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Theorem seqf1olem1 12702

Description: Lemma for seqf1o 12704. (Contributed by Mario Carneiro, 26-Feb-2014.) (Revised by Mario Carneiro, 27-May-2014.)

**Hypotheses**
Ref	Expression
seqf1o.1	⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)
seqf1o.2	⊢ ((𝜑 ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) → (𝑥 + 𝑦) = (𝑦 + 𝑥))
seqf1o.3	⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)))
seqf1o.4	⊢ (𝜑 → 𝑁 ∈ (ℤ_≥‘𝑀))
seqf1o.5	⊢ (𝜑 → 𝐶 ⊆ 𝑆)
seqf1olem.5	⊢ (𝜑 → 𝐹:(𝑀...(𝑁 + 1))–1-1-onto→(𝑀...(𝑁 + 1)))
seqf1olem.6	⊢ (𝜑 → 𝐺:(𝑀...(𝑁 + 1))⟶𝐶)
seqf1olem.7	⊢ 𝐽 = (𝑘 ∈ (𝑀...𝑁) ↦ (𝐹‘if(𝑘 < 𝐾, 𝑘, (𝑘 + 1))))
seqf1olem.8	⊢ 𝐾 = (^◡𝐹‘(𝑁 + 1))

**Assertion**
Ref	Expression
seqf1olem1	⊢ (𝜑 → 𝐽:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁))

Distinct variable groups: 𝑥,𝑘,𝑦,𝑧,𝐹 𝑘,𝐺,𝑥,𝑦,𝑧 𝑘,𝑀,𝑥,𝑦,𝑧 + ,𝑘,𝑥,𝑦,𝑧 𝑥,𝐽,𝑦,𝑧 𝑘,𝑁,𝑥,𝑦,𝑧 𝑘,𝐾,𝑥,𝑦,𝑧 𝜑,𝑘,𝑥,𝑦,𝑧 𝑆,𝑘,𝑥,𝑦,𝑧 𝐶,𝑘,𝑥,𝑦,𝑧 Allowed substitution hint: 𝐽(𝑘)

**Proof of Theorem seqf1olem1**
Step	Hyp	Ref	Expression
1		seqf1olem.7	. 2 ⊢ 𝐽 = (𝑘 ∈ (𝑀...𝑁) ↦ (𝐹‘if(𝑘 < 𝐾, 𝑘, (𝑘 + 1))))
2		fvex 6113	. . 3 ⊢ (𝐹‘if(𝑘 < 𝐾, 𝑘, (𝑘 + 1))) ∈ V
3	2	a1i 11	. 2 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → (𝐹‘if(𝑘 < 𝐾, 𝑘, (𝑘 + 1))) ∈ V)
4		fvex 6113	. . . 4 ⊢ (^◡𝐹‘𝑥) ∈ V
5		ovex 6577	. . . 4 ⊢ ((^◡𝐹‘𝑥) − 1) ∈ V
6	4, 5	ifex 4106	. . 3 ⊢ if((^◡𝐹‘𝑥) < 𝐾, (^◡𝐹‘𝑥), ((^◡𝐹‘𝑥) − 1)) ∈ V
7	6	a1i 11	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) → if((^◡𝐹‘𝑥) < 𝐾, (^◡𝐹‘𝑥), ((^◡𝐹‘𝑥) − 1)) ∈ V)
8		iftrue 4042	. . . . . . . . 9 ⊢ (𝑘 < 𝐾 → if(𝑘 < 𝐾, 𝑘, (𝑘 + 1)) = 𝑘)
9	8	fveq2d 6107	. . . . . . . 8 ⊢ (𝑘 < 𝐾 → (𝐹‘if(𝑘 < 𝐾, 𝑘, (𝑘 + 1))) = (𝐹‘𝑘))
10	9	eqeq2d 2620	. . . . . . 7 ⊢ (𝑘 < 𝐾 → (𝑥 = (𝐹‘if(𝑘 < 𝐾, 𝑘, (𝑘 + 1))) ↔ 𝑥 = (𝐹‘𝑘)))
11	10	adantl 481	. . . . . 6 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) ∧ 𝑘 < 𝐾) → (𝑥 = (𝐹‘if(𝑘 < 𝐾, 𝑘, (𝑘 + 1))) ↔ 𝑥 = (𝐹‘𝑘)))
12		simprr 792	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) ∧ (𝑘 < 𝐾 ∧ 𝑥 = (𝐹‘𝑘))) → 𝑥 = (𝐹‘𝑘))
13		elfzelz 12213	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ (𝑀...𝑁) → 𝑘 ∈ ℤ)
14	13	zred 11358	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ (𝑀...𝑁) → 𝑘 ∈ ℝ)
15	14	ad2antlr 759	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) ∧ (𝑘 < 𝐾 ∧ 𝑥 = (𝐹‘𝑘))) → 𝑘 ∈ ℝ)
16		simprl 790	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) ∧ (𝑘 < 𝐾 ∧ 𝑥 = (𝐹‘𝑘))) → 𝑘 < 𝐾)
17	15, 16	gtned 10051	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) ∧ (𝑘 < 𝐾 ∧ 𝑥 = (𝐹‘𝑘))) → 𝐾 ≠ 𝑘)
18		seqf1olem.5	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝐹:(𝑀...(𝑁 + 1))–1-1-onto→(𝑀...(𝑁 + 1)))
19		f1of 6050	. . . . . . . . . . . . . . . . 17 ⊢ (𝐹:(𝑀...(𝑁 + 1))–1-1-onto→(𝑀...(𝑁 + 1)) → 𝐹:(𝑀...(𝑁 + 1))⟶(𝑀...(𝑁 + 1)))
20	18, 19	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐹:(𝑀...(𝑁 + 1))⟶(𝑀...(𝑁 + 1)))
21	20	ad2antrr 758	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) ∧ (𝑘 < 𝐾 ∧ 𝑥 = (𝐹‘𝑘))) → 𝐹:(𝑀...(𝑁 + 1))⟶(𝑀...(𝑁 + 1)))
22		fzssp1 12255	. . . . . . . . . . . . . . . 16 ⊢ (𝑀...𝑁) ⊆ (𝑀...(𝑁 + 1))
23		simplr 788	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) ∧ (𝑘 < 𝐾 ∧ 𝑥 = (𝐹‘𝑘))) → 𝑘 ∈ (𝑀...𝑁))
24	22, 23	sseldi 3566	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) ∧ (𝑘 < 𝐾 ∧ 𝑥 = (𝐹‘𝑘))) → 𝑘 ∈ (𝑀...(𝑁 + 1)))
25	21, 24	ffvelrnd 6268	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) ∧ (𝑘 < 𝐾 ∧ 𝑥 = (𝐹‘𝑘))) → (𝐹‘𝑘) ∈ (𝑀...(𝑁 + 1)))
26		seqf1o.4	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑁 ∈ (ℤ_≥‘𝑀))
27		elfzp1 12261	. . . . . . . . . . . . . . . 16 ⊢ (𝑁 ∈ (ℤ_≥‘𝑀) → ((𝐹‘𝑘) ∈ (𝑀...(𝑁 + 1)) ↔ ((𝐹‘𝑘) ∈ (𝑀...𝑁) ∨ (𝐹‘𝑘) = (𝑁 + 1))))
28	26, 27	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((𝐹‘𝑘) ∈ (𝑀...(𝑁 + 1)) ↔ ((𝐹‘𝑘) ∈ (𝑀...𝑁) ∨ (𝐹‘𝑘) = (𝑁 + 1))))
29	28	ad2antrr 758	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) ∧ (𝑘 < 𝐾 ∧ 𝑥 = (𝐹‘𝑘))) → ((𝐹‘𝑘) ∈ (𝑀...(𝑁 + 1)) ↔ ((𝐹‘𝑘) ∈ (𝑀...𝑁) ∨ (𝐹‘𝑘) = (𝑁 + 1))))
30	25, 29	mpbid 221	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) ∧ (𝑘 < 𝐾 ∧ 𝑥 = (𝐹‘𝑘))) → ((𝐹‘𝑘) ∈ (𝑀...𝑁) ∨ (𝐹‘𝑘) = (𝑁 + 1)))
31	30	ord 391	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) ∧ (𝑘 < 𝐾 ∧ 𝑥 = (𝐹‘𝑘))) → (¬ (𝐹‘𝑘) ∈ (𝑀...𝑁) → (𝐹‘𝑘) = (𝑁 + 1)))
