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Theorem fvmptnn04if 20702
Description: The function values of a mapping from the nonnegative integers with four distinct cases. (Contributed by AV, 10-Nov-2019.)
Hypotheses
Ref Expression
fvmptnn04if.g 𝐺 = (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, 𝐴, if(𝑛 = 𝑆, 𝐶, if(𝑆 < 𝑛, 𝐷, 𝐵))))
fvmptnn04if.s (𝜑𝑆 ∈ ℕ)
fvmptnn04if.n (𝜑𝑁 ∈ ℕ0)
fvmptnn04if.y (𝜑𝑌𝑉)
fvmptnn04if.a ((𝜑𝑁 = 0) → 𝑌 = 𝑁 / 𝑛𝐴)
fvmptnn04if.b ((𝜑 ∧ 0 < 𝑁𝑁 < 𝑆) → 𝑌 = 𝑁 / 𝑛𝐵)
fvmptnn04if.c ((𝜑𝑁 = 𝑆) → 𝑌 = 𝑁 / 𝑛𝐶)
fvmptnn04if.d ((𝜑𝑆 < 𝑁) → 𝑌 = 𝑁 / 𝑛𝐷)
Assertion
Ref Expression
fvmptnn04if (𝜑 → (𝐺𝑁) = 𝑌)
Distinct variable groups:   𝑛,𝑁   𝑆,𝑛
Allowed substitution hints:   𝜑(𝑛)   𝐴(𝑛)   𝐵(𝑛)   𝐶(𝑛)   𝐷(𝑛)   𝐺(𝑛)   𝑉(𝑛)   𝑌(𝑛)

Proof of Theorem fvmptnn04if
StepHypRef Expression
1 fvmptnn04if.n . . 3 (𝜑𝑁 ∈ ℕ0)
2 csbif 4171 . . . . 5 𝑁 / 𝑛if(𝑛 = 0, 𝐴, if(𝑛 = 𝑆, 𝐶, if(𝑆 < 𝑛, 𝐷, 𝐵))) = if([𝑁 / 𝑛]𝑛 = 0, 𝑁 / 𝑛𝐴, 𝑁 / 𝑛if(𝑛 = 𝑆, 𝐶, if(𝑆 < 𝑛, 𝐷, 𝐵)))
3 eqsbc3 3508 . . . . . . 7 (𝑁 ∈ ℕ0 → ([𝑁 / 𝑛]𝑛 = 0 ↔ 𝑁 = 0))
41, 3syl 17 . . . . . 6 (𝜑 → ([𝑁 / 𝑛]𝑛 = 0 ↔ 𝑁 = 0))
5 csbif 4171 . . . . . . 7 𝑁 / 𝑛if(𝑛 = 𝑆, 𝐶, if(𝑆 < 𝑛, 𝐷, 𝐵)) = if([𝑁 / 𝑛]𝑛 = 𝑆, 𝑁 / 𝑛𝐶, 𝑁 / 𝑛if(𝑆 < 𝑛, 𝐷, 𝐵))
6 eqsbc3 3508 . . . . . . . . 9 (𝑁 ∈ ℕ0 → ([𝑁 / 𝑛]𝑛 = 𝑆𝑁 = 𝑆))
71, 6syl 17 . . . . . . . 8 (𝜑 → ([𝑁 / 𝑛]𝑛 = 𝑆𝑁 = 𝑆))
