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Theorem cnsubrg 20873
Description: There are no subrings of the complex numbers strictly between ℝ and β„‚. (Contributed by Mario Carneiro, 15-Oct-2015.)
Assertion
Ref Expression
cnsubrg ((𝑅 ∈ (SubRingβ€˜β„‚fld) ∧ ℝ βŠ† 𝑅) β†’ 𝑅 ∈ {ℝ, β„‚})

Proof of Theorem cnsubrg
Dummy variables π‘₯ 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ssdif0 4328 . . . 4 (𝑅 βŠ† ℝ ↔ (𝑅 βˆ– ℝ) = βˆ…)
2 simpr 486 . . . . . 6 (((𝑅 ∈ (SubRingβ€˜β„‚fld) ∧ ℝ βŠ† 𝑅) ∧ 𝑅 βŠ† ℝ) β†’ 𝑅 βŠ† ℝ)
3 simplr 768 . . . . . 6 (((𝑅 ∈ (SubRingβ€˜β„‚fld) ∧ ℝ βŠ† 𝑅) ∧ 𝑅 βŠ† ℝ) β†’ ℝ βŠ† 𝑅)
42, 3eqssd 3966 . . . . 5 (((𝑅 ∈ (SubRingβ€˜β„‚fld) ∧ ℝ βŠ† 𝑅) ∧ 𝑅 βŠ† ℝ) β†’ 𝑅 = ℝ)
54orcd 872 . . . 4 (((𝑅 ∈ (SubRingβ€˜β„‚fld) ∧ ℝ βŠ† 𝑅) ∧ 𝑅 βŠ† ℝ) β†’ (𝑅 = ℝ ∨ 𝑅 = β„‚))
61, 5sylan2br 596 . . 3 (((𝑅 ∈ (SubRingβ€˜β„‚fld) ∧ ℝ βŠ† 𝑅) ∧ (𝑅 βˆ– ℝ) = βˆ…) β†’ (𝑅 = ℝ ∨ 𝑅 = β„‚))
7 n0 4311 . . . 4 ((𝑅 βˆ– ℝ) β‰  βˆ… ↔ βˆƒπ‘₯ π‘₯ ∈ (𝑅 βˆ– ℝ))
8 simpll 766 . . . . . . . . . 10 (((𝑅 ∈ (SubRingβ€˜β„‚fld) ∧ ℝ βŠ† 𝑅) ∧ π‘₯ ∈ (𝑅 βˆ– ℝ)) β†’ 𝑅 ∈ (SubRingβ€˜β„‚fld))
9 cnfldbas 20816 . . . . . . . . . . 11 β„‚ = (Baseβ€˜β„‚fld)
109subrgss 20239 . . . . . . . . . 10 (𝑅 ∈ (SubRingβ€˜β„‚fld) β†’ 𝑅 βŠ† β„‚)
118, 10syl 17 . . . . . . . . 9 (((𝑅 ∈ (SubRingβ€˜β„‚fld) ∧ ℝ βŠ† 𝑅) ∧ π‘₯ ∈ (𝑅 βˆ– ℝ)) β†’ 𝑅 βŠ† β„‚)
12 replim 15008 . . . . . . . . . . . . 13 (𝑦 ∈ β„‚ β†’ 𝑦 = ((β„œβ€˜π‘¦) + (i Β· (β„‘β€˜π‘¦))))
1312ad2antll 728 . . . . . . . . . . . 12 (((𝑅 ∈ (SubRingβ€˜β„‚fld) ∧ ℝ βŠ† 𝑅) ∧ (π‘₯ ∈ (𝑅 βˆ– ℝ) ∧ 𝑦 ∈ β„‚)) β†’ 𝑦 = ((β„œβ€˜π‘¦) + (i Β· (β„‘β€˜π‘¦))))
14 simpll 766 . . . . . . . . . . . . 13 (((𝑅 ∈ (SubRingβ€˜β„‚fld) ∧ ℝ βŠ† 𝑅) ∧ (π‘₯ ∈ (𝑅 βˆ– ℝ) ∧ 𝑦 ∈ β„‚)) β†’ 𝑅 ∈ (SubRingβ€˜β„‚fld))
15 simplr 768 . . . . . . . . . . . . . 14 (((𝑅 ∈ (SubRingβ€˜β„‚fld) ∧ ℝ βŠ† 𝑅) ∧ (π‘₯ ∈ (𝑅 βˆ– ℝ) ∧ 𝑦 ∈ β„‚)) β†’ ℝ βŠ† 𝑅)
