xdivmul - Mathbox for Thierry Arnoux

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Theorem xdivmul 31307

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 31301	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐶 ∈ ℝ ∧ 𝐶 ≠ 0) → (𝐴 /_𝑒 𝐶) = (℩𝑥 ∈ ℝ^* (𝐶 ·_e 𝑥) = 𝐴))
2	1	3expb 1119	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ (𝐶 ∈ ℝ ∧ 𝐶 ≠ 0)) → (𝐴 /_𝑒 𝐶) = (℩𝑥 ∈ ℝ^* (𝐶 ·_e 𝑥) = 𝐴))
3	2	3adant2 1130	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ (𝐶 ∈ ℝ ∧ 𝐶 ≠ 0)) → (𝐴 /_𝑒 𝐶) = (℩𝑥 ∈ ℝ^* (𝐶 ·_e 𝑥) = 𝐴))
4	3	eqeq1d 2739	. 2 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ (𝐶 ∈ ℝ ∧ 𝐶 ≠ 0)) → ((𝐴 /_𝑒 𝐶) = 𝐵 ↔ (℩𝑥 ∈ ℝ^* (𝐶 ·_e 𝑥) = 𝐴) = 𝐵))
5		simp2 1136	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ (𝐶 ∈ ℝ ∧ 𝐶 ≠ 0)) → 𝐵 ∈ ℝ^*)
6		xreceu 31304	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐶 ∈ ℝ ∧ 𝐶 ≠ 0) → ∃!𝑥 ∈ ℝ^* (𝐶 ·_e 𝑥) = 𝐴)
7	6	3expb 1119	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ (𝐶 ∈ ℝ ∧ 𝐶 ≠ 0)) → ∃!𝑥 ∈ ℝ^* (𝐶 ·_e 𝑥) = 𝐴)
8	7	3adant2 1130	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ (𝐶 ∈ ℝ ∧ 𝐶 ≠ 0)) → ∃!𝑥 ∈ ℝ^* (𝐶 ·_e 𝑥) = 𝐴)
9		oveq2 7321	. . . . 5 ⊢ (𝑥 = 𝐵 → (𝐶 ·_e 𝑥) = (𝐶 ·_e 𝐵))
10	9	eqeq1d 2739	. . . 4 ⊢ (𝑥 = 𝐵 → ((𝐶 ·_e 𝑥) = 𝐴 ↔ (𝐶 ·_e 𝐵) = 𝐴))
11	10	riota2 7296	. . 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 584	. 2 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ (𝐶 ∈ ℝ ∧ 𝐶 ≠ 0)) → ((𝐶 ·_e 𝐵) = 𝐴 ↔ (℩𝑥 ∈ ℝ^* (𝐶 ·_e 𝑥) = 𝐴) = 𝐵))
13	4, 12	bitr4d 281	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 205 ∧ wa 396 ∧ w3a 1086 = wceq 1540 ∈ wcel 2105 ≠ wne 2941 ∃!wreu 3348 ℩crio 7269 (class class class)co 7313 ℝcr 10940 0cc0 10941 ℝ^*cxr 11078 ·_e cxmu 12917 /_𝑒 cxdiv 31299

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2708 ax-sep 5236 ax-nul 5243 ax-pow 5301 ax-pr 5365 ax-un 7626 ax-cnex 10997 ax-resscn 10998 ax-1cn 10999 ax-icn 11000 ax-addcl 11001 ax-addrcl 11002 ax-mulcl 11003 ax-mulrcl 11004 ax-mulcom 11005 ax-addass 11006 ax-mulass 11007 ax-distr 11008 ax-i2m1 11009 ax-1ne0 11010 ax-1rid 11011 ax-rnegex 11012 ax-rrecex 11013 ax-cnre 11014 ax-pre-lttri 11015 ax-pre-lttrn 11016 ax-pre-ltadd 11017 ax-pre-mulgt0 11018

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3350 df-reu 3351 df-rab 3405 df-v 3443 df-sbc 3726 df-csb 3842 df-dif 3899 df-un 3901 df-in 3903 df-ss 3913 df-nul 4267 df-if 4470 df-pw 4545 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4849 df-iun 4937 df-br 5086 df-opab 5148 df-mpt 5169 df-id 5505 df-po 5519 df-so 5520 df-xp 5611 df-rel 5612 df-cnv 5613 df-co 5614 df-dm 5615 df-rn 5616 df-res 5617 df-ima 5618 df-iota 6415 df-fun 6465 df-fn 6466 df-f 6467 df-f1 6468 df-fo 6469 df-f1o 6470 df-fv 6471 df-riota 7270 df-ov 7316 df-oprab 7317 df-mpo 7318 df-1st 7874 df-2nd 7875 df-er 8544 df-en 8780 df-dom 8781 df-sdom 8782 df-pnf 11081 df-mnf 11082 df-xr 11083 df-ltxr 11084 df-le 11085 df-sub 11277 df-neg 11278 df-xneg 12918 df-xmul 12920 df-xdiv 31300

This theorem is referenced by: xdivrec 31309

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