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Theorem fvmptnn04if 20473
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 4088 . . . . 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 3442 . . . . . . 7 (𝑁 ∈ ℕ0 → ([𝑁 / 𝑛]𝑛 = 0 ↔ 𝑁 = 0))
41, 3syl 17 . . . . . 6 (𝜑 → ([𝑁 / 𝑛]𝑛 = 0 ↔ 𝑁 = 0))
5 csbif 4088 . . . . . . 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 3442 . . . . . . . . 9 (𝑁 ∈ ℕ0 → ([𝑁 / 𝑛]𝑛 = 𝑆𝑁 = 𝑆))
71, 6syl 17 . . . . . . . 8 (𝜑 → ([𝑁 / 𝑛]𝑛 = 𝑆𝑁 = 𝑆))
8 csbif 4088 . . . . . . . . 9 𝑁 / 𝑛if(𝑆 < 𝑛, 𝐷, 𝐵) = if([𝑁 / 𝑛]𝑆 < 𝑛, 𝑁 / 𝑛𝐷, 𝑁 / 𝑛𝐵)
9 sbcbr2g 4640 . . . . . . . . . . . 12 (𝑁 ∈ ℕ0 → ([𝑁 / 𝑛]𝑆 < 𝑛𝑆 < 𝑁 / 𝑛𝑛))
101, 9syl 17 . . . . . . . . . . 11 (𝜑 → ([𝑁 / 𝑛]𝑆 < 𝑛𝑆 < 𝑁 / 𝑛𝑛))
11 csbvarg 3955 . . . . . . . . . . . . 13 (𝑁 ∈ ℕ0𝑁 / 𝑛𝑛 = 𝑁)
121, 11syl 17 . . . . . . . . . . . 12 (𝜑𝑁 / 𝑛𝑛 = 𝑁)
1312breq2d 4595 . . . . . . . . . . 11 (𝜑 → (𝑆 < 𝑁 / 𝑛𝑛𝑆 < 𝑁))
1410, 13bitrd 267 . . . . . . . . . 10 (𝜑 → ([𝑁 / 𝑛]𝑆 < 𝑛𝑆 < 𝑁))
1514ifbid 4058 . . . . . . . . 9 (𝜑 → if([𝑁 / 𝑛]𝑆 < 𝑛, 𝑁 / 𝑛𝐷, 𝑁 / 𝑛𝐵) = if(𝑆 < 𝑁, 𝑁 / 𝑛𝐷, 𝑁 / 𝑛𝐵))
168, 15syl5eq 2656 . . . . . . . 8 (𝜑𝑁 / 𝑛if(𝑆 < 𝑛, 𝐷, 𝐵) = if(𝑆 < 𝑁, 𝑁 / 𝑛𝐷, 𝑁 / 𝑛𝐵))
177, 16ifbieq2d 4061 . . . . . . 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 2656 . . . . . 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 4061 . . . . 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 2656 . . . 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 2689 . . . . 5 ((𝜑𝑁 = 0) → 𝑁 / 𝑛𝐴𝑉)
25 fvmptnn04if.c . . . . . . . . 9 ((𝜑𝑁 = 𝑆) → 𝑌 = 𝑁 / 𝑛𝐶)
2625eqcomd 2616 . . . . . . . 8 ((𝜑𝑁 = 𝑆) → 𝑁 / 𝑛𝐶 = 𝑌)
2726adantlr 747 . . . . . . 7 (((𝜑 ∧ ¬ 𝑁 = 0) ∧ 𝑁 = 𝑆) → 𝑁 / 𝑛𝐶 = 𝑌)
2822ad2antrr 758 . . . . . . 7 (((𝜑 ∧ ¬ 𝑁 = 0) ∧ 𝑁 = 𝑆) → 𝑌𝑉)
2927, 28eqeltrd 2688 . . . . . 6 (((𝜑 ∧ ¬ 𝑁 = 0) ∧ 𝑁 = 𝑆) → 𝑁 / 𝑛𝐶𝑉)
