xdivmul - Mathbox for Thierry Arnoux

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Theorem xdivmul 33016

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 33010	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐶 ∈ ℝ ∧ 𝐶 ≠ 0) → (𝐴 /_𝑒 𝐶) = (℩𝑥 ∈ ℝ^* (𝐶 ·_e 𝑥) = 𝐴))
2	1	3expb 1121	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ (𝐶 ∈ ℝ ∧ 𝐶 ≠ 0)) → (𝐴 /_𝑒 𝐶) = (℩𝑥 ∈ ℝ^* (𝐶 ·_e 𝑥) = 𝐴))
3	2	3adant2 1132	. . 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 1138	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ (𝐶 ∈ ℝ ∧ 𝐶 ≠ 0)) → 𝐵 ∈ ℝ^*)
6		xreceu 33013	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐶 ∈ ℝ ∧ 𝐶 ≠ 0) → ∃!𝑥 ∈ ℝ^* (𝐶 ·_e 𝑥) = 𝐴)
7	6	3expb 1121	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ (𝐶 ∈ ℝ ∧ 𝐶 ≠ 0)) → ∃!𝑥 ∈ ℝ^* (𝐶 ·_e 𝑥) = 𝐴)
8	7	3adant2 1132	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ (𝐶 ∈ ℝ ∧ 𝐶 ≠ 0)) → ∃!𝑥 ∈ ℝ^* (𝐶 ·_e 𝑥) = 𝐴)
9		oveq2 7376	. . . . 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 7350	. . 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 585	. 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 1087 = wceq 1542 ∈ wcel 2114 ≠ wne 2933 ∃!wreu 3350 ℩crio 7324 (class class class)co 7368 ℝcr 11037 0cc0 11038 ℝ^*cxr 11177 ·_e cxmu 13037 /_𝑒 cxdiv 33008

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-sep 5243 ax-nul 5253 ax-pow 5312 ax-pr 5379 ax-un 7690 ax-cnex 11094 ax-resscn 11095 ax-1cn 11096 ax-icn 11097 ax-addcl 11098 ax-addrcl 11099 ax-mulcl 11100 ax-mulrcl 11101 ax-mulcom 11102 ax-addass 11103 ax-mulass 11104 ax-distr 11105 ax-i2m1 11106 ax-1ne0 11107 ax-1rid 11108 ax-rnegex 11109 ax-rrecex 11110 ax-cnre 11111 ax-pre-lttri 11112 ax-pre-lttrn 11113 ax-pre-ltadd 11114 ax-pre-mulgt0 11115

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rmo 3352 df-reu 3353 df-rab 3402 df-v 3444 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-nul 4288 df-if 4482 df-pw 4558 df-sn 4583 df-pr 4585 df-op 4589 df-uni 4866 df-iun 4950 df-br 5101 df-opab 5163 df-mpt 5182 df-id 5527 df-po 5540 df-so 5541 df-xp 5638 df-rel 5639 df-cnv 5640 df-co 5641 df-dm 5642 df-rn 5643 df-res 5644 df-ima 5645 df-iota 6456 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-riota 7325 df-ov 7371 df-oprab 7372 df-mpo 7373 df-1st 7943 df-2nd 7944 df-er 8645 df-en 8896 df-dom 8897 df-sdom 8898 df-pnf 11180 df-mnf 11181 df-xr 11182 df-ltxr 11183 df-le 11184 df-sub 11378 df-neg 11379 df-xneg 13038 df-xmul 13040 df-xdiv 33009

This theorem is referenced by: xdivrec 33018

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