Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > divmuldivapd | GIF version |
Description: Multiplication of two ratios. (Contributed by Jim Kingdon, 30-Jul-2021.) |
Ref | Expression |
---|---|
divcld.1 | ⊢ (𝜑 → 𝐴 ∈ ℂ) |
divcld.2 | ⊢ (𝜑 → 𝐵 ∈ ℂ) |
divmuld.3 | ⊢ (𝜑 → 𝐶 ∈ ℂ) |
divmuldivapd.4 | ⊢ (𝜑 → 𝐷 ∈ ℂ) |
divmuldivapd.5 | ⊢ (𝜑 → 𝐵 # 0) |
divmuldivapd.6 | ⊢ (𝜑 → 𝐷 # 0) |
Ref | Expression |
---|---|
divmuldivapd | ⊢ (𝜑 → ((𝐴 / 𝐵) · (𝐶 / 𝐷)) = ((𝐴 · 𝐶) / (𝐵 · 𝐷))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | divcld.1 | . 2 ⊢ (𝜑 → 𝐴 ∈ ℂ) | |
2 | divmuld.3 | . 2 ⊢ (𝜑 → 𝐶 ∈ ℂ) | |
3 | divcld.2 | . . 3 ⊢ (𝜑 → 𝐵 ∈ ℂ) | |
4 | divmuldivapd.5 | . . 3 ⊢ (𝜑 → 𝐵 # 0) | |
5 | 3, 4 | jca 300 | . 2 ⊢ (𝜑 → (𝐵 ∈ ℂ ∧ 𝐵 # 0)) |
6 | divmuldivapd.4 | . . 3 ⊢ (𝜑 → 𝐷 ∈ ℂ) | |
7 | divmuldivapd.6 | . . 3 ⊢ (𝜑 → 𝐷 # 0) | |
8 | 6, 7 | jca 300 | . 2 ⊢ (𝜑 → (𝐷 ∈ ℂ ∧ 𝐷 # 0)) |
9 | divmuldivap 7800 | . 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 | syl22anc 1170 | 1 ⊢ (𝜑 → ((𝐴 / 𝐵) · (𝐶 / 𝐷)) = ((𝐴 · 𝐶) / (𝐵 · 𝐷))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 102 = wceq 1284 ∈ wcel 1433 class class class wbr 3785 (class class class)co 5532 ℂcc 6979 0cc0 6981 · cmul 6986 # cap 7681 / cdiv 7760 |
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 576 ax-in2 577 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-13 1444 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-sep 3896 ax-pow 3948 ax-pr 3964 ax-un 4188 ax-setind 4280 ax-cnex 7067 ax-resscn 7068 ax-1cn 7069 ax-1re 7070 ax-icn 7071 ax-addcl 7072 ax-addrcl 7073 ax-mulcl 7074 ax-mulrcl 7075 ax-addcom 7076 ax-mulcom 7077 ax-addass 7078 ax-mulass 7079 ax-distr 7080 ax-i2m1 7081 ax-0lt1 7082 ax-1rid 7083 ax-0id 7084 ax-rnegex 7085 ax-precex 7086 ax-cnre 7087 ax-pre-ltirr 7088 ax-pre-ltwlin 7089 ax-pre-lttrn 7090 ax-pre-apti 7091 ax-pre-ltadd 7092 ax-pre-mulgt0 7093 ax-pre-mulext 7094 |
This theorem depends on definitions: df-bi 115 df-3an 921 df-tru 1287 df-fal 1290 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ne 2246 df-nel 2340 df-ral 2353 df-rex 2354 df-reu 2355 df-rmo 2356 df-rab 2357 df-v 2603 df-sbc 2816 df-dif 2975 df-un 2977 df-in 2979 df-ss 2986 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-br 3786 df-opab 3840 df-id 4048 df-po 4051 df-iso 4052 df-xp 4369 df-rel 4370 df-cnv 4371 df-co 4372 df-dm 4373 df-iota 4887 df-fun 4924 df-fv 4930 df-riota 5488 df-ov 5535 df-oprab 5536 df-mpt2 5537 df-pnf 7155 df-mnf 7156 df-xr 7157 df-ltxr 7158 df-le 7159 df-sub 7281 df-neg 7282 df-reap 7675 df-ap 7682 df-div 7761 |
This theorem is referenced by: faclbnd2 9669 bcm1k 9687 bcp1n 9688 resqrexlemcalc2 9901 |
Copyright terms: Public domain | W3C validator |