32	18	ad2antrr 758	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) ∧ (𝑘 < 𝐾 ∧ 𝑥 = (𝐹‘𝑘))) → 𝐹:(𝑀...(𝑁 + 1))–1-1-onto→(𝑀...(𝑁 + 1)))
33		f1ocnvfv 6434	. . . . . . . . . . . . . 14 ⊢ ((𝐹:(𝑀...(𝑁 + 1))–1-1-onto→(𝑀...(𝑁 + 1)) ∧ 𝑘 ∈ (𝑀...(𝑁 + 1))) → ((𝐹‘𝑘) = (𝑁 + 1) → (^◡𝐹‘(𝑁 + 1)) = 𝑘))
34	32, 24, 33	syl2anc 691	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) ∧ (𝑘 < 𝐾 ∧ 𝑥 = (𝐹‘𝑘))) → ((𝐹‘𝑘) = (𝑁 + 1) → (^◡𝐹‘(𝑁 + 1)) = 𝑘))
35		seqf1olem.8	. . . . . . . . . . . . . 14 ⊢ 𝐾 = (^◡𝐹‘(𝑁 + 1))
36	35	eqeq1i 2615	. . . . . . . . . . . . 13 ⊢ (𝐾 = 𝑘 ↔ (^◡𝐹‘(𝑁 + 1)) = 𝑘)
37	34, 36	syl6ibr 241	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) ∧ (𝑘 < 𝐾 ∧ 𝑥 = (𝐹‘𝑘))) → ((𝐹‘𝑘) = (𝑁 + 1) → 𝐾 = 𝑘))
38	31, 37	syld 46	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) ∧ (𝑘 < 𝐾 ∧ 𝑥 = (𝐹‘𝑘))) → (¬ (𝐹‘𝑘) ∈ (𝑀...𝑁) → 𝐾 = 𝑘))
39	38	necon1ad 2799	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) ∧ (𝑘 < 𝐾 ∧ 𝑥 = (𝐹‘𝑘))) → (𝐾 ≠ 𝑘 → (𝐹‘𝑘) ∈ (𝑀...𝑁)))
40	17, 39	mpd 15	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) ∧ (𝑘 < 𝐾 ∧ 𝑥 = (𝐹‘𝑘))) → (𝐹‘𝑘) ∈ (𝑀...𝑁))
41	12, 40	eqeltrd 2688	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) ∧ (𝑘 < 𝐾 ∧ 𝑥 = (𝐹‘𝑘))) → 𝑥 ∈ (𝑀...𝑁))
42	12	eqcomd 2616	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) ∧ (𝑘 < 𝐾 ∧ 𝑥 = (𝐹‘𝑘))) → (𝐹‘𝑘) = 𝑥)
43		f1ocnvfv 6434	. . . . . . . . . . . . 13 ⊢ ((𝐹:(𝑀...(𝑁 + 1))–1-1-onto→(𝑀...(𝑁 + 1)) ∧ 𝑘 ∈ (𝑀...(𝑁 + 1))) → ((𝐹‘𝑘) = 𝑥 → (^◡𝐹‘𝑥) = 𝑘))
44	32, 24, 43	syl2anc 691	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) ∧ (𝑘 < 𝐾 ∧ 𝑥 = (𝐹‘𝑘))) → ((𝐹‘𝑘) = 𝑥 → (^◡𝐹‘𝑥) = 𝑘))
45	42, 44	mpd 15	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) ∧ (𝑘 < 𝐾 ∧ 𝑥 = (𝐹‘𝑘))) → (^◡𝐹‘𝑥) = 𝑘)
46	45, 16	eqbrtrd 4605	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) ∧ (𝑘 < 𝐾 ∧ 𝑥 = (𝐹‘𝑘))) → (^◡𝐹‘𝑥) < 𝐾)
47		iftrue 4042	. . . . . . . . . 10 ⊢ ((^◡𝐹‘𝑥) < 𝐾 → if((^◡𝐹‘𝑥) < 𝐾, (^◡𝐹‘𝑥), ((^◡𝐹‘𝑥) − 1)) = (^◡𝐹‘𝑥))
48	46, 47	syl 17	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) ∧ (𝑘 < 𝐾 ∧ 𝑥 = (𝐹‘𝑘))) → if((^◡𝐹‘𝑥) < 𝐾, (^◡𝐹‘𝑥), ((^◡𝐹‘𝑥) − 1)) = (^◡𝐹‘𝑥))
49	48, 45	eqtr2d 2645	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) ∧ (𝑘 < 𝐾 ∧ 𝑥 = (𝐹‘𝑘))) → 𝑘 = if((^◡𝐹‘𝑥) < 𝐾, (^◡𝐹‘𝑥), ((^◡𝐹‘𝑥) − 1)))
50	41, 49	jca 553	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) ∧ (𝑘 < 𝐾 ∧ 𝑥 = (𝐹‘𝑘))) → (𝑥 ∈ (𝑀...𝑁) ∧ 𝑘 = if((^◡𝐹‘𝑥) < 𝐾, (^◡𝐹‘𝑥), ((^◡𝐹‘𝑥) − 1))))
51	50	expr 641	. . . . . 6 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) ∧ 𝑘 < 𝐾) → (𝑥 = (𝐹‘𝑘) → (𝑥 ∈ (𝑀...𝑁) ∧ 𝑘 = if((^◡𝐹‘𝑥) < 𝐾, (^◡𝐹‘𝑥), ((^◡𝐹‘𝑥) − 1)))))
52	11, 51	sylbid 229	. . . . 5 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) ∧ 𝑘 < 𝐾) → (𝑥 = (𝐹‘if(𝑘 < 𝐾, 𝑘, (𝑘 + 1))) → (𝑥 ∈ (𝑀...𝑁) ∧ 𝑘 = if((^◡𝐹‘𝑥) < 𝐾, (^◡𝐹‘𝑥), ((^◡𝐹‘𝑥) − 1)))))
53		iffalse 4045	. . . . . . . . 9 ⊢ (¬ 𝑘 < 𝐾 → if(𝑘 < 𝐾, 𝑘, (𝑘 + 1)) = (𝑘 + 1))
54	53	fveq2d 6107	. . . . . . . 8 ⊢ (¬ 𝑘 < 𝐾 → (𝐹‘if(𝑘 < 𝐾, 𝑘, (𝑘 + 1))) = (𝐹‘(𝑘 + 1)))
55	54	eqeq2d 2620	. . . . . . 7 ⊢ (¬ 𝑘 < 𝐾 → (𝑥 = (𝐹‘if(𝑘 < 𝐾, 𝑘, (𝑘 + 1))) ↔ 𝑥 = (𝐹‘(𝑘 + 1))))
56	55	adantl 481	. . . . . 6 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) ∧ ¬ 𝑘 < 𝐾) → (𝑥 = (𝐹‘if(𝑘 < 𝐾, 𝑘, (𝑘 + 1))) ↔ 𝑥 = (𝐹‘(𝑘 + 1))))
57		simprr 792	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) ∧ (¬ 𝑘 < 𝐾 ∧ 𝑥 = (𝐹‘(𝑘 + 1)))) → 𝑥 = (𝐹‘(𝑘 + 1)))
58		f1ocnv 6062	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐹:(𝑀...(𝑁 + 1))–1-1-onto→(𝑀...(𝑁 + 1)) → ^◡𝐹:(𝑀...(𝑁 + 1))–1-1-onto→(𝑀...(𝑁 + 1)))
59	18, 58	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → ^◡𝐹:(𝑀...(𝑁 + 1))–1-1-onto→(𝑀...(𝑁 + 1)))
60		f1of1 6049	. . . . . . . . . . . . . . . . . 18 ⊢ (^◡𝐹:(𝑀...(𝑁 + 1))–1-1-onto→(𝑀...(𝑁 + 1)) → ^◡𝐹:(𝑀...(𝑁 + 1))–1-1→(𝑀...(𝑁 + 1)))
61	59, 60	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ^◡𝐹:(𝑀...(𝑁 + 1))–1-1→(𝑀...(𝑁 + 1)))
62		f1f 6014	. . . . . . . . . . . . . . . . 17 ⊢ (^◡𝐹:(𝑀...(𝑁 + 1))–1-1→(𝑀...(𝑁 + 1)) → ^◡𝐹:(𝑀...(𝑁 + 1))⟶(𝑀...(𝑁 + 1)))
63	61, 62	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ^◡𝐹:(𝑀...(𝑁 + 1))⟶(𝑀...(𝑁 + 1)))
64		peano2uz 11617	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑁 ∈ (ℤ_≥‘𝑀) → (𝑁 + 1) ∈ (ℤ_≥‘𝑀))
