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Theorem cnsubrg 21367
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 4367 . . . 4 (𝑅 βŠ† ℝ ↔ (𝑅 βˆ– ℝ) = βˆ…)
2 simpr 483 . . . . . 6 (((𝑅 ∈ (SubRingβ€˜β„‚fld) ∧ ℝ βŠ† 𝑅) ∧ 𝑅 βŠ† ℝ) β†’ 𝑅 βŠ† ℝ)
3 simplr 767 . . . . . 6 (((𝑅 ∈ (SubRingβ€˜β„‚fld) ∧ ℝ βŠ† 𝑅) ∧ 𝑅 βŠ† ℝ) β†’ ℝ βŠ† 𝑅)
42, 3eqssd 3999 . . . . 5 (((𝑅 ∈ (SubRingβ€˜β„‚fld) ∧ ℝ βŠ† 𝑅) ∧ 𝑅 βŠ† ℝ) β†’ 𝑅 = ℝ)
54orcd 871 . . . 4 (((𝑅 ∈ (SubRingβ€˜β„‚fld) ∧ ℝ βŠ† 𝑅) ∧ 𝑅 βŠ† ℝ) β†’ (𝑅 = ℝ ∨ 𝑅 = β„‚))
61, 5sylan2br 593 . . 3 (((𝑅 ∈ (SubRingβ€˜β„‚fld) ∧ ℝ βŠ† 𝑅) ∧ (𝑅 βˆ– ℝ) = βˆ…) β†’ (𝑅 = ℝ ∨ 𝑅 = β„‚))
7 n0 4350 . . . 4 ((𝑅 βˆ– ℝ) β‰  βˆ… ↔ βˆƒπ‘₯ π‘₯ ∈ (𝑅 βˆ– ℝ))
8 simpll 765 . . . . . . . . . 10 (((𝑅 ∈ (SubRingβ€˜β„‚fld) ∧ ℝ βŠ† 𝑅) ∧ π‘₯ ∈ (𝑅 βˆ– ℝ)) β†’ 𝑅 ∈ (SubRingβ€˜β„‚fld))
9 cnfldbas 21290 . . . . . . . . . . 11 β„‚ = (Baseβ€˜β„‚fld)
109subrgss 20518 . . . . . . . . . 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 15103 . . . . . . . . . . . . 13 (𝑦 ∈ β„‚ β†’ 𝑦 = ((β„œβ€˜π‘¦) + (i Β· (β„‘β€˜π‘¦))))
1312ad2antll 727 . . . . . . . . . . . 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 765 . . . . . . . . . . . . 13 (((𝑅 ∈ (SubRingβ€˜β„‚fld) ∧ ℝ βŠ† 𝑅) ∧ (π‘₯ ∈ (𝑅 βˆ– ℝ) ∧ 𝑦 ∈ β„‚)) β†’ 𝑅 ∈ (SubRingβ€˜β„‚fld))
15 simplr 767 . . . . . . . . . . . . . 14 (((𝑅 ∈ (SubRingβ€˜β„‚fld) ∧ ℝ βŠ† 𝑅) ∧ (π‘₯ ∈ (𝑅 βˆ– ℝ) ∧ 𝑦 ∈ β„‚)) β†’ ℝ βŠ† 𝑅)
16 recl 15097 . . . . . . . . . . . . . . 15 (𝑦 ∈ β„‚ β†’ (β„œβ€˜π‘¦) ∈ ℝ)
1716ad2antll 727 . . . . . . . . . . . . . 14 (((𝑅 ∈ (SubRingβ€˜β„‚fld) ∧ ℝ βŠ† 𝑅) ∧ (π‘₯ ∈ (𝑅 βˆ– ℝ) ∧ 𝑦 ∈ β„‚)) β†’ (β„œβ€˜π‘¦) ∈ ℝ)
1815, 17sseldd 3983 . . . . . . . . . . . . 13 (((𝑅 ∈ (SubRingβ€˜β„‚fld) ∧ ℝ βŠ† 𝑅) ∧ (π‘₯ ∈ (𝑅 βˆ– ℝ) ∧ 𝑦 ∈ β„‚)) β†’ (β„œβ€˜π‘¦) ∈ 𝑅)
19 ax-icn 11205 . . . . . . . . . . . . . . . . . . 19 i ∈ β„‚
2019a1i 11 . . . . . . . . . . . . . . . . . 18 (((𝑅 ∈ (SubRingβ€˜β„‚fld) ∧ ℝ βŠ† 𝑅) ∧ π‘₯ ∈ (𝑅 βˆ– ℝ)) β†’ i ∈ β„‚)
21 eldifi 4127 . . . . . . . . . . . . . . . . . . . . . 22 (π‘₯ ∈ (𝑅 βˆ– ℝ) β†’ π‘₯ ∈ 𝑅)
2221adantl 480 . . . . . . . . . . . . . . . . . . . . 21 (((𝑅 ∈ (SubRingβ€˜β„‚fld) ∧ ℝ βŠ† 𝑅) ∧ π‘₯ ∈ (𝑅 βˆ– ℝ)) β†’ π‘₯ ∈ 𝑅)
