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Theorem cnsubrg 21321
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 4358 . . . 4 (𝑅 βŠ† ℝ ↔ (𝑅 βˆ– ℝ) = βˆ…)
2 simpr 484 . . . . . 6 (((𝑅 ∈ (SubRingβ€˜β„‚fld) ∧ ℝ βŠ† 𝑅) ∧ 𝑅 βŠ† ℝ) β†’ 𝑅 βŠ† ℝ)
3 simplr 766 . . . . . 6 (((𝑅 ∈ (SubRingβ€˜β„‚fld) ∧ ℝ βŠ† 𝑅) ∧ 𝑅 βŠ† ℝ) β†’ ℝ βŠ† 𝑅)
42, 3eqssd 3994 . . . . 5 (((𝑅 ∈ (SubRingβ€˜β„‚fld) ∧ ℝ βŠ† 𝑅) ∧ 𝑅 βŠ† ℝ) β†’ 𝑅 = ℝ)
54orcd 870 . . . 4 (((𝑅 ∈ (SubRingβ€˜β„‚fld) ∧ ℝ βŠ† 𝑅) ∧ 𝑅 βŠ† ℝ) β†’ (𝑅 = ℝ ∨ 𝑅 = β„‚))
61, 5sylan2br 594 . . 3 (((𝑅 ∈ (SubRingβ€˜β„‚fld) ∧ ℝ βŠ† 𝑅) ∧ (𝑅 βˆ– ℝ) = βˆ…) β†’ (𝑅 = ℝ ∨ 𝑅 = β„‚))
7 n0 4341 . . . 4 ((𝑅 βˆ– ℝ) β‰  βˆ… ↔ βˆƒπ‘₯ π‘₯ ∈ (𝑅 βˆ– ℝ))
8 simpll 764 . . . . . . . . . 10 (((𝑅 ∈ (SubRingβ€˜β„‚fld) ∧ ℝ βŠ† 𝑅) ∧ π‘₯ ∈ (𝑅 βˆ– ℝ)) β†’ 𝑅 ∈ (SubRingβ€˜β„‚fld))
9 cnfldbas 21244 . . . . . . . . . . 11 β„‚ = (Baseβ€˜β„‚fld)
109subrgss 20474 . . . . . . . . . 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 15069 . . . . . . . . . . . . 13 (𝑦 ∈ β„‚ β†’ 𝑦 = ((β„œβ€˜π‘¦) + (i Β· (β„‘β€˜π‘¦))))
1312ad2antll 726 . . . . . . . . . . . 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 764 . . . . . . . . . . . . 13 (((𝑅 ∈ (SubRingβ€˜β„‚fld) ∧ ℝ βŠ† 𝑅) ∧ (π‘₯ ∈ (𝑅 βˆ– ℝ) ∧ 𝑦 ∈ β„‚)) β†’ 𝑅 ∈ (SubRingβ€˜β„‚fld))
15 simplr 766 . . . . . . . . . . . . . 14 (((𝑅 ∈ (SubRingβ€˜β„‚fld) ∧ ℝ βŠ† 𝑅) ∧ (π‘₯ ∈ (𝑅 βˆ– ℝ) ∧ 𝑦 ∈ β„‚)) β†’ ℝ βŠ† 𝑅)
16 recl 15063 . . . . . . . . . . . . . . 15 (𝑦 ∈ β„‚ β†’ (β„œβ€˜π‘¦) ∈ ℝ)
1716ad2antll 726 . . . . . . . . . . . . . 14 (((𝑅 ∈ (SubRingβ€˜β„‚fld) ∧ ℝ βŠ† 𝑅) ∧ (π‘₯ ∈ (𝑅 βˆ– ℝ) ∧ 𝑦 ∈ β„‚)) β†’ (β„œβ€˜π‘¦) ∈ ℝ)
1815, 17sseldd 3978 . . . . . . . . . . . . 13 (((𝑅 ∈ (SubRingβ€˜β„‚fld) ∧ ℝ βŠ† 𝑅) ∧ (π‘₯ ∈ (𝑅 βˆ– ℝ) ∧ 𝑦 ∈ β„‚)) β†’ (β„œβ€˜π‘¦) ∈ 𝑅)
19 ax-icn 11171 . . . . . . . . . . . . . . . . . . 19 i ∈ β„‚
2019a1i 11 . . . . . . . . . . . . . . . . . 18 (((𝑅 ∈ (SubRingβ€˜β„‚fld) ∧ ℝ βŠ† 𝑅) ∧ π‘₯ ∈ (𝑅 βˆ– ℝ)) β†’ i ∈ β„‚)
21 eldifi 4121 . . . . . . . . . . . . . . . . . . . . . 22 (π‘₯ ∈ (𝑅 βˆ– ℝ) β†’ π‘₯ ∈ 𝑅)
2221adantl 481 . . . . . . . . . . . . . . . . . . . . 21 (((𝑅 ∈ (SubRingβ€˜β„‚fld) ∧ ℝ βŠ† 𝑅) ∧ π‘₯ ∈ (𝑅 βˆ– ℝ)) β†’ π‘₯ ∈ 𝑅)
2311, 22sseldd 3978 . . . . . . . . . . . . . . . . . . . 20 (((𝑅 ∈ (SubRingβ€˜β„‚fld) ∧ ℝ βŠ† 𝑅) ∧ π‘₯ ∈ (𝑅 βˆ– ℝ)) β†’ π‘₯ ∈ β„‚)
24 imcl 15064 . . . . . . . . . . . . . . . . . . . 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 11246 . . . . . . . . . . . . . . . . . 18 (((𝑅 ∈ (SubRingβ€˜β„‚fld) ∧ ℝ βŠ† 𝑅) ∧ π‘₯ ∈ (𝑅 βˆ– ℝ)) β†’ (β„‘β€˜π‘₯) ∈ β„‚)