65	26, 64	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝑁 + 1) ∈ (ℤ_≥‘𝑀))
66		eluzfz2 12220	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑁 + 1) ∈ (ℤ_≥‘𝑀) → (𝑁 + 1) ∈ (𝑀...(𝑁 + 1)))
67	65, 66	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝑁 + 1) ∈ (𝑀...(𝑁 + 1)))
68	63, 67	ffvelrnd 6268	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (^◡𝐹‘(𝑁 + 1)) ∈ (𝑀...(𝑁 + 1)))
69	35, 68	syl5eqel 2692	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐾 ∈ (𝑀...(𝑁 + 1)))
70		elfzelz 12213	. . . . . . . . . . . . . 14 ⊢ (𝐾 ∈ (𝑀...(𝑁 + 1)) → 𝐾 ∈ ℤ)
71	69, 70	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐾 ∈ ℤ)
72	71	zred 11358	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐾 ∈ ℝ)
73	72	ad2antrr 758	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) ∧ (¬ 𝑘 < 𝐾 ∧ 𝑥 = (𝐹‘(𝑘 + 1)))) → 𝐾 ∈ ℝ)
74	14	ad2antlr 759	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) ∧ (¬ 𝑘 < 𝐾 ∧ 𝑥 = (𝐹‘(𝑘 + 1)))) → 𝑘 ∈ ℝ)
75		peano2re 10088	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ ℝ → (𝑘 + 1) ∈ ℝ)
76	74, 75	syl 17	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) ∧ (¬ 𝑘 < 𝐾 ∧ 𝑥 = (𝐹‘(𝑘 + 1)))) → (𝑘 + 1) ∈ ℝ)
77		simprl 790	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) ∧ (¬ 𝑘 < 𝐾 ∧ 𝑥 = (𝐹‘(𝑘 + 1)))) → ¬ 𝑘 < 𝐾)
78	73, 74	lenltd 10062	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) ∧ (¬ 𝑘 < 𝐾 ∧ 𝑥 = (𝐹‘(𝑘 + 1)))) → (𝐾 ≤ 𝑘 ↔ ¬ 𝑘 < 𝐾))
79	77, 78	mpbird 246	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) ∧ (¬ 𝑘 < 𝐾 ∧ 𝑥 = (𝐹‘(𝑘 + 1)))) → 𝐾 ≤ 𝑘)
80	74	ltp1d 10833	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) ∧ (¬ 𝑘 < 𝐾 ∧ 𝑥 = (𝐹‘(𝑘 + 1)))) → 𝑘 < (𝑘 + 1))
81	73, 74, 76, 79, 80	lelttrd 10074	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) ∧ (¬ 𝑘 < 𝐾 ∧ 𝑥 = (𝐹‘(𝑘 + 1)))) → 𝐾 < (𝑘 + 1))
82	73, 81	ltned 10052	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) ∧ (¬ 𝑘 < 𝐾 ∧ 𝑥 = (𝐹‘(𝑘 + 1)))) → 𝐾 ≠ (𝑘 + 1))
83	20	ad2antrr 758	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) ∧ (¬ 𝑘 < 𝐾 ∧ 𝑥 = (𝐹‘(𝑘 + 1)))) → 𝐹:(𝑀...(𝑁 + 1))⟶(𝑀...(𝑁 + 1)))
84		fzp1elp1 12264	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 ∈ (𝑀...𝑁) → (𝑘 + 1) ∈ (𝑀...(𝑁 + 1)))
85	84	ad2antlr 759	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) ∧ (¬ 𝑘 < 𝐾 ∧ 𝑥 = (𝐹‘(𝑘 + 1)))) → (𝑘 + 1) ∈ (𝑀...(𝑁 + 1)))
86	83, 85	ffvelrnd 6268	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) ∧ (¬ 𝑘 < 𝐾 ∧ 𝑥 = (𝐹‘(𝑘 + 1)))) → (𝐹‘(𝑘 + 1)) ∈ (𝑀...(𝑁 + 1)))
87		elfzp1 12261	. . . . . . . . . . . . . . . 16 ⊢ (𝑁 ∈ (ℤ_≥‘𝑀) → ((𝐹‘(𝑘 + 1)) ∈ (𝑀...(𝑁 + 1)) ↔ ((𝐹‘(𝑘 + 1)) ∈ (𝑀...𝑁) ∨ (𝐹‘(𝑘 + 1)) = (𝑁 + 1))))
88	26, 87	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((𝐹‘(𝑘 + 1)) ∈ (𝑀...(𝑁 + 1)) ↔ ((𝐹‘(𝑘 + 1)) ∈ (𝑀...𝑁) ∨ (𝐹‘(𝑘 + 1)) = (𝑁 + 1))))
89	88	ad2antrr 758	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) ∧ (¬ 𝑘 < 𝐾 ∧ 𝑥 = (𝐹‘(𝑘 + 1)))) → ((𝐹‘(𝑘 + 1)) ∈ (𝑀...(𝑁 + 1)) ↔ ((𝐹‘(𝑘 + 1)) ∈ (𝑀...𝑁) ∨ (𝐹‘(𝑘 + 1)) = (𝑁 + 1))))
90	86, 89	mpbid 221	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) ∧ (¬ 𝑘 < 𝐾 ∧ 𝑥 = (𝐹‘(𝑘 + 1)))) → ((𝐹‘(𝑘 + 1)) ∈ (𝑀...𝑁) ∨ (𝐹‘(𝑘 + 1)) = (𝑁 + 1)))
91	90	ord 391	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) ∧ (¬ 𝑘 < 𝐾 ∧ 𝑥 = (𝐹‘(𝑘 + 1)))) → (¬ (𝐹‘(𝑘 + 1)) ∈ (𝑀...𝑁) → (𝐹‘(𝑘 + 1)) = (𝑁 + 1)))
92	18	ad2antrr 758	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) ∧ (¬ 𝑘 < 𝐾 ∧ 𝑥 = (𝐹‘(𝑘 + 1)))) → 𝐹:(𝑀...(𝑁 + 1))–1-1-onto→(𝑀...(𝑁 + 1)))
93		f1ocnvfv 6434	. . . . . . . . . . . . . 14 ⊢ ((𝐹:(𝑀...(𝑁 + 1))–1-1-onto→(𝑀...(𝑁 + 1)) ∧ (𝑘 + 1) ∈ (𝑀...(𝑁 + 1))) → ((𝐹‘(𝑘 + 1)) = (𝑁 + 1) → (^◡𝐹‘(𝑁 + 1)) = (𝑘 + 1)))
94	92, 85, 93	syl2anc 691	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) ∧ (¬ 𝑘 < 𝐾 ∧ 𝑥 = (𝐹‘(𝑘 + 1)))) → ((𝐹‘(𝑘 + 1)) = (𝑁 + 1) → (^◡𝐹‘(𝑁 + 1)) = (𝑘 + 1)))
95	35	eqeq1i 2615	. . . . . . . . . . . . 13 ⊢ (𝐾 = (𝑘 + 1) ↔ (^◡𝐹‘(𝑁 + 1)) = (𝑘 + 1))
96	94, 95	syl6ibr 241	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) ∧ (¬ 𝑘 < 𝐾 ∧ 𝑥 = (𝐹‘(𝑘 + 1)))) → ((𝐹‘(𝑘 + 1)) = (𝑁 + 1) → 𝐾 = (𝑘 + 1)))
97	91, 96	syld 46	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) ∧ (¬ 𝑘 < 𝐾 ∧ 𝑥 = (𝐹‘(𝑘 + 1)))) → (¬ (𝐹‘(𝑘 + 1)) ∈ (𝑀...𝑁) → 𝐾 = (𝑘 + 1)))