16 recl 15002 . . . . . . . . . . . . . . 15 (𝑦 ∈ β„‚ β†’ (β„œβ€˜π‘¦) ∈ ℝ)
1716ad2antll 728 . . . . . . . . . . . . . 14 (((𝑅 ∈ (SubRingβ€˜β„‚fld) ∧ ℝ βŠ† 𝑅) ∧ (π‘₯ ∈ (𝑅 βˆ– ℝ) ∧ 𝑦 ∈ β„‚)) β†’ (β„œβ€˜π‘¦) ∈ ℝ)
1815, 17sseldd 3950 . . . . . . . . . . . . 13 (((𝑅 ∈ (SubRingβ€˜β„‚fld) ∧ ℝ βŠ† 𝑅) ∧ (π‘₯ ∈ (𝑅 βˆ– ℝ) ∧ 𝑦 ∈ β„‚)) β†’ (β„œβ€˜π‘¦) ∈ 𝑅)
19 ax-icn 11117 . . . . . . . . . . . . . . . . . . 19 i ∈ β„‚
2019a1i 11 . . . . . . . . . . . . . . . . . 18 (((𝑅 ∈ (SubRingβ€˜β„‚fld) ∧ ℝ βŠ† 𝑅) ∧ π‘₯ ∈ (𝑅 βˆ– ℝ)) β†’ i ∈ β„‚)
21 eldifi 4091 . . . . . . . . . . . . . . . . . . . . . 22 (π‘₯ ∈ (𝑅 βˆ– ℝ) β†’ π‘₯ ∈ 𝑅)
2221adantl 483 . . . . . . . . . . . . . . . . . . . . 21 (((𝑅 ∈ (SubRingβ€˜β„‚fld) ∧ ℝ βŠ† 𝑅) ∧ π‘₯ ∈ (𝑅 βˆ– ℝ)) β†’ π‘₯ ∈ 𝑅)
2311, 22sseldd 3950 . . . . . . . . . . . . . . . . . . . 20 (((𝑅 ∈ (SubRingβ€˜β„‚fld) ∧ ℝ βŠ† 𝑅) ∧ π‘₯ ∈ (𝑅 βˆ– ℝ)) β†’ π‘₯ ∈ β„‚)
24 imcl 15003 . . . . . . . . . . . . . . . . . . . 20 (π‘₯ ∈ β„‚ β†’ (β„‘β€˜π‘₯) ∈ ℝ)
2523, 24syl 17 . . . . . . . . . . . . . . . . . . 19 (((𝑅 ∈ (SubRingβ€˜β„‚fld) ∧ ℝ βŠ† 𝑅) ∧ π‘₯ ∈ (𝑅 βˆ– ℝ)) β†’ (β„‘β€˜π‘₯) ∈ ℝ)
2625recnd 11190 . . . . . . . . . . . . . . . . . 18 (((𝑅 ∈ (SubRingβ€˜β„‚fld) ∧ ℝ βŠ† 𝑅) ∧ π‘₯ ∈ (𝑅 βˆ– ℝ)) β†’ (β„‘β€˜π‘₯) ∈ β„‚)
27 eldifn 4092 . . . . . . . . . . . . . . . . . . . 20 (π‘₯ ∈ (𝑅 βˆ– ℝ) β†’ Β¬ π‘₯ ∈ ℝ)
2827adantl 483 . . . . . . . . . . . . . . . . . . 19 (((𝑅 ∈ (SubRingβ€˜β„‚fld) ∧ ℝ βŠ† 𝑅) ∧ π‘₯ ∈ (𝑅 βˆ– ℝ)) β†’ Β¬ π‘₯ ∈ ℝ)
29 reim0b 15011 . . . . . . . . . . . . . . . . . . . . 21 (π‘₯ ∈ β„‚ β†’ (π‘₯ ∈ ℝ ↔ (β„‘β€˜π‘₯) = 0))
3029necon3bbid 2982 . . . . . . . . . . . . . . . . . . . 20 (π‘₯ ∈ β„‚ β†’ (Β¬ π‘₯ ∈ ℝ ↔ (β„‘β€˜π‘₯) β‰  0))
3123, 30syl 17 . . . . . . . . . . . . . . . . . . 19 (((𝑅 ∈ (SubRingβ€˜β„‚fld) ∧ ℝ βŠ† 𝑅) ∧ π‘₯ ∈ (𝑅 βˆ– ℝ)) β†’ (Β¬ π‘₯ ∈ ℝ ↔ (β„‘β€˜π‘₯) β‰  0))
