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Theorem efgh 23979
Description: The exponential function of a scaled complex number is a group homomorphism from the group of complex numbers under addition to the set of complex numbers under multiplication. (Contributed by Paul Chapman, 25-Apr-2008.) (Revised by Mario Carneiro, 11-May-2014.) (Revised by Thierry Arnoux, 26-Jan-2020.)
Hypothesis
Ref Expression
efgh.1 𝐹 = (𝑥𝑋 ↦ (exp‘(𝐴 · 𝑥)))
Assertion
Ref Expression
efgh (((𝐴 ∈ ℂ ∧ 𝑋 ∈ (SubGrp‘ℂfld)) ∧ 𝐵𝑋𝐶𝑋) → (𝐹‘(𝐵 + 𝐶)) = ((𝐹𝐵) · (𝐹𝐶)))
Distinct variable groups:   𝑥,𝐴   𝑥,𝑋
Allowed substitution hints:   𝐵(𝑥)   𝐶(𝑥)   𝐹(𝑥)

Proof of Theorem efgh
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 simp1l 1077 . . . . 5 (((𝐴 ∈ ℂ ∧ 𝑋 ∈ (SubGrp‘ℂfld)) ∧ 𝐵𝑋𝐶𝑋) → 𝐴 ∈ ℂ)
2 simp1r 1078 . . . . . . 7 (((𝐴 ∈ ℂ ∧ 𝑋 ∈ (SubGrp‘ℂfld)) ∧ 𝐵𝑋𝐶𝑋) → 𝑋 ∈ (SubGrp‘ℂfld))
3 cnfldbas 19473 . . . . . . . 8 ℂ = (Base‘ℂfld)
43subgss 17308 . . . . . . 7 (𝑋 ∈ (SubGrp‘ℂfld) → 𝑋 ⊆ ℂ)
52, 4syl 17 . . . . . 6 (((𝐴 ∈ ℂ ∧ 𝑋 ∈ (SubGrp‘ℂfld)) ∧ 𝐵𝑋𝐶𝑋) → 𝑋 ⊆ ℂ)
6 simp2 1054 . . . . . 6 (((𝐴 ∈ ℂ ∧ 𝑋 ∈ (SubGrp‘ℂfld)) ∧ 𝐵𝑋𝐶𝑋) → 𝐵𝑋)
75, 6sseldd 3473 . . . . 5 (((𝐴 ∈ ℂ ∧ 𝑋 ∈ (SubGrp‘ℂfld)) ∧ 𝐵𝑋𝐶𝑋) → 𝐵 ∈ ℂ)
8 simp3 1055 . . . . . 6 (((𝐴 ∈ ℂ ∧ 𝑋 ∈ (SubGrp‘ℂfld)) ∧ 𝐵𝑋𝐶𝑋) → 𝐶𝑋)
95, 8sseldd 3473 . . . . 5 (((𝐴 ∈ ℂ ∧ 𝑋 ∈ (SubGrp‘ℂfld)) ∧ 𝐵𝑋𝐶𝑋) → 𝐶 ∈ ℂ)
101, 7, 9adddid 9818 . . . 4 (((𝐴 ∈ ℂ ∧ 𝑋 ∈ (SubGrp‘ℂfld)) ∧ 𝐵𝑋𝐶𝑋) → (𝐴 · (𝐵 + 𝐶)) = ((𝐴 · 𝐵) + (𝐴 · 𝐶)))
1110fveq2d 5990 . . 3 (((𝐴 ∈ ℂ ∧ 𝑋 ∈ (SubGrp‘ℂfld)) ∧ 𝐵𝑋𝐶𝑋) → (exp‘(𝐴 · (𝐵 + 𝐶))) = (exp‘((𝐴 · 𝐵) + (𝐴 · 𝐶))))
121, 7mulcld 9814 . . . 4 (((𝐴 ∈ ℂ ∧ 𝑋 ∈ (SubGrp‘ℂfld)) ∧ 𝐵𝑋𝐶𝑋) → (𝐴 · 𝐵) ∈ ℂ)
131, 9mulcld 9814 . . . 4 (((𝐴 ∈ ℂ ∧ 𝑋 ∈ (SubGrp‘ℂfld)) ∧ 𝐵𝑋𝐶𝑋) → (𝐴 · 𝐶) ∈ ℂ)
14 efadd 14530 . . . 4 (((𝐴 · 𝐵) ∈ ℂ ∧ (𝐴 · 𝐶) ∈ ℂ) → (exp‘((𝐴 · 𝐵) + (𝐴 · 𝐶))) = ((exp‘(𝐴 · 𝐵)) · (exp‘(𝐴 · 𝐶))))