98	97	necon1ad 2799	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) ∧ (¬ 𝑘 < 𝐾 ∧ 𝑥 = (𝐹‘(𝑘 + 1)))) → (𝐾 ≠ (𝑘 + 1) → (𝐹‘(𝑘 + 1)) ∈ (𝑀...𝑁)))
99	82, 98	mpd 15	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) ∧ (¬ 𝑘 < 𝐾 ∧ 𝑥 = (𝐹‘(𝑘 + 1)))) → (𝐹‘(𝑘 + 1)) ∈ (𝑀...𝑁))
100	57, 99	eqeltrd 2688	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) ∧ (¬ 𝑘 < 𝐾 ∧ 𝑥 = (𝐹‘(𝑘 + 1)))) → 𝑥 ∈ (𝑀...𝑁))
101	57	eqcomd 2616	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) ∧ (¬ 𝑘 < 𝐾 ∧ 𝑥 = (𝐹‘(𝑘 + 1)))) → (𝐹‘(𝑘 + 1)) = 𝑥)
102		f1ocnvfv 6434	. . . . . . . . . . . . . . 15 ⊢ ((𝐹:(𝑀...(𝑁 + 1))–1-1-onto→(𝑀...(𝑁 + 1)) ∧ (𝑘 + 1) ∈ (𝑀...(𝑁 + 1))) → ((𝐹‘(𝑘 + 1)) = 𝑥 → (^◡𝐹‘𝑥) = (𝑘 + 1)))
103	92, 85, 102	syl2anc 691	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) ∧ (¬ 𝑘 < 𝐾 ∧ 𝑥 = (𝐹‘(𝑘 + 1)))) → ((𝐹‘(𝑘 + 1)) = 𝑥 → (^◡𝐹‘𝑥) = (𝑘 + 1)))
104	101, 103	mpd 15	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) ∧ (¬ 𝑘 < 𝐾 ∧ 𝑥 = (𝐹‘(𝑘 + 1)))) → (^◡𝐹‘𝑥) = (𝑘 + 1))
105	104	breq1d 4593	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) ∧ (¬ 𝑘 < 𝐾 ∧ 𝑥 = (𝐹‘(𝑘 + 1)))) → ((^◡𝐹‘𝑥) < 𝐾 ↔ (𝑘 + 1) < 𝐾))
106		lttr 9993	. . . . . . . . . . . . . 14 ⊢ ((𝑘 ∈ ℝ ∧ (𝑘 + 1) ∈ ℝ ∧ 𝐾 ∈ ℝ) → ((𝑘 < (𝑘 + 1) ∧ (𝑘 + 1) < 𝐾) → 𝑘 < 𝐾))
107	74, 76, 73, 106	syl3anc 1318	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) ∧ (¬ 𝑘 < 𝐾 ∧ 𝑥 = (𝐹‘(𝑘 + 1)))) → ((𝑘 < (𝑘 + 1) ∧ (𝑘 + 1) < 𝐾) → 𝑘 < 𝐾))
108	80, 107	mpand 707	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) ∧ (¬ 𝑘 < 𝐾 ∧ 𝑥 = (𝐹‘(𝑘 + 1)))) → ((𝑘 + 1) < 𝐾 → 𝑘 < 𝐾))
109	105, 108	sylbid 229	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) ∧ (¬ 𝑘 < 𝐾 ∧ 𝑥 = (𝐹‘(𝑘 + 1)))) → ((^◡𝐹‘𝑥) < 𝐾 → 𝑘 < 𝐾))
110	77, 109	mtod 188	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) ∧ (¬ 𝑘 < 𝐾 ∧ 𝑥 = (𝐹‘(𝑘 + 1)))) → ¬ (^◡𝐹‘𝑥) < 𝐾)
111		iffalse 4045	. . . . . . . . . 10 ⊢ (¬ (^◡𝐹‘𝑥) < 𝐾 → if((^◡𝐹‘𝑥) < 𝐾, (^◡𝐹‘𝑥), ((^◡𝐹‘𝑥) − 1)) = ((^◡𝐹‘𝑥) − 1))
112	110, 111	syl 17	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) ∧ (¬ 𝑘 < 𝐾 ∧ 𝑥 = (𝐹‘(𝑘 + 1)))) → if((^◡𝐹‘𝑥) < 𝐾, (^◡𝐹‘𝑥), ((^◡𝐹‘𝑥) − 1)) = ((^◡𝐹‘𝑥) − 1))
113	104	oveq1d 6564	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) ∧ (¬ 𝑘 < 𝐾 ∧ 𝑥 = (𝐹‘(𝑘 + 1)))) → ((^◡𝐹‘𝑥) − 1) = ((𝑘 + 1) − 1))
114	74	recnd 9947	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) ∧ (¬ 𝑘 < 𝐾 ∧ 𝑥 = (𝐹‘(𝑘 + 1)))) → 𝑘 ∈ ℂ)
115		ax-1cn 9873	. . . . . . . . . 10 ⊢ 1 ∈ ℂ
116		pncan 10166	. . . . . . . . . 10 ⊢ ((𝑘 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑘 + 1) − 1) = 𝑘)
117	114, 115, 116	sylancl 693	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) ∧ (¬ 𝑘 < 𝐾 ∧ 𝑥 = (𝐹‘(𝑘 + 1)))) → ((𝑘 + 1) − 1) = 𝑘)
118	112, 113, 117	3eqtrrd 2649	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) ∧ (¬ 𝑘 < 𝐾 ∧ 𝑥 = (𝐹‘(𝑘 + 1)))) → 𝑘 = if((^◡𝐹‘𝑥) < 𝐾, (^◡𝐹‘𝑥), ((^◡𝐹‘𝑥) − 1)))
119	100, 118	jca 553	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) ∧ (¬ 𝑘 < 𝐾 ∧ 𝑥 = (𝐹‘(𝑘 + 1)))) → (𝑥 ∈ (𝑀...𝑁) ∧ 𝑘 = if((^◡𝐹‘𝑥) < 𝐾, (^◡𝐹‘𝑥), ((^◡𝐹‘𝑥) − 1))))
120	119	expr 641	. . . . . 6 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) ∧ ¬ 𝑘 < 𝐾) → (𝑥 = (𝐹‘(𝑘 + 1)) → (𝑥 ∈ (𝑀...𝑁) ∧ 𝑘 = if((^◡𝐹‘𝑥) < 𝐾, (^◡𝐹‘𝑥), ((^◡𝐹‘𝑥) − 1)))))
121	56, 120	sylbid 229	. . . . 5 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) ∧ ¬ 𝑘 < 𝐾) → (𝑥 = (𝐹‘if(𝑘 < 𝐾, 𝑘, (𝑘 + 1))) → (𝑥 ∈ (𝑀...𝑁) ∧ 𝑘 = if((^◡𝐹‘𝑥) < 𝐾, (^◡𝐹‘𝑥), ((^◡𝐹‘𝑥) − 1)))))
122	52, 121	pm2.61dan 828	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → (𝑥 = (𝐹‘if(𝑘 < 𝐾, 𝑘, (𝑘 + 1))) → (𝑥 ∈ (𝑀...𝑁) ∧ 𝑘 = if((^◡𝐹‘𝑥) < 𝐾, (^◡𝐹‘𝑥), ((^◡𝐹‘𝑥) − 1)))))
123	122	expimpd 627	. . 3 ⊢ (𝜑 → ((𝑘 ∈ (𝑀...𝑁) ∧ 𝑥 = (𝐹‘if(𝑘 < 𝐾, 𝑘, (𝑘 + 1)))) → (𝑥 ∈ (𝑀...𝑁) ∧ 𝑘 = if((^◡𝐹‘𝑥) < 𝐾, (^◡𝐹‘𝑥), ((^◡𝐹‘𝑥) − 1)))))
124	47	eqeq2d 2620	. . . . . . 7 ⊢ ((^◡𝐹‘𝑥) < 𝐾 → (𝑘 = if((^◡𝐹‘𝑥) < 𝐾, (^◡𝐹‘𝑥), ((^◡𝐹‘𝑥) − 1)) ↔ 𝑘 = (^◡𝐹‘𝑥)))
125	124	adantl 481	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) ∧ (^◡𝐹‘𝑥) < 𝐾) → (𝑘 = if((^◡𝐹‘𝑥) < 𝐾, (^◡𝐹‘𝑥), ((^◡𝐹‘𝑥) − 1)) ↔ 𝑘 = (^◡𝐹‘𝑥)))
126		simprr 792	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) ∧ ((^◡𝐹‘𝑥) < 𝐾 ∧ 𝑘 = (^◡𝐹‘𝑥))) → 𝑘 = (^◡𝐹‘𝑥))