3228, 31mpbid 231 . . . . . . . . . . . . . . . . . 18 (((𝑅 ∈ (SubRingβ€˜β„‚fld) ∧ ℝ βŠ† 𝑅) ∧ π‘₯ ∈ (𝑅 βˆ– ℝ)) β†’ (β„‘β€˜π‘₯) β‰  0)
3320, 26, 32divcan4d 11944 . . . . . . . . . . . . . . . . 17 (((𝑅 ∈ (SubRingβ€˜β„‚fld) ∧ ℝ βŠ† 𝑅) ∧ π‘₯ ∈ (𝑅 βˆ– ℝ)) β†’ ((i Β· (β„‘β€˜π‘₯)) / (β„‘β€˜π‘₯)) = i)
34 mulcl 11142 . . . . . . . . . . . . . . . . . . 19 ((i ∈ β„‚ ∧ (β„‘β€˜π‘₯) ∈ β„‚) β†’ (i Β· (β„‘β€˜π‘₯)) ∈ β„‚)
3519, 26, 34sylancr 588 . . . . . . . . . . . . . . . . . 18 (((𝑅 ∈ (SubRingβ€˜β„‚fld) ∧ ℝ βŠ† 𝑅) ∧ π‘₯ ∈ (𝑅 βˆ– ℝ)) β†’ (i Β· (β„‘β€˜π‘₯)) ∈ β„‚)
3635, 26, 32divrecd 11941 . . . . . . . . . . . . . . . . 17 (((𝑅 ∈ (SubRingβ€˜β„‚fld) ∧ ℝ βŠ† 𝑅) ∧ π‘₯ ∈ (𝑅 βˆ– ℝ)) β†’ ((i Β· (β„‘β€˜π‘₯)) / (β„‘β€˜π‘₯)) = ((i Β· (β„‘β€˜π‘₯)) Β· (1 / (β„‘β€˜π‘₯))))
3733, 36eqtr3d 2779 . . . . . . . . . . . . . . . 16 (((𝑅 ∈ (SubRingβ€˜β„‚fld) ∧ ℝ βŠ† 𝑅) ∧ π‘₯ ∈ (𝑅 βˆ– ℝ)) β†’ i = ((i Β· (β„‘β€˜π‘₯)) Β· (1 / (β„‘β€˜π‘₯))))
3823recld 15086 . . . . . . . . . . . . . . . . . . . . 21 (((𝑅 ∈ (SubRingβ€˜β„‚fld) ∧ ℝ βŠ† 𝑅) ∧ π‘₯ ∈ (𝑅 βˆ– ℝ)) β†’ (β„œβ€˜π‘₯) ∈ ℝ)
3938recnd 11190 . . . . . . . . . . . . . . . . . . . 20 (((𝑅 ∈ (SubRingβ€˜β„‚fld) ∧ ℝ βŠ† 𝑅) ∧ π‘₯ ∈ (𝑅 βˆ– ℝ)) β†’ (β„œβ€˜π‘₯) ∈ β„‚)
4023, 39negsubd 11525 . . . . . . . . . . . . . . . . . . 19 (((𝑅 ∈ (SubRingβ€˜β„‚fld) ∧ ℝ βŠ† 𝑅) ∧ π‘₯ ∈ (𝑅 βˆ– ℝ)) β†’ (π‘₯ + -(β„œβ€˜π‘₯)) = (π‘₯ βˆ’ (β„œβ€˜π‘₯)))
41 replim 15008 . . . . . . . . . . . . . . . . . . . . 21 (π‘₯ ∈ β„‚ β†’ π‘₯ = ((β„œβ€˜π‘₯) + (i Β· (β„‘β€˜π‘₯))))
4223, 41syl 17 . . . . . . . . . . . . . . . . . . . 20 (((𝑅 ∈ (SubRingβ€˜β„‚fld) ∧ ℝ βŠ† 𝑅) ∧ π‘₯ ∈ (𝑅 βˆ– ℝ)) β†’ π‘₯ = ((β„œβ€˜π‘₯) + (i Β· (β„‘β€˜π‘₯))))
4342oveq1d 7377 . . . . . . . . . . . . . . . . . . 19 (((𝑅 ∈ (SubRingβ€˜β„‚fld) ∧ ℝ βŠ† 𝑅) ∧ π‘₯ ∈ (𝑅 βˆ– ℝ)) β†’ (π‘₯ βˆ’ (β„œβ€˜π‘₯)) = (((β„œβ€˜π‘₯) + (i Β· (β„‘β€˜π‘₯))) βˆ’ (β„œβ€˜π‘₯)))
