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Mirrors > Home > ILE Home > Th. List > mulcanapd | GIF version |
Description: Cancellation law for multiplication. (Contributed by Jim Kingdon, 21-Feb-2020.) |
Ref | Expression |
---|---|
mulcand.1 | ⊢ (𝜑 → 𝐴 ∈ ℂ) |
mulcand.2 | ⊢ (𝜑 → 𝐵 ∈ ℂ) |
mulcand.3 | ⊢ (𝜑 → 𝐶 ∈ ℂ) |
mulcand.4 | ⊢ (𝜑 → 𝐶 # 0) |
Ref | Expression |
---|---|
mulcanapd | ⊢ (𝜑 → ((𝐶 · 𝐴) = (𝐶 · 𝐵) ↔ 𝐴 = 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mulcand.3 | . . . 4 ⊢ (𝜑 → 𝐶 ∈ ℂ) | |
2 | mulcand.4 | . . . 4 ⊢ (𝜑 → 𝐶 # 0) | |
3 | recexap 7810 | . . . 4 ⊢ ((𝐶 ∈ ℂ ∧ 𝐶 # 0) → ∃𝑥 ∈ ℂ (𝐶 · 𝑥) = 1) | |
4 | 1, 2, 3 | syl2anc 403 | . . 3 ⊢ (𝜑 → ∃𝑥 ∈ ℂ (𝐶 · 𝑥) = 1) |
5 | oveq2 5551 | . . . 4 ⊢ ((𝐶 · 𝐴) = (𝐶 · 𝐵) → (𝑥 · (𝐶 · 𝐴)) = (𝑥 · (𝐶 · 𝐵))) | |
6 | simprl 498 | . . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ ℂ ∧ (𝐶 · 𝑥) = 1)) → 𝑥 ∈ ℂ) | |
7 | 1 | adantr 270 | . . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ ℂ ∧ (𝐶 · 𝑥) = 1)) → 𝐶 ∈ ℂ) |
8 | 6, 7 | mulcomd 7202 | . . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ ℂ ∧ (𝐶 · 𝑥) = 1)) → (𝑥 · 𝐶) = (𝐶 · 𝑥)) |
9 | simprr 499 | . . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ ℂ ∧ (𝐶 · 𝑥) = 1)) → (𝐶 · 𝑥) = 1) | |
10 | 8, 9 | eqtrd 2114 | . . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ ℂ ∧ (𝐶 · 𝑥) = 1)) → (𝑥 · 𝐶) = 1) |
11 | 10 | oveq1d 5558 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ ℂ ∧ (𝐶 · 𝑥) = 1)) → ((𝑥 · 𝐶) · 𝐴) = (1 · 𝐴)) |
12 | mulcand.1 | . . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ ℂ) | |
13 | 12 | adantr 270 | . . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ ℂ ∧ (𝐶 · 𝑥) = 1)) → 𝐴 ∈ ℂ) |
14 | 6, 7, 13 | mulassd 7204 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ ℂ ∧ (𝐶 · 𝑥) = 1)) → ((𝑥 · 𝐶) · 𝐴) = (𝑥 · (𝐶 · 𝐴))) |
15 | 13 | mulid2d 7199 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ ℂ ∧ (𝐶 · 𝑥) = 1)) → (1 · 𝐴) = 𝐴) |
16 | 11, 14, 15 | 3eqtr3d 2122 | . . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ ℂ ∧ (𝐶 · 𝑥) = 1)) → (𝑥 · (𝐶 · 𝐴)) = 𝐴) |
17 | 10 | oveq1d 5558 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ ℂ ∧ (𝐶 · 𝑥) = 1)) → ((𝑥 · 𝐶) · 𝐵) = (1 · 𝐵)) |
18 | mulcand.2 | . . . . . . . 8 ⊢ (𝜑 → 𝐵 ∈ ℂ) | |
19 | 18 | adantr 270 | . . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ ℂ ∧ (𝐶 · 𝑥) = 1)) → 𝐵 ∈ ℂ) |
20 | 6, 7, 19 | mulassd 7204 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ ℂ ∧ (𝐶 · 𝑥) = 1)) → ((𝑥 · 𝐶) · 𝐵) = (𝑥 · (𝐶 · 𝐵))) |
