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Theorem seqf1olem1 14064

Description: Lemma for seqf1o 14066. (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		fvexd 6896	. 2 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → (𝐹‘if(𝑘 < 𝐾, 𝑘, (𝑘 + 1))) ∈ V)
3		fvex 6894	. . . 4 ⊢ (^◡𝐹‘𝑥) ∈ V
4		ovex 7443	. . . 4 ⊢ ((^◡𝐹‘𝑥) − 1) ∈ V
5	3, 4	ifex 4556	. . 3 ⊢ if((^◡𝐹‘𝑥) < 𝐾, (^◡𝐹‘𝑥), ((^◡𝐹‘𝑥) − 1)) ∈ V
6	5	a1i 11	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) → if((^◡𝐹‘𝑥) < 𝐾, (^◡𝐹‘𝑥), ((^◡𝐹‘𝑥) − 1)) ∈ V)
7		iftrue 4511	. . . . . . . . 9 ⊢ (𝑘 < 𝐾 → if(𝑘 < 𝐾, 𝑘, (𝑘 + 1)) = 𝑘)
8	7	fveq2d 6885	. . . . . . . 8 ⊢ (𝑘 < 𝐾 → (𝐹‘if(𝑘 < 𝐾, 𝑘, (𝑘 + 1))) = (𝐹‘𝑘))
9	8	eqeq2d 2747	. . . . . . 7 ⊢ (𝑘 < 𝐾 → (𝑥 = (𝐹‘if(𝑘 < 𝐾, 𝑘, (𝑘 + 1))) ↔ 𝑥 = (𝐹‘𝑘)))
10	9	adantl 481	. . . . . 6 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) ∧ 𝑘 < 𝐾) → (𝑥 = (𝐹‘if(𝑘 < 𝐾, 𝑘, (𝑘 + 1))) ↔ 𝑥 = (𝐹‘𝑘)))
11		simprr 772	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) ∧ (𝑘 < 𝐾 ∧ 𝑥 = (𝐹‘𝑘))) → 𝑥 = (𝐹‘𝑘))
12		elfzelz 13546	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ (𝑀...𝑁) → 𝑘 ∈ ℤ)
13	12	zred 12702	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ (𝑀...𝑁) → 𝑘 ∈ ℝ)
14	13	ad2antlr 727	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) ∧ (𝑘 < 𝐾 ∧ 𝑥 = (𝐹‘𝑘))) → 𝑘 ∈ ℝ)
15		simprl 770	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) ∧ (𝑘 < 𝐾 ∧ 𝑥 = (𝐹‘𝑘))) → 𝑘 < 𝐾)
16	14, 15	gtned 11375	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) ∧ (𝑘 < 𝐾 ∧ 𝑥 = (𝐹‘𝑘))) → 𝐾 ≠ 𝑘)
17		seqf1olem.5	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝐹:(𝑀...(𝑁 + 1))–1-1-onto→(𝑀...(𝑁 + 1)))
18		f1of 6823	. . . . . . . . . . . . . . . . 17 ⊢ (𝐹:(𝑀...(𝑁 + 1))–1-1-onto→(𝑀...(𝑁 + 1)) → 𝐹:(𝑀...(𝑁 + 1))⟶(𝑀...(𝑁 + 1)))
19	17, 18	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐹:(𝑀...(𝑁 + 1))⟶(𝑀...(𝑁 + 1)))
20	19	ad2antrr 726	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) ∧ (𝑘 < 𝐾 ∧ 𝑥 = (𝐹‘𝑘))) → 𝐹:(𝑀...(𝑁 + 1))⟶(𝑀...(𝑁 + 1)))
21		fzssp1 13589	. . . . . . . . . . . . . . . 16 ⊢ (𝑀...𝑁) ⊆ (𝑀...(𝑁 + 1))
22		simplr 768	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) ∧ (𝑘 < 𝐾 ∧ 𝑥 = (𝐹‘𝑘))) → 𝑘 ∈ (𝑀...𝑁))
23	21, 22	sselid 3961	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) ∧ (𝑘 < 𝐾 ∧ 𝑥 = (𝐹‘𝑘))) → 𝑘 ∈ (𝑀...(𝑁 + 1)))
24	20, 23	ffvelcdmd 7080	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) ∧ (𝑘 < 𝐾 ∧ 𝑥 = (𝐹‘𝑘))) → (𝐹‘𝑘) ∈ (𝑀...(𝑁 + 1)))
25		seqf1o.4	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑁 ∈ (ℤ_≥‘𝑀))
26		elfzp1 13596	. . . . . . . . . . . . . . . 16 ⊢ (𝑁 ∈ (ℤ_≥‘𝑀) → ((𝐹‘𝑘) ∈ (𝑀...(𝑁 + 1)) ↔ ((𝐹‘𝑘) ∈ (𝑀...𝑁) ∨ (𝐹‘𝑘) = (𝑁 + 1))))
27	25, 26	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((𝐹‘𝑘) ∈ (𝑀...(𝑁 + 1)) ↔ ((𝐹‘𝑘) ∈ (𝑀...𝑁) ∨ (𝐹‘𝑘) = (𝑁 + 1))))
28	27	ad2antrr 726	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) ∧ (𝑘 < 𝐾 ∧ 𝑥 = (𝐹‘𝑘))) → ((𝐹‘𝑘) ∈ (𝑀...(𝑁 + 1)) ↔ ((𝐹‘𝑘) ∈ (𝑀...𝑁) ∨ (𝐹‘𝑘) = (𝑁 + 1))))
29	24, 28	mpbid 232	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) ∧ (𝑘 < 𝐾 ∧ 𝑥 = (𝐹‘𝑘))) → ((𝐹‘𝑘) ∈ (𝑀...𝑁) ∨ (𝐹‘𝑘) = (𝑁 + 1)))
30	29	ord 864	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) ∧ (𝑘 < 𝐾 ∧ 𝑥 = (𝐹‘𝑘))) → (¬ (𝐹‘𝑘) ∈ (𝑀...𝑁) → (𝐹‘𝑘) = (𝑁 + 1)))
31	17	ad2antrr 726	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) ∧ (𝑘 < 𝐾 ∧ 𝑥 = (𝐹‘𝑘))) → 𝐹:(𝑀...(𝑁 + 1))–1-1-onto→(𝑀...(𝑁 + 1)))
32		f1ocnvfv 7276	. . . . . . . . . . . . . 14 ⊢ ((𝐹:(𝑀...(𝑁 + 1))–1-1-onto→(𝑀...(𝑁 + 1)) ∧ 𝑘 ∈ (𝑀...(𝑁 + 1))) → ((𝐹‘𝑘) = (𝑁 + 1) → (^◡𝐹‘(𝑁 + 1)) = 𝑘))
33	31, 23, 32	syl2anc 584	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) ∧ (𝑘 < 𝐾 ∧ 𝑥 = (𝐹‘𝑘))) → ((𝐹‘𝑘) = (𝑁 + 1) → (^◡𝐹‘(𝑁 + 1)) = 𝑘))
34		seqf1olem.8	. . . . . . . . . . . . . 14 ⊢ 𝐾 = (^◡𝐹‘(𝑁 + 1))
35	34	eqeq1i 2741	. . . . . . . . . . . . 13 ⊢ (𝐾 = 𝑘 ↔ (^◡𝐹‘(𝑁 + 1)) = 𝑘)