4439, 35pncan2d 11521 . . . . . . . . . . . . . . . . . . 19 (((𝑅 ∈ (SubRingβ€˜β„‚fld) ∧ ℝ βŠ† 𝑅) ∧ π‘₯ ∈ (𝑅 βˆ– ℝ)) β†’ (((β„œβ€˜π‘₯) + (i Β· (β„‘β€˜π‘₯))) βˆ’ (β„œβ€˜π‘₯)) = (i Β· (β„‘β€˜π‘₯)))
4540, 43, 443eqtrd 2781 . . . . . . . . . . . . . . . . . 18 (((𝑅 ∈ (SubRingβ€˜β„‚fld) ∧ ℝ βŠ† 𝑅) ∧ π‘₯ ∈ (𝑅 βˆ– ℝ)) β†’ (π‘₯ + -(β„œβ€˜π‘₯)) = (i Β· (β„‘β€˜π‘₯)))
46 simplr 768 . . . . . . . . . . . . . . . . . . . 20 (((𝑅 ∈ (SubRingβ€˜β„‚fld) ∧ ℝ βŠ† 𝑅) ∧ π‘₯ ∈ (𝑅 βˆ– ℝ)) β†’ ℝ βŠ† 𝑅)
4738renegcld 11589 . . . . . . . . . . . . . . . . . . . 20 (((𝑅 ∈ (SubRingβ€˜β„‚fld) ∧ ℝ βŠ† 𝑅) ∧ π‘₯ ∈ (𝑅 βˆ– ℝ)) β†’ -(β„œβ€˜π‘₯) ∈ ℝ)
4846, 47sseldd 3950 . . . . . . . . . . . . . . . . . . 19 (((𝑅 ∈ (SubRingβ€˜β„‚fld) ∧ ℝ βŠ† 𝑅) ∧ π‘₯ ∈ (𝑅 βˆ– ℝ)) β†’ -(β„œβ€˜π‘₯) ∈ 𝑅)
49 cnfldadd 20817 . . . . . . . . . . . . . . . . . . . 20 + = (+gβ€˜β„‚fld)
5049subrgacl 20249 . . . . . . . . . . . . . . . . . . 19 ((𝑅 ∈ (SubRingβ€˜β„‚fld) ∧ π‘₯ ∈ 𝑅 ∧ -(β„œβ€˜π‘₯) ∈ 𝑅) β†’ (π‘₯ + -(β„œβ€˜π‘₯)) ∈ 𝑅)
518, 22, 48, 50syl3anc 1372 . . . . . . . . . . . . . . . . . 18 (((𝑅 ∈ (SubRingβ€˜β„‚fld) ∧ ℝ βŠ† 𝑅) ∧ π‘₯ ∈ (𝑅 βˆ– ℝ)) β†’ (π‘₯ + -(β„œβ€˜π‘₯)) ∈ 𝑅)
5245, 51eqeltrrd 2839 . . . . . . . . . . . . . . . . 17 (((𝑅 ∈ (SubRingβ€˜β„‚fld) ∧ ℝ βŠ† 𝑅) ∧ π‘₯ ∈ (𝑅 βˆ– ℝ)) β†’ (i Β· (β„‘β€˜π‘₯)) ∈ 𝑅)
5325, 32rereccld 11989 . . . . . . . . . . . . . . . . . 18 (((𝑅 ∈ (SubRingβ€˜β„‚fld) ∧ ℝ βŠ† 𝑅) ∧ π‘₯ ∈ (𝑅 βˆ– ℝ)) β†’ (1 / (β„‘β€˜π‘₯)) ∈ ℝ)
5446, 53sseldd 3950 . . . . . . . . . . . . . . . . 17 (((𝑅 ∈ (SubRingβ€˜β„‚fld) ∧ ℝ βŠ† 𝑅) ∧ π‘₯ ∈ (𝑅 βˆ– ℝ)) β†’ (1 / (β„‘β€˜π‘₯)) ∈ 𝑅)
55 cnfldmul 20818 . . . . . . . . . . . . . . . . . 18 Β· = (.rβ€˜β„‚fld)
5655subrgmcl 20250 . . . . . . . . . . . . . . . . 17 ((𝑅 ∈ (SubRingβ€˜β„‚fld) ∧ (i Β· (β„‘β€˜π‘₯)) ∈ 𝑅 ∧ (1 / (β„‘β€˜π‘₯)) ∈ 𝑅) β†’ ((i Β· (β„‘β€˜π‘₯)) Β· (1 / (β„‘β€˜π‘₯))) ∈ 𝑅)