30 fvmptnn04if.d . . . . . . . . . . . 12 ((𝜑𝑆 < 𝑁) → 𝑌 = 𝑁 / 𝑛𝐷)
3130eqcomd 2616 . . . . . . . . . . 11 ((𝜑𝑆 < 𝑁) → 𝑁 / 𝑛𝐷 = 𝑌)
3231ex 449 . . . . . . . . . 10 (𝜑 → (𝑆 < 𝑁𝑁 / 𝑛𝐷 = 𝑌))
3332ad2antrr 758 . . . . . . . . 9 (((𝜑 ∧ ¬ 𝑁 = 0) ∧ ¬ 𝑁 = 𝑆) → (𝑆 < 𝑁𝑁 / 𝑛𝐷 = 𝑌))
3433imp 444 . . . . . . . 8 ((((𝜑 ∧ ¬ 𝑁 = 0) ∧ ¬ 𝑁 = 𝑆) ∧ 𝑆 < 𝑁) → 𝑁 / 𝑛𝐷 = 𝑌)
3522ad3antrrr 762 . . . . . . . 8 ((((𝜑 ∧ ¬ 𝑁 = 0) ∧ ¬ 𝑁 = 𝑆) ∧ 𝑆 < 𝑁) → 𝑌𝑉)
3634, 35eqeltrd 2688 . . . . . . 7 ((((𝜑 ∧ ¬ 𝑁 = 0) ∧ ¬ 𝑁 = 𝑆) ∧ 𝑆 < 𝑁) → 𝑁 / 𝑛𝐷𝑉)
37 simplll 794 . . . . . . . . 9 ((((𝜑 ∧ ¬ 𝑁 = 0) ∧ ¬ 𝑁 = 𝑆) ∧ ¬ 𝑆 < 𝑁) → 𝜑)
38 ancom 465 . . . . . . . . . . . . 13 ((¬ 𝑆 < 𝑁𝜑) ↔ (𝜑 ∧ ¬ 𝑆 < 𝑁))
3938anbi2i 726 . . . . . . . . . . . 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 679 . . . . . . . . . . . . . . 15 (((¬ 𝑁 = 0 ∧ ¬ 𝑁 = 𝑆) ∧ ¬ 𝑆 < 𝑁) ↔ (¬ 𝑁 = 0 ∧ (¬ 𝑁 = 𝑆 ∧ ¬ 𝑆 < 𝑁)))
4241bicomi 213 . . . . . . . . . . . . . 14 ((¬ 𝑁 = 0 ∧ (¬ 𝑁 = 𝑆 ∧ ¬ 𝑆 < 𝑁)) ↔ ((¬ 𝑁 = 0 ∧ ¬ 𝑁 = 𝑆) ∧ ¬ 𝑆 < 𝑁))
4342anbi1i 727 . . . . . . . . . . . . 13 (((¬ 𝑁 = 0 ∧ (¬ 𝑁 = 𝑆 ∧ ¬ 𝑆 < 𝑁)) ∧ 𝜑) ↔ (((¬ 𝑁 = 0 ∧ ¬ 𝑁 = 𝑆) ∧ ¬ 𝑆 < 𝑁) ∧ 𝜑))
44 anass 679 . . . . . . . . . . . . 13 ((((¬ 𝑁 = 0 ∧ ¬ 𝑁 = 𝑆) ∧ ¬ 𝑆 < 𝑁) ∧ 𝜑) ↔ ((¬ 𝑁 = 0 ∧ ¬ 𝑁 = 𝑆) ∧ (¬ 𝑆 < 𝑁𝜑)))
4540, 43, 443bitri 285 . . . . . . . . . . . 12 ((𝜑 ∧ (¬ 𝑁 = 0 ∧ (¬ 𝑁 = 𝑆 ∧ ¬ 𝑆 < 𝑁))) ↔ ((¬ 𝑁 = 0 ∧ ¬ 𝑁 = 𝑆) ∧ (¬ 𝑆 < 𝑁𝜑)))
46 anass 679 . . . . . . . . . . . 12 ((((¬ 𝑁 = 0 ∧ ¬ 𝑁 = 𝑆) ∧ 𝜑) ∧ ¬ 𝑆 < 𝑁) ↔ ((¬ 𝑁 = 0 ∧ ¬ 𝑁 = 𝑆) ∧ (𝜑 ∧ ¬ 𝑆 < 𝑁)))
4739, 45, 463bitr4i 291 . . . . . . . . . . 11 ((𝜑 ∧ (¬ 𝑁 = 0 ∧ (¬ 𝑁 = 𝑆 ∧ ¬ 𝑆 < 𝑁))) ↔ (((¬ 𝑁 = 0 ∧ ¬ 𝑁 = 𝑆) ∧ 𝜑) ∧ ¬ 𝑆 < 𝑁))
48 an32 835 . . . . . . . . . . . . 13 (((¬ 𝑁 = 0 ∧ ¬ 𝑁 = 𝑆) ∧ 𝜑) ↔ ((¬ 𝑁 = 0 ∧ 𝜑) ∧ ¬ 𝑁 = 𝑆))
49 ancom 465 . . . . . . . . . . . . . 14 ((¬ 𝑁 = 0 ∧ 𝜑) ↔ (𝜑 ∧ ¬ 𝑁 = 0))
5049anbi1i 727 . . . . . . . . . . . . 13 (((¬ 𝑁 = 0 ∧ 𝜑) ∧ ¬ 𝑁 = 𝑆) ↔ ((𝜑 ∧ ¬ 𝑁 = 0) ∧ ¬ 𝑁 = 𝑆))
