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Theorem fvmptnn04if 20635
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 4129 . . . . 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 3469 . . . . . . 7 (𝑁 ∈ ℕ0 → ([𝑁 / 𝑛]𝑛 = 0 ↔ 𝑁 = 0))
41, 3syl 17 . . . . . 6 (𝜑 → ([𝑁 / 𝑛]𝑛 = 0 ↔ 𝑁 = 0))
5 csbif 4129 . . . . . . 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 3469 . . . . . . . . 9 (𝑁 ∈ ℕ0 → ([𝑁 / 𝑛]𝑛 = 𝑆𝑁 = 𝑆))
71, 6syl 17 . . . . . . . 8 (𝜑 → ([𝑁 / 𝑛]𝑛 = 𝑆𝑁 = 𝑆))
8 csbif 4129 . . . . . . . . 9 𝑁 / 𝑛if(𝑆 < 𝑛, 𝐷, 𝐵) = if([𝑁 / 𝑛]𝑆 < 𝑛, 𝑁 / 𝑛𝐷, 𝑁 / 𝑛𝐵)
9 sbcbr2g 4701 . . . . . . . . . . . 12 (𝑁 ∈ ℕ0 → ([𝑁 / 𝑛]𝑆 < 𝑛𝑆 < 𝑁 / 𝑛𝑛))
101, 9syl 17 . . . . . . . . . . 11 (𝜑 → ([𝑁 / 𝑛]𝑆 < 𝑛𝑆 < 𝑁 / 𝑛𝑛))
11 csbvarg 3994 . . . . . . . . . . . . 13 (𝑁 ∈ ℕ0𝑁 / 𝑛𝑛 = 𝑁)
121, 11syl 17 . . . . . . . . . . . 12 (𝜑𝑁 / 𝑛𝑛 = 𝑁)
1312breq2d 4656 . . . . . . . . . . 11 (𝜑 → (𝑆 < 𝑁 / 𝑛𝑛𝑆 < 𝑁))
1410, 13bitrd 268 . . . . . . . . . 10 (𝜑 → ([𝑁 / 𝑛]𝑆 < 𝑛𝑆 < 𝑁))
1514ifbid 4099 . . . . . . . . 9 (𝜑 → if([𝑁 / 𝑛]𝑆 < 𝑛, 𝑁 / 𝑛𝐷, 𝑁 / 𝑛𝐵) = if(𝑆 < 𝑁, 𝑁 / 𝑛𝐷, 𝑁 / 𝑛𝐵))
168, 15syl5eq 2666 . . . . . . . 8 (𝜑𝑁 / 𝑛if(𝑆 < 𝑛, 𝐷, 𝐵) = if(𝑆 < 𝑁, 𝑁 / 𝑛𝐷, 𝑁 / 𝑛𝐵))
177, 16ifbieq2d 4102 . . . . . . 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 2666 . . . . . 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 4102 . . . . 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 2666 . . . 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 481 . . . . . 6 ((𝜑𝑁 = 0) → 𝑌𝑉)
2421, 23eqeltrrd 2700 . . . . 5 ((𝜑𝑁 = 0) → 𝑁 / 𝑛𝐴𝑉)
25 fvmptnn04if.c . . . . . . . . 9 ((𝜑𝑁 = 𝑆) → 𝑌 = 𝑁 / 𝑛𝐶)
2625eqcomd 2626 . . . . . . . 8 ((𝜑𝑁 = 𝑆) → 𝑁 / 𝑛𝐶 = 𝑌)
2726adantlr 750 . . . . . . 7 (((𝜑 ∧ ¬ 𝑁 = 0) ∧ 𝑁 = 𝑆) → 𝑁 / 𝑛𝐶 = 𝑌)
2822ad2antrr 761 . . . . . . 7 (((𝜑 ∧ ¬ 𝑁 = 0) ∧ 𝑁 = 𝑆) → 𝑌𝑉)
2927, 28eqeltrd 2699 . . . . . 6 (((𝜑 ∧ ¬ 𝑁 = 0) ∧ 𝑁 = 𝑆) → 𝑁 / 𝑛𝐶𝑉)
30 fvmptnn04if.d . . . . . . . . . . . 12 ((𝜑𝑆 < 𝑁) → 𝑌 = 𝑁 / 𝑛𝐷)
3130eqcomd 2626 . . . . . . . . . . 11 ((𝜑𝑆 < 𝑁) → 𝑁 / 𝑛𝐷 = 𝑌)
3231ex 450 . . . . . . . . . 10 (𝜑 → (𝑆 < 𝑁𝑁 / 𝑛𝐷 = 𝑌))