27 eldifn 4122 . . . . . . . . . . . . . . . . . . . 20 (π‘₯ ∈ (𝑅 βˆ– ℝ) β†’ Β¬ π‘₯ ∈ ℝ)
2827adantl 481 . . . . . . . . . . . . . . . . . . 19 (((𝑅 ∈ (SubRingβ€˜β„‚fld) ∧ ℝ βŠ† 𝑅) ∧ π‘₯ ∈ (𝑅 βˆ– ℝ)) β†’ Β¬ π‘₯ ∈ ℝ)
29 reim0b 15072 . . . . . . . . . . . . . . . . . . . . 21 (π‘₯ ∈ β„‚ β†’ (π‘₯ ∈ ℝ ↔ (β„‘β€˜π‘₯) = 0))
3029necon3bbid 2972 . . . . . . . . . . . . . . . . . . . 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 12000 . . . . . . . . . . . . . . . . 17 (((𝑅 ∈ (SubRingβ€˜β„‚fld) ∧ ℝ βŠ† 𝑅) ∧ π‘₯ ∈ (𝑅 βˆ– ℝ)) β†’ ((i Β· (β„‘β€˜π‘₯)) / (β„‘β€˜π‘₯)) = i)
34 mulcl 11196 . . . . . . . . . . . . . . . . . . 19 ((i ∈ β„‚ ∧ (β„‘β€˜π‘₯) ∈ β„‚) β†’ (i Β· (β„‘β€˜π‘₯)) ∈ β„‚)
3519, 26, 34sylancr 586 . . . . . . . . . . . . . . . . . 18 (((𝑅 ∈ (SubRingβ€˜β„‚fld) ∧ ℝ βŠ† 𝑅) ∧ π‘₯ ∈ (𝑅 βˆ– ℝ)) β†’ (i Β· (β„‘β€˜π‘₯)) ∈ β„‚)
3635, 26, 32divrecd 11997 . . . . . . . . . . . . . . . . 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 2768 . . . . . . . . . . . . . . . 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 15147 . . . . . . . . . . . . . . . . . . . . 21 (((𝑅 ∈ (SubRingβ€˜β„‚fld) ∧ ℝ βŠ† 𝑅) ∧ π‘₯ ∈ (𝑅 βˆ– ℝ)) β†’ (β„œβ€˜π‘₯) ∈ ℝ)
3938recnd 11246 . . . . . . . . . . . . . . . . . . . 20 (((𝑅 ∈ (SubRingβ€˜β„‚fld) ∧ ℝ βŠ† 𝑅) ∧ π‘₯ ∈ (𝑅 βˆ– ℝ)) β†’ (β„œβ€˜π‘₯) ∈ β„‚)
4023, 39negsubd 11581 . . . . . . . . . . . . . . . . . . 19 (((𝑅 ∈ (SubRingβ€˜β„‚fld) ∧ ℝ βŠ† 𝑅) ∧ π‘₯ ∈ (𝑅 βˆ– ℝ)) β†’ (π‘₯ + -(β„œβ€˜π‘₯)) = (π‘₯ βˆ’ (β„œβ€˜π‘₯)))
41 replim 15069 . . . . . . . . . . . . . . . . . . . . 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 7420 . . . . . . . . . . . . . . . . . . 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 11577 . . . . . . . . . . . . . . . . . . 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 2770 . . . . . . . . . . . . . . . . . 18 (((𝑅 ∈ (SubRingβ€˜β„‚fld) ∧ ℝ βŠ† 𝑅) ∧ π‘₯ ∈ (𝑅 βˆ– ℝ)) β†’ (π‘₯ + -(β„œβ€˜π‘₯)) = (i Β· (β„‘β€˜π‘₯)))
46 simplr 766 . . . . . . . . . . . . . . . . . . . 20 (((𝑅 ∈ (SubRingβ€˜β„‚fld) ∧ ℝ βŠ† 𝑅) ∧ π‘₯ ∈ (𝑅 βˆ– ℝ)) β†’ ℝ βŠ† 𝑅)
4738renegcld 11645 . . . . . . . . . . . . . . . . . . . 20 (((𝑅 ∈ (SubRingβ€˜β„‚fld) ∧ ℝ βŠ† 𝑅) ∧ π‘₯ ∈ (𝑅 βˆ– ℝ)) β†’ -(β„œβ€˜π‘₯) ∈ ℝ)
4846, 47sseldd 3978 . . . . . . . . . . . . . . . . . . 19 (((𝑅 ∈ (SubRingβ€˜β„‚fld) ∧ ℝ βŠ† 𝑅) ∧ π‘₯ ∈ (𝑅 βˆ– ℝ)) β†’ -(β„œβ€˜π‘₯) ∈ 𝑅)
