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Theorem fvmptnn04if 22150
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 4541 . . . . 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 eqsbc1 3786 . . . . . . 7 (𝑁 ∈ ℕ0 → ([𝑁 / 𝑛]𝑛 = 0 ↔ 𝑁 = 0))
41, 3syl 17 . . . . . 6 (𝜑 → ([𝑁 / 𝑛]𝑛 = 0 ↔ 𝑁 = 0))
5 csbif 4541 . . . . . . 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 eqsbc1 3786 . . . . . . . . 9 (𝑁 ∈ ℕ0 → ([𝑁 / 𝑛]𝑛 = 𝑆𝑁 = 𝑆))
71, 6syl 17 . . . . . . . 8 (𝜑 → ([𝑁 / 𝑛]𝑛 = 𝑆𝑁 = 𝑆))
8 csbif 4541 . . . . . . . . 9 𝑁 / 𝑛if(𝑆 < 𝑛, 𝐷, 𝐵) = if([𝑁 / 𝑛]𝑆 < 𝑛, 𝑁 / 𝑛𝐷, 𝑁 / 𝑛𝐵)
9 sbcbr2g 5161 . . . . . . . . . . . 12 (𝑁 ∈ ℕ0 → ([𝑁 / 𝑛]𝑆 < 𝑛𝑆 < 𝑁 / 𝑛𝑛))
101, 9syl 17 . . . . . . . . . . 11 (𝜑 → ([𝑁 / 𝑛]𝑆 < 𝑛𝑆 < 𝑁 / 𝑛𝑛))
11 csbvarg 4389 . . . . . . . . . . . . 13 (𝑁 ∈ ℕ0𝑁 / 𝑛𝑛 = 𝑁)
121, 11syl 17 . . . . . . . . . . . 12 (𝜑𝑁 / 𝑛𝑛 = 𝑁)
1312breq2d 5115 . . . . . . . . . . 11 (𝜑 → (𝑆 < 𝑁 / 𝑛𝑛𝑆 < 𝑁))
1410, 13bitrd 278 . . . . . . . . . 10 (𝜑 → ([𝑁 / 𝑛]𝑆 < 𝑛𝑆 < 𝑁))
1514ifbid 4507 . . . . . . . . 9 (𝜑 → if([𝑁 / 𝑛]𝑆 < 𝑛, 𝑁 / 𝑛𝐷, 𝑁 / 𝑛𝐵) = if(𝑆 < 𝑁, 𝑁 / 𝑛𝐷, 𝑁 / 𝑛𝐵))
168, 15eqtrid 2789 . . . . . . . 8 (𝜑𝑁 / 𝑛if(𝑆 < 𝑛, 𝐷, 𝐵) = if(𝑆 < 𝑁, 𝑁 / 𝑛𝐷, 𝑁 / 𝑛𝐵))
177, 16ifbieq2d 4510 . . . . . . 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, 17eqtrid 2789 . . . . . 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 4510 . . . . 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, 19eqtrid 2789 . . . 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 2839 . . . . 5 ((𝜑𝑁 = 0) → 𝑁 / 𝑛𝐴𝑉)
25 fvmptnn04if.c . . . . . . . . 9 ((𝜑𝑁 = 𝑆) → 𝑌 = 𝑁 / 𝑛𝐶)
2625eqcomd 2743 . . . . . . . 8 ((𝜑𝑁 = 𝑆) → 𝑁 / 𝑛𝐶 = 𝑌)