3332ad2antrr 761 . . . . . . . . 9 (((𝜑 ∧ ¬ 𝑁 = 0) ∧ ¬ 𝑁 = 𝑆) → (𝑆 < 𝑁𝑁 / 𝑛𝐷 = 𝑌))
3433imp 445 . . . . . . . 8 ((((𝜑 ∧ ¬ 𝑁 = 0) ∧ ¬ 𝑁 = 𝑆) ∧ 𝑆 < 𝑁) → 𝑁 / 𝑛𝐷 = 𝑌)
3522ad3antrrr 765 . . . . . . . 8 ((((𝜑 ∧ ¬ 𝑁 = 0) ∧ ¬ 𝑁 = 𝑆) ∧ 𝑆 < 𝑁) → 𝑌𝑉)
3634, 35eqeltrd 2699 . . . . . . 7 ((((𝜑 ∧ ¬ 𝑁 = 0) ∧ ¬ 𝑁 = 𝑆) ∧ 𝑆 < 𝑁) → 𝑁 / 𝑛𝐷𝑉)
37 simplll 797 . . . . . . . . 9 ((((𝜑 ∧ ¬ 𝑁 = 0) ∧ ¬ 𝑁 = 𝑆) ∧ ¬ 𝑆 < 𝑁) → 𝜑)
38 ancom 466 . . . . . . . . . . . . 13 ((¬ 𝑆 < 𝑁𝜑) ↔ (𝜑 ∧ ¬ 𝑆 < 𝑁))
3938anbi2i 729 . . . . . . . . . . . 12 (((¬ 𝑁 = 0 ∧ ¬ 𝑁 = 𝑆) ∧ (¬ 𝑆 < 𝑁𝜑)) ↔ ((¬ 𝑁 = 0 ∧ ¬ 𝑁 = 𝑆) ∧ (𝜑 ∧ ¬ 𝑆 < 𝑁)))
40 ancom 466 . . . . . . . . . . . . 13 ((𝜑 ∧ (¬ 𝑁 = 0 ∧ (¬ 𝑁 = 𝑆 ∧ ¬ 𝑆 < 𝑁))) ↔ ((¬ 𝑁 = 0 ∧ (¬ 𝑁 = 𝑆 ∧ ¬ 𝑆 < 𝑁)) ∧ 𝜑))
41 anass 680 . . . . . . . . . . . . . . 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 730 . . . . . . . . . . . . 13 (((¬ 𝑁 = 0 ∧ (¬ 𝑁 = 𝑆 ∧ ¬ 𝑆 < 𝑁)) ∧ 𝜑) ↔ (((¬ 𝑁 = 0 ∧ ¬ 𝑁 = 𝑆) ∧ ¬ 𝑆 < 𝑁) ∧ 𝜑))
44 anass 680 . . . . . . . . . . . . 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 680 . . . . . . . . . . . 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 838 . . . . . . . . . . . . 13 (((¬ 𝑁 = 0 ∧ ¬ 𝑁 = 𝑆) ∧ 𝜑) ↔ ((¬ 𝑁 = 0 ∧ 𝜑) ∧ ¬ 𝑁 = 𝑆))
49 ancom 466 . . . . . . . . . . . . . 14 ((¬ 𝑁 = 0 ∧ 𝜑) ↔ (𝜑 ∧ ¬ 𝑁 = 0))
5049anbi1i 730 . . . . . . . . . . . . 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 730 . . . . . . . . . . 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 2792 . . . . . . . . . . . . 13 (𝑁 ≠ 0 ↔ ¬ 𝑁 = 0)
55 elnnne0 11291 . . . . . . . . . . . . . . 15 (𝑁 ∈ ℕ ↔ (𝑁 ∈ ℕ0𝑁 ≠ 0))
56 nngt0 11034 . . . . . . . . . . . . . . 15 (𝑁 ∈ ℕ → 0 < 𝑁)
5755, 56sylbir 225 . . . . . . . . . . . . . 14 ((𝑁 ∈ ℕ0𝑁 ≠ 0) → 0 < 𝑁)
5857expcom 451 . . . . . . . . . . . . 13 (𝑁 ≠ 0 → (𝑁 ∈ ℕ0 → 0 < 𝑁))
5954, 58sylbir 225 . . . . . . . . . . . 12 𝑁 = 0 → (𝑁 ∈ ℕ0 → 0 < 𝑁))
6059adantr 481 . . . . . . . . . . 11 ((¬ 𝑁 = 0 ∧ (¬ 𝑁 = 𝑆 ∧ ¬ 𝑆 < 𝑁)) → (𝑁 ∈ ℕ0 → 0 < 𝑁))
611, 60mpan9 486 . . . . . . . . . 10 ((𝜑 ∧ (¬ 𝑁 = 0 ∧ (¬ 𝑁 = 𝑆 ∧ ¬ 𝑆 < 𝑁))) → 0 < 𝑁)
6253, 61sylbir 225 . . . . . . . . 9 ((((𝜑 ∧ ¬ 𝑁 = 0) ∧ ¬ 𝑁 = 𝑆) ∧ ¬ 𝑆 < 𝑁) → 0 < 𝑁)
631nn0red 11337 . . . . . . . . . . . . . . . 16 (𝜑𝑁 ∈ ℝ)