578, 52, 54, 56syl3anc 1372 . . . . . . . . . . . . . . . 16 (((𝑅 ∈ (SubRingβ€˜β„‚fld) ∧ ℝ βŠ† 𝑅) ∧ π‘₯ ∈ (𝑅 βˆ– ℝ)) β†’ ((i Β· (β„‘β€˜π‘₯)) Β· (1 / (β„‘β€˜π‘₯))) ∈ 𝑅)
5837, 57eqeltrd 2838 . . . . . . . . . . . . . . 15 (((𝑅 ∈ (SubRingβ€˜β„‚fld) ∧ ℝ βŠ† 𝑅) ∧ π‘₯ ∈ (𝑅 βˆ– ℝ)) β†’ i ∈ 𝑅)
5958adantrr 716 . . . . . . . . . . . . . 14 (((𝑅 ∈ (SubRingβ€˜β„‚fld) ∧ ℝ βŠ† 𝑅) ∧ (π‘₯ ∈ (𝑅 βˆ– ℝ) ∧ 𝑦 ∈ β„‚)) β†’ i ∈ 𝑅)
60 imcl 15003 . . . . . . . . . . . . . . . 16 (𝑦 ∈ β„‚ β†’ (β„‘β€˜π‘¦) ∈ ℝ)
6160ad2antll 728 . . . . . . . . . . . . . . 15 (((𝑅 ∈ (SubRingβ€˜β„‚fld) ∧ ℝ βŠ† 𝑅) ∧ (π‘₯ ∈ (𝑅 βˆ– ℝ) ∧ 𝑦 ∈ β„‚)) β†’ (β„‘β€˜π‘¦) ∈ ℝ)
6215, 61sseldd 3950 . . . . . . . . . . . . . 14 (((𝑅 ∈ (SubRingβ€˜β„‚fld) ∧ ℝ βŠ† 𝑅) ∧ (π‘₯ ∈ (𝑅 βˆ– ℝ) ∧ 𝑦 ∈ β„‚)) β†’ (β„‘β€˜π‘¦) ∈ 𝑅)
6355subrgmcl 20250 . . . . . . . . . . . . . 14 ((𝑅 ∈ (SubRingβ€˜β„‚fld) ∧ i ∈ 𝑅 ∧ (β„‘β€˜π‘¦) ∈ 𝑅) β†’ (i Β· (β„‘β€˜π‘¦)) ∈ 𝑅)
6414, 59, 62, 63syl3anc 1372 . . . . . . . . . . . . 13 (((𝑅 ∈ (SubRingβ€˜β„‚fld) ∧ ℝ βŠ† 𝑅) ∧ (π‘₯ ∈ (𝑅 βˆ– ℝ) ∧ 𝑦 ∈ β„‚)) β†’ (i Β· (β„‘β€˜π‘¦)) ∈ 𝑅)
6549subrgacl 20249 . . . . . . . . . . . . 13 ((𝑅 ∈ (SubRingβ€˜β„‚fld) ∧ (β„œβ€˜π‘¦) ∈ 𝑅 ∧ (i Β· (β„‘β€˜π‘¦)) ∈ 𝑅) β†’ ((β„œβ€˜π‘¦) + (i Β· (β„‘β€˜π‘¦))) ∈ 𝑅)
6614, 18, 64, 65syl3anc 1372 . . . . . . . . . . . 12 (((𝑅 ∈ (SubRingβ€˜β„‚fld) ∧ ℝ βŠ† 𝑅) ∧ (π‘₯ ∈ (𝑅 βˆ– ℝ) ∧ 𝑦 ∈ β„‚)) β†’ ((β„œβ€˜π‘¦) + (i Β· (β„‘β€˜π‘¦))) ∈ 𝑅)
6713, 66eqeltrd 2838 . . . . . . . . . . 11 (((𝑅 ∈ (SubRingβ€˜β„‚fld) ∧ ℝ βŠ† 𝑅) ∧ (π‘₯ ∈ (𝑅 βˆ– ℝ) ∧ 𝑦 ∈ β„‚)) β†’ 𝑦 ∈ 𝑅)
6867expr 458 . . . . . . . . . 10 (((𝑅 ∈ (SubRingβ€˜β„‚fld) ∧ ℝ βŠ† 𝑅) ∧ π‘₯ ∈ (𝑅 βˆ– ℝ)) β†’ (𝑦 ∈ β„‚ β†’ 𝑦 ∈ 𝑅))
6968ssrdv 3955 . . . . . . . . 9 (((𝑅 ∈ (SubRingβ€˜β„‚fld) ∧ ℝ βŠ† 𝑅) ∧ π‘₯ ∈ (𝑅 βˆ– ℝ)) β†’ β„‚ βŠ† 𝑅)
