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Mirrors > Home > ILE Home > Th. List > divmuldivapi | GIF version |
Description: Multiplication of two ratios. (Contributed by Jim Kingdon, 9-Mar-2020.) |
Ref | Expression |
---|---|
divclz.1 | ⊢ 𝐴 ∈ ℂ |
divclz.2 | ⊢ 𝐵 ∈ ℂ |
divmulz.3 | ⊢ 𝐶 ∈ ℂ |
divmuldivap.4 | ⊢ 𝐷 ∈ ℂ |
divmuldivap.5 | ⊢ 𝐵 # 0 |
divmuldivap.6 | ⊢ 𝐷 # 0 |
Ref | Expression |
---|---|
divmuldivapi | ⊢ ((𝐴 / 𝐵) · (𝐶 / 𝐷)) = ((𝐴 · 𝐶) / (𝐵 · 𝐷)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | divclz.1 | . 2 ⊢ 𝐴 ∈ ℂ | |
2 | divmulz.3 | . 2 ⊢ 𝐶 ∈ ℂ | |
3 | divclz.2 | . . 3 ⊢ 𝐵 ∈ ℂ | |
4 | divmuldivap.5 | . . 3 ⊢ 𝐵 # 0 | |
5 | 3, 4 | pm3.2i 266 | . 2 ⊢ (𝐵 ∈ ℂ ∧ 𝐵 # 0) |
6 | divmuldivap.4 | . . 3 ⊢ 𝐷 ∈ ℂ | |
7 | divmuldivap.6 | . . 3 ⊢ 𝐷 # 0 | |
8 | 6, 7 | pm3.2i 266 | . 2 ⊢ (𝐷 ∈ ℂ ∧ 𝐷 # 0) |
9 | divmuldivap 8095 | . 2 ⊢ (((𝐴 ∈ ℂ ∧ 𝐶 ∈ ℂ) ∧ ((𝐵 ∈ ℂ ∧ 𝐵 # 0) ∧ (𝐷 ∈ ℂ ∧ 𝐷 # 0))) → ((𝐴 / 𝐵) · (𝐶 / 𝐷)) = ((𝐴 · 𝐶) / (𝐵 · 𝐷))) | |
10 | 1, 2, 5, 8, 9 | mp4an 418 | 1 ⊢ ((𝐴 / 𝐵) · (𝐶 / 𝐷)) = ((𝐴 · 𝐶) / (𝐵 · 𝐷)) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 102 = wceq 1287 ∈ wcel 1436 class class class wbr 3814 (class class class)co 5594 ℂcc 7269 0cc0 7271 · cmul 7276 # cap 7976 / cdiv 8055 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 577 ax-in2 578 ax-io 663 ax-5 1379 ax-7 1380 ax-gen 1381 ax-ie1 1425 ax-ie2 1426 ax-8 1438 ax-10 1439 ax-11 1440 ax-i12 1441 ax-bndl 1442 ax-4 1443 ax-13 1447 ax-14 1448 ax-17 1462 ax-i9 1466 ax-ial 1470 ax-i5r 1471 ax-ext 2067 ax-sep 3925 ax-pow 3977 ax-pr 4003 ax-un 4227 ax-setind 4319 ax-cnex 7357 ax-resscn 7358 ax-1cn 7359 ax-1re 7360 ax-icn 7361 ax-addcl 7362 ax-addrcl 7363 ax-mulcl 7364 ax-mulrcl 7365 ax-addcom 7366 ax-mulcom 7367 ax-addass 7368 ax-mulass 7369 ax-distr 7370 ax-i2m1 7371 ax-0lt1 7372 ax-1rid 7373 ax-0id 7374 ax-rnegex 7375 ax-precex 7376 ax-cnre 7377 ax-pre-ltirr 7378 ax-pre-ltwlin 7379 ax-pre-lttrn 7380 ax-pre-apti 7381 ax-pre-ltadd 7382 ax-pre-mulgt0 7383 ax-pre-mulext 7384 |
This theorem depends on definitions: df-bi 115 df-3an 924 df-tru 1290 df-fal 1293 df-nf 1393 df-sb 1690 df-eu 1948 df-mo 1949 df-clab 2072 df-cleq 2078 df-clel 2081 df-nfc 2214 df-ne 2252 df-nel 2347 df-ral 2360 df-rex 2361 df-reu 2362 df-rmo 2363 df-rab 2364 df-v 2616 df-sbc 2829 df-dif 2988 df-un 2990 df-in 2992 df-ss 2999 df-pw 3411 df-sn 3431 df-pr 3432 df-op 3434 df-uni 3631 df-br 3815 df-opab 3869 df-id 4087 df-po 4090 df-iso 4091 df-xp 4410 df-rel 4411 df-cnv 4412 df-co 4413 df-dm 4414 df-iota 4937 df-fun 4974 df-fv 4980 df-riota 5550 df-ov 5597 df-oprab 5598 df-mpt2 5599 df-pnf 7445 df-mnf 7446 df-xr 7447 df-ltxr 7448 df-le 7449 df-sub 7576 df-neg 7577 df-reap 7970 df-ap 7977 df-div 8056 |
This theorem is referenced by: 8th4div3 8545 halfpm6th 8546 sqdivapi 9889 |
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