xdivmul - Mathbox for Thierry Arnoux

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Theorem xdivmul 30209

Description: Relationship between division and multiplication. (Contributed by Thierry Arnoux, 24-Dec-2016.)

**Assertion**
Ref	Expression
xdivmul	⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ (𝐶 ∈ ℝ ∧ 𝐶 ≠ 0)) → ((𝐴 /_𝑒 𝐶) = 𝐵 ↔ (𝐶 ·_e 𝐵) = 𝐴))

Proof of Theorem xdivmul Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		xdivval 30203	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐶 ∈ ℝ ∧ 𝐶 ≠ 0) → (𝐴 /_𝑒 𝐶) = (℩𝑥 ∈ ℝ^* (𝐶 ·_e 𝑥) = 𝐴))
2	1	3expb 1110	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ (𝐶 ∈ ℝ ∧ 𝐶 ≠ 0)) → (𝐴 /_𝑒 𝐶) = (℩𝑥 ∈ ℝ^* (𝐶 ·_e 𝑥) = 𝐴))
3	2	3adant2 1122	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ (𝐶 ∈ ℝ ∧ 𝐶 ≠ 0)) → (𝐴 /_𝑒 𝐶) = (℩𝑥 ∈ ℝ^* (𝐶 ·_e 𝑥) = 𝐴))
4	3	eqeq1d 2780	. 2 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ (𝐶 ∈ ℝ ∧ 𝐶 ≠ 0)) → ((𝐴 /_𝑒 𝐶) = 𝐵 ↔ (℩𝑥 ∈ ℝ^* (𝐶 ·_e 𝑥) = 𝐴) = 𝐵))
5		simp2 1128	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ (𝐶 ∈ ℝ ∧ 𝐶 ≠ 0)) → 𝐵 ∈ ℝ^*)
6		xreceu 30206	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐶 ∈ ℝ ∧ 𝐶 ≠ 0) → ∃!𝑥 ∈ ℝ^* (𝐶 ·_e 𝑥) = 𝐴)
7	6	3expb 1110	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ (𝐶 ∈ ℝ ∧ 𝐶 ≠ 0)) → ∃!𝑥 ∈ ℝ^* (𝐶 ·_e 𝑥) = 𝐴)
8	7	3adant2 1122	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ (𝐶 ∈ ℝ ∧ 𝐶 ≠ 0)) → ∃!𝑥 ∈ ℝ^* (𝐶 ·_e 𝑥) = 𝐴)
9		oveq2 6932	. . . . 5 ⊢ (𝑥 = 𝐵 → (𝐶 ·_e 𝑥) = (𝐶 ·_e 𝐵))
10	9	eqeq1d 2780	. . . 4 ⊢ (𝑥 = 𝐵 → ((𝐶 ·_e 𝑥) = 𝐴 ↔ (𝐶 ·_e 𝐵) = 𝐴))
11	10	riota2 6907	. . 3 ⊢ ((𝐵 ∈ ℝ^* ∧ ∃!𝑥 ∈ ℝ^* (𝐶 ·_e 𝑥) = 𝐴) → ((𝐶 ·_e 𝐵) = 𝐴 ↔ (℩𝑥 ∈ ℝ^* (𝐶 ·_e 𝑥) = 𝐴) = 𝐵))
12	5, 8, 11	syl2anc 579	. 2 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ (𝐶 ∈ ℝ ∧ 𝐶 ≠ 0)) → ((𝐶 ·_e 𝐵) = 𝐴 ↔ (℩𝑥 ∈ ℝ^* (𝐶 ·_e 𝑥) = 𝐴) = 𝐵))
13	4, 12	bitr4d 274	1 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ (𝐶 ∈ ℝ ∧ 𝐶 ≠ 0)) → ((𝐴 /_𝑒 𝐶) = 𝐵 ↔ (𝐶 ·_e 𝐵) = 𝐴))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 198 ∧ wa 386 ∧ w3a 1071 = wceq 1601 ∈ wcel 2107 ≠ wne 2969 ∃!wreu 3092 ℩crio 6884 (class class class)co 6924 ℝcr 10273 0cc0 10274 ℝ^*cxr 10412 ·_e cxmu 12261 /_𝑒 cxdiv 30201

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-sep 5019 ax-nul 5027 ax-pow 5079 ax-pr 5140 ax-un 7228 ax-cnex 10330 ax-resscn 10331 ax-1cn 10332 ax-icn 10333 ax-addcl 10334 ax-addrcl 10335 ax-mulcl 10336 ax-mulrcl 10337 ax-mulcom 10338 ax-addass 10339 ax-mulass 10340 ax-distr 10341 ax-i2m1 10342 ax-1ne0 10343 ax-1rid 10344 ax-rnegex 10345 ax-rrecex 10346 ax-cnre 10347 ax-pre-lttri 10348 ax-pre-lttrn 10349 ax-pre-ltadd 10350 ax-pre-mulgt0 10351

This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-nel 3076 df-ral 3095 df-rex 3096 df-reu 3097 df-rmo 3098 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-op 4405 df-uni 4674 df-iun 4757 df-br 4889 df-opab 4951 df-mpt 4968 df-id 5263 df-po 5276 df-so 5277 df-xp 5363 df-rel 5364 df-cnv 5365 df-co 5366 df-dm 5367 df-rn 5368 df-res 5369 df-ima 5370 df-iota 6101 df-fun 6139 df-fn 6140 df-f 6141 df-f1 6142 df-fo 6143 df-f1o 6144 df-fv 6145 df-riota 6885 df-ov 6927 df-oprab 6928 df-mpt2 6929 df-1st 7447 df-2nd 7448 df-er 8028 df-en 8244 df-dom 8245 df-sdom 8246 df-pnf 10415 df-mnf 10416 df-xr 10417 df-ltxr 10418 df-le 10419 df-sub 10610 df-neg 10611 df-xneg 12262 df-xmul 12264 df-xdiv 30202

This theorem is referenced by: xdivrec 30211

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