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Theorem rabren3dioph 41855
Description: Change variable numbers in a 3-variable Diophantine class abstraction. (Contributed by Stefan O'Rear, 17-Oct-2014.)
Hypotheses
Ref Expression
rabren3dioph.a (((π‘Žβ€˜1) = (π‘β€˜π‘‹) ∧ (π‘Žβ€˜2) = (π‘β€˜π‘Œ) ∧ (π‘Žβ€˜3) = (π‘β€˜π‘)) β†’ (πœ‘ ↔ πœ“))
rabren3dioph.b 𝑋 ∈ (1...𝑁)
rabren3dioph.c π‘Œ ∈ (1...𝑁)
rabren3dioph.d 𝑍 ∈ (1...𝑁)
Assertion
Ref Expression
rabren3dioph ((𝑁 ∈ β„•0 ∧ {π‘Ž ∈ (β„•0 ↑m (1...3)) ∣ πœ‘} ∈ (Diophβ€˜3)) β†’ {𝑏 ∈ (β„•0 ↑m (1...𝑁)) ∣ πœ“} ∈ (Diophβ€˜π‘))
Distinct variable groups:   πœ“,π‘Ž   πœ‘,𝑏   𝑋,π‘Ž,𝑏   π‘Œ,π‘Ž,𝑏   𝑍,π‘Ž,𝑏   𝑁,π‘Ž,𝑏
Allowed substitution hints:   πœ‘(π‘Ž)   πœ“(𝑏)

Proof of Theorem rabren3dioph
StepHypRef Expression
1 vex 3476 . . . . 5 𝑏 ∈ V
2 tpex 7736 . . . . 5 {⟨1, π‘‹βŸ©, ⟨2, π‘ŒβŸ©, ⟨3, π‘βŸ©} ∈ V
31, 2coex 7923 . . . 4 (𝑏 ∘ {⟨1, π‘‹βŸ©, ⟨2, π‘ŒβŸ©, ⟨3, π‘βŸ©}) ∈ V
4 1ne2 12424 . . . . . . . . . 10 1 β‰  2
5 1re 11218 . . . . . . . . . . 11 1 ∈ ℝ
6 1lt3 12389 . . . . . . . . . . 11 1 < 3
75, 6ltneii 11331 . . . . . . . . . 10 1 β‰  3
8 2re 12290 . . . . . . . . . . 11 2 ∈ ℝ
9 2lt3 12388 . . . . . . . . . . 11 2 < 3
108, 9ltneii 11331 . . . . . . . . . 10 2 β‰  3
11 1ex 11214 . . . . . . . . . . 11 1 ∈ V
12 2ex 12293 . . . . . . . . . . 11 2 ∈ V
13 3ex 12298 . . . . . . . . . . 11 3 ∈ V
14 rabren3dioph.b . . . . . . . . . . . 12 𝑋 ∈ (1...𝑁)
1514elexi 3492 . . . . . . . . . . 11 𝑋 ∈ V
16 rabren3dioph.c . . . . . . . . . . . 12 π‘Œ ∈ (1...𝑁)
1716elexi 3492 . . . . . . . . . . 11 π‘Œ ∈ V
18 rabren3dioph.d . . . . . . . . . . . 12 𝑍 ∈ (1...𝑁)
1918elexi 3492 . . . . . . . . . . 11 𝑍 ∈ V