8 csbif 4171 . . . . . . . . 9 𝑁 / 𝑛if(𝑆 < 𝑛, 𝐷, 𝐵) = if([𝑁 / 𝑛]𝑆 < 𝑛, 𝑁 / 𝑛𝐷, 𝑁 / 𝑛𝐵)
9 sbcbr2g 4743 . . . . . . . . . . . 12 (𝑁 ∈ ℕ0 → ([𝑁 / 𝑛]𝑆 < 𝑛𝑆 < 𝑁 / 𝑛𝑛))
101, 9syl 17 . . . . . . . . . . 11 (𝜑 → ([𝑁 / 𝑛]𝑆 < 𝑛𝑆 < 𝑁 / 𝑛𝑛))
11 csbvarg 4036 . . . . . . . . . . . . 13 (𝑁 ∈ ℕ0𝑁 / 𝑛𝑛 = 𝑁)
121, 11syl 17 . . . . . . . . . . . 12 (𝜑𝑁 / 𝑛𝑛 = 𝑁)
1312breq2d 4697 . . . . . . . . . . 11 (𝜑 → (𝑆 < 𝑁 / 𝑛𝑛𝑆 < 𝑁))
1410, 13bitrd 268 . . . . . . . . . 10 (𝜑 → ([𝑁 / 𝑛]𝑆 < 𝑛𝑆 < 𝑁))
1514ifbid 4141 . . . . . . . . 9 (𝜑 → if([𝑁 / 𝑛]𝑆 < 𝑛, 𝑁 / 𝑛𝐷, 𝑁 / 𝑛𝐵) = if(𝑆 < 𝑁, 𝑁 / 𝑛𝐷, 𝑁 / 𝑛𝐵))
168, 15syl5eq 2697 . . . . . . . 8 (𝜑𝑁 / 𝑛if(𝑆 < 𝑛, 𝐷, 𝐵) = if(𝑆 < 𝑁, 𝑁 / 𝑛𝐷, 𝑁 / 𝑛𝐵))
177, 16ifbieq2d 4144 . . . . . . 7 (𝜑 → if([𝑁 / 𝑛]𝑛 = 𝑆, 𝑁 / 𝑛𝐶, 𝑁 / 𝑛if(𝑆 < 𝑛, 𝐷, 𝐵)) = if(𝑁 = 𝑆, 𝑁 / 𝑛𝐶, if(𝑆 < 𝑁, 𝑁 / 𝑛𝐷, 𝑁 / 𝑛𝐵)))
185, 17syl5eq 2697 . . . . . 6 (𝜑𝑁 / 𝑛if(𝑛 = 𝑆, 𝐶, if(𝑆 < 𝑛, 𝐷, 𝐵)) = if(𝑁 = 𝑆, 𝑁 / 𝑛𝐶, if(𝑆 < 𝑁, 𝑁 / 𝑛𝐷, 𝑁 / 𝑛𝐵)))
194, 18ifbieq2d 4144 . . . . 5 (𝜑 → if([𝑁 / 𝑛]𝑛 = 0, 𝑁 / 𝑛𝐴, 𝑁 / 𝑛if(𝑛 = 𝑆, 𝐶, if(𝑆 < 𝑛, 𝐷, 𝐵))) = if(𝑁 = 0, 𝑁 / 𝑛𝐴, if(𝑁 = 𝑆, 𝑁 / 𝑛𝐶, if(𝑆 < 𝑁, 𝑁 / 𝑛𝐷, 𝑁 / 𝑛𝐵))))
202, 19syl5eq 2697 . . . 4 (𝜑𝑁 / 𝑛if(𝑛 = 0, 𝐴, if(𝑛 = 𝑆, 𝐶, if(𝑆 < 𝑛, 𝐷, 𝐵))) = if(𝑁 = 0, 𝑁 / 𝑛𝐴, if(𝑁 = 𝑆, 𝑁 / 𝑛𝐶, if(𝑆 < 𝑁, 𝑁 / 𝑛𝐷, 𝑁 / 𝑛𝐵))))
21 fvmptnn04if.a . . . . . 6 ((𝜑𝑁 = 0) → 𝑌 = 𝑁 / 𝑛𝐴)
22 fvmptnn04if.y . . . . . . 7 (𝜑𝑌𝑉)
2322adantr 480 . . . . . 6 ((𝜑𝑁 = 0) → 𝑌𝑉)
2421, 23eqeltrrd 2731 . . . . 5 ((𝜑𝑁 = 0) → 𝑁 / 𝑛𝐴𝑉)
