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 6920 | . 2
⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → (𝐹‘if(𝑘 < 𝐾, 𝑘, (𝑘 + 1))) ∈ V) | 
| 3 |  | fvex 6918 | . . . 4
⊢ (◡𝐹‘𝑥) ∈ V | 
| 4 |  | ovex 7465 | . . . 4
⊢ ((◡𝐹‘𝑥) − 1) ∈ V | 
| 5 | 3, 4 | ifex 4575 | . . 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 4530 | . . . . . . . . 9
⊢ (𝑘 < 𝐾 → if(𝑘 < 𝐾, 𝑘, (𝑘 + 1)) = 𝑘) | 
| 8 | 7 | fveq2d 6909 | . . . . . . . 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 13565 | . . . . . . . . . . . . 13
⊢ (𝑘 ∈ (𝑀...𝑁) → 𝑘 ∈ ℤ) | 
| 13 | 12 | zred 12724 | . . . . . . . . . . . 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 11397 | . . . . . . . . . 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 6847 | . . . . . . . . . . . . . . . . 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 13608 | . . . . . . . . . . . . . . . 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 3980 | . . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) ∧ (𝑘 < 𝐾 ∧ 𝑥 = (𝐹‘𝑘))) → 𝑘 ∈ (𝑀...(𝑁 + 1))) | 
| 24 | 20, 23 | ffvelcdmd 7104 | . . . . . . . . . . . . . 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 13615 | . . . . . . . . . . . . . . . 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 7299 | . . . . . . . . . . . . . 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 2956 | . . . . . . . . . 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 2840 | . . . . . . . 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 7299 | . . . . . . . . . . . . 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 5164 | . . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) ∧ (𝑘 < 𝐾 ∧ 𝑥 = (𝐹‘𝑘))) → (◡𝐹‘𝑥) < 𝐾) | 
| 46 |  | iftrue 4530 | . . . . . . . . . 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 2777 | . . . . . . . 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 4533 | . . . . . . . . 9
⊢ (¬
𝑘 < 𝐾 → if(𝑘 < 𝐾, 𝑘, (𝑘 + 1)) = (𝑘 + 1)) | 
| 53 | 52 | fveq2d 6909 | . . . . . . . 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 6859 | . . . . . . . . . . . . . . . . . . 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 6846 | . . . . . . . . . . . . . . . . . 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 6803 | . . . . . . . . . . . . . . . . 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 12944 | . . . . . . . . . . . . . . . . . 18
⊢ (𝑁 ∈
(ℤ≥‘𝑀) → (𝑁 + 1) ∈
(ℤ≥‘𝑀)) | 
| 64 | 25, 63 | syl 17 | . . . . . . . . . . . . . . . . 17
⊢ (𝜑 → (𝑁 + 1) ∈
(ℤ≥‘𝑀)) | 
| 65 |  | eluzfz2 13573 | . . . . . . . . . . . . . . . . 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 7104 | . . . . . . . . . . . . . . 15
⊢ (𝜑 → (◡𝐹‘(𝑁 + 1)) ∈ (𝑀...(𝑁 + 1))) | 
| 68 | 34, 67 | eqeltrid 2844 | . . . . . . . . . . . . . 14
⊢ (𝜑 → 𝐾 ∈ (𝑀...(𝑁 + 1))) | 
| 69 | 68 | elfzelzd 13566 | . . . . . . . . . . . . 13
⊢ (𝜑 → 𝐾 ∈ ℤ) | 
| 70 | 69 | zred 12724 | . . . . . . . . . . . 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 11435 | . . . . . . . . . . . . 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 11412 | . . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) ∧ (¬ 𝑘 < 𝐾 ∧ 𝑥 = (𝐹‘(𝑘 + 1)))) → 𝐾 ≤ 𝑘) | 
| 77 | 72 | ltp1d 12199 | . . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) ∧ (¬ 𝑘 < 𝐾 ∧ 𝑥 = (𝐹‘(𝑘 + 1)))) → 𝑘 < (𝑘 + 1)) | 
| 78 | 71, 72, 74, 76, 77 | lelttrd 11420 | . . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) ∧ (¬ 𝑘 < 𝐾 ∧ 𝑥 = (𝐹‘(𝑘 + 1)))) → 𝐾 < (𝑘 + 1)) | 
| 79 | 71, 78 | ltned 11398 | . . . . . . . . . 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 13618 | . . . . . . . . . . . . . . . 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 7104 | . . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) ∧ (¬ 𝑘 < 𝐾 ∧ 𝑥 = (𝐹‘(𝑘 + 1)))) → (𝐹‘(𝑘 + 1)) ∈ (𝑀...(𝑁 + 1))) | 