2011, 12, 13, 15, 17, 19fntp 6608 . . . . . . . . . 10 ((1 β‰  2 ∧ 1 β‰  3 ∧ 2 β‰  3) β†’ {⟨1, π‘‹βŸ©, ⟨2, π‘ŒβŸ©, ⟨3, π‘βŸ©} Fn {1, 2, 3})
214, 7, 10, 20mp3an 1459 . . . . . . . . 9 {⟨1, π‘‹βŸ©, ⟨2, π‘ŒβŸ©, ⟨3, π‘βŸ©} Fn {1, 2, 3}
2211tpid1 4771 . . . . . . . . 9 1 ∈ {1, 2, 3}
23 fvco2 6987 . . . . . . . . 9 (({⟨1, π‘‹βŸ©, ⟨2, π‘ŒβŸ©, ⟨3, π‘βŸ©} Fn {1, 2, 3} ∧ 1 ∈ {1, 2, 3}) β†’ ((𝑏 ∘ {⟨1, π‘‹βŸ©, ⟨2, π‘ŒβŸ©, ⟨3, π‘βŸ©})β€˜1) = (π‘β€˜({⟨1, π‘‹βŸ©, ⟨2, π‘ŒβŸ©, ⟨3, π‘βŸ©}β€˜1)))
2421, 22, 23mp2an 688 . . . . . . . 8 ((𝑏 ∘ {⟨1, π‘‹βŸ©, ⟨2, π‘ŒβŸ©, ⟨3, π‘βŸ©})β€˜1) = (π‘β€˜({⟨1, π‘‹βŸ©, ⟨2, π‘ŒβŸ©, ⟨3, π‘βŸ©}β€˜1))
2511, 15fvtp1 7197 . . . . . . . . . 10 ((1 β‰  2 ∧ 1 β‰  3) β†’ ({⟨1, π‘‹βŸ©, ⟨2, π‘ŒβŸ©, ⟨3, π‘βŸ©}β€˜1) = 𝑋)
264, 7, 25mp2an 688 . . . . . . . . 9 ({⟨1, π‘‹βŸ©, ⟨2, π‘ŒβŸ©, ⟨3, π‘βŸ©}β€˜1) = 𝑋
2726fveq2i 6893 . . . . . . . 8 (π‘β€˜({⟨1, π‘‹βŸ©, ⟨2, π‘ŒβŸ©, ⟨3, π‘βŸ©}β€˜1)) = (π‘β€˜π‘‹)
2824, 27eqtri 2758 . . . . . . 7 ((𝑏 ∘ {⟨1, π‘‹βŸ©, ⟨2, π‘ŒβŸ©, ⟨3, π‘βŸ©})β€˜1) = (π‘β€˜π‘‹)
2912tpid2 4773 . . . . . . . . 9 2 ∈ {1, 2, 3}
30 fvco2 6987 . . . . . . . . 9 (({⟨1, π‘‹βŸ©, ⟨2, π‘ŒβŸ©, ⟨3, π‘βŸ©} Fn {1, 2, 3} ∧ 2 ∈ {1, 2, 3}) β†’ ((𝑏 ∘ {⟨1, π‘‹βŸ©, ⟨2, π‘ŒβŸ©, ⟨3, π‘βŸ©})β€˜2) = (π‘β€˜({⟨1, π‘‹βŸ©, ⟨2, π‘ŒβŸ©, ⟨3, π‘βŸ©}β€˜2)))
3121, 29, 30mp2an 688 . . . . . . . 8 ((𝑏 ∘ {⟨1, π‘‹βŸ©, ⟨2, π‘ŒβŸ©, ⟨3, π‘βŸ©})β€˜2) = (π‘β€˜({⟨1, π‘‹βŸ©, ⟨2, π‘ŒβŸ©, ⟨3, π‘βŸ©}β€˜2))
3212, 17fvtp2 7198 . . . . . . . . . 10 ((1 β‰  2 ∧ 2 β‰  3) β†’ ({⟨1, π‘‹βŸ©, ⟨2, π‘ŒβŸ©, ⟨3, π‘βŸ©}β€˜2) = π‘Œ)
334, 10, 32mp2an 688 . . . . . . . . 9 ({⟨1, π‘‹βŸ©, ⟨2, π‘ŒβŸ©, ⟨3, π‘βŸ©}β€˜2) = π‘Œ
3433fveq2i 6893 . . . . . . . 8 (π‘β€˜({⟨1, π‘‹βŸ©, ⟨2, π‘ŒβŸ©, ⟨3, π‘βŸ©}β€˜2)) = (π‘β€˜π‘Œ)
3531, 34eqtri 2758 . . . . . . 7 ((𝑏 ∘ {⟨1, π‘‹βŸ©, ⟨2, π‘ŒβŸ©, ⟨3, π‘βŸ©})β€˜2) = (π‘β€˜π‘Œ)