36	33, 35	imbitrrdi 252	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) ∧ (𝑘 < 𝐾 ∧ 𝑥 = (𝐹‘𝑘))) → ((𝐹‘𝑘) = (𝑁 + 1) → 𝐾 = 𝑘))
37	30, 36	syld 47	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) ∧ (𝑘 < 𝐾 ∧ 𝑥 = (𝐹‘𝑘))) → (¬ (𝐹‘𝑘) ∈ (𝑀...𝑁) → 𝐾 = 𝑘))
38	37	necon1ad 2950	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) ∧ (𝑘 < 𝐾 ∧ 𝑥 = (𝐹‘𝑘))) → (𝐾 ≠ 𝑘 → (𝐹‘𝑘) ∈ (𝑀...𝑁)))
39	16, 38	mpd 15	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) ∧ (𝑘 < 𝐾 ∧ 𝑥 = (𝐹‘𝑘))) → (𝐹‘𝑘) ∈ (𝑀...𝑁))
40	11, 39	eqeltrd 2835	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) ∧ (𝑘 < 𝐾 ∧ 𝑥 = (𝐹‘𝑘))) → 𝑥 ∈ (𝑀...𝑁))
41	11	eqcomd 2742	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) ∧ (𝑘 < 𝐾 ∧ 𝑥 = (𝐹‘𝑘))) → (𝐹‘𝑘) = 𝑥)
42		f1ocnvfv 7276	. . . . . . . . . . . . 13 ⊢ ((𝐹:(𝑀...(𝑁 + 1))–1-1-onto→(𝑀...(𝑁 + 1)) ∧ 𝑘 ∈ (𝑀...(𝑁 + 1))) → ((𝐹‘𝑘) = 𝑥 → (^◡𝐹‘𝑥) = 𝑘))
43	31, 23, 42	syl2anc 584	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) ∧ (𝑘 < 𝐾 ∧ 𝑥 = (𝐹‘𝑘))) → ((𝐹‘𝑘) = 𝑥 → (^◡𝐹‘𝑥) = 𝑘))
44	41, 43	mpd 15	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) ∧ (𝑘 < 𝐾 ∧ 𝑥 = (𝐹‘𝑘))) → (^◡𝐹‘𝑥) = 𝑘)
45	44, 15	eqbrtrd 5146	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) ∧ (𝑘 < 𝐾 ∧ 𝑥 = (𝐹‘𝑘))) → (^◡𝐹‘𝑥) < 𝐾)
46		iftrue 4511	. . . . . . . . . 10 ⊢ ((^◡𝐹‘𝑥) < 𝐾 → if((^◡𝐹‘𝑥) < 𝐾, (^◡𝐹‘𝑥), ((^◡𝐹‘𝑥) − 1)) = (^◡𝐹‘𝑥))
47	45, 46	syl 17	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) ∧ (𝑘 < 𝐾 ∧ 𝑥 = (𝐹‘𝑘))) → if((^◡𝐹‘𝑥) < 𝐾, (^◡𝐹‘𝑥), ((^◡𝐹‘𝑥) − 1)) = (^◡𝐹‘𝑥))
48	47, 44	eqtr2d 2772	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) ∧ (𝑘 < 𝐾 ∧ 𝑥 = (𝐹‘𝑘))) → 𝑘 = if((^◡𝐹‘𝑥) < 𝐾, (^◡𝐹‘𝑥), ((^◡𝐹‘𝑥) − 1)))
49	40, 48	jca 511	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) ∧ (𝑘 < 𝐾 ∧ 𝑥 = (𝐹‘𝑘))) → (𝑥 ∈ (𝑀...𝑁) ∧ 𝑘 = if((^◡𝐹‘𝑥) < 𝐾, (^◡𝐹‘𝑥), ((^◡𝐹‘𝑥) − 1))))
50	49	expr 456	. . . . . 6 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) ∧ 𝑘 < 𝐾) → (𝑥 = (𝐹‘𝑘) → (𝑥 ∈ (𝑀...𝑁) ∧ 𝑘 = if((^◡𝐹‘𝑥) < 𝐾, (^◡𝐹‘𝑥), ((^◡𝐹‘𝑥) − 1)))))
51	10, 50	sylbid 240	. . . . 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)))))
52		iffalse 4514	. . . . . . . . 9 ⊢ (¬ 𝑘 < 𝐾 → if(𝑘 < 𝐾, 𝑘, (𝑘 + 1)) = (𝑘 + 1))
53	52	fveq2d 6885	. . . . . . . 8 ⊢ (¬ 𝑘 < 𝐾 → (𝐹‘if(𝑘 < 𝐾, 𝑘, (𝑘 + 1))) = (𝐹‘(𝑘 + 1)))
54	53	eqeq2d 2747	. . . . . . 7 ⊢ (¬ 𝑘 < 𝐾 → (𝑥 = (𝐹‘if(𝑘 < 𝐾, 𝑘, (𝑘 + 1))) ↔ 𝑥 = (𝐹‘(𝑘 + 1))))
55	54	adantl 481	. . . . . 6 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) ∧ ¬ 𝑘 < 𝐾) → (𝑥 = (𝐹‘if(𝑘 < 𝐾, 𝑘, (𝑘 + 1))) ↔ 𝑥 = (𝐹‘(𝑘 + 1))))
56		simprr 772	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) ∧ (¬ 𝑘 < 𝐾 ∧ 𝑥 = (𝐹‘(𝑘 + 1)))) → 𝑥 = (𝐹‘(𝑘 + 1)))
57		f1ocnv 6835	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐹:(𝑀...(𝑁 + 1))–1-1-onto→(𝑀...(𝑁 + 1)) → ^◡𝐹:(𝑀...(𝑁 + 1))–1-1-onto→(𝑀...(𝑁 + 1)))
58	17, 57	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → ^◡𝐹:(𝑀...(𝑁 + 1))–1-1-onto→(𝑀...(𝑁 + 1)))
59		f1of1 6822	. . . . . . . . . . . . . . . . . 18 ⊢ (^◡𝐹:(𝑀...(𝑁 + 1))–1-1-onto→(𝑀...(𝑁 + 1)) → ^◡𝐹:(𝑀...(𝑁 + 1))–1-1→(𝑀...(𝑁 + 1)))
60	58, 59	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ^◡𝐹:(𝑀...(𝑁 + 1))–1-1→(𝑀...(𝑁 + 1)))
61		f1f 6779	. . . . . . . . . . . . . . . . 17 ⊢ (^◡𝐹:(𝑀...(𝑁 + 1))–1-1→(𝑀...(𝑁 + 1)) → ^◡𝐹:(𝑀...(𝑁 + 1))⟶(𝑀...(𝑁 + 1)))
62	60, 61	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ^◡𝐹:(𝑀...(𝑁 + 1))⟶(𝑀...(𝑁 + 1)))
63		peano2uz 12922	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑁 ∈ (ℤ_≥‘𝑀) → (𝑁 + 1) ∈ (ℤ_≥‘𝑀))
64	25, 63	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝑁 + 1) ∈ (ℤ_≥‘𝑀))
65		eluzfz2 13554	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑁 + 1) ∈ (ℤ_≥‘𝑀) → (𝑁 + 1) ∈ (𝑀...(𝑁 + 1)))
66	64, 65	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝑁 + 1) ∈ (𝑀...(𝑁 + 1)))
67	62, 66	ffvelcdmd 7080	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (^◡𝐹‘(𝑁 + 1)) ∈ (𝑀...(𝑁 + 1)))
68	34, 67	eqeltrid 2839	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐾 ∈ (𝑀...(𝑁 + 1)))