2311, 22sseldd 3983 . . . . . . . . . . . . . . . . . . . 20 (((𝑅 ∈ (SubRingβ€˜β„‚fld) ∧ ℝ βŠ† 𝑅) ∧ π‘₯ ∈ (𝑅 βˆ– ℝ)) β†’ π‘₯ ∈ β„‚)
24 imcl 15098 . . . . . . . . . . . . . . . . . . . 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 11280 . . . . . . . . . . . . . . . . . 18 (((𝑅 ∈ (SubRingβ€˜β„‚fld) ∧ ℝ βŠ† 𝑅) ∧ π‘₯ ∈ (𝑅 βˆ– ℝ)) β†’ (β„‘β€˜π‘₯) ∈ β„‚)
27 eldifn 4128 . . . . . . . . . . . . . . . . . . . 20 (π‘₯ ∈ (𝑅 βˆ– ℝ) β†’ Β¬ π‘₯ ∈ ℝ)
2827adantl 480 . . . . . . . . . . . . . . . . . . 19 (((𝑅 ∈ (SubRingβ€˜β„‚fld) ∧ ℝ βŠ† 𝑅) ∧ π‘₯ ∈ (𝑅 βˆ– ℝ)) β†’ Β¬ π‘₯ ∈ ℝ)
29 reim0b 15106 . . . . . . . . . . . . . . . . . . . . 21 (π‘₯ ∈ β„‚ β†’ (π‘₯ ∈ ℝ ↔ (β„‘β€˜π‘₯) = 0))
3029necon3bbid 2975 . . . . . . . . . . . . . . . . . . . 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 12034 . . . . . . . . . . . . . . . . 17 (((𝑅 ∈ (SubRingβ€˜β„‚fld) ∧ ℝ βŠ† 𝑅) ∧ π‘₯ ∈ (𝑅 βˆ– ℝ)) β†’ ((i Β· (β„‘β€˜π‘₯)) / (β„‘β€˜π‘₯)) = i)
34 mulcl 11230 . . . . . . . . . . . . . . . . . . 19 ((i ∈ β„‚ ∧ (β„‘β€˜π‘₯) ∈ β„‚) β†’ (i Β· (β„‘β€˜π‘₯)) ∈ β„‚)
3519, 26, 34sylancr 585 . . . . . . . . . . . . . . . . . 18 (((𝑅 ∈ (SubRingβ€˜β„‚fld) ∧ ℝ βŠ† 𝑅) ∧ π‘₯ ∈ (𝑅 βˆ– ℝ)) β†’ (i Β· (β„‘β€˜π‘₯)) ∈ β„‚)
3635, 26, 32divrecd 12031 . . . . . . . . . . . . . . . . 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 2770 . . . . . . . . . . . . . . . 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 15181 . . . . . . . . . . . . . . . . . . . . 21 (((𝑅 ∈ (SubRingβ€˜β„‚fld) ∧ ℝ βŠ† 𝑅) ∧ π‘₯ ∈ (𝑅 βˆ– ℝ)) β†’ (β„œβ€˜π‘₯) ∈ ℝ)
3938recnd 11280 . . . . . . . . . . . . . . . . . . . 20 (((𝑅 ∈ (SubRingβ€˜β„‚fld) ∧ ℝ βŠ† 𝑅) ∧ π‘₯ ∈ (𝑅 βˆ– ℝ)) β†’ (β„œβ€˜π‘₯) ∈ β„‚)
4023, 39negsubd 11615 . . . . . . . . . . . . . . . . . . 19 (((𝑅 ∈ (SubRingβ€˜β„‚fld) ∧ ℝ βŠ† 𝑅) ∧ π‘₯ ∈ (𝑅 βˆ– ℝ)) β†’ (π‘₯ + -(β„œβ€˜π‘₯)) = (π‘₯ βˆ’ (β„œβ€˜π‘₯)))
41 replim 15103 . . . . . . . . . . . . . . . . . . . . 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 7441 . . . . . . . . . . . . . . . . . . 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 11611 . . . . . . . . . . . . . . . . . . 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 2772 . . . . . . . . . . . . . . . . . 18 (((𝑅 ∈ (SubRingβ€˜β„‚fld) ∧ ℝ βŠ† 𝑅) ∧ π‘₯ ∈ (𝑅 βˆ– ℝ)) β†’ (π‘₯ + -(β„œβ€˜π‘₯)) = (i Β· (β„‘β€˜π‘₯)))
