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Mirrors > Home > MPE Home > Th. List > mulcxplem | Structured version Visualization version GIF version |
Description: Lemma for mulcxp 25195. (Contributed by Mario Carneiro, 2-Aug-2014.) |
Ref | Expression |
---|---|
mulcxp.1 | ⊢ (𝜑 → 𝐴 ∈ ℂ) |
mulcxp.2 | ⊢ (𝜑 → 𝐶 ∈ ℂ) |
Ref | Expression |
---|---|
mulcxplem | ⊢ (𝜑 → (0↑𝑐𝐶) = ((𝐴↑𝑐𝐶) · (0↑𝑐𝐶))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oveq2 7153 | . . . 4 ⊢ (𝐶 = 0 → (0↑𝑐𝐶) = (0↑𝑐0)) | |
2 | 0cn 10621 | . . . . 5 ⊢ 0 ∈ ℂ | |
3 | cxp0 25180 | . . . . 5 ⊢ (0 ∈ ℂ → (0↑𝑐0) = 1) | |
4 | 2, 3 | ax-mp 5 | . . . 4 ⊢ (0↑𝑐0) = 1 |
5 | 1, 4 | syl6eq 2869 | . . 3 ⊢ (𝐶 = 0 → (0↑𝑐𝐶) = 1) |
6 | oveq2 7153 | . . . 4 ⊢ (𝐶 = 0 → (𝐴↑𝑐𝐶) = (𝐴↑𝑐0)) | |
7 | 6, 5 | oveq12d 7163 | . . 3 ⊢ (𝐶 = 0 → ((𝐴↑𝑐𝐶) · (0↑𝑐𝐶)) = ((𝐴↑𝑐0) · 1)) |
8 | 5, 7 | eqeq12d 2834 | . 2 ⊢ (𝐶 = 0 → ((0↑𝑐𝐶) = ((𝐴↑𝑐𝐶) · (0↑𝑐𝐶)) ↔ 1 = ((𝐴↑𝑐0) · 1))) |
9 | mulcxp.1 | . . . . . 6 ⊢ (𝜑 → 𝐴 ∈ ℂ) | |
10 | mulcxp.2 | . . . . . 6 ⊢ (𝜑 → 𝐶 ∈ ℂ) | |
11 | cxpcl 25184 | . . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝐶 ∈ ℂ) → (𝐴↑𝑐𝐶) ∈ ℂ) | |
12 | 9, 10, 11 | syl2anc 584 | . . . . 5 ⊢ (𝜑 → (𝐴↑𝑐𝐶) ∈ ℂ) |
13 | 12 | adantr 481 | . . . 4 ⊢ ((𝜑 ∧ 𝐶 ≠ 0) → (𝐴↑𝑐𝐶) ∈ ℂ) |
14 | 13 | mul01d 10827 | . . 3 ⊢ ((𝜑 ∧ 𝐶 ≠ 0) → ((𝐴↑𝑐𝐶) · 0) = 0) |
15 | 0cxp 25176 | . . . . 5 ⊢ ((𝐶 ∈ ℂ ∧ 𝐶 ≠ 0) → (0↑𝑐𝐶) = 0) | |
16 | 10, 15 | sylan 580 | . . . 4 ⊢ ((𝜑 ∧ 𝐶 ≠ 0) → (0↑𝑐𝐶) = 0) |
17 | 16 | oveq2d 7161 | . . 3 ⊢ ((𝜑 ∧ 𝐶 ≠ 0) → ((𝐴↑𝑐𝐶) · (0↑𝑐𝐶)) = ((𝐴↑𝑐𝐶) · 0)) |
18 | 14, 17, 16 | 3eqtr4rd 2864 | . 2 ⊢ ((𝜑 ∧ 𝐶 ≠ 0) → (0↑𝑐𝐶) = ((𝐴↑𝑐𝐶) · (0↑𝑐𝐶))) |
19 | cxp0 25180 | . . . . 5 ⊢ (𝐴 ∈ ℂ → (𝐴↑𝑐0) = 1) | |
20 | 9, 19 | syl 17 | . . . 4 ⊢ (𝜑 → (𝐴↑𝑐0) = 1) |
21 | 20 | oveq1d 7160 | . . 3 ⊢ (𝜑 → ((𝐴↑𝑐0) · 1) = (1 · 1)) |
22 | 1t1e1 11787 | . . 3 ⊢ (1 · 1) = 1 | |
23 | 21, 22 | syl6req 2870 | . 2 ⊢ (𝜑 → 1 = ((𝐴↑𝑐0) · 1)) |
24 | 8, 18, 23 | pm2.61ne 3099 | 1 ⊢ (𝜑 → (0↑𝑐𝐶) = ((𝐴↑𝑐𝐶) · (0↑𝑐𝐶))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1528 ∈ wcel 2105 ≠ wne 3013 (class class class)co 7145 ℂcc 10523 0cc0 10525 1c1 10526 · cmul 10530 ↑𝑐ccxp 25066 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-inf2 9092 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 ax-pre-sup 10603 ax-addf 10604 ax-mulf 10605 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-fal 