25 fvmptnn04if.c . . . . . . . . 9 ((𝜑𝑁 = 𝑆) → 𝑌 = 𝑁 / 𝑛𝐶)
2625eqcomd 2657 . . . . . . . 8 ((𝜑𝑁 = 𝑆) → 𝑁 / 𝑛𝐶 = 𝑌)
2726adantlr 751 . . . . . . 7 (((𝜑 ∧ ¬ 𝑁 = 0) ∧ 𝑁 = 𝑆) → 𝑁 / 𝑛𝐶 = 𝑌)
2822ad2antrr 762 . . . . . . 7 (((𝜑 ∧ ¬ 𝑁 = 0) ∧ 𝑁 = 𝑆) → 𝑌𝑉)
2927, 28eqeltrd 2730 . . . . . 6 (((𝜑 ∧ ¬ 𝑁 = 0) ∧ 𝑁 = 𝑆) → 𝑁 / 𝑛𝐶𝑉)
30 fvmptnn04if.d . . . . . . . . . . . 12 ((𝜑𝑆 < 𝑁) → 𝑌 = 𝑁 / 𝑛𝐷)
3130eqcomd 2657 . . . . . . . . . . 11 ((𝜑𝑆 < 𝑁) → 𝑁 / 𝑛𝐷 = 𝑌)
3231ex 449 . . . . . . . . . 10 (𝜑 → (𝑆 < 𝑁𝑁 / 𝑛𝐷 = 𝑌))
3332ad2antrr 762 . . . . . . . . 9 (((𝜑 ∧ ¬ 𝑁 = 0) ∧ ¬ 𝑁 = 𝑆) → (𝑆 < 𝑁𝑁 / 𝑛𝐷 = 𝑌))
3433imp 444 . . . . . . . 8 ((((𝜑 ∧ ¬ 𝑁 = 0) ∧ ¬ 𝑁 = 𝑆) ∧ 𝑆 < 𝑁) → 𝑁 / 𝑛𝐷 = 𝑌)
3522ad3antrrr 766 . . . . . . . 8 ((((𝜑 ∧ ¬ 𝑁 = 0) ∧ ¬ 𝑁 = 𝑆) ∧ 𝑆 < 𝑁) → 𝑌𝑉)
3634, 35eqeltrd 2730 . . . . . . 7 ((((𝜑 ∧ ¬ 𝑁 = 0) ∧ ¬ 𝑁 = 𝑆) ∧ 𝑆 < 𝑁) → 𝑁 / 𝑛𝐷𝑉)
37 simplll 813 . . . . . . . . 9 ((((𝜑 ∧ ¬ 𝑁 = 0) ∧ ¬ 𝑁 = 𝑆) ∧ ¬ 𝑆 < 𝑁) → 𝜑)
38 ancom 465 . . . . . . . . . . . . 13 ((¬ 𝑆 < 𝑁𝜑) ↔ (𝜑 ∧ ¬ 𝑆 < 𝑁))
3938anbi2i 730 . . . . . . . . . . . 12 (((¬ 𝑁 = 0 ∧ ¬ 𝑁 = 𝑆) ∧ (¬ 𝑆 < 𝑁𝜑)) ↔ ((¬ 𝑁 = 0 ∧ ¬ 𝑁 = 𝑆) ∧ (𝜑 ∧ ¬ 𝑆 < 𝑁)))
40 ancom 465 . . . . . . . . . . . . 13 ((𝜑 ∧ (¬ 𝑁 = 0 ∧ (¬ 𝑁 = 𝑆 ∧ ¬ 𝑆 < 𝑁))) ↔ ((¬ 𝑁 = 0 ∧ (¬ 𝑁 = 𝑆 ∧ ¬ 𝑆 < 𝑁)) ∧ 𝜑))
41 anass 682 . . . . . . . . . . . . . . 15 (((¬ 𝑁 = 0 ∧ ¬ 𝑁 = 𝑆) ∧ ¬ 𝑆 < 𝑁) ↔ (¬ 𝑁 = 0 ∧ (¬ 𝑁 = 𝑆 ∧ ¬ 𝑆 < 𝑁)))