2726adantlr 713 . . . . . . 7 (((𝜑 ∧ ¬ 𝑁 = 0) ∧ 𝑁 = 𝑆) → 𝑁 / 𝑛𝐶 = 𝑌)
2822ad2antrr 724 . . . . . . 7 (((𝜑 ∧ ¬ 𝑁 = 0) ∧ 𝑁 = 𝑆) → 𝑌𝑉)
2927, 28eqeltrd 2838 . . . . . 6 (((𝜑 ∧ ¬ 𝑁 = 0) ∧ 𝑁 = 𝑆) → 𝑁 / 𝑛𝐶𝑉)
30 fvmptnn04if.d . . . . . . . . . 10 ((𝜑𝑆 < 𝑁) → 𝑌 = 𝑁 / 𝑛𝐷)
3130eqcomd 2743 . . . . . . . . 9 ((𝜑𝑆 < 𝑁) → 𝑁 / 𝑛𝐷 = 𝑌)
3231ad4ant14 750 . . . . . . . 8 ((((𝜑 ∧ ¬ 𝑁 = 0) ∧ ¬ 𝑁 = 𝑆) ∧ 𝑆 < 𝑁) → 𝑁 / 𝑛𝐷 = 𝑌)
3322ad3antrrr 728 . . . . . . . 8 ((((𝜑 ∧ ¬ 𝑁 = 0) ∧ ¬ 𝑁 = 𝑆) ∧ 𝑆 < 𝑁) → 𝑌𝑉)
3432, 33eqeltrd 2838 . . . . . . 7 ((((𝜑 ∧ ¬ 𝑁 = 0) ∧ ¬ 𝑁 = 𝑆) ∧ 𝑆 < 𝑁) → 𝑁 / 𝑛𝐷𝑉)
35 simplll 773 . . . . . . . . 9 ((((𝜑 ∧ ¬ 𝑁 = 0) ∧ ¬ 𝑁 = 𝑆) ∧ ¬ 𝑆 < 𝑁) → 𝜑)
36 anass 469 . . . . . . . . . . . . 13 (((¬ 𝑁 = 0 ∧ ¬ 𝑁 = 𝑆) ∧ ¬ 𝑆 < 𝑁) ↔ (¬ 𝑁 = 0 ∧ (¬ 𝑁 = 𝑆 ∧ ¬ 𝑆 < 𝑁)))
3736bicomi 223 . . . . . . . . . . . 12 ((¬ 𝑁 = 0 ∧ (¬ 𝑁 = 𝑆 ∧ ¬ 𝑆 < 𝑁)) ↔ ((¬ 𝑁 = 0 ∧ ¬ 𝑁 = 𝑆) ∧ ¬ 𝑆 < 𝑁))
3837bianassc 641 . . . . . . . . . . 11 ((𝜑 ∧ (¬ 𝑁 = 0 ∧ (¬ 𝑁 = 𝑆 ∧ ¬ 𝑆 < 𝑁))) ↔ (((¬ 𝑁 = 0 ∧ ¬ 𝑁 = 𝑆) ∧ 𝜑) ∧ ¬ 𝑆 < 𝑁))
39 an32 644 . . . . . . . . . . . . 13 (((¬ 𝑁 = 0 ∧ ¬ 𝑁 = 𝑆) ∧ 𝜑) ↔ ((¬ 𝑁 = 0 ∧ 𝜑) ∧ ¬ 𝑁 = 𝑆))
40 ancom 461 . . . . . . . . . . . . . 14 ((¬ 𝑁 = 0 ∧ 𝜑) ↔ (𝜑 ∧ ¬ 𝑁 = 0))
4140anbi1i 624 . . . . . . . . . . . . 13 (((¬ 𝑁 = 0 ∧ 𝜑) ∧ ¬ 𝑁 = 𝑆) ↔ ((𝜑 ∧ ¬ 𝑁 = 0) ∧ ¬ 𝑁 = 𝑆))
4239, 41bitri 274 . . . . . . . . . . . 12 (((¬ 𝑁 = 0 ∧ ¬ 𝑁 = 𝑆) ∧ 𝜑) ↔ ((𝜑 ∧ ¬ 𝑁 = 0) ∧ ¬ 𝑁 = 𝑆))
4342anbi1i 624 . . . . . . . . . . 11 ((((¬ 𝑁 = 0 ∧ ¬ 𝑁 = 𝑆) ∧ 𝜑) ∧ ¬ 𝑆 < 𝑁) ↔ (((𝜑 ∧ ¬ 𝑁 = 0) ∧ ¬ 𝑁 = 𝑆) ∧ ¬ 𝑆 < 𝑁))
4438, 43bitri 274 . . . . . . . . . 10 ((𝜑 ∧ (¬ 𝑁 = 0 ∧ (¬ 𝑁 = 𝑆 ∧ ¬ 𝑆 < 𝑁))) ↔ (((𝜑 ∧ ¬ 𝑁 = 0) ∧ ¬ 𝑁 = 𝑆) ∧ ¬ 𝑆 < 𝑁))
45 df-ne 2942 . . . . . . . . . . . . 13 (𝑁 ≠ 0 ↔ ¬ 𝑁 = 0)
46 elnnne0 12385 . . . . . . . . . . . . . . 15 (𝑁 ∈ ℕ ↔ (𝑁 ∈ ℕ0𝑁 ≠ 0))