3613tpid3 4776 . . . . . . . . 9 3 ∈ {1, 2, 3}
37 fvco2 6987 . . . . . . . . 9 (({⟨1, π‘‹βŸ©, ⟨2, π‘ŒβŸ©, ⟨3, π‘βŸ©} Fn {1, 2, 3} ∧ 3 ∈ {1, 2, 3}) β†’ ((𝑏 ∘ {⟨1, π‘‹βŸ©, ⟨2, π‘ŒβŸ©, ⟨3, π‘βŸ©})β€˜3) = (π‘β€˜({⟨1, π‘‹βŸ©, ⟨2, π‘ŒβŸ©, ⟨3, π‘βŸ©}β€˜3)))
3821, 36, 37mp2an 688 . . . . . . . 8 ((𝑏 ∘ {⟨1, π‘‹βŸ©, ⟨2, π‘ŒβŸ©, ⟨3, π‘βŸ©})β€˜3) = (π‘β€˜({⟨1, π‘‹βŸ©, ⟨2, π‘ŒβŸ©, ⟨3, π‘βŸ©}β€˜3))
3913, 19fvtp3 7199 . . . . . . . . . 10 ((1 β‰  3 ∧ 2 β‰  3) β†’ ({⟨1, π‘‹βŸ©, ⟨2, π‘ŒβŸ©, ⟨3, π‘βŸ©}β€˜3) = 𝑍)
407, 10, 39mp2an 688 . . . . . . . . 9 ({⟨1, π‘‹βŸ©, ⟨2, π‘ŒβŸ©, ⟨3, π‘βŸ©}β€˜3) = 𝑍
4140fveq2i 6893 . . . . . . . 8 (π‘β€˜({⟨1, π‘‹βŸ©, ⟨2, π‘ŒβŸ©, ⟨3, π‘βŸ©}β€˜3)) = (π‘β€˜π‘)
4238, 41eqtri 2758 . . . . . . 7 ((𝑏 ∘ {⟨1, π‘‹βŸ©, ⟨2, π‘ŒβŸ©, ⟨3, π‘βŸ©})β€˜3) = (π‘β€˜π‘)
4328, 35, 423pm3.2i 1337 . . . . . 6 (((𝑏 ∘ {⟨1, π‘‹βŸ©, ⟨2, π‘ŒβŸ©, ⟨3, π‘βŸ©})β€˜1) = (π‘β€˜π‘‹) ∧ ((𝑏 ∘ {⟨1, π‘‹βŸ©, ⟨2, π‘ŒβŸ©, ⟨3, π‘βŸ©})β€˜2) = (π‘β€˜π‘Œ) ∧ ((𝑏 ∘ {⟨1, π‘‹βŸ©, ⟨2, π‘ŒβŸ©, ⟨3, π‘βŸ©})β€˜3) = (π‘β€˜π‘))
44 fveq1 6889 . . . . . . . 8 (π‘Ž = (𝑏 ∘ {⟨1, π‘‹βŸ©, ⟨2, π‘ŒβŸ©, ⟨3, π‘βŸ©}) β†’ (π‘Žβ€˜1) = ((𝑏 ∘ {⟨1, π‘‹βŸ©, ⟨2, π‘ŒβŸ©, ⟨3, π‘βŸ©})β€˜1))
4544eqeq1d 2732 . . . . . . 7 (π‘Ž = (𝑏 ∘ {⟨1, π‘‹βŸ©, ⟨2, π‘ŒβŸ©, ⟨3, π‘βŸ©}) β†’ ((π‘Žβ€˜1) = (π‘β€˜π‘‹) ↔ ((𝑏 ∘ {⟨1, π‘‹βŸ©, ⟨2, π‘ŒβŸ©, ⟨3, π‘βŸ©})β€˜1) = (π‘β€˜π‘‹)))
46 fveq1 6889 . . . . . . . 8 (π‘Ž = (𝑏 ∘ {⟨1, π‘‹βŸ©, ⟨2, π‘ŒβŸ©, ⟨3, π‘βŸ©}) β†’ (π‘Žβ€˜2) = ((𝑏 ∘ {⟨1, π‘‹βŸ©, ⟨2, π‘ŒβŸ©, ⟨3, π‘βŸ©})β€˜2))
4746eqeq1d 2732 . . . . . . 7 (π‘Ž = (𝑏 ∘ {⟨1, π‘‹βŸ©, ⟨2, π‘ŒβŸ©, ⟨3, π‘βŸ©}) β†’ ((π‘Žβ€˜2) = (π‘β€˜π‘Œ) ↔ ((𝑏 ∘ {⟨1, π‘‹βŸ©, ⟨2, π‘ŒβŸ©, ⟨3, π‘βŸ©})β€˜2) = (π‘β€˜π‘Œ)))