1541 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-int 4868 df-iun 4912 df-iin 4913 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-se 5508 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-isom 6357 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-of 7398 df-om 7570 df-1st 7678 df-2nd 7679 df-supp 7820 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-1o 8091 df-2o 8092 df-oadd 8095 df-er 8278 df-map 8397 df-pm 8398 df-ixp 8450 df-en 8498 df-dom 8499 df-sdom 8500 df-fin 8501 df-fsupp 8822 df-fi 8863 df-sup 8894 df-inf 8895 df-oi 8962 df-card 9356 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-div 11286 df-nn 11627 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-7 11693 df-8 11694 df-9 11695 df-n0 11886 df-z 11970 df-dec 12087 df-uz 12232 df-q 12337 df-rp 12378 df-xneg 12495 df-xadd 12496 df-xmul 12497 df-ioo 12730 df-ioc 12731 df-ico 12732 df-icc 12733 df-fz 12881 df-fzo 13022 df-fl 13150 df-mod 13226 df-seq 13358 df-exp 13418 df-fac 13622 df-bc 13651 df-hash 13679 df-shft 14414 df-cj 14446 df-re 14447 df-im 14448 df-sqrt 14582 df-abs 14583 df-limsup 14816 df-clim 14833 df-rlim 14834 df-sum 15031 df-ef 15409 df-sin 15411 df-cos 15412 df-pi 15414 df-struct 16473 df-ndx 16474 df-slot 16475 df-base 16477 df-sets 16478 df-ress 16479 df-plusg 16566 df-mulr 16567 df-starv 16568 df-sca 16569 df-vsca 16570 df-ip 16571 df-tset 16572 df-ple 16573 df-ds 16575 df-unif 16576 df-hom 16577 df-cco 16578 df-rest 16684 df-topn 16685 df-0g 16703 df-gsum 16704 df-topgen 16705 df-pt 16706 df-prds 16709 df-xrs 16763 df-qtop 16768 df-imas 16769 df-xps 16771 df-mre 16845 df-mrc 16846 df-acs 16848 df-mgm 17840 df-sgrp 17889 df-mnd 17900 df-submnd 17945 df-mulg 18163 df-cntz 18385 df-cmn 18837 df-psmet 20465 df-xmet 20466 df-met 20467 df-bl 20468 df-mopn 20469 df-fbas 20470 df-fg 20471 df-cnfld 20474 df-top 21430 df-topon 21447 df-topsp 21469 df-bases 21482 df-cld 21555 df-ntr 21556 df-cls 21557 df-nei 21634 df-lp 21672 df-perf 21673 df-cn 21763 df-cnp 21764 df-haus 21851 df-tx 22098 df-hmeo 22291 df-fil 22382 df-fm 22474 df-flim 22475 df-flf 22476 df-xms 22857 df-ms 22858 df-tms 22859 df-cncf 23413 df-limc 24391 df-dv 24392 df-log 25067 df-cxp 25068 |
This theorem is referenced by: mulcxp 25195 |
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