69	68	elfzelzd 13547	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐾 ∈ ℤ)
70	69	zred 12702	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐾 ∈ ℝ)
71	70	ad2antrr 726	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) ∧ (¬ 𝑘 < 𝐾 ∧ 𝑥 = (𝐹‘(𝑘 + 1)))) → 𝐾 ∈ ℝ)
72	13	ad2antlr 727	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) ∧ (¬ 𝑘 < 𝐾 ∧ 𝑥 = (𝐹‘(𝑘 + 1)))) → 𝑘 ∈ ℝ)
73		peano2re 11413	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ ℝ → (𝑘 + 1) ∈ ℝ)
74	72, 73	syl 17	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) ∧ (¬ 𝑘 < 𝐾 ∧ 𝑥 = (𝐹‘(𝑘 + 1)))) → (𝑘 + 1) ∈ ℝ)
75		simprl 770	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) ∧ (¬ 𝑘 < 𝐾 ∧ 𝑥 = (𝐹‘(𝑘 + 1)))) → ¬ 𝑘 < 𝐾)
76	71, 72, 75	nltled 11390	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) ∧ (¬ 𝑘 < 𝐾 ∧ 𝑥 = (𝐹‘(𝑘 + 1)))) → 𝐾 ≤ 𝑘)
77	72	ltp1d 12177	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) ∧ (¬ 𝑘 < 𝐾 ∧ 𝑥 = (𝐹‘(𝑘 + 1)))) → 𝑘 < (𝑘 + 1))
78	71, 72, 74, 76, 77	lelttrd 11398	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) ∧ (¬ 𝑘 < 𝐾 ∧ 𝑥 = (𝐹‘(𝑘 + 1)))) → 𝐾 < (𝑘 + 1))
79	71, 78	ltned 11376	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) ∧ (¬ 𝑘 < 𝐾 ∧ 𝑥 = (𝐹‘(𝑘 + 1)))) → 𝐾 ≠ (𝑘 + 1))
80	19	ad2antrr 726	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) ∧ (¬ 𝑘 < 𝐾 ∧ 𝑥 = (𝐹‘(𝑘 + 1)))) → 𝐹:(𝑀...(𝑁 + 1))⟶(𝑀...(𝑁 + 1)))
81		fzp1elp1 13599	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 ∈ (𝑀...𝑁) → (𝑘 + 1) ∈ (𝑀...(𝑁 + 1)))
82	81	ad2antlr 727	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) ∧ (¬ 𝑘 < 𝐾 ∧ 𝑥 = (𝐹‘(𝑘 + 1)))) → (𝑘 + 1) ∈ (𝑀...(𝑁 + 1)))
83	80, 82	ffvelcdmd 7080	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) ∧ (¬ 𝑘 < 𝐾 ∧ 𝑥 = (𝐹‘(𝑘 + 1)))) → (𝐹‘(𝑘 + 1)) ∈ (𝑀...(𝑁 + 1)))
84		elfzp1 13596	. . . . . . . . . . . . . . . 16 ⊢ (𝑁 ∈ (ℤ_≥‘𝑀) → ((𝐹‘(𝑘 + 1)) ∈ (𝑀...(𝑁 + 1)) ↔ ((𝐹‘(𝑘 + 1)) ∈ (𝑀...𝑁) ∨ (𝐹‘(𝑘 + 1)) = (𝑁 + 1))))
85	25, 84	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((𝐹‘(𝑘 + 1)) ∈ (𝑀...(𝑁 + 1)) ↔ ((𝐹‘(𝑘 + 1)) ∈ (𝑀...𝑁) ∨ (𝐹‘(𝑘 + 1)) = (𝑁 + 1))))
86	85	ad2antrr 726	. . . . . . . . . . . . . 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))))
87	83, 86	mpbid 232	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) ∧ (¬ 𝑘 < 𝐾 ∧ 𝑥 = (𝐹‘(𝑘 + 1)))) → ((𝐹‘(𝑘 + 1)) ∈ (𝑀...𝑁) ∨ (𝐹‘(𝑘 + 1)) = (𝑁 + 1)))
88	87	ord 864	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) ∧ (¬ 𝑘 < 𝐾 ∧ 𝑥 = (𝐹‘(𝑘 + 1)))) → (¬ (𝐹‘(𝑘 + 1)) ∈ (𝑀...𝑁) → (𝐹‘(𝑘 + 1)) = (𝑁 + 1)))
89	17	ad2antrr 726	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) ∧ (¬ 𝑘 < 𝐾 ∧ 𝑥 = (𝐹‘(𝑘 + 1)))) → 𝐹:(𝑀...(𝑁 + 1))–1-1-onto→(𝑀...(𝑁 + 1)))
90		f1ocnvfv 7276	. . . . . . . . . . . . . 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)))
91	89, 82, 90	syl2anc 584	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) ∧ (¬ 𝑘 < 𝐾 ∧ 𝑥 = (𝐹‘(𝑘 + 1)))) → ((𝐹‘(𝑘 + 1)) = (𝑁 + 1) → (^◡𝐹‘(𝑁 + 1)) = (𝑘 + 1)))
92	34	eqeq1i 2741	. . . . . . . . . . . . 13 ⊢ (𝐾 = (𝑘 + 1) ↔ (^◡𝐹‘(𝑁 + 1)) = (𝑘 + 1))
93	91, 92	imbitrrdi 252	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) ∧ (¬ 𝑘 < 𝐾 ∧ 𝑥 = (𝐹‘(𝑘 + 1)))) → ((𝐹‘(𝑘 + 1)) = (𝑁 + 1) → 𝐾 = (𝑘 + 1)))
94	88, 93	syld 47	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) ∧ (¬ 𝑘 < 𝐾 ∧ 𝑥 = (𝐹‘(𝑘 + 1)))) → (¬ (𝐹‘(𝑘 + 1)) ∈ (𝑀...𝑁) → 𝐾 = (𝑘 + 1)))
95	94	necon1ad 2950	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) ∧ (¬ 𝑘 < 𝐾 ∧ 𝑥 = (𝐹‘(𝑘 + 1)))) → (𝐾 ≠ (𝑘 + 1) → (𝐹‘(𝑘 + 1)) ∈ (𝑀...𝑁)))
96	79, 95	mpd 15	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) ∧ (¬ 𝑘 < 𝐾 ∧ 𝑥 = (𝐹‘(𝑘 + 1)))) → (𝐹‘(𝑘 + 1)) ∈ (𝑀...𝑁))
97	56, 96	eqeltrd 2835	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) ∧ (¬ 𝑘 < 𝐾 ∧ 𝑥 = (𝐹‘(𝑘 + 1)))) → 𝑥 ∈ (𝑀...𝑁))
98	56	eqcomd 2742	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) ∧ (¬ 𝑘 < 𝐾 ∧ 𝑥 = (𝐹‘(𝑘 + 1)))) → (𝐹‘(𝑘 + 1)) = 𝑥)