127	63	ad2antrr 758	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) ∧ ((^◡𝐹‘𝑥) < 𝐾 ∧ 𝑘 = (^◡𝐹‘𝑥))) → ^◡𝐹:(𝑀...(𝑁 + 1))⟶(𝑀...(𝑁 + 1)))
128		simplr 788	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) ∧ ((^◡𝐹‘𝑥) < 𝐾 ∧ 𝑘 = (^◡𝐹‘𝑥))) → 𝑥 ∈ (𝑀...𝑁))
129	22, 128	sseldi 3566	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) ∧ ((^◡𝐹‘𝑥) < 𝐾 ∧ 𝑘 = (^◡𝐹‘𝑥))) → 𝑥 ∈ (𝑀...(𝑁 + 1)))
130	127, 129	ffvelrnd 6268	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) ∧ ((^◡𝐹‘𝑥) < 𝐾 ∧ 𝑘 = (^◡𝐹‘𝑥))) → (^◡𝐹‘𝑥) ∈ (𝑀...(𝑁 + 1)))
131	126, 130	eqeltrd 2688	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) ∧ ((^◡𝐹‘𝑥) < 𝐾 ∧ 𝑘 = (^◡𝐹‘𝑥))) → 𝑘 ∈ (𝑀...(𝑁 + 1)))
132		elfzle1 12215	. . . . . . . . . 10 ⊢ (𝑘 ∈ (𝑀...(𝑁 + 1)) → 𝑀 ≤ 𝑘)
133	131, 132	syl 17	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) ∧ ((^◡𝐹‘𝑥) < 𝐾 ∧ 𝑘 = (^◡𝐹‘𝑥))) → 𝑀 ≤ 𝑘)
134		elfzelz 12213	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ (𝑀...(𝑁 + 1)) → 𝑘 ∈ ℤ)
135	131, 134	syl 17	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) ∧ ((^◡𝐹‘𝑥) < 𝐾 ∧ 𝑘 = (^◡𝐹‘𝑥))) → 𝑘 ∈ ℤ)
136	135	zred 11358	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) ∧ ((^◡𝐹‘𝑥) < 𝐾 ∧ 𝑘 = (^◡𝐹‘𝑥))) → 𝑘 ∈ ℝ)
137	72	ad2antrr 758	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) ∧ ((^◡𝐹‘𝑥) < 𝐾 ∧ 𝑘 = (^◡𝐹‘𝑥))) → 𝐾 ∈ ℝ)
138		eluzelz 11573	. . . . . . . . . . . . . . 15 ⊢ (𝑁 ∈ (ℤ_≥‘𝑀) → 𝑁 ∈ ℤ)
139	26, 138	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑁 ∈ ℤ)
140	139	peano2zd 11361	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑁 + 1) ∈ ℤ)
141	140	zred 11358	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑁 + 1) ∈ ℝ)
142	141	ad2antrr 758	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) ∧ ((^◡𝐹‘𝑥) < 𝐾 ∧ 𝑘 = (^◡𝐹‘𝑥))) → (𝑁 + 1) ∈ ℝ)
143		simprl 790	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) ∧ ((^◡𝐹‘𝑥) < 𝐾 ∧ 𝑘 = (^◡𝐹‘𝑥))) → (^◡𝐹‘𝑥) < 𝐾)
144	126, 143	eqbrtrd 4605	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) ∧ ((^◡𝐹‘𝑥) < 𝐾 ∧ 𝑘 = (^◡𝐹‘𝑥))) → 𝑘 < 𝐾)
145		elfzle2 12216	. . . . . . . . . . . . 13 ⊢ (𝐾 ∈ (𝑀...(𝑁 + 1)) → 𝐾 ≤ (𝑁 + 1))
146	69, 145	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐾 ≤ (𝑁 + 1))
147	146	ad2antrr 758	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) ∧ ((^◡𝐹‘𝑥) < 𝐾 ∧ 𝑘 = (^◡𝐹‘𝑥))) → 𝐾 ≤ (𝑁 + 1))
148	136, 137, 142, 144, 147	ltletrd 10076	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) ∧ ((^◡𝐹‘𝑥) < 𝐾 ∧ 𝑘 = (^◡𝐹‘𝑥))) → 𝑘 < (𝑁 + 1))
149	139	ad2antrr 758	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) ∧ ((^◡𝐹‘𝑥) < 𝐾 ∧ 𝑘 = (^◡𝐹‘𝑥))) → 𝑁 ∈ ℤ)
150		zleltp1 11305	. . . . . . . . . . 11 ⊢ ((𝑘 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑘 ≤ 𝑁 ↔ 𝑘 < (𝑁 + 1)))
151	135, 149, 150	syl2anc 691	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) ∧ ((^◡𝐹‘𝑥) < 𝐾 ∧ 𝑘 = (^◡𝐹‘𝑥))) → (𝑘 ≤ 𝑁 ↔ 𝑘 < (𝑁 + 1)))
152	148, 151	mpbird 246	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) ∧ ((^◡𝐹‘𝑥) < 𝐾 ∧ 𝑘 = (^◡𝐹‘𝑥))) → 𝑘 ≤ 𝑁)
153		eluzel2 11568	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ (ℤ_≥‘𝑀) → 𝑀 ∈ ℤ)
154	26, 153	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑀 ∈ ℤ)
155	154	ad2antrr 758	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) ∧ ((^◡𝐹‘𝑥) < 𝐾 ∧ 𝑘 = (^◡𝐹‘𝑥))) → 𝑀 ∈ ℤ)
156		elfz 12203	. . . . . . . . . 10 ⊢ ((𝑘 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑘 ∈ (𝑀...𝑁) ↔ (𝑀 ≤ 𝑘 ∧ 𝑘 ≤ 𝑁)))
157	135, 155, 149, 156	syl3anc 1318	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) ∧ ((^◡𝐹‘𝑥) < 𝐾 ∧ 𝑘 = (^◡𝐹‘𝑥))) → (𝑘 ∈ (𝑀...𝑁) ↔ (𝑀 ≤ 𝑘 ∧ 𝑘 ≤ 𝑁)))
158	133, 152, 157	mpbir2and 959	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) ∧ ((^◡𝐹‘𝑥) < 𝐾 ∧ 𝑘 = (^◡𝐹‘𝑥))) → 𝑘 ∈ (𝑀...𝑁))
159	144, 9	syl 17	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) ∧ ((^◡𝐹‘𝑥) < 𝐾 ∧ 𝑘 = (^◡𝐹‘𝑥))) → (𝐹‘if(𝑘 < 𝐾, 𝑘, (𝑘 + 1))) = (𝐹‘𝑘))
160	126	fveq2d 6107	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) ∧ ((^◡𝐹‘𝑥) < 𝐾 ∧ 𝑘 = (^◡𝐹‘𝑥))) → (𝐹‘𝑘) = (𝐹‘(^◡𝐹‘𝑥)))
161	18	ad2antrr 758	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) ∧ ((^◡𝐹‘𝑥) < 𝐾 ∧ 𝑘 = (^◡𝐹‘𝑥))) → 𝐹:(𝑀...(𝑁 + 1))–1-1-onto→(𝑀...(𝑁 + 1)))
162		f1ocnvfv2 6433	. . . . . . . . . 10 ⊢ ((𝐹:(𝑀...(𝑁 + 1))–1-1-onto→(𝑀...(𝑁 + 1)) ∧ 𝑥 ∈ (𝑀...(𝑁 + 1))) → (𝐹‘(^◡𝐹‘𝑥)) = 𝑥)