1512, 13, 14syl2anc 690 . . 3 (((𝐴 ∈ ℂ ∧ 𝑋 ∈ (SubGrp‘ℂfld)) ∧ 𝐵𝑋𝐶𝑋) → (exp‘((𝐴 · 𝐵) + (𝐴 · 𝐶))) = ((exp‘(𝐴 · 𝐵)) · (exp‘(𝐴 · 𝐶))))
1611, 15eqtrd 2548 . 2 (((𝐴 ∈ ℂ ∧ 𝑋 ∈ (SubGrp‘ℂfld)) ∧ 𝐵𝑋𝐶𝑋) → (exp‘(𝐴 · (𝐵 + 𝐶))) = ((exp‘(𝐴 · 𝐵)) · (exp‘(𝐴 · 𝐶))))
17 efgh.1 . . . . 5 𝐹 = (𝑥𝑋 ↦ (exp‘(𝐴 · 𝑥)))
18 oveq2 6433 . . . . . . 7 (𝑥 = 𝑦 → (𝐴 · 𝑥) = (𝐴 · 𝑦))
1918fveq2d 5990 . . . . . 6 (𝑥 = 𝑦 → (exp‘(𝐴 · 𝑥)) = (exp‘(𝐴 · 𝑦)))
2019cbvmptv 4576 . . . . 5 (𝑥𝑋 ↦ (exp‘(𝐴 · 𝑥))) = (𝑦𝑋 ↦ (exp‘(𝐴 · 𝑦)))
2117, 20eqtri 2536 . . . 4 𝐹 = (𝑦𝑋 ↦ (exp‘(𝐴 · 𝑦)))
2221a1i 11 . . 3 (((𝐴 ∈ ℂ ∧ 𝑋 ∈ (SubGrp‘ℂfld)) ∧ 𝐵𝑋𝐶𝑋) → 𝐹 = (𝑦𝑋 ↦ (exp‘(𝐴 · 𝑦))))
23 oveq2 6433 . . . . 5 (𝑦 = (𝐵 + 𝐶) → (𝐴 · 𝑦) = (𝐴 · (𝐵 + 𝐶)))
2423fveq2d 5990 . . . 4 (𝑦 = (𝐵 + 𝐶) → (exp‘(𝐴 · 𝑦)) = (exp‘(𝐴 · (𝐵 + 𝐶))))
2524adantl 480 . . 3 ((((𝐴 ∈ ℂ ∧ 𝑋 ∈ (SubGrp‘ℂfld)) ∧ 𝐵𝑋𝐶𝑋) ∧ 𝑦 = (𝐵 + 𝐶)) → (exp‘(𝐴 · 𝑦)) = (exp‘(𝐴 · (𝐵 + 𝐶))))
26 cnfldadd 19474 . . . . 5 + = (+g‘ℂfld)
2726subgcl 17317 . . . 4 ((𝑋 ∈ (SubGrp‘ℂfld) ∧ 𝐵𝑋𝐶𝑋) → (𝐵 + 𝐶) ∈ 𝑋)
28273adant1l 1309 . . 3 (((𝐴 ∈ ℂ ∧ 𝑋 ∈ (SubGrp‘ℂfld)) ∧ 𝐵𝑋𝐶𝑋) → (𝐵 + 𝐶) ∈ 𝑋)
29 fvex 5996 . . . 4 (exp‘(𝐴 · (𝐵 + 𝐶))) ∈ V
3029a1i 11 . . 3 (((𝐴 ∈ ℂ ∧ 𝑋 ∈ (SubGrp‘ℂfld)) ∧ 𝐵𝑋𝐶𝑋) → (exp‘(𝐴 · (𝐵 + 𝐶))) ∈ V)
3122, 25, 28, 30fvmptd 6080 . 2 (((𝐴 ∈ ℂ ∧ 𝑋 ∈ (SubGrp‘ℂfld)) ∧ 𝐵𝑋𝐶𝑋) → (𝐹‘(𝐵 + 𝐶)) = (exp‘(𝐴 · (𝐵 + 𝐶))))
32 oveq2 6433 . . . . . 6 (𝑦 = 𝐵 → (𝐴 · 𝑦) = (𝐴 · 𝐵))
3332fveq2d 5990 . . . . 5 (𝑦 = 𝐵 → (exp‘(𝐴 · 𝑦)) = (exp‘(𝐴 · 𝐵)))
3433adantl 480 . . . 4 ((((𝐴 ∈ ℂ ∧ 𝑋 ∈ (SubGrp‘ℂfld)) ∧ 𝐵𝑋𝐶𝑋) ∧ 𝑦 = 𝐵) → (exp‘(𝐴 · 𝑦)) = (exp‘(𝐴 · 𝐵)))
35 fvex 5996 . . . . 5 (exp‘(𝐴 · 𝐵)) ∈ V
3635a1i 11 . . . 4 (((𝐴 ∈ ℂ ∧ 𝑋 ∈ (SubGrp‘ℂfld)) ∧ 𝐵𝑋𝐶𝑋) → (exp‘(𝐴 · 𝐵)) ∈ V)
3722, 34, 6, 36fvmptd 6080 . . 3 (((𝐴 ∈ ℂ ∧ 𝑋 ∈ (SubGrp‘ℂfld)) ∧ 𝐵𝑋𝐶𝑋) → (𝐹𝐵) = (exp‘(𝐴 · 𝐵)))
38 oveq2 6433 . . . . . 6 (𝑦 = 𝐶 → (𝐴 · 𝑦) = (𝐴 · 𝐶))
