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Theorem rabren3dioph 41553
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 3479 . . . . 5 𝑏 ∈ V
2 tpex 7734 . . . . 5 {⟨1, π‘‹βŸ©, ⟨2, π‘ŒβŸ©, ⟨3, π‘βŸ©} ∈ V
31, 2coex 7921 . . . 4 (𝑏 ∘ {⟨1, π‘‹βŸ©, ⟨2, π‘ŒβŸ©, ⟨3, π‘βŸ©}) ∈ V
4 1ne2 12420 . . . . . . . . . 10 1 β‰  2
5 1re 11214 . . . . . . . . . . 11 1 ∈ ℝ
6 1lt3 12385 . . . . . . . . . . 11 1 < 3
75, 6ltneii 11327 . . . . . . . . . 10 1 β‰  3
8 2re 12286 . . . . . . . . . . 11 2 ∈ ℝ
9 2lt3 12384 . . . . . . . . . . 11 2 < 3
108, 9ltneii 11327 . . . . . . . . . 10 2 β‰  3
11 1ex 11210 . . . . . . . . . . 11 1 ∈ V
12 2ex 12289 . . . . . . . . . . 11 2 ∈ V
13 3ex 12294 . . . . . . . . . . 11 3 ∈ V
14 rabren3dioph.b . . . . . . . . . . . 12 𝑋 ∈ (1...𝑁)
1514elexi 3494 . . . . . . . . . . 11 𝑋 ∈ V
16 rabren3dioph.c . . . . . . . . . . . 12 π‘Œ ∈ (1...𝑁)
1716elexi 3494 . . . . . . . . . . 11 π‘Œ ∈ V
18 rabren3dioph.d . . . . . . . . . . . 12 𝑍 ∈ (1...𝑁)
1918elexi 3494 . . . . . . . . . . 11 𝑍 ∈ V
2011, 12, 13, 15, 17, 19fntp 6610 . . . . . . . . . 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 1462 . . . . . . . . 9 {⟨1, π‘‹βŸ©, ⟨2, π‘ŒβŸ©, ⟨3, π‘βŸ©} Fn {1, 2, 3}
2211tpid1 4773 . . . . . . . . 9 1 ∈ {1, 2, 3}
23 fvco2 6989 . . . . . . . . 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 691 . . . . . . . 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 7196 . . . . . . . . . 10 ((1 β‰  2 ∧ 1 β‰  3) β†’ ({⟨1, π‘‹βŸ©, ⟨2, π‘ŒβŸ©, ⟨3, π‘βŸ©}β€˜1) = 𝑋)
264, 7, 25mp2an 691 . . . . . . . . 9 ({⟨1, π‘‹βŸ©, ⟨2, π‘ŒβŸ©, ⟨3, π‘βŸ©}β€˜1) = 𝑋
2726fveq2i 6895 . . . . . . . 8 (π‘β€˜({⟨1, π‘‹βŸ©, ⟨2, π‘ŒβŸ©, ⟨3, π‘βŸ©}β€˜1)) = (π‘β€˜π‘‹)
2824, 27eqtri 2761 . . . . . . 7 ((𝑏 ∘ {⟨1, π‘‹βŸ©, ⟨2, π‘ŒβŸ©, ⟨3, π‘βŸ©})β€˜1) = (π‘β€˜π‘‹)
2912tpid2 4775 . . . . . . . . 9 2 ∈ {1, 2, 3}
30 fvco2 6989 . . . . . . . . 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 691 . . . . . . . 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 7197 . . . . . . . . . 10 ((1 β‰  2 ∧ 2 β‰  3) β†’ ({⟨1, π‘‹βŸ©, ⟨2, π‘ŒβŸ©, ⟨3, π‘βŸ©}β€˜2) = π‘Œ)
334, 10, 32mp2an 691 . . . . . . . . 9 ({⟨1, π‘‹βŸ©, ⟨2, π‘ŒβŸ©, ⟨3, π‘βŸ©}β€˜2) = π‘Œ
3433fveq2i 6895 . . . . . . . 8 (π‘β€˜({⟨1, π‘‹βŸ©, ⟨2, π‘ŒβŸ©, ⟨3, π‘βŸ©}β€˜2)) = (π‘β€˜π‘Œ)
3531, 34eqtri 2761 . . . . . . 7 ((𝑏 ∘ {⟨1, π‘‹βŸ©, ⟨2, π‘ŒβŸ©, ⟨3, π‘βŸ©})β€˜2) = (π‘β€˜π‘Œ)
3613tpid3 4778 . . . . . . . . 9 3 ∈ {1, 2, 3}
37 fvco2 6989 . . . . . . . . 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 691 . . . . . . . 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 7198 . . . . . . . . . 10 ((1 β‰  3 ∧ 2 β‰  3) β†’ ({⟨1, π‘‹βŸ©, ⟨2, π‘ŒβŸ©, ⟨3, π‘βŸ©}β€˜3) = 𝑍)
407, 10, 39mp2an 691 . . . . . . . . 9 ({⟨1, π‘‹βŸ©, ⟨2, π‘ŒβŸ©, ⟨3, π‘βŸ©}β€˜3) = 𝑍
4140fveq2i 6895 . . . . . . . 8 (π‘β€˜({⟨1, π‘‹βŸ©, ⟨2, π‘ŒβŸ©, ⟨3, π‘βŸ©}β€˜3)) = (π‘β€˜π‘)
4238, 41eqtri 2761 . . . . . . 7 ((𝑏 ∘ {⟨1, π‘‹βŸ©, ⟨2, π‘ŒβŸ©, ⟨3, π‘βŸ©})β€˜3) = (π‘β€˜π‘)