4241bicomi 214 . . . . . . . . . . . . . 14 ((¬ 𝑁 = 0 ∧ (¬ 𝑁 = 𝑆 ∧ ¬ 𝑆 < 𝑁)) ↔ ((¬ 𝑁 = 0 ∧ ¬ 𝑁 = 𝑆) ∧ ¬ 𝑆 < 𝑁))
4342anbi1i 731 . . . . . . . . . . . . 13 (((¬ 𝑁 = 0 ∧ (¬ 𝑁 = 𝑆 ∧ ¬ 𝑆 < 𝑁)) ∧ 𝜑) ↔ (((¬ 𝑁 = 0 ∧ ¬ 𝑁 = 𝑆) ∧ ¬ 𝑆 < 𝑁) ∧ 𝜑))
44 anass 682 . . . . . . . . . . . . 13 ((((¬ 𝑁 = 0 ∧ ¬ 𝑁 = 𝑆) ∧ ¬ 𝑆 < 𝑁) ∧ 𝜑) ↔ ((¬ 𝑁 = 0 ∧ ¬ 𝑁 = 𝑆) ∧ (¬ 𝑆 < 𝑁𝜑)))
4540, 43, 443bitri 286 . . . . . . . . . . . 12 ((𝜑 ∧ (¬ 𝑁 = 0 ∧ (¬ 𝑁 = 𝑆 ∧ ¬ 𝑆 < 𝑁))) ↔ ((¬ 𝑁 = 0 ∧ ¬ 𝑁 = 𝑆) ∧ (¬ 𝑆 < 𝑁𝜑)))
46 anass 682 . . . . . . . . . . . 12 ((((¬ 𝑁 = 0 ∧ ¬ 𝑁 = 𝑆) ∧ 𝜑) ∧ ¬ 𝑆 < 𝑁) ↔ ((¬ 𝑁 = 0 ∧ ¬ 𝑁 = 𝑆) ∧ (𝜑 ∧ ¬ 𝑆 < 𝑁)))
4739, 45, 463bitr4i 292 . . . . . . . . . . 11 ((𝜑 ∧ (¬ 𝑁 = 0 ∧ (¬ 𝑁 = 𝑆 ∧ ¬ 𝑆 < 𝑁))) ↔ (((¬ 𝑁 = 0 ∧ ¬ 𝑁 = 𝑆) ∧ 𝜑) ∧ ¬ 𝑆 < 𝑁))
48 an32 856 . . . . . . . . . . . . 13 (((¬ 𝑁 = 0 ∧ ¬ 𝑁 = 𝑆) ∧ 𝜑) ↔ ((¬ 𝑁 = 0 ∧ 𝜑) ∧ ¬ 𝑁 = 𝑆))
49 ancom 465 . . . . . . . . . . . . . 14 ((¬ 𝑁 = 0 ∧ 𝜑) ↔ (𝜑 ∧ ¬ 𝑁 = 0))
5049anbi1i 731 . . . . . . . . . . . . 13 (((¬ 𝑁 = 0 ∧ 𝜑) ∧ ¬ 𝑁 = 𝑆) ↔ ((𝜑 ∧ ¬ 𝑁 = 0) ∧ ¬ 𝑁 = 𝑆))
5148, 50bitri 264 . . . . . . . . . . . 12 (((¬ 𝑁 = 0 ∧ ¬ 𝑁 = 𝑆) ∧ 𝜑) ↔ ((𝜑 ∧ ¬ 𝑁 = 0) ∧ ¬ 𝑁 = 𝑆))
5251anbi1i 731 . . . . . . . . . . 11 ((((¬ 𝑁 = 0 ∧ ¬ 𝑁 = 𝑆) ∧ 𝜑) ∧ ¬ 𝑆 < 𝑁) ↔ (((𝜑 ∧ ¬ 𝑁 = 0) ∧ ¬ 𝑁 = 𝑆) ∧ ¬ 𝑆 < 𝑁))
5347, 52bitri 264 . . . . . . . . . 10 ((𝜑 ∧ (¬ 𝑁 = 0 ∧ (¬ 𝑁 = 𝑆 ∧ ¬ 𝑆 < 𝑁))) ↔ (((𝜑 ∧ ¬ 𝑁 = 0) ∧ ¬ 𝑁 = 𝑆) ∧ ¬ 𝑆 < 𝑁))
54 df-ne 2824 . . . . . . . . . . . . 13 (𝑁 ≠ 0 ↔ ¬ 𝑁 = 0)