3938fveq2d 5990 . . . . 5 (𝑦 = 𝐶 → (exp‘(𝐴 · 𝑦)) = (exp‘(𝐴 · 𝐶)))
4039adantl 480 . . . 4 ((((𝐴 ∈ ℂ ∧ 𝑋 ∈ (SubGrp‘ℂfld)) ∧ 𝐵𝑋𝐶𝑋) ∧ 𝑦 = 𝐶) → (exp‘(𝐴 · 𝑦)) = (exp‘(𝐴 · 𝐶)))
41 fvex 5996 . . . . 5 (exp‘(𝐴 · 𝐶)) ∈ V
4241a1i 11 . . . 4 (((𝐴 ∈ ℂ ∧ 𝑋 ∈ (SubGrp‘ℂfld)) ∧ 𝐵𝑋𝐶𝑋) → (exp‘(𝐴 · 𝐶)) ∈ V)
4322, 40, 8, 42fvmptd 6080 . . 3 (((𝐴 ∈ ℂ ∧ 𝑋 ∈ (SubGrp‘ℂfld)) ∧ 𝐵𝑋𝐶𝑋) → (𝐹𝐶) = (exp‘(𝐴 · 𝐶)))
4437, 43oveq12d 6443 . 2 (((𝐴 ∈ ℂ ∧ 𝑋 ∈ (SubGrp‘ℂfld)) ∧ 𝐵𝑋𝐶𝑋) → ((𝐹𝐵) · (𝐹𝐶)) = ((exp‘(𝐴 · 𝐵)) · (exp‘(𝐴 · 𝐶))))
4516, 31, 443eqtr4d 2558 1 (((𝐴 ∈ ℂ ∧ 𝑋 ∈ (SubGrp‘ℂfld)) ∧ 𝐵𝑋𝐶𝑋) → (𝐹‘(𝐵 + 𝐶)) = ((𝐹𝐵) · (𝐹𝐶)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382  w3a 1030   = wceq 1474  wcel 1938  Vcvv 3077  wss 3444  cmpt 4541  cfv 5689  (class class class)co 6425  cc 9688   + caddc 9693   · cmul 9695  expce 14498  SubGrpcsubg 17301  fldccnfld 19469
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1700  ax-4 1713  ax-5 1793  ax-6 1838  ax-7 1885  ax-8 1940  ax-9 1947  ax-10 1966  ax-11 1971  ax-12 1983  ax-13 2137  ax-ext 2494  ax-rep 4597  ax-sep 4607  ax-nul 4616  ax-pow 4668  ax-pr 4732  ax-un 6722  ax-inf2 8296  ax-cnex 9746  ax-resscn 9747  ax-1cn 9748  ax-icn 9749  ax-addcl 9750  ax-addrcl 9751  ax-mulcl 9752  ax-mulrcl 9753  ax-mulcom 9754  ax-addass 9755  ax-mulass 9756  ax-distr 9757  ax-i2m1 9758  ax-1ne0 9759  ax-1rid 9760  ax-rnegex 9761  ax-rrecex 9762  ax-cnre 9763  ax-pre-lttri 9764  ax-pre-lttrn 9765  ax-pre-ltadd 9766  ax-pre-mulgt0 9767  ax-pre-sup 9768  ax-addf 9769  ax-mulf 9770
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-fal 1480  df-ex 1695  df-nf 1699  df-sb 1831  df-eu 2366  df-mo 2367  df-clab 2501  df-cleq 2507  df-clel 2510  df-nfc 2644  df-ne 2686  df-nel 2687  df-ral 2805  df-rex 2806  df-reu 2807  df-rmo 2808  df-rab 2809  df-v 3079  df-sbc 3307  df-csb 3404  df-dif 3447  df-un 