64 fvmptnn04if.s . . . . . . . . . . . . . . . . 17 (𝜑𝑆 ∈ ℕ)
6564nnred 11020 . . . . . . . . . . . . . . . 16 (𝜑𝑆 ∈ ℝ)
6663, 65lenltd 10168 . . . . . . . . . . . . . . 15 (𝜑 → (𝑁𝑆 ↔ ¬ 𝑆 < 𝑁))
6766biimprd 238 . . . . . . . . . . . . . 14 (𝜑 → (¬ 𝑆 < 𝑁𝑁𝑆))
6867adantld 483 . . . . . . . . . . . . 13 (𝜑 → ((¬ 𝑁 = 𝑆 ∧ ¬ 𝑆 < 𝑁) → 𝑁𝑆))
6968adantld 483 . . . . . . . . . . . 12 (𝜑 → ((¬ 𝑁 = 0 ∧ (¬ 𝑁 = 𝑆 ∧ ¬ 𝑆 < 𝑁)) → 𝑁𝑆))
7069imp 445 . . . . . . . . . . 11 ((𝜑 ∧ (¬ 𝑁 = 0 ∧ (¬ 𝑁 = 𝑆 ∧ ¬ 𝑆 < 𝑁))) → 𝑁𝑆)
71 nesym 2847 . . . . . . . . . . . . . 14 (𝑆𝑁 ↔ ¬ 𝑁 = 𝑆)
7271biimpri 218 . . . . . . . . . . . . 13 𝑁 = 𝑆𝑆𝑁)
7372adantr 481 . . . . . . . . . . . 12 ((¬ 𝑁 = 𝑆 ∧ ¬ 𝑆 < 𝑁) → 𝑆𝑁)
7473ad2antll 764 . . . . . . . . . . 11 ((𝜑 ∧ (¬ 𝑁 = 0 ∧ (¬ 𝑁 = 𝑆 ∧ ¬ 𝑆 < 𝑁))) → 𝑆𝑁)
7563adantr 481 . . . . . . . . . . . 12 ((𝜑 ∧ (¬ 𝑁 = 0 ∧ (¬ 𝑁 = 𝑆 ∧ ¬ 𝑆 < 𝑁))) → 𝑁 ∈ ℝ)
7665adantr 481 . . . . . . . . . . . 12 ((𝜑 ∧ (¬ 𝑁 = 0 ∧ (¬ 𝑁 = 𝑆 ∧ ¬ 𝑆 < 𝑁))) → 𝑆 ∈ ℝ)
7775, 76ltlend 10167 . . . . . . . . . . 11 ((𝜑 ∧ (¬ 𝑁 = 0 ∧ (¬ 𝑁 = 𝑆 ∧ ¬ 𝑆 < 𝑁))) → (𝑁 < 𝑆 ↔ (𝑁𝑆𝑆𝑁)))
7870, 74, 77mpbir2and 956 . . . . . . . . . 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 2626 . . . . . . . . 9 ((𝜑 ∧ 0 < 𝑁𝑁 < 𝑆) → 𝑁 / 𝑛𝐵 = 𝑌)
8237, 62, 79, 81syl3anc 1324 . . . . . . . 8 ((((𝜑 ∧ ¬ 𝑁 = 0) ∧ ¬ 𝑁 = 𝑆) ∧ ¬ 𝑆 < 𝑁) → 𝑁 / 𝑛𝐵 = 𝑌)
8322ad3antrrr 765 . . . . . . . 8 ((((𝜑 ∧ ¬ 𝑁 = 0) ∧ ¬ 𝑁 = 𝑆) ∧ ¬ 𝑆 < 𝑁) → 𝑌𝑉)
8482, 83eqeltrd 2699 . . . . . . 7 ((((𝜑 ∧ ¬ 𝑁 = 0) ∧ ¬ 𝑁 = 𝑆) ∧ ¬ 𝑆 < 𝑁) → 𝑁 / 𝑛𝐵𝑉)
8536, 84ifclda 4111 . . . . . 6 (((𝜑 ∧ ¬ 𝑁 = 0) ∧ ¬ 𝑁 = 𝑆) → if(𝑆 < 𝑁, 𝑁 / 𝑛𝐷, 𝑁 / 𝑛𝐵) ∈ 𝑉)
8629, 85ifclda 4111 . . . . 5 ((𝜑 ∧ ¬ 𝑁 = 0) → if(𝑁 = 𝑆, 𝑁 / 𝑛𝐶, if(𝑆 < 𝑁, 𝑁 / 𝑛𝐷, 𝑁 / 𝑛𝐵)) ∈ 𝑉)
8724, 86ifclda 4111 . . . 4 (𝜑 → if(𝑁 = 0, 𝑁 / 𝑛𝐴, if(𝑁 = 𝑆, 𝑁 / 𝑛𝐶, if(𝑆 < 𝑁, 𝑁 / 𝑛𝐷, 𝑁 / 𝑛𝐵))) ∈ 𝑉)
8820, 87eqeltrd 2699 . . 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 6272 . . 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 692 . 2 (𝜑 → (𝐺𝑁) = 𝑁 / 𝑛if(𝑛 = 0, 𝐴, if(𝑛 = 𝑆, 𝐶, if(𝑆 < 𝑛, 𝐷, 𝐵))))
9221eqcomd 2626 . . 3 ((𝜑𝑁 = 0) → 𝑁 / 𝑛𝐴 = 𝑌)