47 nngt0 12142 . . . . . . . . . . . . . . 15 (𝑁 ∈ ℕ → 0 < 𝑁)
4846, 47sylbir 234 . . . . . . . . . . . . . 14 ((𝑁 ∈ ℕ0𝑁 ≠ 0) → 0 < 𝑁)
4948expcom 414 . . . . . . . . . . . . 13 (𝑁 ≠ 0 → (𝑁 ∈ ℕ0 → 0 < 𝑁))
5045, 49sylbir 234 . . . . . . . . . . . 12 𝑁 = 0 → (𝑁 ∈ ℕ0 → 0 < 𝑁))
5150adantr 481 . . . . . . . . . . 11 ((¬ 𝑁 = 0 ∧ (¬ 𝑁 = 𝑆 ∧ ¬ 𝑆 < 𝑁)) → (𝑁 ∈ ℕ0 → 0 < 𝑁))
521, 51mpan9 507 . . . . . . . . . 10 ((𝜑 ∧ (¬ 𝑁 = 0 ∧ (¬ 𝑁 = 𝑆 ∧ ¬ 𝑆 < 𝑁))) → 0 < 𝑁)
5344, 52sylbir 234 . . . . . . . . 9 ((((𝜑 ∧ ¬ 𝑁 = 0) ∧ ¬ 𝑁 = 𝑆) ∧ ¬ 𝑆 < 𝑁) → 0 < 𝑁)
541nn0red 12432 . . . . . . . . . . . 12 (𝜑𝑁 ∈ ℝ)
5554adantr 481 . . . . . . . . . . 11 ((𝜑 ∧ (¬ 𝑁 = 0 ∧ (¬ 𝑁 = 𝑆 ∧ ¬ 𝑆 < 𝑁))) → 𝑁 ∈ ℝ)
56 fvmptnn04if.s . . . . . . . . . . . . 13 (𝜑𝑆 ∈ ℕ)
5756nnred 12126 . . . . . . . . . . . 12 (𝜑𝑆 ∈ ℝ)
5857adantr 481 . . . . . . . . . . 11 ((𝜑 ∧ (¬ 𝑁 = 0 ∧ (¬ 𝑁 = 𝑆 ∧ ¬ 𝑆 < 𝑁))) → 𝑆 ∈ ℝ)
5954, 57lenltd 11259 . . . . . . . . . . . . . . 15 (𝜑 → (𝑁𝑆 ↔ ¬ 𝑆 < 𝑁))
6059biimprd 247 . . . . . . . . . . . . . 14 (𝜑 → (¬ 𝑆 < 𝑁𝑁𝑆))
6160adantld 491 . . . . . . . . . . . . 13 (𝜑 → ((¬ 𝑁 = 𝑆 ∧ ¬ 𝑆 < 𝑁) → 𝑁𝑆))
6261adantld 491 . . . . . . . . . . . 12 (𝜑 → ((¬ 𝑁 = 0 ∧ (¬ 𝑁 = 𝑆 ∧ ¬ 𝑆 < 𝑁)) → 𝑁𝑆))
6362imp 407 . . . . . . . . . . 11 ((𝜑 ∧ (¬ 𝑁 = 0 ∧ (¬ 𝑁 = 𝑆 ∧ ¬ 𝑆 < 𝑁))) → 𝑁𝑆)
64 nesym 2998 . . . . . . . . . . . . . 14 (𝑆𝑁 ↔ ¬ 𝑁 = 𝑆)
6564biimpri 227 . . . . . . . . . . . . 13 𝑁 = 𝑆𝑆𝑁)
6665adantr 481 . . . . . . . . . . . 12 ((¬ 𝑁 = 𝑆 ∧ ¬ 𝑆 < 𝑁) → 𝑆𝑁)
6766ad2antll 727 . . . . . . . . . . 11 ((𝜑 ∧ (¬ 𝑁 = 0 ∧ (¬ 𝑁 = 𝑆 ∧ ¬ 𝑆 < 𝑁))) → 𝑆𝑁)
6855, 58, 63, 67leneltd 11267 . . . . . . . . . 10 ((𝜑 ∧ (¬ 𝑁 = 0 ∧ (¬ 𝑁 = 𝑆 ∧ ¬ 𝑆 < 𝑁))) → 𝑁 < 𝑆)
6944, 68sylbir 234 . . . . . . . . 9 ((((𝜑 ∧ ¬ 𝑁 = 0) ∧ ¬ 𝑁 = 𝑆) ∧ ¬ 𝑆 < 𝑁) → 𝑁 < 𝑆)