55 elnnne0 11344 . . . . . . . . . . . . . . 15 (𝑁 ∈ ℕ ↔ (𝑁 ∈ ℕ0𝑁 ≠ 0))
56 nngt0 11087 . . . . . . . . . . . . . . 15 (𝑁 ∈ ℕ → 0 < 𝑁)
5755, 56sylbir 225 . . . . . . . . . . . . . 14 ((𝑁 ∈ ℕ0𝑁 ≠ 0) → 0 < 𝑁)
5857expcom 450 . . . . . . . . . . . . 13 (𝑁 ≠ 0 → (𝑁 ∈ ℕ0 → 0 < 𝑁))
5954, 58sylbir 225 . . . . . . . . . . . 12 𝑁 = 0 → (𝑁 ∈ ℕ0 → 0 < 𝑁))
6059adantr 480 . . . . . . . . . . 11 ((¬ 𝑁 = 0 ∧ (¬ 𝑁 = 𝑆 ∧ ¬ 𝑆 < 𝑁)) → (𝑁 ∈ ℕ0 → 0 < 𝑁))
611, 60mpan9 485 . . . . . . . . . 10 ((𝜑 ∧ (¬ 𝑁 = 0 ∧ (¬ 𝑁 = 𝑆 ∧ ¬ 𝑆 < 𝑁))) → 0 < 𝑁)
6253, 61sylbir 225 . . . . . . . . 9 ((((𝜑 ∧ ¬ 𝑁 = 0) ∧ ¬ 𝑁 = 𝑆) ∧ ¬ 𝑆 < 𝑁) → 0 < 𝑁)
631nn0red 11390 . . . . . . . . . . . . . . . 16 (𝜑𝑁 ∈ ℝ)
64 fvmptnn04if.s . . . . . . . . . . . . . . . . 17 (𝜑𝑆 ∈ ℕ)
6564nnred 11073 . . . . . . . . . . . . . . . 16 (𝜑𝑆 ∈ ℝ)
6663, 65lenltd 10221 . . . . . . . . . . . . . . 15 (𝜑 → (𝑁𝑆 ↔ ¬ 𝑆 < 𝑁))
6766biimprd 238 . . . . . . . . . . . . . 14 (𝜑 → (¬ 𝑆 < 𝑁𝑁𝑆))
6867adantld 482 . . . . . . . . . . . . 13 (𝜑 → ((¬ 𝑁 = 𝑆 ∧ ¬ 𝑆 < 𝑁) → 𝑁𝑆))
6968adantld 482 . . . . . . . . . . . 12 (𝜑 → ((¬ 𝑁 = 0 ∧ (¬ 𝑁 = 𝑆 ∧ ¬ 𝑆 < 𝑁)) → 𝑁𝑆))
7069imp 444 . . . . . . . . . . 11 ((𝜑 ∧ (¬ 𝑁 = 0 ∧ (¬ 𝑁 = 𝑆 ∧ ¬ 𝑆 < 𝑁))) → 𝑁𝑆)
71 nesym 2879 . . . . . . . . . . . . . 14 (𝑆𝑁 ↔ ¬ 𝑁 = 𝑆)
7271biimpri 218 . . . . . . . . . . . . 13 𝑁 = 𝑆𝑆𝑁)
7372adantr 480 . . . . . . . . . . . 12 ((¬ 𝑁 = 𝑆 ∧ ¬ 𝑆 < 𝑁) → 𝑆𝑁)
