xdivmul - Mathbox for Thierry Arnoux

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Theorem xdivmul 32916

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 32910	. . . . 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 2735	. 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 32913	. . . . 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 7363	. . . . 5 ⊢ (𝑥 = 𝐵 → (𝐶 ·_e 𝑥) = (𝐶 ·_e 𝐵))
10	9	eqeq1d 2735	. . . 4 ⊢ (𝑥 = 𝐵 → ((𝐶 ·_e 𝑥) = 𝐴 ↔ (𝐶 ·_e 𝐵) = 𝐴))
11	10	riota2 7337	. . 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 1541 ∈ wcel 2113 ≠ wne 2930 ∃!wreu 3346 ℩crio 7311 (class class class)co 7355 ℝcr 11015 0cc0 11016 ℝ^*cxr 11155 ·_e cxmu 13020 /_𝑒 cxdiv 32908

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 1911 ax-6 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2182 ax-ext 2705 ax-sep 5238 ax-nul 5248 ax-pow 5307 ax-pr 5374 ax-un 7677 ax-cnex 11072 ax-resscn 11073 ax-1cn 11074 ax-icn 11075 ax-addcl 11076 ax-addrcl 11077 ax-mulcl 11078 ax-mulrcl 11079 ax-mulcom 11080 ax-addass 11081 ax-mulass 11082 ax-distr 11083 ax-i2m1 11084 ax-1ne0 11085 ax-1rid 11086 ax-rnegex 11087 ax-rrecex 11088 ax-cnre 11089 ax-pre-lttri 11090 ax-pre-lttrn 11091 ax-pre-ltadd 11092 ax-pre-mulgt0 11093

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2725 df-clel 2808 df-nfc 2883 df-ne 2931 df-nel 3035 df-ral 3050 df-rex 3059 df-rmo 3348 df-reu 3349 df-rab 3398 df-v 3440 df-sbc 3739 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-nul 4285 df-if 4477 df-pw 4553 df-sn 4578 df-pr 4580 df-op 4584 df-uni 4861 df-iun 4945 df-br 5096 df-opab 5158 df-mpt 5177 df-id 5516 df-po 5529 df-so 5530 df-xp 5627 df-rel 5628 df-cnv 5629 df-co 5630 df-dm 5631 df-rn 5632 df-res 5633 df-ima 5634 df-iota 6445 df-fun 6491 df-fn 6492 df-f 6493 df-f1 6494 df-fo 6495 df-f1o 6496 df-fv 6497 df-riota 7312 df-ov 7358 df-oprab 7359 df-mpo 7360 df-1st 7930 df-2nd 7931 df-er 8631 df-en 8879 df-dom 8880 df-sdom 8881 df-pnf 11158 df-mnf 11159 df-xr 11160 df-ltxr 11161 df-le 11162 df-sub 11356 df-neg 11357 df-xneg 13021 df-xmul 13023 df-xdiv 32909

This theorem is referenced by: xdivrec 32918

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