5148, 50bitri 263 . . . . . . . . . . . 12 (((¬ 𝑁 = 0 ∧ ¬ 𝑁 = 𝑆) ∧ 𝜑) ↔ ((𝜑 ∧ ¬ 𝑁 = 0) ∧ ¬ 𝑁 = 𝑆))
5251anbi1i 727 . . . . . . . . . . 11 ((((¬ 𝑁 = 0 ∧ ¬ 𝑁 = 𝑆) ∧ 𝜑) ∧ ¬ 𝑆 < 𝑁) ↔ (((𝜑 ∧ ¬ 𝑁 = 0) ∧ ¬ 𝑁 = 𝑆) ∧ ¬ 𝑆 < 𝑁))
5347, 52bitri 263 . . . . . . . . . 10 ((𝜑 ∧ (¬ 𝑁 = 0 ∧ (¬ 𝑁 = 𝑆 ∧ ¬ 𝑆 < 𝑁))) ↔ (((𝜑 ∧ ¬ 𝑁 = 0) ∧ ¬ 𝑁 = 𝑆) ∧ ¬ 𝑆 < 𝑁))
54 df-ne 2782 . . . . . . . . . . . . 13 (𝑁 ≠ 0 ↔ ¬ 𝑁 = 0)
55 elnnne0 11183 . . . . . . . . . . . . . . 15 (𝑁 ∈ ℕ ↔ (𝑁 ∈ ℕ0𝑁 ≠ 0))
56 nngt0 10926 . . . . . . . . . . . . . . 15 (𝑁 ∈ ℕ → 0 < 𝑁)
5755, 56sylbir 224 . . . . . . . . . . . . . 14 ((𝑁 ∈ ℕ0𝑁 ≠ 0) → 0 < 𝑁)
5857expcom 450 . . . . . . . . . . . . 13 (𝑁 ≠ 0 → (𝑁 ∈ ℕ0 → 0 < 𝑁))
5954, 58sylbir 224 . . . . . . . . . . . 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 224 . . . . . . . . 9 ((((𝜑 ∧ ¬ 𝑁 = 0) ∧ ¬ 𝑁 = 𝑆) ∧ ¬ 𝑆 < 𝑁) → 0 < 𝑁)
631nn0red 11229 . . . . . . . . . . . . . . . 16 (𝜑𝑁 ∈ ℝ)
64 fvmptnn04if.s . . . . . . . . . . . . . . . . 17 (𝜑𝑆 ∈ ℕ)
6564nnred 10912 . . . . . . . . . . . . . . . 16 (𝜑𝑆 ∈ ℝ)
6663, 65lenltd 10062 . . . . . . . . . . . . . . 15 (𝜑 → (𝑁𝑆 ↔ ¬ 𝑆 < 𝑁))
6766biimprd 237 . . . . . . . . . . . . . 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 2838 . . . . . . . . . . . . . 14 (𝑆𝑁 ↔ ¬ 𝑁 = 𝑆)
7271biimpri 217 . . . . . . . . . . . . 13 𝑁 = 𝑆𝑆𝑁)
7372adantr 480 . . . . . . . . . . . 12 ((¬ 𝑁 = 𝑆 ∧ ¬ 𝑆 < 𝑁) → 𝑆𝑁)
7473ad2antll 761 . . . . . . . . . . 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 10061 . . . . . . . . . . 11 ((𝜑 ∧ (¬ 𝑁 = 0 ∧ (¬ 𝑁 = 𝑆 ∧ ¬ 𝑆 < 𝑁))) → (𝑁 < 𝑆 ↔ (𝑁𝑆𝑆𝑁)))
7870, 74, 77mpbir2and 959 . . . . . . . . . 10 ((𝜑 ∧ (¬ 𝑁 = 0 ∧ (¬ 𝑁 = 𝑆 ∧ ¬ 𝑆 < 𝑁))) → 𝑁 < 𝑆)
7953, 78sylbir 224 . . . . . . . . 9 ((((𝜑 ∧ ¬ 𝑁 = 0) ∧ ¬ 𝑁 = 𝑆) ∧ ¬ 𝑆 < 𝑁) → 𝑁 < 𝑆)
80 fvmptnn04if.b . . . . . . . . . 10 ((𝜑 ∧ 0 < 𝑁𝑁 < 𝑆) → 𝑌 = 𝑁 / 𝑛𝐵)
8180eqcomd 2616 . . . . . . . . 9 ((𝜑 ∧ 0 < 𝑁𝑁 < 𝑆) → 𝑁 / 𝑛𝐵 = 𝑌)