46 simplr 767 . . . . . . . . . . . . . . . . . . . 20 (((𝑅 ∈ (SubRingβ€˜β„‚fld) ∧ ℝ βŠ† 𝑅) ∧ π‘₯ ∈ (𝑅 βˆ– ℝ)) β†’ ℝ βŠ† 𝑅)
4738renegcld 11679 . . . . . . . . . . . . . . . . . . . 20 (((𝑅 ∈ (SubRingβ€˜β„‚fld) ∧ ℝ βŠ† 𝑅) ∧ π‘₯ ∈ (𝑅 βˆ– ℝ)) β†’ -(β„œβ€˜π‘₯) ∈ ℝ)
4846, 47sseldd 3983 . . . . . . . . . . . . . . . . . . 19 (((𝑅 ∈ (SubRingβ€˜β„‚fld) ∧ ℝ βŠ† 𝑅) ∧ π‘₯ ∈ (𝑅 βˆ– ℝ)) β†’ -(β„œβ€˜π‘₯) ∈ 𝑅)
49 cnfldadd 21292 . . . . . . . . . . . . . . . . . . . 20 + = (+gβ€˜β„‚fld)
5049subrgacl 20529 . . . . . . . . . . . . . . . . . . 19 ((𝑅 ∈ (SubRingβ€˜β„‚fld) ∧ π‘₯ ∈ 𝑅 ∧ -(β„œβ€˜π‘₯) ∈ 𝑅) β†’ (π‘₯ + -(β„œβ€˜π‘₯)) ∈ 𝑅)
518, 22, 48, 50syl3anc 1368 . . . . . . . . . . . . . . . . . 18 (((𝑅 ∈ (SubRingβ€˜β„‚fld) ∧ ℝ βŠ† 𝑅) ∧ π‘₯ ∈ (𝑅 βˆ– ℝ)) β†’ (π‘₯ + -(β„œβ€˜π‘₯)) ∈ 𝑅)
5245, 51eqeltrrd 2830 . . . . . . . . . . . . . . . . 17 (((𝑅 ∈ (SubRingβ€˜β„‚fld) ∧ ℝ βŠ† 𝑅) ∧ π‘₯ ∈ (𝑅 βˆ– ℝ)) β†’ (i Β· (β„‘β€˜π‘₯)) ∈ 𝑅)
5325, 32rereccld 12079 . . . . . . . . . . . . . . . . . 18 (((𝑅 ∈ (SubRingβ€˜β„‚fld) ∧ ℝ βŠ† 𝑅) ∧ π‘₯ ∈ (𝑅 βˆ– ℝ)) β†’ (1 / (β„‘β€˜π‘₯)) ∈ ℝ)
5446, 53sseldd 3983 . . . . . . . . . . . . . . . . 17 (((𝑅 ∈ (SubRingβ€˜β„‚fld) ∧ ℝ βŠ† 𝑅) ∧ π‘₯ ∈ (𝑅 βˆ– ℝ)) β†’ (1 / (β„‘β€˜π‘₯)) ∈ 𝑅)
55 cnfldmul 21294 . . . . . . . . . . . . . . . . . 18 Β· = (.rβ€˜β„‚fld)
5655subrgmcl 20530 . . . . . . . . . . . . . . . . 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 1368 . . . . . . . . . . . . . . . 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 2829 . . . . . . . . . . . . . . 15 (((𝑅 ∈ (SubRingβ€˜β„‚fld) ∧ ℝ βŠ† 𝑅) ∧ π‘₯ ∈ (𝑅 βˆ– ℝ)) β†’ i ∈ 𝑅)
5958adantrr 715 . . . . . . . . . . . . . 14 (((𝑅 ∈ (SubRingβ€˜β„‚fld) ∧ ℝ βŠ† 𝑅) ∧ (π‘₯ ∈ (𝑅 βˆ– ℝ) ∧ 𝑦 ∈ β„‚)) β†’ i ∈ 𝑅)
60 imcl 15098 . . . . . . . . . . . . . . . 16 (𝑦 ∈ β„‚ β†’ (β„‘β€˜π‘¦) ∈ ℝ)
6160ad2antll 727 . . . . . . . . . . . . . . 15 (((𝑅 ∈ (SubRingβ€˜β„‚fld) ∧ ℝ βŠ† 𝑅) ∧ (π‘₯ ∈ (𝑅 βˆ– ℝ) ∧ 𝑦 ∈ β„‚)) β†’ (β„‘β€˜π‘¦) ∈ ℝ)