| 84 |  | elfzp1 13615 | . . . . . . . . . . . . . . . 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 7299 | . . . . . . . . . . . . . 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 2956 | . . . . . . . . . 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 2840 | . . . . . . . 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 7299 | . . . . . . . . . . . . . . 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 5152 | . . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) ∧ (¬ 𝑘 < 𝐾 ∧ 𝑥 = (𝐹‘(𝑘 + 1)))) → ((◡𝐹‘𝑥) < 𝐾 ↔ (𝑘 + 1) < 𝐾)) | 
| 103 |  | lttr 11338 | . . . . . . . . . . . . . 14
⊢ ((𝑘 ∈ ℝ ∧ (𝑘 + 1) ∈ ℝ ∧ 𝐾 ∈ ℝ) → ((𝑘 < (𝑘 + 1) ∧ (𝑘 + 1) < 𝐾) → 𝑘 < 𝐾)) | 
| 104 | 72, 74, 71, 103 | syl3anc 1372 | . . . . . . . . . . . . 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 4533 | . . . . . . . . . 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 7447 | . . . . . . . . 9
⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) ∧ (¬ 𝑘 < 𝐾 ∧ 𝑥 = (𝐹‘(𝑘 + 1)))) → ((◡𝐹‘𝑥) − 1) = ((𝑘 + 1) − 1)) | 
| 111 | 72 | recnd 11290 | . . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) ∧ (¬ 𝑘 < 𝐾 ∧ 𝑥 = (𝐹‘(𝑘 + 1)))) → 𝑘 ∈ ℂ) | 
| 112 |  | ax-1cn 11214 | . . . . . . . . . 10
⊢ 1 ∈
ℂ | 
| 113 |  | pncan 11515 | . . . . . . . . . 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 2781 | . . . . . . . 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 12884 | . . . . . . . . . . 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 12889 | . . . . . . . . . . 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 3980 | . . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) ∧ ((◡𝐹‘𝑥) < 𝐾 ∧ 𝑘 = (◡𝐹‘𝑥))) → 𝑥 ∈ (𝑀...(𝑁 + 1))) | 
| 133 | 130, 132 | ffvelcdmd 7104 | . . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) ∧ ((◡𝐹‘𝑥) < 𝐾 ∧ 𝑘 = (◡𝐹‘𝑥))) → (◡𝐹‘𝑥) ∈ (𝑀...(𝑁 + 1))) | 
| 134 | 129, 133 | eqeltrd 2840 | . . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) ∧ ((◡𝐹‘𝑥) < 𝐾 ∧ 𝑘 = (◡𝐹‘𝑥))) → 𝑘 ∈ (𝑀...(𝑁 + 1))) | 
| 135 | 134 | elfzelzd 13566 | . . . . . . . . 9
⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) ∧ ((◡𝐹‘𝑥) < 𝐾 ∧ 𝑘 = (◡𝐹‘𝑥))) → 𝑘 ∈ ℤ) | 
| 136 |  | elfzle1 13568 | . . . . . . . . . 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 12724 | . . . . . . . . . . 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 12727 | . . . . . . . . . . . . 13
⊢ (𝜑 → (𝑁 + 1) ∈ ℤ) | 
| 141 | 140 | zred 12724 | . . . . . . . . . . . 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 5164 | . . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) ∧ ((◡𝐹‘𝑥) < 𝐾 ∧ 𝑘 = (◡𝐹‘𝑥))) → 𝑘 < 𝐾) | 
| 145 |  | elfzle2 13569 | . . . . . . . . . . . . 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 11422 | . . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) ∧ ((◡𝐹‘𝑥) < 𝐾 ∧ 𝑘 = (◡𝐹‘𝑥))) → 𝑘 < (𝑁 + 1)) | 
| 149 |  | zleltp1 12670 | . . . . . . . . . . 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 13556 | . . . . . . . 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 6909 | . . . . . . . . 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 7298 | . . . . . . . . . 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 2781 | . . . . . . . 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 3980 | . . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) ∧ (¬ (◡𝐹‘𝑥) < 𝐾 ∧ 𝑘 = ((◡𝐹‘𝑥) − 1))) → 𝑥 ∈ (𝑀...(𝑁 + 1))) | 
| 170 | 167, 169 | ffvelcdmd 7104 | . . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) ∧ (¬ (◡𝐹‘𝑥) < 𝐾 ∧ 𝑘 = ((◡𝐹‘𝑥) − 1))) → (◡𝐹‘𝑥) ∈ (𝑀...(𝑁 + 1))) | 
| 171 | 170 | elfzelzd 13566 | . . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) ∧ (¬ (◡𝐹‘𝑥) < 𝐾 ∧ 𝑘 = ((◡𝐹‘𝑥) − 1))) → (◡𝐹‘𝑥) ∈ ℤ) | 
| 172 |  | peano2zm 12662 | . . . . . . . . . . 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 2840 | . . . . . . . . 9
⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) ∧ (¬ (◡𝐹‘𝑥) < 𝐾 ∧ 𝑘 = ((◡𝐹‘𝑥) − 1))) → 𝑘 ∈ ℤ) | 