4328, 35, 423pm3.2i 1340 . . . . . 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 6891 . . . . . . . 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 2735 . . . . . . 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 6891 . . . . . . . 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 2735 . . . . . . 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 6891 . . . . . . . 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 2735 . . . . . . 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 1437 . . . . . 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 258 . . . . 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 3821 . . 3 ([(𝑏 ∘ {⟨1, π‘‹βŸ©, ⟨2, π‘ŒβŸ©, ⟨3, π‘βŸ©}) / π‘Ž]πœ‘ ↔ πœ“)
5554rabbii 3439 . 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 7155 . . . . 5 {⟨1, π‘‹βŸ©, ⟨2, π‘ŒβŸ©, ⟨3, π‘βŸ©}:{1, 2, 3}⟢{𝑋, π‘Œ, 𝑍}
57 1z 12592 . . . . . . . 8 1 ∈ β„€
58 fztp 13557 . . . . . . . 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 12355 . . . . . . . 8 (1 + 2) = 3
6160oveq2i 7420 . . . . . . 7 (1...(1 + 2)) = (1...3)
62 eqidd 2734 . . . . . . . . 9 (1 ∈ β„€ β†’ 1 = 1)
63 1p1e2 12337 . . . . . . . . . 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 4753 . . . . . . . 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 2769 . . . . . 6 (1...3) = {1, 2, 3}
6968feq2i 6710 . . . . 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 1340 . . . . 5 (𝑋 ∈ (1...𝑁) ∧ π‘Œ ∈ (1...𝑁) ∧ 𝑍 ∈ (1...𝑁))
7215, 17, 19tpss 4839 . . . . 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 6735 . . . 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 691 . . 3 {⟨1, π‘‹βŸ©, ⟨2, π‘ŒβŸ©, ⟨3, π‘βŸ©}:(1...3)⟢(1...𝑁)
76 rabrenfdioph 41552 . . 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 1450 . 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 2839 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 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107   β‰  wne 2941  {crab 3433  [wsbc 3778   βŠ† wss 3949  {ctp 4633  βŸ¨cop 4635   ∘ ccom 5681   Fn wfn 6539  βŸΆwf 6540  β€˜cfv 6544  (class class class)co 7409   ↑m cmap 8820  1c1 11111   + caddc 11113  2c2 12267  3c3 12268  β„•0cn0 12472  β„€cz 12558  ...cfz 13484  Diophcdioph 41493
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725  ax-inf2 9636  ax-cnex 11166  ax-resscn 11167  ax-1cn 11168  ax-icn 11169  ax-addcl 11170  ax-addrcl 11171  ax-mulcl 11172  ax-mulrcl 11173  ax-mulcom 11174  ax-addass 11175  ax-mulass 11176  ax-distr 11177  ax-i2m1 11178  ax-1ne0 11179  ax-1rid 11180  ax-rnegex 11181  ax-rrecex 11182  ax-cnre 11183  ax-pre-lttri 11184  ax-pre-lttrn 11185  ax-pre-ltadd 11186  ax-pre-mulgt0 11187
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-tp 4634  df-op 4636  df-uni 4910  df-int 4952  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-riota 7365  df-ov 7412  df-oprab 7413  df-mpo 7414  df-of 7670  df-om 7856  df-1st 7975  df-2nd 7976  df-frecs 8266  df-wrecs 8297  df-recs 8371  df-rdg 8410  df-1o 8466  df-oadd 8470  df-er 8703  df-map 8822  df-en 8940  df-dom 8941  df-sdom 8942  df-fin 8943  df-dju 9896  df-card 9934  df-pnf 11250  df-mnf 11251  df-xr 11252  df-ltxr 11253  df-le 11254  df-sub 11446  df-neg 11447  df-nn 12213  df-2 12275  df-3 12276  df-n0 12473  df-z 12559  df-uz 12823  df-fz 13485  df-hash 14291  df-mzpcl 41461  df-mzp 41462  df-dioph 41494
This theorem is referenced by:  rmxdioph  41755  expdiophlem2  41761
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