163	161, 129, 162	syl2anc 691	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) ∧ ((^◡𝐹‘𝑥) < 𝐾 ∧ 𝑘 = (^◡𝐹‘𝑥))) → (𝐹‘(^◡𝐹‘𝑥)) = 𝑥)
164	159, 160, 163	3eqtrrd 2649	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) ∧ ((^◡𝐹‘𝑥) < 𝐾 ∧ 𝑘 = (^◡𝐹‘𝑥))) → 𝑥 = (𝐹‘if(𝑘 < 𝐾, 𝑘, (𝑘 + 1))))
165	158, 164	jca 553	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) ∧ ((^◡𝐹‘𝑥) < 𝐾 ∧ 𝑘 = (^◡𝐹‘𝑥))) → (𝑘 ∈ (𝑀...𝑁) ∧ 𝑥 = (𝐹‘if(𝑘 < 𝐾, 𝑘, (𝑘 + 1)))))
166	165	expr 641	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) ∧ (^◡𝐹‘𝑥) < 𝐾) → (𝑘 = (^◡𝐹‘𝑥) → (𝑘 ∈ (𝑀...𝑁) ∧ 𝑥 = (𝐹‘if(𝑘 < 𝐾, 𝑘, (𝑘 + 1))))))
167	125, 166	sylbid 229	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) ∧ (^◡𝐹‘𝑥) < 𝐾) → (𝑘 = if((^◡𝐹‘𝑥) < 𝐾, (^◡𝐹‘𝑥), ((^◡𝐹‘𝑥) − 1)) → (𝑘 ∈ (𝑀...𝑁) ∧ 𝑥 = (𝐹‘if(𝑘 < 𝐾, 𝑘, (𝑘 + 1))))))
168	111	eqeq2d 2620	. . . . . . 7 ⊢ (¬ (^◡𝐹‘𝑥) < 𝐾 → (𝑘 = if((^◡𝐹‘𝑥) < 𝐾, (^◡𝐹‘𝑥), ((^◡𝐹‘𝑥) − 1)) ↔ 𝑘 = ((^◡𝐹‘𝑥) − 1)))
169	168	adantl 481	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) ∧ ¬ (^◡𝐹‘𝑥) < 𝐾) → (𝑘 = if((^◡𝐹‘𝑥) < 𝐾, (^◡𝐹‘𝑥), ((^◡𝐹‘𝑥) − 1)) ↔ 𝑘 = ((^◡𝐹‘𝑥) − 1)))
170	154	zred 11358	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑀 ∈ ℝ)
171	170	ad2antrr 758	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) ∧ (¬ (^◡𝐹‘𝑥) < 𝐾 ∧ 𝑘 = ((^◡𝐹‘𝑥) − 1))) → 𝑀 ∈ ℝ)
172	72	ad2antrr 758	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) ∧ (¬ (^◡𝐹‘𝑥) < 𝐾 ∧ 𝑘 = ((^◡𝐹‘𝑥) − 1))) → 𝐾 ∈ ℝ)
173		simprr 792	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) ∧ (¬ (^◡𝐹‘𝑥) < 𝐾 ∧ 𝑘 = ((^◡𝐹‘𝑥) − 1))) → 𝑘 = ((^◡𝐹‘𝑥) − 1))
174	63	ad2antrr 758	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) ∧ (¬ (^◡𝐹‘𝑥) < 𝐾 ∧ 𝑘 = ((^◡𝐹‘𝑥) − 1))) → ^◡𝐹:(𝑀...(𝑁 + 1))⟶(𝑀...(𝑁 + 1)))
175		simplr 788	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) ∧ (¬ (^◡𝐹‘𝑥) < 𝐾 ∧ 𝑘 = ((^◡𝐹‘𝑥) − 1))) → 𝑥 ∈ (𝑀...𝑁))
176	22, 175	sseldi 3566	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) ∧ (¬ (^◡𝐹‘𝑥) < 𝐾 ∧ 𝑘 = ((^◡𝐹‘𝑥) − 1))) → 𝑥 ∈ (𝑀...(𝑁 + 1)))
177	174, 176	ffvelrnd 6268	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) ∧ (¬ (^◡𝐹‘𝑥) < 𝐾 ∧ 𝑘 = ((^◡𝐹‘𝑥) − 1))) → (^◡𝐹‘𝑥) ∈ (𝑀...(𝑁 + 1)))
178		elfzelz 12213	. . . . . . . . . . . . . 14 ⊢ ((^◡𝐹‘𝑥) ∈ (𝑀...(𝑁 + 1)) → (^◡𝐹‘𝑥) ∈ ℤ)
179	177, 178	syl 17	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) ∧ (¬ (^◡𝐹‘𝑥) < 𝐾 ∧ 𝑘 = ((^◡𝐹‘𝑥) − 1))) → (^◡𝐹‘𝑥) ∈ ℤ)
180		peano2zm 11297	. . . . . . . . . . . . 13 ⊢ ((^◡𝐹‘𝑥) ∈ ℤ → ((^◡𝐹‘𝑥) − 1) ∈ ℤ)
181	179, 180	syl 17	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) ∧ (¬ (^◡𝐹‘𝑥) < 𝐾 ∧ 𝑘 = ((^◡𝐹‘𝑥) − 1))) → ((^◡𝐹‘𝑥) − 1) ∈ ℤ)
182	173, 181	eqeltrd 2688	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) ∧ (¬ (^◡𝐹‘𝑥) < 𝐾 ∧ 𝑘 = ((^◡𝐹‘𝑥) − 1))) → 𝑘 ∈ ℤ)
183	182	zred 11358	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) ∧ (¬ (^◡𝐹‘𝑥) < 𝐾 ∧ 𝑘 = ((^◡𝐹‘𝑥) − 1))) → 𝑘 ∈ ℝ)
184		elfzle1 12215	. . . . . . . . . . . 12 ⊢ (𝐾 ∈ (𝑀...(𝑁 + 1)) → 𝑀 ≤ 𝐾)
185	69, 184	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑀 ≤ 𝐾)
186	185	ad2antrr 758	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) ∧ (¬ (^◡𝐹‘𝑥) < 𝐾 ∧ 𝑘 = ((^◡𝐹‘𝑥) − 1))) → 𝑀 ≤ 𝐾)
187		simprl 790	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) ∧ (¬ (^◡𝐹‘𝑥) < 𝐾 ∧ 𝑘 = ((^◡𝐹‘𝑥) − 1))) → ¬ (^◡𝐹‘𝑥) < 𝐾)
188	179	zred 11358	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) ∧ (¬ (^◡𝐹‘𝑥) < 𝐾 ∧ 𝑘 = ((^◡𝐹‘𝑥) − 1))) → (^◡𝐹‘𝑥) ∈ ℝ)
189	172, 188	lenltd 10062	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) ∧ (¬ (^◡𝐹‘𝑥) < 𝐾 ∧ 𝑘 = ((^◡𝐹‘𝑥) − 1))) → (𝐾 ≤ (^◡𝐹‘𝑥) ↔ ¬ (^◡𝐹‘𝑥) < 𝐾))
190	187, 189	mpbird 246	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) ∧ (¬ (^◡𝐹‘𝑥) < 𝐾 ∧ 𝑘 = ((^◡𝐹‘𝑥) − 1))) → 𝐾 ≤ (^◡𝐹‘𝑥))
191		elfzelz 12213	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥 ∈ (𝑀...𝑁) → 𝑥 ∈ ℤ)
192	191	adantl 481	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) → 𝑥 ∈ ℤ)
193	192	zred 11358	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) → 𝑥 ∈ ℝ)
194	139	zred 11358	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → 𝑁 ∈ ℝ)
195	194	adantr 480	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) → 𝑁 ∈ ℝ)
196	141	adantr 480	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) → (𝑁 + 1) ∈ ℝ)