6215, 61sseldd 3983 . . . . . . . . . . . . . 14 (((𝑅 ∈ (SubRingβ€˜β„‚fld) ∧ ℝ βŠ† 𝑅) ∧ (π‘₯ ∈ (𝑅 βˆ– ℝ) ∧ 𝑦 ∈ β„‚)) β†’ (β„‘β€˜π‘¦) ∈ 𝑅)
6355subrgmcl 20530 . . . . . . . . . . . . . 14 ((𝑅 ∈ (SubRingβ€˜β„‚fld) ∧ i ∈ 𝑅 ∧ (β„‘β€˜π‘¦) ∈ 𝑅) β†’ (i Β· (β„‘β€˜π‘¦)) ∈ 𝑅)
6414, 59, 62, 63syl3anc 1368 . . . . . . . . . . . . 13 (((𝑅 ∈ (SubRingβ€˜β„‚fld) ∧ ℝ βŠ† 𝑅) ∧ (π‘₯ ∈ (𝑅 βˆ– ℝ) ∧ 𝑦 ∈ β„‚)) β†’ (i Β· (β„‘β€˜π‘¦)) ∈ 𝑅)
6549subrgacl 20529 . . . . . . . . . . . . 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 1368 . . . . . . . . . . . 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 2829 . . . . . . . . . . 11 (((𝑅 ∈ (SubRingβ€˜β„‚fld) ∧ ℝ βŠ† 𝑅) ∧ (π‘₯ ∈ (𝑅 βˆ– ℝ) ∧ 𝑦 ∈ β„‚)) β†’ 𝑦 ∈ 𝑅)
6867expr 455 . . . . . . . . . 10 (((𝑅 ∈ (SubRingβ€˜β„‚fld) ∧ ℝ βŠ† 𝑅) ∧ π‘₯ ∈ (𝑅 βˆ– ℝ)) β†’ (𝑦 ∈ β„‚ β†’ 𝑦 ∈ 𝑅))
6968ssrdv 3988 . . . . . . . . 9 (((𝑅 ∈ (SubRingβ€˜β„‚fld) ∧ ℝ βŠ† 𝑅) ∧ π‘₯ ∈ (𝑅 βˆ– ℝ)) β†’ β„‚ βŠ† 𝑅)
7011, 69eqssd 3999 . . . . . . . 8 (((𝑅 ∈ (SubRingβ€˜β„‚fld) ∧ ℝ βŠ† 𝑅) ∧ π‘₯ ∈ (𝑅 βˆ– ℝ)) β†’ 𝑅 = β„‚)
7170olcd 872 . . . . . . 7 (((𝑅 ∈ (SubRingβ€˜β„‚fld) ∧ ℝ βŠ† 𝑅) ∧ π‘₯ ∈ (𝑅 βˆ– ℝ)) β†’ (𝑅 = ℝ ∨ 𝑅 = β„‚))
7271ex 411 . . . . . 6 ((𝑅 ∈ (SubRingβ€˜β„‚fld) ∧ ℝ βŠ† 𝑅) β†’ (π‘₯ ∈ (𝑅 βˆ– ℝ) β†’ (𝑅 = ℝ ∨ 𝑅 = β„‚)))
7372exlimdv 1928 . . . . 5 ((𝑅 ∈ (SubRingβ€˜β„‚fld) ∧ ℝ βŠ† 𝑅) β†’ (βˆƒπ‘₯ π‘₯ ∈ (𝑅 βˆ– ℝ) β†’ (𝑅 = ℝ ∨ 𝑅 = β„‚)))
7473imp 405 . . . 4 (((𝑅 ∈ (SubRingβ€˜β„‚fld) ∧ ℝ βŠ† 𝑅) ∧ βˆƒπ‘₯ π‘₯ ∈ (𝑅 βˆ– ℝ)) β†’ (𝑅 = ℝ ∨ 𝑅 = β„‚))
757, 74sylan2b 592 . . 3 (((𝑅 ∈ (SubRingβ€˜β„‚fld) ∧ ℝ βŠ† 𝑅) ∧ (𝑅 βˆ– ℝ) β‰  βˆ…) β†’ (𝑅 = ℝ ∨ 𝑅 = β„‚))
766, 75pm2.61dane 3026 . 2 ((𝑅 ∈ (SubRingβ€˜β„‚fld) ∧ ℝ βŠ† 𝑅) β†’ (𝑅 = ℝ ∨ 𝑅 = β„‚))
77 elprg 4654 . . 3 (𝑅 ∈ (SubRingβ€˜β„‚fld) β†’ (𝑅 ∈ {ℝ, β„‚} ↔ (𝑅 = ℝ ∨ 𝑅 = β„‚)))
7877adantr 479 . 2 ((𝑅 ∈ (SubRingβ€˜β„‚fld) ∧ ℝ βŠ† 𝑅) β†’ (𝑅 ∈ {ℝ, β„‚} ↔ (𝑅 = ℝ ∨ 𝑅 = β„‚)))
7976, 78mpbird 256 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 394   ∨ wo 845   = wceq 1533  βˆƒwex 1773   ∈ wcel 2098   β‰  wne 2937   βˆ– cdif 3946   βŠ† wss 3949  βˆ…c0 4326  {cpr 4634  β€˜cfv 6553  (class class class)co 7426  β„‚cc 11144  β„cr 11145  0cc0 11146  1c1 11147  ici 11148   + caddc 11149   Β· cmul 11151   βˆ’ cmin 11482  -cneg 11483   / cdiv 11909  β„œcre 15084  β„‘cim 15085  SubRingcsubrg 20513  β„‚fldccnfld 21286