7011, 69eqssd 3966 . . . . . . . 8 (((𝑅 ∈ (SubRingβ€˜β„‚fld) ∧ ℝ βŠ† 𝑅) ∧ π‘₯ ∈ (𝑅 βˆ– ℝ)) β†’ 𝑅 = β„‚)
7170olcd 873 . . . . . . 7 (((𝑅 ∈ (SubRingβ€˜β„‚fld) ∧ ℝ βŠ† 𝑅) ∧ π‘₯ ∈ (𝑅 βˆ– ℝ)) β†’ (𝑅 = ℝ ∨ 𝑅 = β„‚))
7271ex 414 . . . . . 6 ((𝑅 ∈ (SubRingβ€˜β„‚fld) ∧ ℝ βŠ† 𝑅) β†’ (π‘₯ ∈ (𝑅 βˆ– ℝ) β†’ (𝑅 = ℝ ∨ 𝑅 = β„‚)))
7372exlimdv 1937 . . . . 5 ((𝑅 ∈ (SubRingβ€˜β„‚fld) ∧ ℝ βŠ† 𝑅) β†’ (βˆƒπ‘₯ π‘₯ ∈ (𝑅 βˆ– ℝ) β†’ (𝑅 = ℝ ∨ 𝑅 = β„‚)))
7473imp 408 . . . 4 (((𝑅 ∈ (SubRingβ€˜β„‚fld) ∧ ℝ βŠ† 𝑅) ∧ βˆƒπ‘₯ π‘₯ ∈ (𝑅 βˆ– ℝ)) β†’ (𝑅 = ℝ ∨ 𝑅 = β„‚))
757, 74sylan2b 595 . . 3 (((𝑅 ∈ (SubRingβ€˜β„‚fld) ∧ ℝ βŠ† 𝑅) ∧ (𝑅 βˆ– ℝ) β‰  βˆ…) β†’ (𝑅 = ℝ ∨ 𝑅 = β„‚))
766, 75pm2.61dane 3033 . 2 ((𝑅 ∈ (SubRingβ€˜β„‚fld) ∧ ℝ βŠ† 𝑅) β†’ (𝑅 = ℝ ∨ 𝑅 = β„‚))
77 elprg 4612 . . 3 (𝑅 ∈ (SubRingβ€˜β„‚fld) β†’ (𝑅 ∈ {ℝ, β„‚} ↔ (𝑅 = ℝ ∨ 𝑅 = β„‚)))
7877adantr 482 . 2 ((𝑅 ∈ (SubRingβ€˜β„‚fld) ∧ ℝ βŠ† 𝑅) β†’ (𝑅 ∈ {ℝ, β„‚} ↔ (𝑅 = ℝ ∨ 𝑅 = β„‚)))
7976, 78mpbird 257 1 ((𝑅 ∈ (SubRingβ€˜β„‚fld) ∧ ℝ βŠ† 𝑅) β†’ 𝑅 ∈ {ℝ, β„‚})
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ↔ wb 205   ∧ wa 397   ∨ wo 846   = wceq 1542  βˆƒwex 1782   ∈ wcel 2107   β‰  wne 2944   βˆ– cdif 3912   βŠ† wss 3915  βˆ…c0 4287  {cpr 4593  β€˜cfv 6501  (class class class)co 7362  β„‚cc 11056  β„cr 11057  0cc0 11058  1c1 11059  ici 11060   + caddc 11061   Β· cmul 11063   βˆ’ cmin 11392  -cneg 11393   / cdiv 11819  β„œcre 14989  β„‘cim 14990  SubRingcsubrg 20234  β„‚fldccnfld 20812
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 2708  ax-sep 5261  ax-nul 5268  ax-pow 5325  ax-pr 5389  ax-un 7677  ax-cnex 11114  ax-resscn 11115  ax-1cn 11116  ax-icn 11117  ax-addcl 11118  ax-addrcl 11119  ax-mulcl 11120  ax-mulrcl 11121  ax-mulcom 11122  ax-addass 11123  ax-mulass 11124  ax-distr 11125  ax-i2m1 11126  ax-1ne0 11127  ax-1rid 11128  ax-rnegex 11129  ax-rrecex 11130  ax-cnre 11131  ax-pre-lttri 11132  ax-pre-lttrn 11133  ax-pre-ltadd 11134  ax-pre-mulgt0 11135  ax-addf 11137  ax-mulf 11138