| 175 | 124 | zred 12724 | . . . . . . . . . . 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 12724 | . . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) ∧ (¬ (◡𝐹‘𝑥) < 𝐾 ∧ 𝑘 = ((◡𝐹‘𝑥) − 1))) → 𝑘 ∈ ℝ) | 
| 179 |  | elfzle1 13568 | . . . . . . . . . . . 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 12724 | . . . . . . . . . . . . 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 11412 | . . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) ∧ (¬ (◡𝐹‘𝑥) < 𝐾 ∧ 𝑘 = ((◡𝐹‘𝑥) − 1))) → 𝐾 ≤ (◡𝐹‘𝑥)) | 
| 185 |  | elfzelz 13565 | . . . . . . . . . . . . . . . . . . . 20
⊢ (𝑥 ∈ (𝑀...𝑁) → 𝑥 ∈ ℤ) | 
| 186 | 185 | adantl 481 | . . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) → 𝑥 ∈ ℤ) | 
| 187 | 186 | zred 12724 | . . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) → 𝑥 ∈ ℝ) | 
| 188 | 127 | zred 12724 | . . . . . . . . . . . . . . . . . . . 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 13569 | . . . . . . . . . . . . . . . . . . . 20
⊢ (𝑥 ∈ (𝑀...𝑁) → 𝑥 ≤ 𝑁) | 
| 192 | 191 | adantl 481 | . . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) → 𝑥 ≤ 𝑁) | 
| 193 | 189 | ltp1d 12199 | . . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) → 𝑁 < (𝑁 + 1)) | 
| 194 | 187, 189,
190, 192, 193 | lelttrd 11420 | . . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) → 𝑥 < (𝑁 + 1)) | 
| 195 | 187, 194 | gtned 11397 | . . . . . . . . . . . . . . . . 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 7283 | . . . . . . . . . . . . . . . . . 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 2984 | . . . . . . . . . . . . . . . 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 3004 | . . . . . . . . . . . . . . 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 2995 | . . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) ∧ (¬ (◡𝐹‘𝑥) < 𝐾 ∧ 𝑘 = ((◡𝐹‘𝑥) − 1))) → (◡𝐹‘𝑥) ≠ 𝐾) | 
| 206 | 177, 182,
184, 205 | leneltd 11416 | . . . . . . . . . . . 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 12672 | . . . . . . . . . . . . 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 5170 | . . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) ∧ (¬ (◡𝐹‘𝑥) < 𝐾 ∧ 𝑘 = ((◡𝐹‘𝑥) − 1))) → 𝐾 ≤ 𝑘) | 
| 212 | 176, 177,
178, 181, 211 | letrd 11419 | . . . . . . . . 9
⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) ∧ (¬ (◡𝐹‘𝑥) < 𝐾 ∧ 𝑘 = ((◡𝐹‘𝑥) − 1))) → 𝑀 ≤ 𝑘) | 
| 213 |  | elfzle2 13569 | . . . . . . . . . . . 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 11262 | . . . . . . . . . . . . 13
⊢ 1 ∈
ℝ | 
| 217 |  | lesubadd 11736 | . . . . . . . . . . . . 13
⊢ (((◡𝐹‘𝑥) ∈ ℝ ∧ 1 ∈ ℝ ∧
𝑁 ∈ ℝ) →
(((◡𝐹‘𝑥) − 1) ≤ 𝑁 ↔ (◡𝐹‘𝑥) ≤ (𝑁 + 1))) | 
| 218 | 216, 217 | mp3an2 1450 | . . . . . . . . . . . 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 5164 | . . . . . . . . 9
⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) ∧ (¬ (◡𝐹‘𝑥) < 𝐾 ∧ 𝑘 = ((◡𝐹‘𝑥) − 1))) → 𝑘 ≤ 𝑁) | 
| 222 | 164, 165,
174, 212, 221 | elfzd 13556 | . . . . . . . 8
⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) ∧ (¬ (◡𝐹‘𝑥) < 𝐾 ∧ 𝑘 = ((◡𝐹‘𝑥) − 1))) → 𝑘 ∈ (𝑀...𝑁)) | 
| 223 | 177, 178,
211 | lensymd 11413 | . . . . . . . . . 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 7447 | . . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) ∧ (¬ (◡𝐹‘𝑥) < 𝐾 ∧ 𝑘 = ((◡𝐹‘𝑥) − 1))) → (𝑘 + 1) = (((◡𝐹‘𝑥) − 1) + 1)) | 
| 226 | 171 | zcnd 12725 | . . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) ∧ (¬ (◡𝐹‘𝑥) < 𝐾 ∧ 𝑘 = ((◡𝐹‘𝑥) − 1))) → (◡𝐹‘𝑥) ∈ ℂ) | 
| 227 |  | npcan 11518 | . . . . . . . . . . . 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 2776 | . . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) ∧ (¬ (◡𝐹‘𝑥) < 𝐾 ∧ 𝑘 = ((◡𝐹‘𝑥) − 1))) → (𝑘 + 1) = (◡𝐹‘𝑥)) | 
| 230 | 229 | fveq2d 6909 | . . . . . . . . 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 2781 | . . . . . . . 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 7686 | 1
⊢ (𝜑 → 𝐽:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁)) |