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pow 5369  ax-pr 5433  ax-un 7746  ax-cnex 11202  ax-resscn 11203  ax-1cn 11204  ax-icn 11205  ax-addcl 11206  ax-addrcl 11207  ax-mulcl 11208  ax-mulrcl 11209  ax-mulcom 11210  ax-addass 11211  ax-mulass 11212  ax-distr 11213  ax-i2m1 11214  ax-1ne0 11215  ax-1rid 11216  ax-rnegex 11217  ax-rrecex 11218  ax-cnre 11219  ax-pre-lttri 11220  ax-pre-lttrn 11221  ax-pre-ltadd 11222  ax-pre-mulgt0 11223  ax-addf 11225  ax-mulf 11226
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-rmo 3374  df-reu 3375  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-tp 4637  df-op 4639  df-uni 4913  df-iun 5002  df-br 5153  df-opab 5215  df-mpt 5236  df-tr 5270  df-id 5580  df-eprel 5586  df-po 5594  df-so 5595  df-fr 5637  df-we 5639  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-pred 6310  df-ord 6377  df-on 6378  df-lim 6379  df-suc 6380  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-riota 7382  df-ov 7429  df-oprab 7430  df-mpo 7431  df-om 7877  df-1st 7999  df-2nd 8000  df-frecs 8293  df-wrecs 8324  df-recs 8398  df-rdg 8437  df-1o 8493  df-er 8731  df-en 8971  df-dom 8972  df-sdom 8973  df-fin 8974  df-pnf 11288  df-mnf 11289  df-xr 11290  df-ltxr 11291  df-le 11292  df-sub 11484  df-neg 11485  df-div 11910  df-nn 12251  df-2 12313  df-3 12314  df-4 12315  df-5 12316  df-6 12317  df-7 12318  df-8 12319  df-9 12320  df-n0 12511  df-z 12597  df-dec 12716  df-uz 12861  df-fz 13525  df-cj 15086  df-re 15087  df-im 15088  df-struct 17123  df-sets 17140  df-slot 17158  df-ndx 17170  df-base 17188  df-ress 17217  df-plusg 17253  df-mulr 17254  df-starv 17255  df-tset 17259  df-ple 17260  df-ds 17262  df-unif 17263  df-0g 17430  df-mgm 18607  df-sgrp 18686  df-mnd 18702  df-grp 18900  df-minusg 18901  df-subg 19085  df-cmn 19744  df-abl 19745  df-mgp 20082  df-rng 20100  df-ur 20129  df-ring 20182  df-subrng 20490  df-subrg 20515  df-cnfld 21287
This theorem is referenced by:  cncdrg  25307
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