70 fvmptnn04if.b . . . . . . . . . 10 ((𝜑 ∧ 0 < 𝑁𝑁 < 𝑆) → 𝑌 = 𝑁 / 𝑛𝐵)
7170eqcomd 2743 . . . . . . . . 9 ((𝜑 ∧ 0 < 𝑁𝑁 < 𝑆) → 𝑁 / 𝑛𝐵 = 𝑌)
7235, 53, 69, 71syl3anc 1371 . . . . . . . 8 ((((𝜑 ∧ ¬ 𝑁 = 0) ∧ ¬ 𝑁 = 𝑆) ∧ ¬ 𝑆 < 𝑁) → 𝑁 / 𝑛𝐵 = 𝑌)
7322ad3antrrr 728 . . . . . . . 8 ((((𝜑 ∧ ¬ 𝑁 = 0) ∧ ¬ 𝑁 = 𝑆) ∧ ¬ 𝑆 < 𝑁) → 𝑌𝑉)
7472, 73eqeltrd 2838 . . . . . . 7 ((((𝜑 ∧ ¬ 𝑁 = 0) ∧ ¬ 𝑁 = 𝑆) ∧ ¬ 𝑆 < 𝑁) → 𝑁 / 𝑛𝐵𝑉)
7534, 74ifclda 4519 . . . . . 6 (((𝜑 ∧ ¬ 𝑁 = 0) ∧ ¬ 𝑁 = 𝑆) → if(𝑆 < 𝑁, 𝑁 / 𝑛𝐷, 𝑁 / 𝑛𝐵) ∈ 𝑉)
7629, 75ifclda 4519 . . . . 5 ((𝜑 ∧ ¬ 𝑁 = 0) → if(𝑁 = 𝑆, 𝑁 / 𝑛𝐶, if(𝑆 < 𝑁, 𝑁 / 𝑛𝐷, 𝑁 / 𝑛𝐵)) ∈ 𝑉)
7724, 76ifclda 4519 . . . 4 (𝜑 → if(𝑁 = 0, 𝑁 / 𝑛𝐴, if(𝑁 = 𝑆, 𝑁 / 𝑛𝐶, if(𝑆 < 𝑁, 𝑁 / 𝑛𝐷, 𝑁 / 𝑛𝐵))) ∈ 𝑉)
7820, 77eqeltrd 2838 . . 3 (𝜑𝑁 / 𝑛if(𝑛 = 0, 𝐴, if(𝑛 = 𝑆, 𝐶, if(𝑆 < 𝑛, 𝐷, 𝐵))) ∈ 𝑉)
79 fvmptnn04if.g . . . 4 𝐺 = (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, 𝐴, if(𝑛 = 𝑆, 𝐶, if(𝑆 < 𝑛, 𝐷, 𝐵))))
8079fvmpts 6948 . . 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))))
811, 78, 80syl2anc 584 . 2 (𝜑 → (𝐺𝑁) = 𝑁 / 𝑛if(𝑛 = 0, 𝐴, if(𝑛 = 𝑆, 𝐶, if(𝑆 < 𝑛, 𝐷, 𝐵))))
8221eqcomd 2743 . . 3 ((𝜑𝑁 = 0) → 𝑁 / 𝑛𝐴 = 𝑌)
8332, 72ifeqda 4520 . . . 4 (((𝜑 ∧ ¬ 𝑁 = 0) ∧ ¬ 𝑁 = 𝑆) → if(𝑆 < 𝑁, 𝑁 / 𝑛𝐷, 𝑁 / 𝑛𝐵) = 𝑌)
8427, 83ifeqda 4520 . . 3 ((𝜑 ∧ ¬ 𝑁 = 0) → if(𝑁 = 𝑆, 𝑁 / 𝑛𝐶, if(𝑆 < 𝑁, 𝑁 / 𝑛𝐷, 𝑁 / 𝑛𝐵)) = 𝑌)
8582, 84ifeqda 4520 . 2 (𝜑 → if(𝑁 = 0, 𝑁 / 𝑛𝐴, if(𝑁 = 𝑆, 𝑁 / 𝑛𝐶, if(𝑆 < 𝑁, 𝑁 / 𝑛𝐷, 𝑁 / 𝑛𝐵))) = 𝑌)
8681, 20, 853eqtrd 2781 1 (𝜑 → (𝐺𝑁) = 𝑌)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 396  w3a 1087   = wceq 1541  wcel 2106  wne 2941  [wsbc 3737  csb 3853  ifcif 4484   class class class wbr 5103  cmpt 5186  cfv 6493  cr 11008  0cc0 11009   < clt 11147  cle 11148  cn 12111  0cn0 12371