7473ad2antll 765 . . . . . . . . . . 11 ((𝜑 ∧ (¬ 𝑁 = 0 ∧ (¬ 𝑁 = 𝑆 ∧ ¬ 𝑆 < 𝑁))) → 𝑆𝑁)
7563adantr 480 . . . . . . . . . . . 12 ((𝜑 ∧ (¬ 𝑁 = 0 ∧ (¬ 𝑁 = 𝑆 ∧ ¬ 𝑆 < 𝑁))) → 𝑁 ∈ ℝ)
7665adantr 480 . . . . . . . . . . . 12 ((𝜑 ∧ (¬ 𝑁 = 0 ∧ (¬ 𝑁 = 𝑆 ∧ ¬ 𝑆 < 𝑁))) → 𝑆 ∈ ℝ)
7775, 76ltlend 10220 . . . . . . . . . . 11 ((𝜑 ∧ (¬ 𝑁 = 0 ∧ (¬ 𝑁 = 𝑆 ∧ ¬ 𝑆 < 𝑁))) → (𝑁 < 𝑆 ↔ (𝑁𝑆𝑆𝑁)))
7870, 74, 77mpbir2and 977 . . . . . . . . . 10 ((𝜑 ∧ (¬ 𝑁 = 0 ∧ (¬ 𝑁 = 𝑆 ∧ ¬ 𝑆 < 𝑁))) → 𝑁 < 𝑆)
7953, 78sylbir 225 . . . . . . . . 9 ((((𝜑 ∧ ¬ 𝑁 = 0) ∧ ¬ 𝑁 = 𝑆) ∧ ¬ 𝑆 < 𝑁) → 𝑁 < 𝑆)
80 fvmptnn04if.b . . . . . . . . . 10 ((𝜑 ∧ 0 < 𝑁𝑁 < 𝑆) → 𝑌 = 𝑁 / 𝑛𝐵)
8180eqcomd 2657 . . . . . . . . 9 ((𝜑 ∧ 0 < 𝑁𝑁 < 𝑆) → 𝑁 / 𝑛𝐵 = 𝑌)
8237, 62, 79, 81syl3anc 1366 . . . . . . . 8 ((((𝜑 ∧ ¬ 𝑁 = 0) ∧ ¬ 𝑁 = 𝑆) ∧ ¬ 𝑆 < 𝑁) → 𝑁 / 𝑛𝐵 = 𝑌)
8322ad3antrrr 766 . . . . . . . 8 ((((𝜑 ∧ ¬ 𝑁 = 0) ∧ ¬ 𝑁 = 𝑆) ∧ ¬ 𝑆 < 𝑁) → 𝑌𝑉)
8482, 83eqeltrd 2730 . . . . . . 7 ((((𝜑 ∧ ¬ 𝑁 = 0) ∧ ¬ 𝑁 = 𝑆) ∧ ¬ 𝑆 < 𝑁) → 𝑁 / 𝑛𝐵𝑉)
8536, 84ifclda 4153 . . . . . 6 (((𝜑 ∧ ¬ 𝑁 = 0) ∧ ¬ 𝑁 = 𝑆) → if(𝑆 < 𝑁, 𝑁 / 𝑛𝐷, 𝑁 / 𝑛𝐵) ∈ 𝑉)
8629, 85ifclda 4153 . . . . 5 ((𝜑 ∧ ¬ 𝑁 = 0) → if(𝑁 = 𝑆, 𝑁 / 𝑛𝐶, if(𝑆 < 𝑁, 𝑁 / 𝑛𝐷, 𝑁 / 𝑛𝐵)) ∈ 𝑉)
8724, 86ifclda 4153 . . . 4 (𝜑 → if(𝑁 = 0, 𝑁 / 𝑛𝐴, if(𝑁 = 𝑆, 𝑁 / 𝑛𝐶, if(𝑆 < 𝑁, 𝑁 / 𝑛𝐷, 𝑁 / 𝑛𝐵))) ∈ 𝑉)
8820, 87eqeltrd 2730 . . 3 (𝜑𝑁 / 𝑛if(𝑛 = 0, 𝐴, if(𝑛 = 𝑆, 𝐶, if(𝑆 < 𝑛, 𝐷, 𝐵))) ∈ 𝑉)
