xdivmul - Mathbox for Thierry Arnoux

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Theorem xdivmul 32853

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 32847	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐶 ∈ ℝ ∧ 𝐶 ≠ 0) → (𝐴 /_𝑒 𝐶) = (℩𝑥 ∈ ℝ^* (𝐶 ·_e 𝑥) = 𝐴))
2	1	3expb 1120	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ (𝐶 ∈ ℝ ∧ 𝐶 ≠ 0)) → (𝐴 /_𝑒 𝐶) = (℩𝑥 ∈ ℝ^* (𝐶 ·_e 𝑥) = 𝐴))
3	2	3adant2 1131	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ (𝐶 ∈ ℝ ∧ 𝐶 ≠ 0)) → (𝐴 /_𝑒 𝐶) = (℩𝑥 ∈ ℝ^* (𝐶 ·_e 𝑥) = 𝐴))
4	3	eqeq1d 2732	. 2 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ (𝐶 ∈ ℝ ∧ 𝐶 ≠ 0)) → ((𝐴 /_𝑒 𝐶) = 𝐵 ↔ (℩𝑥 ∈ ℝ^* (𝐶 ·_e 𝑥) = 𝐴) = 𝐵))
5		simp2 1137	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ (𝐶 ∈ ℝ ∧ 𝐶 ≠ 0)) → 𝐵 ∈ ℝ^*)
6		xreceu 32850	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐶 ∈ ℝ ∧ 𝐶 ≠ 0) → ∃!𝑥 ∈ ℝ^* (𝐶 ·_e 𝑥) = 𝐴)
7	6	3expb 1120	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ (𝐶 ∈ ℝ ∧ 𝐶 ≠ 0)) → ∃!𝑥 ∈ ℝ^* (𝐶 ·_e 𝑥) = 𝐴)
8	7	3adant2 1131	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ (𝐶 ∈ ℝ ∧ 𝐶 ≠ 0)) → ∃!𝑥 ∈ ℝ^* (𝐶 ·_e 𝑥) = 𝐴)
9		oveq2 7402	. . . . 5 ⊢ (𝑥 = 𝐵 → (𝐶 ·_e 𝑥) = (𝐶 ·_e 𝐵))
10	9	eqeq1d 2732	. . . 4 ⊢ (𝑥 = 𝐵 → ((𝐶 ·_e 𝑥) = 𝐴 ↔ (𝐶 ·_e 𝐵) = 𝐴))
11	10	riota2 7376	. . 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 282	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 206 ∧ wa 395 ∧ w3a 1086 = wceq 1540 ∈ wcel 2109 ≠ wne 2927 ∃!wreu 3355 ℩crio 7350 (class class class)co 7394 ℝcr 11085 0cc0 11086 ℝ^*cxr 11225 ·_e cxmu 13084 /_𝑒 cxdiv 32845

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 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2702 ax-sep 5259 ax-nul 5269 ax-pow 5328 ax-pr 5395 ax-un 7718 ax-cnex 11142 ax-resscn 11143 ax-1cn 11144 ax-icn 11145 ax-addcl 11146 ax-addrcl 11147 ax-mulcl 11148 ax-mulrcl 11149 ax-mulcom 11150 ax-addass 11151 ax-mulass 11152 ax-distr 11153 ax-i2m1 11154 ax-1ne0 11155 ax-1rid 11156 ax-rnegex 11157 ax-rrecex 11158 ax-cnre 11159 ax-pre-lttri 11160 ax-pre-lttrn 11161 ax-pre-ltadd 11162 ax-pre-mulgt0 11163

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2534 df-eu 2563 df-clab 2709 df-cleq 2722 df-clel 2804 df-nfc 2880 df-ne 2928 df-nel 3032 df-ral 3047 df-rex 3056 df-rmo 3357 df-reu 3358 df-rab 3412 df-v 3457 df-sbc 3762 df-csb 3871 df-dif 3925 df-un 3927 df-in 3929 df-ss 3939 df-nul 4305 df-if 4497 df-pw 4573 df-sn 4598 df-pr 4600 df-op 4604 df-uni 4880 df-iun 4965 df-br 5116 df-opab 5178 df-mpt 5197 df-id 5541 df-po 5554 df-so 5555 df-xp 5652 df-rel 5653 df-cnv 5654 df-co 5655 df-dm 5656 df-rn 5657 df-res 5658 df-ima 5659 df-iota 6472 df-fun 6521 df-fn 6522 df-f 6523 df-f1 6524 df-fo 6525 df-f1o 6526 df-fv 6527 df-riota 7351 df-ov 7397 df-oprab 7398 df-mpo 7399 df-1st 7977 df-2nd 7978 df-er 8682 df-en 8923 df-dom 8924 df-sdom 8925 df-pnf 11228 df-mnf 11229 df-xr 11230 df-ltxr 11231 df-le 11232 df-sub 11425 df-neg 11426 df-xneg 13085 df-xmul 13087 df-xdiv 32846

This theorem is referenced by: xdivrec 32855

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