49 cnfldadd 21246 . . . . . . . . . . . . . . . . . . . 20 + = (+gβ€˜β„‚fld)
5049subrgacl 20485 . . . . . . . . . . . . . . . . . . 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 2828 . . . . . . . . . . . . . . . . 17 (((𝑅 ∈ (SubRingβ€˜β„‚fld) ∧ ℝ βŠ† 𝑅) ∧ π‘₯ ∈ (𝑅 βˆ– ℝ)) β†’ (i Β· (β„‘β€˜π‘₯)) ∈ 𝑅)
5325, 32rereccld 12045 . . . . . . . . . . . . . . . . . 18 (((𝑅 ∈ (SubRingβ€˜β„‚fld) ∧ ℝ βŠ† 𝑅) ∧ π‘₯ ∈ (𝑅 βˆ– ℝ)) β†’ (1 / (β„‘β€˜π‘₯)) ∈ ℝ)
5446, 53sseldd 3978 . . . . . . . . . . . . . . . . 17 (((𝑅 ∈ (SubRingβ€˜β„‚fld) ∧ ℝ βŠ† 𝑅) ∧ π‘₯ ∈ (𝑅 βˆ– ℝ)) β†’ (1 / (β„‘β€˜π‘₯)) ∈ 𝑅)
55 cnfldmul 21248 . . . . . . . . . . . . . . . . . 18 Β· = (.rβ€˜β„‚fld)
5655subrgmcl 20486 . . . . . . . . . . . . . . . . 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 2827 . . . . . . . . . . . . . . 15 (((𝑅 ∈ (SubRingβ€˜β„‚fld) ∧ ℝ βŠ† 𝑅) ∧ π‘₯ ∈ (𝑅 βˆ– ℝ)) β†’ i ∈ 𝑅)
5958adantrr 714 . . . . . . . . . . . . . 14 (((𝑅 ∈ (SubRingβ€˜β„‚fld) ∧ ℝ βŠ† 𝑅) ∧ (π‘₯ ∈ (𝑅 βˆ– ℝ) ∧ 𝑦 ∈ β„‚)) β†’ i ∈ 𝑅)
60 imcl 15064 . . . . . . . . . . . . . . . 16 (𝑦 ∈ β„‚ β†’ (β„‘β€˜π‘¦) ∈ ℝ)
6160ad2antll 726 . . . . . . . . . . . . . . 15 (((𝑅 ∈ (SubRingβ€˜β„‚fld) ∧ ℝ βŠ† 𝑅) ∧ (π‘₯ ∈ (𝑅 βˆ– ℝ) ∧ 𝑦 ∈ β„‚)) β†’ (β„‘β€˜π‘¦) ∈ ℝ)
6215, 61sseldd 3978 . . . . . . . . . . . . . 14 (((𝑅 ∈ (SubRingβ€˜β„‚fld) ∧ ℝ βŠ† 𝑅) ∧ (π‘₯ ∈ (𝑅 βˆ– ℝ) ∧ 𝑦 ∈ β„‚)) β†’ (β„‘β€˜π‘¦) ∈ 𝑅)
6355subrgmcl 20486 . . . . . . . . . . . . . 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 20485 . . . . . . . . . . . . 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 2827 . . . . . . . . . . 11 (((𝑅 ∈ (SubRingβ€˜β„‚fld) ∧ ℝ βŠ† 𝑅) ∧ (π‘₯ ∈ (𝑅 βˆ– ℝ) ∧ 𝑦 ∈ β„‚)) β†’ 𝑦 ∈ 𝑅)
6867expr 456 . . . . . . . . . 10 (((𝑅 ∈ (SubRingβ€˜β„‚fld) ∧ ℝ βŠ† 𝑅) ∧ π‘₯ ∈ (𝑅 βˆ– ℝ)) β†’ (𝑦 ∈ β„‚ β†’ 𝑦 ∈ 𝑅))
6968ssrdv 3983 . . . . . . . . 9 (((𝑅 ∈ (SubRingβ€˜β„‚fld) ∧ ℝ βŠ† 𝑅) ∧ π‘₯ ∈ (𝑅 βˆ– ℝ)) β†’ β„‚ βŠ† 𝑅)
7011, 69eqssd 3994 . . . . . . . 8 (((𝑅 ∈ (SubRingβ€˜β„‚fld) ∧ ℝ βŠ† 𝑅) ∧ π‘₯ ∈ (𝑅 βˆ– ℝ)) β†’ 𝑅 = β„‚)
7170olcd 871 . . . . . . 7 (((𝑅 ∈ (SubRingβ€˜β„‚fld) ∧ ℝ βŠ† 𝑅) ∧ π‘₯ ∈ (𝑅 βˆ– ℝ)) β†’ (𝑅 = ℝ ∨ 𝑅 = β„‚))
7271ex 412 . . . . . 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 406 . . . 4 (((𝑅 ∈ (SubRingβ€˜β„‚fld) ∧ ℝ βŠ† 𝑅) ∧ βˆƒπ‘₯ π‘₯ ∈ (𝑅 βˆ– ℝ)) β†’ (𝑅 = ℝ ∨ 𝑅 = β„‚))
757, 74sylan2b 593 . . 3 (((𝑅 ∈ (SubRingβ€˜β„‚fld) ∧ ℝ βŠ† 𝑅) ∧ (𝑅 βˆ– ℝ) β‰  βˆ…) β†’ (𝑅 = ℝ ∨ 𝑅 = β„‚))