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 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-nel 3051  df-ral 3066  df-rex 3075  df-rmo 3356  df-reu 3357  df-rab 3411  df-v 3450  df-sbc 3745  df-csb 3861  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-pss 3934  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-tp 4596  df-op 4598  df-uni 4871  df-iun 4961  df-br 5111  df-opab 5173  df-mpt 5194  df-tr 5228  df-id 5536  df-eprel 5542  df-po 5550  df-so 5551  df-fr 5593  df-we 5595  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6258  df-ord 6325  df-on 6326  df-lim 6327  df-suc 6328  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-f1 6506  df-fo 6507  df-f1o 6508  df-fv 6509  df-riota 7318  df-ov 7365  df-oprab 7366  df-mpo 7367  df-om 7808  df-1st 7926  df-2nd 7927  df-frecs 8217  df-wrecs 8248  df-recs 8322  df-rdg 8361  df-1o 8417  df-er 8655  df-en 8891  df-dom 8892  df-sdom 8893  df-fin 8894  df-pnf 11198  df-mnf 11199  df-xr 11200  df-ltxr 11201  df-le 11202  df-sub 11394  df-neg 11395  df-div 11820  df-nn 12161  df-2 12223  df-3 12224  df-4 12225  df-5 12226  df-6 12227  df-7 12228  df-8 12229  df-9 12230  df-n0 12421  df-z 12507  df-dec 12626  df-uz 12771  df-fz 13432  df-cj 14991  df-re 14992  df-im 14993  df-struct 17026  df-sets 17043  df-slot 17061  df-ndx 17073  df-base 17091  df-ress 17120  df-plusg 17153  df-mulr 17154  df-starv 17155  df-tset 17159  df-ple 17160  df-ds 17162  df-unif 17163  df-mgm 18504  df-sgrp 18553  df-mnd 18564  df-grp 18758  df-subg 18932  df-mgp 19904  df-ring 19973  df-subrg 20236  df-cnfld 20813
This theorem is referenced by:  cncdrg  24739
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