99		f1ocnvfv 7276	. . . . . . . . . . . . . . 15 ⊢ ((𝐹:(𝑀...(𝑁 + 1))–1-1-onto→(𝑀...(𝑁 + 1)) ∧ (𝑘 + 1) ∈ (𝑀...(𝑁 + 1))) → ((𝐹‘(𝑘 + 1)) = 𝑥 → (^◡𝐹‘𝑥) = (𝑘 + 1)))
100	89, 82, 99	syl2anc 584	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) ∧ (¬ 𝑘 < 𝐾 ∧ 𝑥 = (𝐹‘(𝑘 + 1)))) → ((𝐹‘(𝑘 + 1)) = 𝑥 → (^◡𝐹‘𝑥) = (𝑘 + 1)))
101	98, 100	mpd 15	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) ∧ (¬ 𝑘 < 𝐾 ∧ 𝑥 = (𝐹‘(𝑘 + 1)))) → (^◡𝐹‘𝑥) = (𝑘 + 1))
102	101	breq1d 5134	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) ∧ (¬ 𝑘 < 𝐾 ∧ 𝑥 = (𝐹‘(𝑘 + 1)))) → ((^◡𝐹‘𝑥) < 𝐾 ↔ (𝑘 + 1) < 𝐾))
103		lttr 11316	. . . . . . . . . . . . . 14 ⊢ ((𝑘 ∈ ℝ ∧ (𝑘 + 1) ∈ ℝ ∧ 𝐾 ∈ ℝ) → ((𝑘 < (𝑘 + 1) ∧ (𝑘 + 1) < 𝐾) → 𝑘 < 𝐾))
104	72, 74, 71, 103	syl3anc 1373	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) ∧ (¬ 𝑘 < 𝐾 ∧ 𝑥 = (𝐹‘(𝑘 + 1)))) → ((𝑘 < (𝑘 + 1) ∧ (𝑘 + 1) < 𝐾) → 𝑘 < 𝐾))
105	77, 104	mpand 695	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) ∧ (¬ 𝑘 < 𝐾 ∧ 𝑥 = (𝐹‘(𝑘 + 1)))) → ((𝑘 + 1) < 𝐾 → 𝑘 < 𝐾))
106	102, 105	sylbid 240	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) ∧ (¬ 𝑘 < 𝐾 ∧ 𝑥 = (𝐹‘(𝑘 + 1)))) → ((^◡𝐹‘𝑥) < 𝐾 → 𝑘 < 𝐾))
107	75, 106	mtod 198	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) ∧ (¬ 𝑘 < 𝐾 ∧ 𝑥 = (𝐹‘(𝑘 + 1)))) → ¬ (^◡𝐹‘𝑥) < 𝐾)
108		iffalse 4514	. . . . . . . . . 10 ⊢ (¬ (^◡𝐹‘𝑥) < 𝐾 → if((^◡𝐹‘𝑥) < 𝐾, (^◡𝐹‘𝑥), ((^◡𝐹‘𝑥) − 1)) = ((^◡𝐹‘𝑥) − 1))
109	107, 108	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))
110	101	oveq1d 7425	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) ∧ (¬ 𝑘 < 𝐾 ∧ 𝑥 = (𝐹‘(𝑘 + 1)))) → ((^◡𝐹‘𝑥) − 1) = ((𝑘 + 1) − 1))
111	72	recnd 11268	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) ∧ (¬ 𝑘 < 𝐾 ∧ 𝑥 = (𝐹‘(𝑘 + 1)))) → 𝑘 ∈ ℂ)
112		ax-1cn 11192	. . . . . . . . . 10 ⊢ 1 ∈ ℂ
113		pncan 11493	. . . . . . . . . 10 ⊢ ((𝑘 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑘 + 1) − 1) = 𝑘)
114	111, 112, 113	sylancl 586	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) ∧ (¬ 𝑘 < 𝐾 ∧ 𝑥 = (𝐹‘(𝑘 + 1)))) → ((𝑘 + 1) − 1) = 𝑘)
115	109, 110, 114	3eqtrrd 2776	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) ∧ (¬ 𝑘 < 𝐾 ∧ 𝑥 = (𝐹‘(𝑘 + 1)))) → 𝑘 = if((^◡𝐹‘𝑥) < 𝐾, (^◡𝐹‘𝑥), ((^◡𝐹‘𝑥) − 1)))
116	97, 115	jca 511	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) ∧ (¬ 𝑘 < 𝐾 ∧ 𝑥 = (𝐹‘(𝑘 + 1)))) → (𝑥 ∈ (𝑀...𝑁) ∧ 𝑘 = if((^◡𝐹‘𝑥) < 𝐾, (^◡𝐹‘𝑥), ((^◡𝐹‘𝑥) − 1))))
117	116	expr 456	. . . . . 6 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) ∧ ¬ 𝑘 < 𝐾) → (𝑥 = (𝐹‘(𝑘 + 1)) → (𝑥 ∈ (𝑀...𝑁) ∧ 𝑘 = if((^◡𝐹‘𝑥) < 𝐾, (^◡𝐹‘𝑥), ((^◡𝐹‘𝑥) − 1)))))
118	55, 117	sylbid 240	. . . . 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)))))
119	51, 118	pm2.61dan 812	. . . 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)))))
120	119	expimpd 453	. . 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)))))
121	46	eqeq2d 2747	. . . . . . 7 ⊢ ((^◡𝐹‘𝑥) < 𝐾 → (𝑘 = if((^◡𝐹‘𝑥) < 𝐾, (^◡𝐹‘𝑥), ((^◡𝐹‘𝑥) − 1)) ↔ 𝑘 = (^◡𝐹‘𝑥)))
122	121	adantl 481	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) ∧ (^◡𝐹‘𝑥) < 𝐾) → (𝑘 = if((^◡𝐹‘𝑥) < 𝐾, (^◡𝐹‘𝑥), ((^◡𝐹‘𝑥) − 1)) ↔ 𝑘 = (^◡𝐹‘𝑥)))
123		eluzel2 12862	. . . . . . . . . . 11 ⊢ (𝑁 ∈ (ℤ_≥‘𝑀) → 𝑀 ∈ ℤ)
124	25, 123	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝑀 ∈ ℤ)
125	124	ad2antrr 726	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) ∧ ((^◡𝐹‘𝑥) < 𝐾 ∧ 𝑘 = (^◡𝐹‘𝑥))) → 𝑀 ∈ ℤ)
126		eluzelz 12867	. . . . . . . . . . 11 ⊢ (𝑁 ∈ (ℤ_≥‘𝑀) → 𝑁 ∈ ℤ)
127	25, 126	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝑁 ∈ ℤ)
128	127	ad2antrr 726	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) ∧ ((^◡𝐹‘𝑥) < 𝐾 ∧ 𝑘 = (^◡𝐹‘𝑥))) → 𝑁 ∈ ℤ)
129		simprr 772	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) ∧ ((^◡𝐹‘𝑥) < 𝐾 ∧ 𝑘 = (^◡𝐹‘𝑥))) → 𝑘 = (^◡𝐹‘𝑥))
130	62	ad2antrr 726	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) ∧ ((^◡𝐹‘𝑥) < 𝐾 ∧ 𝑘 = (^◡𝐹‘𝑥))) → ^◡𝐹:(𝑀...(𝑁 + 1))⟶(𝑀...(𝑁 + 1)))