8237, 62, 79, 81syl3anc 1318 . . . . . . . 8 ((((𝜑 ∧ ¬ 𝑁 = 0) ∧ ¬ 𝑁 = 𝑆) ∧ ¬ 𝑆 < 𝑁) → 𝑁 / 𝑛𝐵 = 𝑌)
8322ad3antrrr 762 . . . . . . . 8 ((((𝜑 ∧ ¬ 𝑁 = 0) ∧ ¬ 𝑁 = 𝑆) ∧ ¬ 𝑆 < 𝑁) → 𝑌𝑉)
8482, 83eqeltrd 2688 . . . . . . 7 ((((𝜑 ∧ ¬ 𝑁 = 0) ∧ ¬ 𝑁 = 𝑆) ∧ ¬ 𝑆 < 𝑁) → 𝑁 / 𝑛𝐵𝑉)
8536, 84ifclda 4070 . . . . . 6 (((𝜑 ∧ ¬ 𝑁 = 0) ∧ ¬ 𝑁 = 𝑆) → if(𝑆 < 𝑁, 𝑁 / 𝑛𝐷, 𝑁 / 𝑛𝐵) ∈ 𝑉)
8629, 85ifclda 4070 . . . . 5 ((𝜑 ∧ ¬ 𝑁 = 0) → if(𝑁 = 𝑆, 𝑁 / 𝑛𝐶, if(𝑆 < 𝑁, 𝑁 / 𝑛𝐷, 𝑁 / 𝑛𝐵)) ∈ 𝑉)
8724, 86ifclda 4070 . . . 4 (𝜑 → if(𝑁 = 0, 𝑁 / 𝑛𝐴, if(𝑁 = 𝑆, 𝑁 / 𝑛𝐶, if(𝑆 < 𝑁, 𝑁 / 𝑛𝐷, 𝑁 / 𝑛𝐵))) ∈ 𝑉)
8820, 87eqeltrd 2688 . . 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 6194 . . 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 691 . 2 (𝜑 → (𝐺𝑁) = 𝑁 / 𝑛if(𝑛 = 0, 𝐴, if(𝑛 = 𝑆, 𝐶, if(𝑆 < 𝑛, 𝐷, 𝐵))))
9221eqcomd 2616 . . 3 ((𝜑𝑁 = 0) → 𝑁 / 𝑛𝐴 = 𝑌)
9334, 82ifeqda 4071 . . . 4 (((𝜑 ∧ ¬ 𝑁 = 0) ∧ ¬ 𝑁 = 𝑆) → if(𝑆 < 𝑁, 𝑁 / 𝑛𝐷, 𝑁 / 𝑛𝐵) = 𝑌)
9427, 93ifeqda 4071 . . 3 ((𝜑 ∧ ¬ 𝑁 = 0) → if(𝑁 = 𝑆, 𝑁 / 𝑛𝐶, if(𝑆 < 𝑁, 𝑁 / 𝑛𝐷, 𝑁 / 𝑛𝐵)) = 𝑌)
9592, 94ifeqda 4071 . 2 (𝜑 → if(𝑁 = 0, 𝑁 / 𝑛𝐴, if(𝑁 = 𝑆, 𝑁 / 𝑛𝐶, if(𝑆 < 𝑁, 𝑁 / 𝑛𝐷, 𝑁 / 𝑛𝐵))) = 𝑌)
9691, 20, 953eqtrd 2648 1 (𝜑 → (𝐺𝑁) = 𝑌)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 195  wa 383  w3a 1031   = wceq 1475  wcel 1977  wne 2780  [wsbc 3402  csb 3499  ifcif 4036   class class class wbr 4583  cmpt 4643  cfv 5804  cr 9814  0cc0 9815   < clt 9953  cle 9954  cn 10897  0cn0 11169
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-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-fal 1481  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-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
This theorem is referenced by:  fvmptnn04ifa  20474  fvmptnn04ifb  20475  fvmptnn04ifc  20476  fvmptnn04ifd  20477
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