48 fveq1 6889 . . . . . . . 8 (π‘Ž = (𝑏 ∘ {⟨1, π‘‹βŸ©, ⟨2, π‘ŒβŸ©, ⟨3, π‘βŸ©}) β†’ (π‘Žβ€˜3) = ((𝑏 ∘ {⟨1, π‘‹βŸ©, ⟨2, π‘ŒβŸ©, ⟨3, π‘βŸ©})β€˜3))
4948eqeq1d 2732 . . . . . . 7 (π‘Ž = (𝑏 ∘ {⟨1, π‘‹βŸ©, ⟨2, π‘ŒβŸ©, ⟨3, π‘βŸ©}) β†’ ((π‘Žβ€˜3) = (π‘β€˜π‘) ↔ ((𝑏 ∘ {⟨1, π‘‹βŸ©, ⟨2, π‘ŒβŸ©, ⟨3, π‘βŸ©})β€˜3) = (π‘β€˜π‘)))
5045, 47, 493anbi123d 1434 . . . . . 6 (π‘Ž = (𝑏 ∘ {⟨1, π‘‹βŸ©, ⟨2, π‘ŒβŸ©, ⟨3, π‘βŸ©}) β†’ (((π‘Žβ€˜1) = (π‘β€˜π‘‹) ∧ (π‘Žβ€˜2) = (π‘β€˜π‘Œ) ∧ (π‘Žβ€˜3) = (π‘β€˜π‘)) ↔ (((𝑏 ∘ {⟨1, π‘‹βŸ©, ⟨2, π‘ŒβŸ©, ⟨3, π‘βŸ©})β€˜1) = (π‘β€˜π‘‹) ∧ ((𝑏 ∘ {⟨1, π‘‹βŸ©, ⟨2, π‘ŒβŸ©, ⟨3, π‘βŸ©})β€˜2) = (π‘β€˜π‘Œ) ∧ ((𝑏 ∘ {⟨1, π‘‹βŸ©, ⟨2, π‘ŒβŸ©, ⟨3, π‘βŸ©})β€˜3) = (π‘β€˜π‘))))
5143, 50mpbiri 257 . . . . 5 (π‘Ž = (𝑏 ∘ {⟨1, π‘‹βŸ©, ⟨2, π‘ŒβŸ©, ⟨3, π‘βŸ©}) β†’ ((π‘Žβ€˜1) = (π‘β€˜π‘‹) ∧ (π‘Žβ€˜2) = (π‘β€˜π‘Œ) ∧ (π‘Žβ€˜3) = (π‘β€˜π‘)))
52 rabren3dioph.a . . . . 5 (((π‘Žβ€˜1) = (π‘β€˜π‘‹) ∧ (π‘Žβ€˜2) = (π‘β€˜π‘Œ) ∧ (π‘Žβ€˜3) = (π‘β€˜π‘)) β†’ (πœ‘ ↔ πœ“))
5351, 52syl 17 . . . 4 (π‘Ž = (𝑏 ∘ {⟨1, π‘‹βŸ©, ⟨2, π‘ŒβŸ©, ⟨3, π‘βŸ©}) β†’ (πœ‘ ↔ πœ“))
543, 53sbcie 3819 . . 3 ([(𝑏 ∘ {⟨1, π‘‹βŸ©, ⟨2, π‘ŒβŸ©, ⟨3, π‘βŸ©}) / π‘Ž]πœ‘ ↔ πœ“)
5554rabbii 3436 . 2 {𝑏 ∈ (β„•0 ↑m (1...𝑁)) ∣ [(𝑏 ∘ {⟨1, π‘‹βŸ©, ⟨2, π‘ŒβŸ©, ⟨3, π‘βŸ©}) / π‘Ž]πœ‘} = {𝑏 ∈ (β„•0 ↑m (1...𝑁)) ∣ πœ“}
5611, 12, 13, 15, 17, 19, 4, 7, 10ftp 7156 . . . . 5 {⟨1, π‘‹βŸ©, ⟨2, π‘ŒβŸ©, ⟨3, π‘βŸ©}:{1, 2, 3}⟢{𝑋, π‘Œ, 𝑍}
57 1z 12596 . . . . . . . 8 1 ∈ β„€
58 fztp 13561 . . . . . . . 8 (1 ∈ β„€ β†’ (1...(1 + 2)) = {1, (1 + 1), (1 + 2)})
5957, 58ax-mp 5 . . . . . . 7 (1...(1 + 2)) = {1, (1 + 1), (1 + 2)}
60 1p2e3 12359 . . . . . . . 8 (1 + 2) = 3
6160oveq2i 7422 . . . . . . 7 (1...(1 + 2)) = (1...3)
62 eqidd 2731 . . . . . . . . 9 (1 ∈ β„€ β†’ 1 = 1)