131		simplr 768	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) ∧ ((^◡𝐹‘𝑥) < 𝐾 ∧ 𝑘 = (^◡𝐹‘𝑥))) → 𝑥 ∈ (𝑀...𝑁))
132	21, 131	sselid 3961	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) ∧ ((^◡𝐹‘𝑥) < 𝐾 ∧ 𝑘 = (^◡𝐹‘𝑥))) → 𝑥 ∈ (𝑀...(𝑁 + 1)))
133	130, 132	ffvelcdmd 7080	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) ∧ ((^◡𝐹‘𝑥) < 𝐾 ∧ 𝑘 = (^◡𝐹‘𝑥))) → (^◡𝐹‘𝑥) ∈ (𝑀...(𝑁 + 1)))
134	129, 133	eqeltrd 2835	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) ∧ ((^◡𝐹‘𝑥) < 𝐾 ∧ 𝑘 = (^◡𝐹‘𝑥))) → 𝑘 ∈ (𝑀...(𝑁 + 1)))
135	134	elfzelzd 13547	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) ∧ ((^◡𝐹‘𝑥) < 𝐾 ∧ 𝑘 = (^◡𝐹‘𝑥))) → 𝑘 ∈ ℤ)
136		elfzle1 13549	. . . . . . . . . 10 ⊢ (𝑘 ∈ (𝑀...(𝑁 + 1)) → 𝑀 ≤ 𝑘)
137	134, 136	syl 17	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) ∧ ((^◡𝐹‘𝑥) < 𝐾 ∧ 𝑘 = (^◡𝐹‘𝑥))) → 𝑀 ≤ 𝑘)
138	135	zred 12702	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) ∧ ((^◡𝐹‘𝑥) < 𝐾 ∧ 𝑘 = (^◡𝐹‘𝑥))) → 𝑘 ∈ ℝ)
139	70	ad2antrr 726	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) ∧ ((^◡𝐹‘𝑥) < 𝐾 ∧ 𝑘 = (^◡𝐹‘𝑥))) → 𝐾 ∈ ℝ)
140	127	peano2zd 12705	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑁 + 1) ∈ ℤ)
141	140	zred 12702	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑁 + 1) ∈ ℝ)
142	141	ad2antrr 726	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) ∧ ((^◡𝐹‘𝑥) < 𝐾 ∧ 𝑘 = (^◡𝐹‘𝑥))) → (𝑁 + 1) ∈ ℝ)
143		simprl 770	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) ∧ ((^◡𝐹‘𝑥) < 𝐾 ∧ 𝑘 = (^◡𝐹‘𝑥))) → (^◡𝐹‘𝑥) < 𝐾)
144	129, 143	eqbrtrd 5146	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) ∧ ((^◡𝐹‘𝑥) < 𝐾 ∧ 𝑘 = (^◡𝐹‘𝑥))) → 𝑘 < 𝐾)
145		elfzle2 13550	. . . . . . . . . . . . 13 ⊢ (𝐾 ∈ (𝑀...(𝑁 + 1)) → 𝐾 ≤ (𝑁 + 1))
146	68, 145	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐾 ≤ (𝑁 + 1))
147	146	ad2antrr 726	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) ∧ ((^◡𝐹‘𝑥) < 𝐾 ∧ 𝑘 = (^◡𝐹‘𝑥))) → 𝐾 ≤ (𝑁 + 1))
148	138, 139, 142, 144, 147	ltletrd 11400	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) ∧ ((^◡𝐹‘𝑥) < 𝐾 ∧ 𝑘 = (^◡𝐹‘𝑥))) → 𝑘 < (𝑁 + 1))
149		zleltp1 12648	. . . . . . . . . . 11 ⊢ ((𝑘 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑘 ≤ 𝑁 ↔ 𝑘 < (𝑁 + 1)))
150	135, 128, 149	syl2anc 584	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) ∧ ((^◡𝐹‘𝑥) < 𝐾 ∧ 𝑘 = (^◡𝐹‘𝑥))) → (𝑘 ≤ 𝑁 ↔ 𝑘 < (𝑁 + 1)))
151	148, 150	mpbird 257	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) ∧ ((^◡𝐹‘𝑥) < 𝐾 ∧ 𝑘 = (^◡𝐹‘𝑥))) → 𝑘 ≤ 𝑁)
152	125, 128, 135, 137, 151	elfzd 13537	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) ∧ ((^◡𝐹‘𝑥) < 𝐾 ∧ 𝑘 = (^◡𝐹‘𝑥))) → 𝑘 ∈ (𝑀...𝑁))
153	144, 8	syl 17	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) ∧ ((^◡𝐹‘𝑥) < 𝐾 ∧ 𝑘 = (^◡𝐹‘𝑥))) → (𝐹‘if(𝑘 < 𝐾, 𝑘, (𝑘 + 1))) = (𝐹‘𝑘))
154	129	fveq2d 6885	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) ∧ ((^◡𝐹‘𝑥) < 𝐾 ∧ 𝑘 = (^◡𝐹‘𝑥))) → (𝐹‘𝑘) = (𝐹‘(^◡𝐹‘𝑥)))
155	17	ad2antrr 726	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) ∧ ((^◡𝐹‘𝑥) < 𝐾 ∧ 𝑘 = (^◡𝐹‘𝑥))) → 𝐹:(𝑀...(𝑁 + 1))–1-1-onto→(𝑀...(𝑁 + 1)))
156		f1ocnvfv2 7275	. . . . . . . . . 10 ⊢ ((𝐹:(𝑀...(𝑁 + 1))–1-1-onto→(𝑀...(𝑁 + 1)) ∧ 𝑥 ∈ (𝑀...(𝑁 + 1))) → (𝐹‘(^◡𝐹‘𝑥)) = 𝑥)
157	155, 132, 156	syl2anc 584	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) ∧ ((^◡𝐹‘𝑥) < 𝐾 ∧ 𝑘 = (^◡𝐹‘𝑥))) → (𝐹‘(^◡𝐹‘𝑥)) = 𝑥)
158	153, 154, 157	3eqtrrd 2776	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) ∧ ((^◡𝐹‘𝑥) < 𝐾 ∧ 𝑘 = (^◡𝐹‘𝑥))) → 𝑥 = (𝐹‘if(𝑘 < 𝐾, 𝑘, (𝑘 + 1))))
159	152, 158	jca 511	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) ∧ ((^◡𝐹‘𝑥) < 𝐾 ∧ 𝑘 = (^◡𝐹‘𝑥))) → (𝑘 ∈ (𝑀...𝑁) ∧ 𝑥 = (𝐹‘if(𝑘 < 𝐾, 𝑘, (𝑘 + 1)))))
160	159	expr 456	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) ∧ (^◡𝐹‘𝑥) < 𝐾) → (𝑘 = (^◡𝐹‘𝑥) → (𝑘 ∈ (𝑀...𝑁) ∧ 𝑥 = (𝐹‘if(𝑘 < 𝐾, 𝑘, (𝑘 + 1))))))
161	122, 160	sylbid 240	. . . . 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))))))
162	108	eqeq2d 2747	. . . . . . 7 ⊢ (¬ (^◡𝐹‘𝑥) < 𝐾 → (𝑘 = if((^◡𝐹‘𝑥) < 𝐾, (^◡𝐹‘𝑥), ((^◡𝐹‘𝑥) − 1)) ↔ 𝑘 = ((^◡𝐹‘𝑥) − 1)))