197		elfzle2 12216	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥 ∈ (𝑀...𝑁) → 𝑥 ≤ 𝑁)
198	197	adantl 481	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) → 𝑥 ≤ 𝑁)
199	195	ltp1d 10833	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) → 𝑁 < (𝑁 + 1))
200	193, 195, 196, 198, 199	lelttrd 10074	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) → 𝑥 < (𝑁 + 1))
201	193, 200	gtned 10051	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) → (𝑁 + 1) ≠ 𝑥)
202	201	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) ∧ (¬ (^◡𝐹‘𝑥) < 𝐾 ∧ 𝑘 = ((^◡𝐹‘𝑥) − 1))) → (𝑁 + 1) ≠ 𝑥)
203	61	ad2antrr 758	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) ∧ (¬ (^◡𝐹‘𝑥) < 𝐾 ∧ 𝑘 = ((^◡𝐹‘𝑥) − 1))) → ^◡𝐹:(𝑀...(𝑁 + 1))–1-1→(𝑀...(𝑁 + 1)))
204	67	ad2antrr 758	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) ∧ (¬ (^◡𝐹‘𝑥) < 𝐾 ∧ 𝑘 = ((^◡𝐹‘𝑥) − 1))) → (𝑁 + 1) ∈ (𝑀...(𝑁 + 1)))
205		f1fveq 6420	. . . . . . . . . . . . . . . . . 18 ⊢ ((^◡𝐹:(𝑀...(𝑁 + 1))–1-1→(𝑀...(𝑁 + 1)) ∧ ((𝑁 + 1) ∈ (𝑀...(𝑁 + 1)) ∧ 𝑥 ∈ (𝑀...(𝑁 + 1)))) → ((^◡𝐹‘(𝑁 + 1)) = (^◡𝐹‘𝑥) ↔ (𝑁 + 1) = 𝑥))
206	203, 204, 176, 205	syl12anc 1316	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) ∧ (¬ (^◡𝐹‘𝑥) < 𝐾 ∧ 𝑘 = ((^◡𝐹‘𝑥) − 1))) → ((^◡𝐹‘(𝑁 + 1)) = (^◡𝐹‘𝑥) ↔ (𝑁 + 1) = 𝑥))
207	206	necon3bid 2826	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) ∧ (¬ (^◡𝐹‘𝑥) < 𝐾 ∧ 𝑘 = ((^◡𝐹‘𝑥) − 1))) → ((^◡𝐹‘(𝑁 + 1)) ≠ (^◡𝐹‘𝑥) ↔ (𝑁 + 1) ≠ 𝑥))
208	202, 207	mpbird 246	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) ∧ (¬ (^◡𝐹‘𝑥) < 𝐾 ∧ 𝑘 = ((^◡𝐹‘𝑥) − 1))) → (^◡𝐹‘(𝑁 + 1)) ≠ (^◡𝐹‘𝑥))
209	35	neeq1i 2846	. . . . . . . . . . . . . . 15 ⊢ (𝐾 ≠ (^◡𝐹‘𝑥) ↔ (^◡𝐹‘(𝑁 + 1)) ≠ (^◡𝐹‘𝑥))
210	208, 209	sylibr 223	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) ∧ (¬ (^◡𝐹‘𝑥) < 𝐾 ∧ 𝑘 = ((^◡𝐹‘𝑥) − 1))) → 𝐾 ≠ (^◡𝐹‘𝑥))
211	210	necomd 2837	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) ∧ (¬ (^◡𝐹‘𝑥) < 𝐾 ∧ 𝑘 = ((^◡𝐹‘𝑥) − 1))) → (^◡𝐹‘𝑥) ≠ 𝐾)
212	172, 188	ltlend 10061	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) ∧ (¬ (^◡𝐹‘𝑥) < 𝐾 ∧ 𝑘 = ((^◡𝐹‘𝑥) − 1))) → (𝐾 < (^◡𝐹‘𝑥) ↔ (𝐾 ≤ (^◡𝐹‘𝑥) ∧ (^◡𝐹‘𝑥) ≠ 𝐾)))
213	190, 211, 212	mpbir2and 959	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) ∧ (¬ (^◡𝐹‘𝑥) < 𝐾 ∧ 𝑘 = ((^◡𝐹‘𝑥) − 1))) → 𝐾 < (^◡𝐹‘𝑥))
214	71	ad2antrr 758	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) ∧ (¬ (^◡𝐹‘𝑥) < 𝐾 ∧ 𝑘 = ((^◡𝐹‘𝑥) − 1))) → 𝐾 ∈ ℤ)
215		zltlem1 11307	. . . . . . . . . . . . 13 ⊢ ((𝐾 ∈ ℤ ∧ (^◡𝐹‘𝑥) ∈ ℤ) → (𝐾 < (^◡𝐹‘𝑥) ↔ 𝐾 ≤ ((^◡𝐹‘𝑥) − 1)))
216	214, 179, 215	syl2anc 691	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) ∧ (¬ (^◡𝐹‘𝑥) < 𝐾 ∧ 𝑘 = ((^◡𝐹‘𝑥) − 1))) → (𝐾 < (^◡𝐹‘𝑥) ↔ 𝐾 ≤ ((^◡𝐹‘𝑥) − 1)))
217	213, 216	mpbid 221	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) ∧ (¬ (^◡𝐹‘𝑥) < 𝐾 ∧ 𝑘 = ((^◡𝐹‘𝑥) − 1))) → 𝐾 ≤ ((^◡𝐹‘𝑥) − 1))
218	217, 173	breqtrrd 4611	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) ∧ (¬ (^◡𝐹‘𝑥) < 𝐾 ∧ 𝑘 = ((^◡𝐹‘𝑥) − 1))) → 𝐾 ≤ 𝑘)
219	171, 172, 183, 186, 218	letrd 10073	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) ∧ (¬ (^◡𝐹‘𝑥) < 𝐾 ∧ 𝑘 = ((^◡𝐹‘𝑥) − 1))) → 𝑀 ≤ 𝑘)
220		elfzle2 12216	. . . . . . . . . . . 12 ⊢ ((^◡𝐹‘𝑥) ∈ (𝑀...(𝑁 + 1)) → (^◡𝐹‘𝑥) ≤ (𝑁 + 1))
221	177, 220	syl 17	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) ∧ (¬ (^◡𝐹‘𝑥) < 𝐾 ∧ 𝑘 = ((^◡𝐹‘𝑥) − 1))) → (^◡𝐹‘𝑥) ≤ (𝑁 + 1))
222	194	ad2antrr 758	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) ∧ (¬ (^◡𝐹‘𝑥) < 𝐾 ∧ 𝑘 = ((^◡𝐹‘𝑥) − 1))) → 𝑁 ∈ ℝ)
223		1re 9918	. . . . . . . . . . . . 13 ⊢ 1 ∈ ℝ
224		lesubadd 10379	. . . . . . . . . . . . 13 ⊢ (((^◡𝐹‘𝑥) ∈ ℝ ∧ 1 ∈ ℝ ∧ 𝑁 ∈ ℝ) → (((^◡𝐹‘𝑥) − 1) ≤ 𝑁 ↔ (^◡𝐹‘𝑥) ≤ (𝑁 + 1)))
225	223, 224	mp3an2 1404	. . . . . . . . . . . 12 ⊢ (((^◡𝐹‘𝑥) ∈ ℝ ∧ 𝑁 ∈ ℝ) → (((^◡𝐹‘𝑥) − 1) ≤ 𝑁 ↔ (^◡𝐹‘𝑥) ≤ (𝑁 + 1)))
226	188, 222, 225	syl2anc 691	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) ∧ (¬ (^◡𝐹‘𝑥) < 𝐾 ∧ 𝑘 = ((^◡𝐹‘𝑥) − 1))) → (((^◡𝐹‘𝑥) − 1) ≤ 𝑁 ↔ (^◡𝐹‘𝑥) ≤ (𝑁 + 1)))
227	221, 226	mpbird 246	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) ∧ (¬ (^◡𝐹‘𝑥) < 𝐾 ∧ 𝑘 = ((^◡𝐹‘𝑥) − 1))) → ((^◡𝐹‘𝑥) − 1) ≤ 𝑁)
228	173, 227	eqbrtrd 4605	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) ∧ (¬ (^◡𝐹‘𝑥) < 𝐾 ∧ 𝑘 = ((^◡𝐹‘𝑥) − 1))) → 𝑘 ≤ 𝑁)
