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Mirrors > Home > MPE Home > Th. List > ldiv | Structured version Visualization version GIF version |
Description: Left-division. (Contributed by BJ, 6-Jun-2019.) |
Ref | Expression |
---|---|
ldiv.a | ⊢ (𝜑 → 𝐴 ∈ ℂ) |
ldiv.b | ⊢ (𝜑 → 𝐵 ∈ ℂ) |
ldiv.c | ⊢ (𝜑 → 𝐶 ∈ ℂ) |
ldiv.bn0 | ⊢ (𝜑 → 𝐵 ≠ 0) |
Ref | Expression |
---|---|
ldiv | ⊢ (𝜑 → ((𝐴 · 𝐵) = 𝐶 ↔ 𝐴 = (𝐶 / 𝐵))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oveq1 7156 | . . 3 ⊢ ((𝐴 · 𝐵) = 𝐶 → ((𝐴 · 𝐵) / 𝐵) = (𝐶 / 𝐵)) | |
2 | ldiv.a | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ ℂ) | |
3 | ldiv.b | . . . . 5 ⊢ (𝜑 → 𝐵 ∈ ℂ) | |
4 | ldiv.bn0 | . . . . 5 ⊢ (𝜑 → 𝐵 ≠ 0) | |
5 | 2, 3, 4 | divcan4d 11415 | . . . 4 ⊢ (𝜑 → ((𝐴 · 𝐵) / 𝐵) = 𝐴) |
6 | 5 | eqeq1d 2822 | . . 3 ⊢ (𝜑 → (((𝐴 · 𝐵) / 𝐵) = (𝐶 / 𝐵) ↔ 𝐴 = (𝐶 / 𝐵))) |
7 | 1, 6 | syl5ib 246 | . 2 ⊢ (𝜑 → ((𝐴 · 𝐵) = 𝐶 → 𝐴 = (𝐶 / 𝐵))) |
8 | oveq1 7156 | . . 3 ⊢ (𝐴 = (𝐶 / 𝐵) → (𝐴 · 𝐵) = ((𝐶 / 𝐵) · 𝐵)) | |
9 | ldiv.c | . . . . 5 ⊢ (𝜑 → 𝐶 ∈ ℂ) | |
10 | 9, 3, 4 | divcan1d 11410 | . . . 4 ⊢ (𝜑 → ((𝐶 / 𝐵) · 𝐵) = 𝐶) |
11 | 10 | eqeq2d 2831 | . . 3 ⊢ (𝜑 → ((𝐴 · 𝐵) = ((𝐶 / 𝐵) · 𝐵) ↔ (𝐴 · 𝐵) = 𝐶)) |
12 | 8, 11 | syl5ib 246 | . 2 ⊢ (𝜑 → (𝐴 = (𝐶 / 𝐵) → (𝐴 · 𝐵) = 𝐶)) |
13 | 7, 12 | impbid 214 | 1 ⊢ (𝜑 → ((𝐴 · 𝐵) = 𝐶 ↔ 𝐴 = (𝐶 / 𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 = wceq 1536 ∈ wcel 2113 ≠ wne 3015 (class class class)co 7149 ℂcc 10528 0cc0 10530 · cmul 10535 / cdiv 11290 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5323 ax-un 7454 ax-resscn 10587 ax-1cn 10588 ax-icn 10589 ax-addcl 10590 ax-addrcl 10591 ax-mulcl 10592 ax-mulrcl 10593 ax-mulcom 10594 ax-addass 10595 ax-mulass 10596 ax-distr 10597 ax-i2m1 10598 ax-1ne0 10599 ax-1rid 10600 ax-rnegex 10601 ax-rrecex 10602 ax-cnre 10603 ax-pre-lttri 10604 ax-pre-lttrn 10605 ax-pre-ltadd 10606 ax-pre-mulgt0 10607 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1083 df-3an 1084 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-ne 3016 df-nel 3123 df-ral 3142 df-rex 3143 df-reu 3144 df-rmo 3145 df-rab 3146 df-v 3493 df-sbc 3769 df-csb 3877 df-dif 3932 df-un 3934 df-in 3936 df-ss 3945 df-nul 4285 df-if 4461 df-pw 4534 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-br 5060 df-opab 5122 df-mpt 5140 df-id 5453 df-po 5467 df-so 5468 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-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7107 df-ov 7152 df-oprab 7153 df-mpo 7154 df-er 8282 df-en 8503 df-dom 8504 df-sdom 8505 df-pnf 10670 df-mnf 10671 df-xr 10672 df-ltxr 10673 df-le 10674 df-sub 10865 df-neg 10866 df-div 11291 |
This theorem is referenced by: rdiv 11468 mdiv 11469 dvdszzq 30529 |
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