3449  df-in 3451  df-ss 3458  df-pss 3460  df-nul 3778  df-if 3940  df-pw 4013  df-sn 4029  df-pr 4031  df-tp 4033  df-op 4035  df-uni 4271  df-int 4309  df-iun 4355  df-br 4482  df-opab 4542  df-mpt 4543  df-tr 4579  df-eprel 4843  df-id 4847  df-po 4853  df-so 4854  df-fr 4891  df-se 4892  df-we 4893  df-xp 4938  df-rel 4939  df-cnv 4940  df-co 4941  df-dm 4942  df-rn 4943  df-res 4944  df-ima 4945  df-pred 5487  df-ord 5533  df-on 5534  df-lim 5535  df-suc 5536  df-iota 5653  df-fun 5691  df-fn 5692  df-f 5693  df-f1 5694  df-fo 5695  df-f1o 5696  df-fv 5697  df-isom 5698  df-riota 6387  df-ov 6428  df-oprab 6429  df-mpt2 6430  df-om 6833  df-1st 6933  df-2nd 6934  df-wrecs 7168  df-recs 7230  df-rdg 7268  df-1o 7322  df-oadd 7326  df-er 7504  df-pm 7622  df-en 7717  df-dom 7718  df-sdom 7719  df-fin 7720  df-sup 8106  df-inf 8107  df-oi 8173  df-card 8523  df-pnf 9830  df-mnf 9831  df-xr 9832  df-ltxr 9833  df-le 9834  df-sub 10018  df-neg 10019  df-div 10433  df-nn 10775  df-2 10833  df-3 10834  df-4 10835  df-5 10836  df-6 10837  df-7 10838  df-8 10839  df-9 10840  df-n0 11047  df-z 11118  df-dec 11233  df-uz 11427  df-rp 11574  df-ico 11920  df-fz 12065  df-fzo 12202  df-fl 12322  df-seq 12531  df-exp 12590  df-fac 12790  df-bc 12819  df-hash 12847  df-shft 13512  df-cj 13544  df-re 13545  df-im 13546  df-sqrt 13680  df-abs 13681  df-limsup 13908  df-clim 13931  df-rlim 13932  df-sum 14132  df-ef 14504  df-struct 15579  df-ndx 15580  df-slot 15581  df-base 15582  df-sets 15583  df-ress 15584  df-plusg 15663  df-mulr 15664  df-starv 15665  df-tset 15669  df-ple 15670  df-ds 15673  df-unif 15674  df-mgm 16955  df-sgrp 16997  df-mnd 17008  df-grp 17138  df-subg 17304  df-cnfld 19470
This theorem is referenced by:  efabl  23988
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