163	162	adantl 481	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) ∧ ¬ (^◡𝐹‘𝑥) < 𝐾) → (𝑘 = if((^◡𝐹‘𝑥) < 𝐾, (^◡𝐹‘𝑥), ((^◡𝐹‘𝑥) − 1)) ↔ 𝑘 = ((^◡𝐹‘𝑥) − 1)))
164	124	ad2antrr 726	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) ∧ (¬ (^◡𝐹‘𝑥) < 𝐾 ∧ 𝑘 = ((^◡𝐹‘𝑥) − 1))) → 𝑀 ∈ ℤ)
165	127	ad2antrr 726	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) ∧ (¬ (^◡𝐹‘𝑥) < 𝐾 ∧ 𝑘 = ((^◡𝐹‘𝑥) − 1))) → 𝑁 ∈ ℤ)
166		simprr 772	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) ∧ (¬ (^◡𝐹‘𝑥) < 𝐾 ∧ 𝑘 = ((^◡𝐹‘𝑥) − 1))) → 𝑘 = ((^◡𝐹‘𝑥) − 1))
167	62	ad2antrr 726	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) ∧ (¬ (^◡𝐹‘𝑥) < 𝐾 ∧ 𝑘 = ((^◡𝐹‘𝑥) − 1))) → ^◡𝐹:(𝑀...(𝑁 + 1))⟶(𝑀...(𝑁 + 1)))
168		simplr 768	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) ∧ (¬ (^◡𝐹‘𝑥) < 𝐾 ∧ 𝑘 = ((^◡𝐹‘𝑥) − 1))) → 𝑥 ∈ (𝑀...𝑁))
169	21, 168	sselid 3961	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) ∧ (¬ (^◡𝐹‘𝑥) < 𝐾 ∧ 𝑘 = ((^◡𝐹‘𝑥) − 1))) → 𝑥 ∈ (𝑀...(𝑁 + 1)))
170	167, 169	ffvelcdmd 7080	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) ∧ (¬ (^◡𝐹‘𝑥) < 𝐾 ∧ 𝑘 = ((^◡𝐹‘𝑥) − 1))) → (^◡𝐹‘𝑥) ∈ (𝑀...(𝑁 + 1)))
171	170	elfzelzd 13547	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) ∧ (¬ (^◡𝐹‘𝑥) < 𝐾 ∧ 𝑘 = ((^◡𝐹‘𝑥) − 1))) → (^◡𝐹‘𝑥) ∈ ℤ)
172		peano2zm 12640	. . . . . . . . . . 11 ⊢ ((^◡𝐹‘𝑥) ∈ ℤ → ((^◡𝐹‘𝑥) − 1) ∈ ℤ)
173	171, 172	syl 17	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) ∧ (¬ (^◡𝐹‘𝑥) < 𝐾 ∧ 𝑘 = ((^◡𝐹‘𝑥) − 1))) → ((^◡𝐹‘𝑥) − 1) ∈ ℤ)
174	166, 173	eqeltrd 2835	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) ∧ (¬ (^◡𝐹‘𝑥) < 𝐾 ∧ 𝑘 = ((^◡𝐹‘𝑥) − 1))) → 𝑘 ∈ ℤ)
175	124	zred 12702	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑀 ∈ ℝ)
176	175	ad2antrr 726	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) ∧ (¬ (^◡𝐹‘𝑥) < 𝐾 ∧ 𝑘 = ((^◡𝐹‘𝑥) − 1))) → 𝑀 ∈ ℝ)
177	70	ad2antrr 726	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) ∧ (¬ (^◡𝐹‘𝑥) < 𝐾 ∧ 𝑘 = ((^◡𝐹‘𝑥) − 1))) → 𝐾 ∈ ℝ)
178	174	zred 12702	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) ∧ (¬ (^◡𝐹‘𝑥) < 𝐾 ∧ 𝑘 = ((^◡𝐹‘𝑥) − 1))) → 𝑘 ∈ ℝ)
179		elfzle1 13549	. . . . . . . . . . . 12 ⊢ (𝐾 ∈ (𝑀...(𝑁 + 1)) → 𝑀 ≤ 𝐾)
180	68, 179	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑀 ≤ 𝐾)
181	180	ad2antrr 726	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) ∧ (¬ (^◡𝐹‘𝑥) < 𝐾 ∧ 𝑘 = ((^◡𝐹‘𝑥) − 1))) → 𝑀 ≤ 𝐾)
182	171	zred 12702	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) ∧ (¬ (^◡𝐹‘𝑥) < 𝐾 ∧ 𝑘 = ((^◡𝐹‘𝑥) − 1))) → (^◡𝐹‘𝑥) ∈ ℝ)
183		simprl 770	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) ∧ (¬ (^◡𝐹‘𝑥) < 𝐾 ∧ 𝑘 = ((^◡𝐹‘𝑥) − 1))) → ¬ (^◡𝐹‘𝑥) < 𝐾)
184	177, 182, 183	nltled 11390	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) ∧ (¬ (^◡𝐹‘𝑥) < 𝐾 ∧ 𝑘 = ((^◡𝐹‘𝑥) − 1))) → 𝐾 ≤ (^◡𝐹‘𝑥))
185		elfzelz 13546	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥 ∈ (𝑀...𝑁) → 𝑥 ∈ ℤ)
186	185	adantl 481	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) → 𝑥 ∈ ℤ)
187	186	zred 12702	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) → 𝑥 ∈ ℝ)
188	127	zred 12702	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → 𝑁 ∈ ℝ)
189	188	adantr 480	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) → 𝑁 ∈ ℝ)
190	141	adantr 480	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) → (𝑁 + 1) ∈ ℝ)
191		elfzle2 13550	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥 ∈ (𝑀...𝑁) → 𝑥 ≤ 𝑁)
192	191	adantl 481	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) → 𝑥 ≤ 𝑁)
193	189	ltp1d 12177	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) → 𝑁 < (𝑁 + 1))
194	187, 189, 190, 192, 193	lelttrd 11398	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) → 𝑥 < (𝑁 + 1))
195	187, 194	gtned 11375	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) → (𝑁 + 1) ≠ 𝑥)
196	195	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) ∧ (¬ (^◡𝐹‘𝑥) < 𝐾 ∧ 𝑘 = ((^◡𝐹‘𝑥) − 1))) → (𝑁 + 1) ≠ 𝑥)