229	154	ad2antrr 758	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) ∧ (¬ (^◡𝐹‘𝑥) < 𝐾 ∧ 𝑘 = ((^◡𝐹‘𝑥) − 1))) → 𝑀 ∈ ℤ)
230	139	ad2antrr 758	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) ∧ (¬ (^◡𝐹‘𝑥) < 𝐾 ∧ 𝑘 = ((^◡𝐹‘𝑥) − 1))) → 𝑁 ∈ ℤ)
231	182, 229, 230, 156	syl3anc 1318	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) ∧ (¬ (^◡𝐹‘𝑥) < 𝐾 ∧ 𝑘 = ((^◡𝐹‘𝑥) − 1))) → (𝑘 ∈ (𝑀...𝑁) ↔ (𝑀 ≤ 𝑘 ∧ 𝑘 ≤ 𝑁)))
232	219, 228, 231	mpbir2and 959	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) ∧ (¬ (^◡𝐹‘𝑥) < 𝐾 ∧ 𝑘 = ((^◡𝐹‘𝑥) − 1))) → 𝑘 ∈ (𝑀...𝑁))
233	172, 183	lenltd 10062	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) ∧ (¬ (^◡𝐹‘𝑥) < 𝐾 ∧ 𝑘 = ((^◡𝐹‘𝑥) − 1))) → (𝐾 ≤ 𝑘 ↔ ¬ 𝑘 < 𝐾))
234	218, 233	mpbid 221	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) ∧ (¬ (^◡𝐹‘𝑥) < 𝐾 ∧ 𝑘 = ((^◡𝐹‘𝑥) − 1))) → ¬ 𝑘 < 𝐾)
235	234, 54	syl 17	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) ∧ (¬ (^◡𝐹‘𝑥) < 𝐾 ∧ 𝑘 = ((^◡𝐹‘𝑥) − 1))) → (𝐹‘if(𝑘 < 𝐾, 𝑘, (𝑘 + 1))) = (𝐹‘(𝑘 + 1)))
236	173	oveq1d 6564	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) ∧ (¬ (^◡𝐹‘𝑥) < 𝐾 ∧ 𝑘 = ((^◡𝐹‘𝑥) − 1))) → (𝑘 + 1) = (((^◡𝐹‘𝑥) − 1) + 1))
237	179	zcnd 11359	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) ∧ (¬ (^◡𝐹‘𝑥) < 𝐾 ∧ 𝑘 = ((^◡𝐹‘𝑥) − 1))) → (^◡𝐹‘𝑥) ∈ ℂ)
238		npcan 10169	. . . . . . . . . . . 12 ⊢ (((^◡𝐹‘𝑥) ∈ ℂ ∧ 1 ∈ ℂ) → (((^◡𝐹‘𝑥) − 1) + 1) = (^◡𝐹‘𝑥))
239	237, 115, 238	sylancl 693	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) ∧ (¬ (^◡𝐹‘𝑥) < 𝐾 ∧ 𝑘 = ((^◡𝐹‘𝑥) − 1))) → (((^◡𝐹‘𝑥) − 1) + 1) = (^◡𝐹‘𝑥))
240	236, 239	eqtrd 2644	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) ∧ (¬ (^◡𝐹‘𝑥) < 𝐾 ∧ 𝑘 = ((^◡𝐹‘𝑥) − 1))) → (𝑘 + 1) = (^◡𝐹‘𝑥))
241	240	fveq2d 6107	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) ∧ (¬ (^◡𝐹‘𝑥) < 𝐾 ∧ 𝑘 = ((^◡𝐹‘𝑥) − 1))) → (𝐹‘(𝑘 + 1)) = (𝐹‘(^◡𝐹‘𝑥)))
242	18	ad2antrr 758	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) ∧ (¬ (^◡𝐹‘𝑥) < 𝐾 ∧ 𝑘 = ((^◡𝐹‘𝑥) − 1))) → 𝐹:(𝑀...(𝑁 + 1))–1-1-onto→(𝑀...(𝑁 + 1)))
243	242, 176, 162	syl2anc 691	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) ∧ (¬ (^◡𝐹‘𝑥) < 𝐾 ∧ 𝑘 = ((^◡𝐹‘𝑥) − 1))) → (𝐹‘(^◡𝐹‘𝑥)) = 𝑥)
244	235, 241, 243	3eqtrrd 2649	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) ∧ (¬ (^◡𝐹‘𝑥) < 𝐾 ∧ 𝑘 = ((^◡𝐹‘𝑥) − 1))) → 𝑥 = (𝐹‘if(𝑘 < 𝐾, 𝑘, (𝑘 + 1))))
245	232, 244	jca 553	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) ∧ (¬ (^◡𝐹‘𝑥) < 𝐾 ∧ 𝑘 = ((^◡𝐹‘𝑥) − 1))) → (𝑘 ∈ (𝑀...𝑁) ∧ 𝑥 = (𝐹‘if(𝑘 < 𝐾, 𝑘, (𝑘 + 1)))))
246	245	expr 641	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) ∧ ¬ (^◡𝐹‘𝑥) < 𝐾) → (𝑘 = ((^◡𝐹‘𝑥) − 1) → (𝑘 ∈ (𝑀...𝑁) ∧ 𝑥 = (𝐹‘if(𝑘 < 𝐾, 𝑘, (𝑘 + 1))))))
247	169, 246	sylbid 229	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) ∧ ¬ (^◡𝐹‘𝑥) < 𝐾) → (𝑘 = if((^◡𝐹‘𝑥) < 𝐾, (^◡𝐹‘𝑥), ((^◡𝐹‘𝑥) − 1)) → (𝑘 ∈ (𝑀...𝑁) ∧ 𝑥 = (𝐹‘if(𝑘 < 𝐾, 𝑘, (𝑘 + 1))))))
248	167, 247	pm2.61dan 828	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) → (𝑘 = if((^◡𝐹‘𝑥) < 𝐾, (^◡𝐹‘𝑥), ((^◡𝐹‘𝑥) − 1)) → (𝑘 ∈ (𝑀...𝑁) ∧ 𝑥 = (𝐹‘if(𝑘 < 𝐾, 𝑘, (𝑘 + 1))))))
249	248	expimpd 627	. . 3 ⊢ (𝜑 → ((𝑥 ∈ (𝑀...𝑁) ∧ 𝑘 = if((^◡𝐹‘𝑥) < 𝐾, (^◡𝐹‘𝑥), ((^◡𝐹‘𝑥) − 1))) → (𝑘 ∈ (𝑀...𝑁) ∧ 𝑥 = (𝐹‘if(𝑘 < 𝐾, 𝑘, (𝑘 + 1))))))
250	123, 249	impbid 201	. 2 ⊢ (𝜑 → ((𝑘 ∈ (𝑀...𝑁) ∧ 𝑥 = (𝐹‘if(𝑘 < 𝐾, 𝑘, (𝑘 + 1)))) ↔ (𝑥 ∈ (𝑀...𝑁) ∧ 𝑘 = if((^◡𝐹‘𝑥) < 𝐾, (^◡𝐹‘𝑥), ((^◡𝐹‘𝑥) − 1)))))
251	1, 3, 7, 250	f1od 6783	1 ⊢ (𝜑 → 𝐽:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 195 ∨ wo 382 ∧ wa 383 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 Vcvv 3173 ⊆ wss 3540 ifcif 4036 class class class wbr 4583 ↦ cmpt 4643 ^◡ccnv 5037 ⟶wf 5800 –1-1→wf1 5801 –1-1-onto→wf1o 5803 ‘cfv 5804 (class class class)co 6549 ℂcc 9813 ℝcr 9814 1c1 9816 + caddc 9818 < clt 9953 ≤ cle 9954 − cmin 10145 ℤcz 11254 ℤ_≥cuz 11563 ...cfz 12197

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-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-uni 4373 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-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 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

This theorem is referenced by: seqf1olem2 12703

W3C validator