9334, 82ifeqda 4112 . . . 4 (((𝜑 ∧ ¬ 𝑁 = 0) ∧ ¬ 𝑁 = 𝑆) → if(𝑆 < 𝑁, 𝑁 / 𝑛𝐷, 𝑁 / 𝑛𝐵) = 𝑌)
9427, 93ifeqda 4112 . . 3 ((𝜑 ∧ ¬ 𝑁 = 0) → if(𝑁 = 𝑆, 𝑁 / 𝑛𝐶, if(𝑆 < 𝑁, 𝑁 / 𝑛𝐷, 𝑁 / 𝑛𝐵)) = 𝑌)
9592, 94ifeqda 4112 . 2 (𝜑 → if(𝑁 = 0, 𝑁 / 𝑛𝐴, if(𝑁 = 𝑆, 𝑁 / 𝑛𝐶, if(𝑆 < 𝑁, 𝑁 / 𝑛𝐷, 𝑁 / 𝑛𝐵))) = 𝑌)
9691, 20, 953eqtrd 2658 1 (𝜑 → (𝐺𝑁) = 𝑌)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 384  w3a 1036   = wceq 1481  wcel 1988  wne 2791  [wsbc 3429  csb 3526  ifcif 4077   class class class wbr 4644  cmpt 4720  cfv 5876  cr 9920  0cc0 9921   < clt 10059  cle 10060  cn 11005  0cn0 11277
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934  ax-resscn 9978  ax-1cn 9979  ax-icn 9980  ax-addcl 9981  ax-addrcl 9982  ax-mulcl 9983  ax-mulrcl 9984  ax-mulcom 9985  ax-addass 9986  ax-mulass 9987  ax-distr 9988  ax-i2m1 9989  ax-1ne0 9990  ax-1rid 9991  ax-rnegex 9992  ax-rrecex 9993  ax-cnre 9994  ax-pre-lttri 9995  ax-pre-lttrn 9996  ax-pre-ltadd 9997  ax-pre-mulgt0 9998
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1484  df-fal 1487  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-nel 2895  df-ral 2914  df-rex 2915  df-reu 2916  df-rab 2918  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-pss 3583  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-tp 4173  df-op 4175  df-uni 4428  df-iun 4513  df-br 4645  df-opab 4704  df-mpt 4721  df-tr 4744  df-id 5014  df-eprel 5019  df-po 5025  df-so 5026  df-fr 5063  df-we 5065  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-pred 5668  df-ord 5714  df-on 5715  df-lim 5716  df-suc 5717  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-f1 5881  df-fo 5882  df-f1o 5883  df-fv 5884  df-riota 6596  df-ov 6638  df-oprab 6639  df-mpt2 6640  df-om 7051  df-wrecs 7392  df-recs 7453  df-rdg 7491  df-er 7727  df-en 7941  df-dom 7942  df-sdom 7943  df-pnf 10061  df-mnf 10062  df-xr 10063  df-ltxr 10064  df-le 10065  df-sub 10253  df-neg 10254  df-nn 11006  df-n0 11278
This theorem is referenced by:  fvmptnn04ifa  20636  fvmptnn04ifb  20637  fvmptnn04ifc  20638  fvmptnn04ifd  20639
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