766, 75pm2.61dane 3023 . 2 ((𝑅 ∈ (SubRingβ€˜β„‚fld) ∧ ℝ βŠ† 𝑅) β†’ (𝑅 = ℝ ∨ 𝑅 = β„‚))
77 elprg 4644 . . 3 (𝑅 ∈ (SubRingβ€˜β„‚fld) β†’ (𝑅 ∈ {ℝ, β„‚} ↔ (𝑅 = ℝ ∨ 𝑅 = β„‚)))
7877adantr 480 . 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 395   ∨ wo 844   = wceq 1533  βˆƒwex 1773   ∈ wcel 2098   β‰  wne 2934   βˆ– cdif 3940   βŠ† wss 3943  βˆ…c0 4317  {cpr 4625  β€˜cfv 6537  (class class class)co 7405  β„‚cc 11110  β„cr 11111  0cc0 11112  1c1 11113  ici 11114   + caddc 11115   Β· cmul 11117   βˆ’ cmin 11448  -cneg 11449   / cdiv 11875  β„œcre 15050  β„‘cim 15051  SubRingcsubrg 20469  β„‚fldccnfld 21240
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 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7722  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  ax-addf 11191  ax-mulf 11192
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-nel 3041  df-ral 3056  df-rex 3065  df-rmo 3370  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-pss 3962  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-tp 4628  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-tr 5259  df-id 5567  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-we 5626  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-pred 6294  df-ord 6361  df-on 6362  df-lim 6363  df-suc 6364  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7361  df-ov 7408  df-oprab 7409  df-mpo 7410  df-om 7853  df-1st 7974  df-2nd 7975  df-frecs 8267  df-wrecs 8298  df-recs 8372  df-rdg 8411  df-1o 8467  df-er 8705  df-en 8942  df-dom 8943  df-sdom 8944  df-fin 8945  df-pnf 11254  df-mnf 11255  df-xr 11256  df-ltxr 11257  df-le 11258  df-sub 11450  df-neg 11451  df-div 11876  df-nn 12217  df-2 12279  df-3 12280  df-4 12281  df-5 12282  df-6 12283  df-7 12284  df-8 12285  df-9 12286  df-n0 12477  df-z 12563  df-dec 12682  df-uz 12827  df-fz 13491  df-cj 15052  df-re 15053  df-im 15054  df-struct 17089  df-sets 17106  df-slot 17124  df-ndx 17136  df-base 17154  df-ress 17183  df-plusg 17219  df-mulr 17220  df-starv 17221  df-tset 17225  df-ple 17226  df-ds 17228  df-unif 17229  df-0g 17396  df-mgm 18573  df-sgrp 18652  df-mnd 18668  df-grp 18866  df-minusg 18867  df-subg 19050  df-cmn 19702  df-abl 19703  df-mgp 20040  df-rng 20058  df-ur 20087  df-ring 20140  df-subrng 20446  df-subrg 20471  df-cnfld 21241
This theorem is referenced by:  cncdrg  25242
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