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Mirrors > Home > MPE Home > Th. List > divdiv32i | Structured version Visualization version GIF version |
Description: Swap denominators in a division. (Contributed by NM, 15-Sep-1999.) |
Ref | Expression |
---|---|
divclz.1 | ⊢ 𝐴 ∈ ℂ |
divclz.2 | ⊢ 𝐵 ∈ ℂ |
divmulz.3 | ⊢ 𝐶 ∈ ℂ |
divmul.4 | ⊢ 𝐵 ≠ 0 |
divdiv23.5 | ⊢ 𝐶 ≠ 0 |
Ref | Expression |
---|---|
divdiv32i | ⊢ ((𝐴 / 𝐵) / 𝐶) = ((𝐴 / 𝐶) / 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | divmul.4 | . 2 ⊢ 𝐵 ≠ 0 | |
2 | divdiv23.5 | . 2 ⊢ 𝐶 ≠ 0 | |
3 | divclz.1 | . . 3 ⊢ 𝐴 ∈ ℂ | |
4 | divclz.2 | . . 3 ⊢ 𝐵 ∈ ℂ | |
5 | divmulz.3 | . . 3 ⊢ 𝐶 ∈ ℂ | |
6 | 3, 4, 5 | divdiv23zi 11200 | . 2 ⊢ ((𝐵 ≠ 0 ∧ 𝐶 ≠ 0) → ((𝐴 / 𝐵) / 𝐶) = ((𝐴 / 𝐶) / 𝐵)) |
7 | 1, 2, 6 | mp2an 680 | 1 ⊢ ((𝐴 / 𝐵) / 𝐶) = ((𝐴 / 𝐶) / 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1508 ∈ wcel 2051 ≠ wne 2969 (class class class)co 6982 ℂcc 10339 0cc0 10341 / cdiv 11104 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1759 ax-4 1773 ax-5 1870 ax-6 1929 ax-7 1966 ax-8 2053 ax-9 2060 ax-10 2080 ax-11 2094 ax-12 2107 ax-13 2302 ax-ext 2752 ax-sep 5064 ax-nul 5071 ax-pow 5123 ax-pr 5190 ax-un 7285 ax-resscn 10398 ax-1cn 10399 ax-icn 10400 ax-addcl 10401 ax-addrcl 10402 ax-mulcl 10403 ax-mulrcl 10404 ax-mulcom 10405 ax-addass 10406 ax-mulass 10407 ax-distr 10408 ax-i2m1 10409 ax-1ne0 10410 ax-1rid 10411 ax-rnegex 10412 ax-rrecex 10413 ax-cnre 10414 ax-pre-lttri 10415 ax-pre-lttrn 10416 ax-pre-ltadd 10417 ax-pre-mulgt0 10418 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 835 df-3or 1070 df-3an 1071 df-tru 1511 df-ex 1744 df-nf 1748 df-sb 2017 df-mo 2551 df-eu 2589 df-clab 2761 df-cleq 2773 df-clel 2848 df-nfc 2920 df-ne 2970 df-nel 3076 df-ral 3095 df-rex 3096 df-reu 3097 df-rmo 3098 df-rab 3099 df-v 3419 df-sbc 3684 df-csb 3789 df-dif 3834 df-un 3836 df-in 3838 df-ss 3845 df-nul 4182 df-if 4354 df-pw 4427 df-sn 4445 df-pr 4447 df-op 4451 df-uni 4718 df-br 4935 df-opab 4997 df-mpt 5014 df-id 5316 df-po 5330 df-so 5331 df-xp 5417 df-rel 5418 df-cnv 5419 df-co 5420 df-dm 5421 df-rn 5422 df-res 5423 df-ima 5424 df-iota 6157 df-fun 6195 df-fn 6196 df-f 6197 df-f1 6198 df-fo 6199 df-f1o 6200 df-fv 6201 df-riota 6943 df-ov 6985 df-oprab 6986 df-mpo 6987 df-er 8095 df-en 8313 df-dom 8314 df-sdom 8315 df-pnf 10482 df-mnf 10483 df-xr 10484 df-ltxr 10485 df-le 10486 df-sub 10678 df-neg 10679 df-div 11105 |
This theorem is referenced by: bposlem8 25584 fourierdlem103 41960 fourierdlem104 41961 fourierswlem 41981 |
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