63 1p1e2 12341 . . . . . . . . . 10 (1 + 1) = 2
6463a1i 11 . . . . . . . . 9 (1 ∈ β„€ β†’ (1 + 1) = 2)
6560a1i 11 . . . . . . . . 9 (1 ∈ β„€ β†’ (1 + 2) = 3)
6662, 64, 65tpeq123d 4751 . . . . . . . 8 (1 ∈ β„€ β†’ {1, (1 + 1), (1 + 2)} = {1, 2, 3})
6757, 66ax-mp 5 . . . . . . 7 {1, (1 + 1), (1 + 2)} = {1, 2, 3}
6859, 61, 673eqtr3i 2766 . . . . . 6 (1...3) = {1, 2, 3}
6968feq2i 6708 . . . . 5 ({⟨1, π‘‹βŸ©, ⟨2, π‘ŒβŸ©, ⟨3, π‘βŸ©}:(1...3)⟢{𝑋, π‘Œ, 𝑍} ↔ {⟨1, π‘‹βŸ©, ⟨2, π‘ŒβŸ©, ⟨3, π‘βŸ©}:{1, 2, 3}⟢{𝑋, π‘Œ, 𝑍})
7056, 69mpbir 230 . . . 4 {⟨1, π‘‹βŸ©, ⟨2, π‘ŒβŸ©, ⟨3, π‘βŸ©}:(1...3)⟢{𝑋, π‘Œ, 𝑍}
7114, 16, 183pm3.2i 1337 . . . . 5 (𝑋 ∈ (1...𝑁) ∧ π‘Œ ∈ (1...𝑁) ∧ 𝑍 ∈ (1...𝑁))
7215, 17, 19tpss 4837 . . . . 5 ((𝑋 ∈ (1...𝑁) ∧ π‘Œ ∈ (1...𝑁) ∧ 𝑍 ∈ (1...𝑁)) ↔ {𝑋, π‘Œ, 𝑍} βŠ† (1...𝑁))
7371, 72mpbi 229 . . . 4 {𝑋, π‘Œ, 𝑍} βŠ† (1...𝑁)
74 fss 6733 . . . 4 (({⟨1, π‘‹βŸ©, ⟨2, π‘ŒβŸ©, ⟨3, π‘βŸ©}:(1...3)⟢{𝑋, π‘Œ, 𝑍} ∧ {𝑋, π‘Œ, 𝑍} βŠ† (1...𝑁)) β†’ {⟨1, π‘‹βŸ©, ⟨2, π‘ŒβŸ©, ⟨3, π‘βŸ©}:(1...3)⟢(1...𝑁))
7570, 73, 74mp2an 688 . . 3 {⟨1, π‘‹βŸ©, ⟨2, π‘ŒβŸ©, ⟨3, π‘βŸ©}:(1...3)⟢(1...𝑁)
76 rabrenfdioph 41854 . . 3 ((𝑁 ∈ β„•0 ∧ {⟨1, π‘‹βŸ©, ⟨2, π‘ŒβŸ©, ⟨3, π‘βŸ©}:(1...3)⟢(1...𝑁) ∧ {π‘Ž ∈ (β„•0 ↑m (1...3)) ∣ πœ‘} ∈ (Diophβ€˜3)) β†’ {𝑏 ∈ (β„•0 ↑m (1...𝑁)) ∣ [(𝑏 ∘ {⟨1, π‘‹βŸ©, ⟨2, π‘ŒβŸ©, ⟨3, π‘βŸ©}) / π‘Ž]πœ‘} ∈ (Diophβ€˜π‘))
7775, 76mp3an2 1447 . 2 ((𝑁 ∈ β„•0 ∧ {π‘Ž ∈ (β„•0 ↑m (1...3)) ∣ πœ‘} ∈ (Diophβ€˜3)) β†’ {𝑏 ∈ (β„•0 ↑m (1...𝑁)) ∣ [(𝑏 ∘ {⟨1, π‘‹βŸ©, ⟨2, π‘ŒβŸ©, ⟨3, π‘βŸ©}) / π‘Ž]πœ‘} ∈ (Diophβ€˜π‘))