197	60	ad2antrr 726	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) ∧ (¬ (^◡𝐹‘𝑥) < 𝐾 ∧ 𝑘 = ((^◡𝐹‘𝑥) − 1))) → ^◡𝐹:(𝑀...(𝑁 + 1))–1-1→(𝑀...(𝑁 + 1)))
198	66	ad2antrr 726	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) ∧ (¬ (^◡𝐹‘𝑥) < 𝐾 ∧ 𝑘 = ((^◡𝐹‘𝑥) − 1))) → (𝑁 + 1) ∈ (𝑀...(𝑁 + 1)))
199		f1fveq 7260	. . . . . . . . . . . . . . . . . 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))
200	197, 198, 169, 199	syl12anc 836	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) ∧ (¬ (^◡𝐹‘𝑥) < 𝐾 ∧ 𝑘 = ((^◡𝐹‘𝑥) − 1))) → ((^◡𝐹‘(𝑁 + 1)) = (^◡𝐹‘𝑥) ↔ (𝑁 + 1) = 𝑥))
201	200	necon3bid 2977	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) ∧ (¬ (^◡𝐹‘𝑥) < 𝐾 ∧ 𝑘 = ((^◡𝐹‘𝑥) − 1))) → ((^◡𝐹‘(𝑁 + 1)) ≠ (^◡𝐹‘𝑥) ↔ (𝑁 + 1) ≠ 𝑥))
202	196, 201	mpbird 257	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) ∧ (¬ (^◡𝐹‘𝑥) < 𝐾 ∧ 𝑘 = ((^◡𝐹‘𝑥) − 1))) → (^◡𝐹‘(𝑁 + 1)) ≠ (^◡𝐹‘𝑥))
203	34	neeq1i 2997	. . . . . . . . . . . . . . 15 ⊢ (𝐾 ≠ (^◡𝐹‘𝑥) ↔ (^◡𝐹‘(𝑁 + 1)) ≠ (^◡𝐹‘𝑥))
204	202, 203	sylibr 234	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) ∧ (¬ (^◡𝐹‘𝑥) < 𝐾 ∧ 𝑘 = ((^◡𝐹‘𝑥) − 1))) → 𝐾 ≠ (^◡𝐹‘𝑥))
205	204	necomd 2988	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) ∧ (¬ (^◡𝐹‘𝑥) < 𝐾 ∧ 𝑘 = ((^◡𝐹‘𝑥) − 1))) → (^◡𝐹‘𝑥) ≠ 𝐾)
206	177, 182, 184, 205	leneltd 11394	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) ∧ (¬ (^◡𝐹‘𝑥) < 𝐾 ∧ 𝑘 = ((^◡𝐹‘𝑥) − 1))) → 𝐾 < (^◡𝐹‘𝑥))
207	69	ad2antrr 726	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) ∧ (¬ (^◡𝐹‘𝑥) < 𝐾 ∧ 𝑘 = ((^◡𝐹‘𝑥) − 1))) → 𝐾 ∈ ℤ)
208		zltlem1 12650	. . . . . . . . . . . . 13 ⊢ ((𝐾 ∈ ℤ ∧ (^◡𝐹‘𝑥) ∈ ℤ) → (𝐾 < (^◡𝐹‘𝑥) ↔ 𝐾 ≤ ((^◡𝐹‘𝑥) − 1)))
209	207, 171, 208	syl2anc 584	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) ∧ (¬ (^◡𝐹‘𝑥) < 𝐾 ∧ 𝑘 = ((^◡𝐹‘𝑥) − 1))) → (𝐾 < (^◡𝐹‘𝑥) ↔ 𝐾 ≤ ((^◡𝐹‘𝑥) − 1)))
210	206, 209	mpbid 232	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) ∧ (¬ (^◡𝐹‘𝑥) < 𝐾 ∧ 𝑘 = ((^◡𝐹‘𝑥) − 1))) → 𝐾 ≤ ((^◡𝐹‘𝑥) − 1))
211	210, 166	breqtrrd 5152	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) ∧ (¬ (^◡𝐹‘𝑥) < 𝐾 ∧ 𝑘 = ((^◡𝐹‘𝑥) − 1))) → 𝐾 ≤ 𝑘)
212	176, 177, 178, 181, 211	letrd 11397	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) ∧ (¬ (^◡𝐹‘𝑥) < 𝐾 ∧ 𝑘 = ((^◡𝐹‘𝑥) − 1))) → 𝑀 ≤ 𝑘)
213		elfzle2 13550	. . . . . . . . . . . 12 ⊢ ((^◡𝐹‘𝑥) ∈ (𝑀...(𝑁 + 1)) → (^◡𝐹‘𝑥) ≤ (𝑁 + 1))
214	170, 213	syl 17	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) ∧ (¬ (^◡𝐹‘𝑥) < 𝐾 ∧ 𝑘 = ((^◡𝐹‘𝑥) − 1))) → (^◡𝐹‘𝑥) ≤ (𝑁 + 1))
215	188	ad2antrr 726	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) ∧ (¬ (^◡𝐹‘𝑥) < 𝐾 ∧ 𝑘 = ((^◡𝐹‘𝑥) − 1))) → 𝑁 ∈ ℝ)
216		1re 11240	. . . . . . . . . . . . 13 ⊢ 1 ∈ ℝ
217		lesubadd 11714	. . . . . . . . . . . . 13 ⊢ (((^◡𝐹‘𝑥) ∈ ℝ ∧ 1 ∈ ℝ ∧ 𝑁 ∈ ℝ) → (((^◡𝐹‘𝑥) − 1) ≤ 𝑁 ↔ (^◡𝐹‘𝑥) ≤ (𝑁 + 1)))
218	216, 217	mp3an2 1451	. . . . . . . . . . . 12 ⊢ (((^◡𝐹‘𝑥) ∈ ℝ ∧ 𝑁 ∈ ℝ) → (((^◡𝐹‘𝑥) − 1) ≤ 𝑁 ↔ (^◡𝐹‘𝑥) ≤ (𝑁 + 1)))
219	182, 215, 218	syl2anc 584	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) ∧ (¬ (^◡𝐹‘𝑥) < 𝐾 ∧ 𝑘 = ((^◡𝐹‘𝑥) − 1))) → (((^◡𝐹‘𝑥) − 1) ≤ 𝑁 ↔ (^◡𝐹‘𝑥) ≤ (𝑁 + 1)))
220	214, 219	mpbird 257	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) ∧ (¬ (^◡𝐹‘𝑥) < 𝐾 ∧ 𝑘 = ((^◡𝐹‘𝑥) − 1))) → ((^◡𝐹‘𝑥) − 1) ≤ 𝑁)
221	166, 220	eqbrtrd 5146	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) ∧ (¬ (^◡𝐹‘𝑥) < 𝐾 ∧ 𝑘 = ((^◡𝐹‘𝑥) − 1))) → 𝑘 ≤ 𝑁)
222	164, 165, 174, 212, 221	elfzd 13537	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) ∧ (¬ (^◡𝐹‘𝑥) < 𝐾 ∧ 𝑘 = ((^◡𝐹‘𝑥) − 1))) → 𝑘 ∈ (𝑀...𝑁))
223	177, 178, 211	lensymd 11391	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) ∧ (¬ (^◡𝐹‘𝑥) < 𝐾 ∧ 𝑘 = ((^◡𝐹‘𝑥) − 1))) → ¬ 𝑘 < 𝐾)
224	223, 53	syl 17	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) ∧ (¬ (^◡𝐹‘𝑥) < 𝐾 ∧ 𝑘 = ((^◡𝐹‘𝑥) − 1))) → (𝐹‘if(𝑘 < 𝐾, 𝑘, (𝑘 + 1))) = (𝐹‘(𝑘 + 1)))
225	166	oveq1d 7425	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) ∧ (¬ (^◡𝐹‘𝑥) < 𝐾 ∧ 𝑘 = ((^◡𝐹‘𝑥) − 1))) → (𝑘 + 1) = (((^◡𝐹‘𝑥) − 1) + 1))
