2g 6977	. . . 4 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝑀 ∈ 𝑍) → ((𝑍 × {𝐴})‘𝑀) = 𝐴)
12	6, 10, 11	syl2anc 587	. . 3 ⊢ (𝜑 → ((𝑍 × {𝐴})‘𝑀) = 𝐴)
13	5, 12	eqtrd 2774	. 2 ⊢ (𝜑 → (seq𝑀((𝐹 ∘ 1^st ), (𝑍 × {𝐴}))‘𝑀) = 𝐴)
14	2, 13	syl5eq 2786	1 ⊢ (𝜑 → (𝑅‘𝑀) = 𝐴)

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2114 {csn 4517 × cxp 5524 ∘ ccom 5530 ‘cfv 6340 1^st c1st 7715 ℤcz 12065 ℤ_≥cuz 12327 seqcseq 13463

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2711 ax-sep 5168 ax-nul 5175 ax-pow 5233 ax-pr 5297 ax-un 7482 ax-cnex 10674 ax-resscn 10675 ax-1cn 10676 ax-icn 10677 ax-addcl 10678 ax-addrcl 10679 ax-mulcl 10680 ax-mulrcl 10681 ax-mulcom 10682 ax-addass 10683 ax-mulass 10684 ax-distr 10685 ax-i2m1 10686 ax-1ne0 10687 ax-1rid 10688 ax-rnegex 10689 ax-rrecex 10690 ax-cnre 10691 ax-pre-lttri 10692 ax-pre-lttrn 10693 ax-pre-ltadd 10694 ax-pre-mulgt0 10695

This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2541 df-eu 2571 df-clab 2718 df-cleq 2731 df-clel 2812 df-nfc 2882 df-ne 2936 df-nel 3040 df-ral 3059 df-rex 3060 df-reu 3061 df-rab 3063 df-v 3401 df-sbc 3682 df-csb 3792 df-dif 3847 df-un 3849 df-in 3851 df-ss 3861 df-pss 3863 df-nul 4213 df-if 4416 df-pw 4491 df-sn 4518 df-pr 4520 df-tp 4522 df-op 4524 df-uni 4798 df-iun 4884 df-br 5032 df-opab 5094 df-mpt 5112 df-tr 5138 df-id 5430 df-eprel 5435 df-po 5443 df-so 5444 df-fr 5484 df-we 5486 df-xp 5532 df-rel 5533 df-cnv 5534 df-co 5535 df-dm 5536 df-rn 5537 df-res 5538 df-ima 5539 df-pred 6130 df-ord 6176 df-on 6177 df-lim 6178 df-suc 6179 df-iota 6298 df-fun 6342 df-fn 6343 df-f 6344 df-f1 6345 df-fo 6346 df-f1o 6347 df-fv 6348 df-riota 7130 df-ov 7176 df-oprab 7177 df-mpo 7178 df-om 7603 df-2nd 7718 df-wrecs 7979 df-recs 8040 df-rdg 8078 df-er 8323 df-en 8559 df-dom 8560 df-sdom 8561 df-pnf 10758 df-mnf 10759 df-xr 10760 df-ltxr 10761 df-le 10762 df-sub 10953 df-neg 10954 df-nn 11720 df-n0 11980 df-z 12066 df-uz 12328 df-seq 13464

This theorem is referenced by: algcvg 16020 eucalg 16031 ovolicc2lem4 24275 bfplem1 35626

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