7855, 77eqeltrrid 2836 1 ((𝑁 ∈ β„•0 ∧ {π‘Ž ∈ (β„•0 ↑m (1...3)) ∣ πœ‘} ∈ (Diophβ€˜3)) β†’ {𝑏 ∈ (β„•0 ↑m (1...𝑁)) ∣ πœ“} ∈ (Diophβ€˜π‘))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 394   ∧ w3a 1085   = wceq 1539   ∈ wcel 2104   β‰  wne 2938  {crab 3430  [wsbc 3776   βŠ† wss 3947  {ctp 4631  βŸ¨cop 4633   ∘ ccom 5679   Fn wfn 6537  βŸΆwf 6538  β€˜cfv 6542  (class class class)co 7411   ↑m cmap 8822  1c1 11113   + caddc 11115  2c2 12271  3c3 12272  β„•0cn0 12476  β„€cz 12562  ...cfz 13488  Diophcdioph 41795
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7727  ax-inf2 9638  ax-cnex 11168  ax-resscn 11169  ax-1cn 11170  ax-icn 11171  ax-addcl 11172  ax-addrcl 11173  ax-mulcl 11174  ax-mulrcl 11175  ax-mulcom 11176  ax-addass 11177  ax-mulass 11178  ax-distr 11179  ax-i2m1 11180  ax-1ne0 11181  ax-1rid 11182  ax-rnegex 11183  ax-rrecex 11184  ax-cnre 11185  ax-pre-lttri 11186  ax-pre-lttrn 11187  ax-pre-ltadd 11188  ax-pre-mulgt0 11189
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-reu 3375  df-rab 3431  df-v 3474  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-tp 4632  df-op 4634  df-uni 4908  df-int 4950  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6299  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-riota 7367  df-ov 7414  df-oprab 7415  df-mpo 7416  df-of 7672  df-om 7858  df-1st 7977  df-2nd 7978  df-frecs 8268  df-wrecs 8299  df-recs 8373  df-rdg 8412  df-1o 8468  df-oadd 8472  df-er 8705  df-map 8824  df-en 8942  df-dom 8943  df-sdom 8944  df-fin 8945  df-dju 9898  df-card 9936  df-pnf 11254  df-mnf 11255  df-xr 11256  df-ltxr 11257  df-le 11258  df-sub 11450  df-neg 11451  df-nn 12217  df-2 12279  df-3 12280  df-n0 12477  df-z 12563  df-uz 12827  df-fz 13489  df-hash 14295  df-mzpcl 41763  df-mzp 41764  df-dioph 41796
This theorem is referenced by:  rmxdioph  42057  expdiophlem2  42063
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