226	171	zcnd 12703	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) ∧ (¬ (^◡𝐹‘𝑥) < 𝐾 ∧ 𝑘 = ((^◡𝐹‘𝑥) − 1))) → (^◡𝐹‘𝑥) ∈ ℂ)
227		npcan 11496	. . . . . . . . . . . 12 ⊢ (((^◡𝐹‘𝑥) ∈ ℂ ∧ 1 ∈ ℂ) → (((^◡𝐹‘𝑥) − 1) + 1) = (^◡𝐹‘𝑥))
228	226, 112, 227	sylancl 586	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) ∧ (¬ (^◡𝐹‘𝑥) < 𝐾 ∧ 𝑘 = ((^◡𝐹‘𝑥) − 1))) → (((^◡𝐹‘𝑥) − 1) + 1) = (^◡𝐹‘𝑥))
229	225, 228	eqtrd 2771	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) ∧ (¬ (^◡𝐹‘𝑥) < 𝐾 ∧ 𝑘 = ((^◡𝐹‘𝑥) − 1))) → (𝑘 + 1) = (^◡𝐹‘𝑥))
230	229	fveq2d 6885	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) ∧ (¬ (^◡𝐹‘𝑥) < 𝐾 ∧ 𝑘 = ((^◡𝐹‘𝑥) − 1))) → (𝐹‘(𝑘 + 1)) = (𝐹‘(^◡𝐹‘𝑥)))
231	17	ad2antrr 726	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) ∧ (¬ (^◡𝐹‘𝑥) < 𝐾 ∧ 𝑘 = ((^◡𝐹‘𝑥) − 1))) → 𝐹:(𝑀...(𝑁 + 1))–1-1-onto→(𝑀...(𝑁 + 1)))
232	231, 169, 156	syl2anc 584	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) ∧ (¬ (^◡𝐹‘𝑥) < 𝐾 ∧ 𝑘 = ((^◡𝐹‘𝑥) − 1))) → (𝐹‘(^◡𝐹‘𝑥)) = 𝑥)
233	224, 230, 232	3eqtrrd 2776	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) ∧ (¬ (^◡𝐹‘𝑥) < 𝐾 ∧ 𝑘 = ((^◡𝐹‘𝑥) − 1))) → 𝑥 = (𝐹‘if(𝑘 < 𝐾, 𝑘, (𝑘 + 1))))
234	222, 233	jca 511	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) ∧ (¬ (^◡𝐹‘𝑥) < 𝐾 ∧ 𝑘 = ((^◡𝐹‘𝑥) − 1))) → (𝑘 ∈ (𝑀...𝑁) ∧ 𝑥 = (𝐹‘if(𝑘 < 𝐾, 𝑘, (𝑘 + 1)))))
235	234	expr 456	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) ∧ ¬ (^◡𝐹‘𝑥) < 𝐾) → (𝑘 = ((^◡𝐹‘𝑥) − 1) → (𝑘 ∈ (𝑀...𝑁) ∧ 𝑥 = (𝐹‘if(𝑘 < 𝐾, 𝑘, (𝑘 + 1))))))
236	163, 235	sylbid 240	. . . . 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))))))
237	161, 236	pm2.61dan 812	. . . 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))))))
238	237	expimpd 453	. . 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))))))
239	120, 238	impbid 212	. 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)))))
240	1, 2, 6, 239	f1od 7664	1 ⊢ (𝜑 → 𝐽:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 ∨ wo 847 ∧ w3a 1086 = wceq 1540 ∈ wcel 2109 ≠ wne 2933 Vcvv 3464 ⊆ wss 3931 ifcif 4505 class class class wbr 5124 ↦ cmpt 5206 ^◡ccnv 5658 ⟶wf 6532 –1-1→wf1 6533 –1-1-onto→wf1o 6535 ‘cfv 6536 (class class class)co 7410 ℂcc 11132 ℝcr 11133 1c1 11135 + caddc 11137 < clt 11274 ≤ cle 11275 − cmin 11471 ℤcz 12593 ℤ_≥cuz 12857 ...cfz 13529

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2708 ax-sep 5271 ax-nul 5281 ax-pow 5340 ax-pr 5407 ax-un 7734 ax-cnex 11190 ax-resscn 11191 ax-1cn 11192 ax-icn 11193 ax-addcl 11194 ax-addrcl 11195 ax-mulcl 11196 ax-mulrcl 11197 ax-mulcom 11198 ax-addass 11199 ax-mulass 11200 ax-distr 11201 ax-i2m1 11202 ax-1ne0 11203 ax-1rid 11204 ax-rnegex 11205 ax-rrecex 11206 ax-cnre 11207 ax-pre-lttri 11208 ax-pre-lttrn 11209 ax-pre-ltadd 11210 ax-pre-mulgt0 11211

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2810 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3062 df-reu 3365 df-rab 3421 df-v 3466 df-sbc 3771 df-csb 3880 df-dif 3934 df-un 3936 df-in 3938 df-ss 3948 df-pss 3951 df-nul 4314 df-if 4506 df-pw 4582 df-sn 4607 df-pr 4609 df-op 4613 df-uni 4889 df-iun 4974 df-br 5125 df-opab 5187 df-mpt 5207 df-tr 5235 df-id 5553 df-eprel 5558 df-po 5566 df-so 5567 df-fr 5611 df-we 5613 df-xp 5665 df-rel 5666 df-cnv 5667 df-co 5668 df-dm 5669 df-rn 5670 df-res 5671 df-ima 5672 df-pred 6295 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6489 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-riota 7367 df-ov 7413 df-oprab 7414 df-mpo 7415 df-om 7867 df-1st 7993 df-2nd 7994 df-frecs 8285 df-wrecs 8316 df-recs 8390 df-rdg 8429 df-er 8724 df-en 8965 df-dom 8966 df-sdom 8967 df-pnf 11276 df-mnf 11277 df-xr 11278 df-ltxr 11279 df-le 11280 df-sub 11473 df-neg 11474 df-nn 12246 df-n0 12507 df-z 12594 df-uz 12858 df-fz 13530

This theorem is referenced by: seqf1olem2 14065

W3C validator