21 | 19 | mulid2d 7199 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ ℂ ∧ (𝐶 · 𝑥) = 1)) → (1 · 𝐵) = 𝐵) |
22 | 17, 20, 21 | 3eqtr3d 2122 | . . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ ℂ ∧ (𝐶 · 𝑥) = 1)) → (𝑥 · (𝐶 · 𝐵)) = 𝐵) |
23 | 16, 22 | eqeq12d 2096 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ ℂ ∧ (𝐶 · 𝑥) = 1)) → ((𝑥 · (𝐶 · 𝐴)) = (𝑥 · (𝐶 · 𝐵)) ↔ 𝐴 = 𝐵)) |
24 | 5, 23 | syl5ib 152 | . . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ ℂ ∧ (𝐶 · 𝑥) = 1)) → ((𝐶 · 𝐴) = (𝐶 · 𝐵) → 𝐴 = 𝐵)) |
25 | 4, 24 | rexlimddv 2482 | . 2 ⊢ (𝜑 → ((𝐶 · 𝐴) = (𝐶 · 𝐵) → 𝐴 = 𝐵)) |
26 | oveq2 5551 | . 2 ⊢ (𝐴 = 𝐵 → (𝐶 · 𝐴) = (𝐶 · 𝐵)) | |
27 | 25, 26 | impbid1 140 | 1 ⊢ (𝜑 → ((𝐶 · 𝐴) = (𝐶 · 𝐵) ↔ 𝐴 = 𝐵)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 102 ↔ wb 103 = wceq 1285 ∈ wcel 1434 ∃wrex 2350 class class class wbr 3793 (class class class)co 5543 ℂcc 7041 0cc0 7043 1c1 7044 · cmul 7048 # cap 7748 |
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 1377 ax-7 1378 ax-gen 1379 ax-ie1 1423 ax-ie2 1424 ax-8 1436 ax-10 1437 ax-11 1438 ax-i12 1439 ax-bndl 1440 ax-4 1441 ax-13 1445 ax-14 1446 ax-17 1460 ax-i9 1464 ax-ial 1468 ax-i5r 1469 ax-ext 2064 ax-sep 3904 ax-pow 3956 ax-pr 3972 ax-un 4196 ax-setind 4288 ax-cnex 7129 ax-resscn 7130 ax-1cn 7131 ax-1re 7132 ax-icn 7133 ax-addcl 7134 ax-addrcl 7135 ax-mulcl 7136 ax-mulrcl 7137 ax-addcom 7138 ax-mulcom 7139 ax-addass 7140 ax-mulass 7141 ax-distr 7142 ax-i2m1 7143 ax-0lt1 7144 ax-1rid 7145 ax-0id 7146 ax-rnegex 7147 ax-precex 7148 ax-cnre 7149 ax-pre-ltirr 7150 ax-pre-ltwlin 7151 ax-pre-lttrn 7152 ax-pre-apti 7153 ax-pre-ltadd 7154 ax-pre-mulgt0 7155 ax-pre-mulext 7156 |
This theorem depends on definitions: df-bi 115 df-3an 922 df-tru 1288 df-fal 1291 df-nf 1391 df-sb 1687 df-eu 1945 df-mo 1946 df-clab 2069 df-cleq 2075 df-clel 2078 df-nfc 2209 df-ne 2247 df-nel 2341 df-ral 2354 df-rex 2355 df-reu 2356 df-rab 2358 df-v 2604 df-sbc 2817 df-dif 2976 df-un 2978 df-in 2980 df-ss 2987 df-pw 3392 df-sn 3412 df-pr 3413 df-op 3415 df-uni 3610 df-br 3794 df-opab 3848 df-id 4056 df-po 4059 df-iso 4060 df-xp 4377 df-rel 4378 df-cnv 4379 df-co 4380 df-dm 4381 df-iota 4897 df-fun 4934 df-fv 4940 df-riota 5499 df-ov 5546 df-oprab 5547 df-mpt2 5548 df-pnf 7217 df-mnf 7218 df-xr 7219 df-ltxr 7220 df-le 7221 df-sub 7348 df-neg 7349 df-reap 7742 df-ap 7749 |
This theorem is referenced by: mulcanap2d 7819 mulcanapad 7820 mulcanap 7822 div11ap 7855 eqneg 7887 dvdscmulr 10369 qredeq 10622 cncongr1 10629 |
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