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2708  ax-sep 5254  ax-nul 5261  ax-pow 5318  ax-pr 5382  ax-un 7664  ax-resscn 11066  ax-1cn 11067  ax-icn 11068  ax-addcl 11069  ax-addrcl 11070  ax-mulcl 11071  ax-mulrcl 11072  ax-mulcom 11073  ax-addass 11074  ax-mulass 11075  ax-distr 11076  ax-i2m1 11077  ax-1ne0 11078  ax-1rid 11079  ax-rnegex 11080  ax-rrecex 11081  ax-cnre 11082  ax-pre-lttri 11083  ax-pre-lttrn 11084  ax-pre-ltadd 11085  ax-pre-mulgt0 11086
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2887  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-reu 3352  df-rab 3406  df-v 3445  df-sbc 3738  df-csb 3854  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-pss 3927  df-nul 4281  df-if 4485  df-pw 4560  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4864  df-iun 4954  df-br 5104  df-opab 5166  df-mpt 5187  df-tr 5221  df-id 5529  df-eprel 5535  df-po 5543  df-so 5544  df-fr 5586  df-we 5588  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6251  df-ord 6318  df-on 6319  df-lim 6320  df-suc 6321  df-iota 6445  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-riota 7307  df-ov 7354  df-oprab 7355  df-mpo 7356  df-om 7795  df-2nd 7914  df-frecs 8204  df-wrecs 8235  df-recs 8309  df-rdg 8348  df-er 8606  df-en 8842  df-dom 8843  df-sdom 8844  df-pnf 11149  df-mnf 11150  df-xr 11151  df-ltxr 11152  df-le 11153  df-sub 11345  df-neg 11346  df-nn 12112  df-n0 12372
This theorem is referenced by:  fvmptnn04ifa  22151  fvmptnn04ifb  22152  fvmptnn04ifc  22153  fvmptnn04ifd  22154
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