Theorem rsqrmo 11771

Description: Uniqueness for the square root function. (Contributed by Jim Kingdon, 10-Aug-2021.)

Assertion

Ref	Expression
rsqrmo	⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → ∃*𝑥 ∈ ℝ ((𝑥↑2) = 𝐴 ∧ 0 ≤ 𝑥))

Distinct variable group:   𝑥,𝐴

Proof of Theorem rsqrmo

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simplrl 541	. . . . 5 ⊢ ((((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ)) ∧ (((𝑥↑2) = 𝐴 ∧ 0 ≤ 𝑥) ∧ ((𝑦↑2) = 𝐴 ∧ 0 ≤ 𝑦))) → 𝑥 ∈ ℝ)
2		simplrr 542	. . . . 5 ⊢ ((((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ)) ∧ (((𝑥↑2) = 𝐴 ∧ 0 ≤ 𝑥) ∧ ((𝑦↑2) = 𝐴 ∧ 0 ≤ 𝑦))) → 𝑦 ∈ ℝ)
3		simprlr 544	. . . . 5 ⊢ ((((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ)) ∧ (((𝑥↑2) = 𝐴 ∧ 0 ≤ 𝑥) ∧ ((𝑦↑2) = 𝐴 ∧ 0 ≤ 𝑦))) → 0 ≤ 𝑥)
4		simprrr 546	. . . . 5 ⊢ ((((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ)) ∧ (((𝑥↑2) = 𝐴 ∧ 0 ≤ 𝑥) ∧ ((𝑦↑2) = 𝐴 ∧ 0 ≤ 𝑦))) → 0 ≤ 𝑦)
5		simprll 543	. . . . . 6 ⊢ ((((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ)) ∧ (((𝑥↑2) = 𝐴 ∧ 0 ≤ 𝑥) ∧ ((𝑦↑2) = 𝐴 ∧ 0 ≤ 𝑦))) → (𝑥↑2) = 𝐴)
6		simprrl 545	. . . . . 6 ⊢ ((((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ)) ∧ (((𝑥↑2) = 𝐴 ∧ 0 ≤ 𝑥) ∧ ((𝑦↑2) = 𝐴 ∧ 0 ≤ 𝑦))) → (𝑦↑2) = 𝐴)
7	5, 6	eqtr4d 2274	. . . . 5 ⊢ ((((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ)) ∧ (((𝑥↑2) = 𝐴 ∧ 0 ≤ 𝑥) ∧ ((𝑦↑2) = 𝐴 ∧ 0 ≤ 𝑦))) → (𝑥↑2) = (𝑦↑2))
8	1, 2, 3, 4, 7	sq11d 11122	. . . 4 ⊢ ((((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ)) ∧ (((𝑥↑2) = 𝐴 ∧ 0 ≤ 𝑥) ∧ ((𝑦↑2) = 𝐴 ∧ 0 ≤ 𝑦))) → 𝑥 = 𝑦)
9	8	ex 115	. . 3 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ)) → ((((𝑥↑2) = 𝐴 ∧ 0 ≤ 𝑥) ∧ ((𝑦↑2) = 𝐴 ∧ 0 ≤ 𝑦)) → 𝑥 = 𝑦))
10	9	ralrimivva 2632	. 2 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → ∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ ((((𝑥↑2) = 𝐴 ∧ 0 ≤ 𝑥) ∧ ((𝑦↑2) = 𝐴 ∧ 0 ≤ 𝑦)) → 𝑥 = 𝑦))
11		oveq1 6082	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝑥↑2) = (𝑦↑2))
12	11	eqeq1d 2247	. . . 4 ⊢ (𝑥 = 𝑦 → ((𝑥↑2) = 𝐴 ↔ (𝑦↑2) = 𝐴))
13		breq2 4129	. . . 4 ⊢ (𝑥 = 𝑦 → (0 ≤ 𝑥 ↔ 0 ≤ 𝑦))
14	12, 13	anbi12d 477	. . 3 ⊢ (𝑥 = 𝑦 → (((𝑥↑2) = 𝐴 ∧ 0 ≤ 𝑥) ↔ ((𝑦↑2) = 𝐴 ∧ 0 ≤ 𝑦)))
15	14	rmo4 3019	. 2 ⊢ (∃*𝑥 ∈ ℝ ((𝑥↑2) = 𝐴 ∧ 0 ≤ 𝑥) ↔ ∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ ((((𝑥↑2) = 𝐴 ∧ 0 ≤ 𝑥) ∧ ((𝑦↑2) = 𝐴 ∧ 0 ≤ 𝑦)) → 𝑥 = 𝑦))
16	10, 15	sylibr 134	1 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → ∃*𝑥 ∈ ℝ ((𝑥↑2) = 𝐴 ∧ 0 ≤ 𝑥))

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1402   ∈ wcel 2209  ∀wral 2528  ∃*wrmo 2531   class class class wbr 4125  (class class class)co 6075  ℝcr 8168  0cc0 8169   ≤ cle 8351  2c2 9334  ↑cexp 10953

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4241  ax-sep 4244  ax-nul 4254  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-iinf 4730  ax-cnex 8260  ax-resscn 8261  ax-1cn 8262  ax-1re 8263  ax-icn 8264  ax-addcl 8265  ax-addrcl 8266  ax-mulcl 8267  ax-mulrcl 8268  ax-addcom 8269  ax-mulcom 8270  ax-addass 8271  ax-mulass 8272  ax-distr 8273  ax-i2m1 8274  ax-0lt1 8275  ax-1rid 8276  ax-0id 8277  ax-rnegex 8278  ax-precex 8279  ax-cnre 8280  ax-pre-ltirr 8281  ax-pre-ltwlin 8282  ax-pre-lttrn 8283  ax-pre-apti 8284  ax-pre-ltadd 8285  ax-pre-mulgt0 8286  ax-pre-mulext 8287

This theorem depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3636  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-iun 4009  df-br 4126  df-opab 4188  df-mpt 4189  df-tr 4225  df-id 4433  df-po 4436  df-iso 4437  df-iord 4506  df-on 4508  df-ilim 4509  df-suc 4511  df-iom 4733  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379  df-fv 5380  df-riota 6028  df-ov 6078  df-oprab 6079  df-mpo 6080  df-1st 6364  df-2nd 6365  df-recs 6566  df-frec 6652  df-pnf 8352  df-mnf 8353  df-xr 8354  df-ltxr 8355  df-le 8356  df-sub 8489  df-neg 8490  df-reap 8893  df-ap 8900  df-div 8993  df-inn 9284  df-2 9342  df-n0 9543  df-z 9624  df-uz 9901  df-seqfrec 10863  df-exp 10954

This theorem is referenced by:  rersqreu  11772