89 fvmptnn04if.g . . . 4 𝐺 = (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, 𝐴, if(𝑛 = 𝑆, 𝐶, if(𝑆 < 𝑛, 𝐷, 𝐵))))
9089fvmpts 6324 . . 3 ((𝑁 ∈ ℕ0𝑁 / 𝑛if(𝑛 = 0, 𝐴, if(𝑛 = 𝑆, 𝐶, if(𝑆 < 𝑛, 𝐷, 𝐵))) ∈ 𝑉) → (𝐺𝑁) = 𝑁 / 𝑛if(𝑛 = 0, 𝐴, if(𝑛 = 𝑆, 𝐶, if(𝑆 < 𝑛, 𝐷, 𝐵))))
911, 88, 90syl2anc 694 . 2 (𝜑 → (𝐺𝑁) = 𝑁 / 𝑛if(𝑛 = 0, 𝐴, if(𝑛 = 𝑆, 𝐶, if(𝑆 < 𝑛, 𝐷, 𝐵))))
9221eqcomd 2657 . . 3 ((𝜑𝑁 = 0) → 𝑁 / 𝑛𝐴 = 𝑌)
9334, 82ifeqda 4154 . . . 4 (((𝜑 ∧ ¬ 𝑁 = 0) ∧ ¬ 𝑁 = 𝑆) → if(𝑆 < 𝑁, 𝑁 / 𝑛𝐷, 𝑁 / 𝑛𝐵) = 𝑌)
9427, 93ifeqda 4154 . . 3 ((𝜑 ∧ ¬ 𝑁 = 0) → if(𝑁 = 𝑆, 𝑁 / 𝑛𝐶, if(𝑆 < 𝑁, 𝑁 / 𝑛𝐷, 𝑁 / 𝑛𝐵)) = 𝑌)
9592, 94ifeqda 4154 . 2 (𝜑 → if(𝑁 = 0, 𝑁 / 𝑛𝐴, if(𝑁 = 𝑆, 𝑁 / 𝑛𝐶, if(𝑆 < 𝑁, 𝑁 / 𝑛𝐷, 𝑁 / 𝑛𝐵))) = 𝑌)
9691, 20, 953eqtrd 2689 1 (𝜑 → (𝐺𝑁) = 𝑌)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 383  w3a 1054   = wceq 1523  wcel 2030  wne 2823  [wsbc 3468  csb 3566  ifcif 4119   class class class wbr 4685  cmpt 4762  cfv 5926  cr 9973  0cc0 9974   < clt 10112  cle 10113  cn 11058  0cn0 11330
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-fal 1529  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-er 7787  df-en 7998  df-dom 7999  df-sdom 8000  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-nn 11059  df-n0 11331
This theorem is referenced by:  fvmptnn04ifa  20703  fvmptnn04ifb  20704  fvmptnn04ifc  20705  fvmptnn04ifd  20706
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