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Theorem addcanprg 7417

Description: Addition cancellation law for positive reals. Proposition 9-3.5(vi) of [Gleason] p. 123. (Contributed by Jim Kingdon, 24-Dec-2019.)

**Assertion**
Ref	Expression
addcanprg	⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) → ((𝐴 +_P 𝐵) = (𝐴 +_P 𝐶) → 𝐵 = 𝐶))

**Proof of Theorem addcanprg**
Step	Hyp	Ref	Expression
1		addcanprleml 7415	. . . . 5 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ (𝐴 +_P 𝐵) = (𝐴 +_P 𝐶)) → (1^st ‘𝐵) ⊆ (1^st ‘𝐶))
2		3ancomb 970	. . . . . . 7 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ↔ (𝐴 ∈ P ∧ 𝐶 ∈ P ∧ 𝐵 ∈ P))
3		eqcom 2139	. . . . . . 7 ⊢ ((𝐴 +_P 𝐵) = (𝐴 +_P 𝐶) ↔ (𝐴 +_P 𝐶) = (𝐴 +_P 𝐵))
4	2, 3	anbi12i 455	. . . . . 6 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ (𝐴 +_P 𝐵) = (𝐴 +_P 𝐶)) ↔ ((𝐴 ∈ P ∧ 𝐶 ∈ P ∧ 𝐵 ∈ P) ∧ (𝐴 +_P 𝐶) = (𝐴 +_P 𝐵)))
5		addcanprleml 7415	. . . . . 6 ⊢ (((𝐴 ∈ P ∧ 𝐶 ∈ P ∧ 𝐵 ∈ P) ∧ (𝐴 +_P 𝐶) = (𝐴 +_P 𝐵)) → (1^st ‘𝐶) ⊆ (1^st ‘𝐵))
6	4, 5	sylbi 120	. . . . 5 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ (𝐴 +_P 𝐵) = (𝐴 +_P 𝐶)) → (1^st ‘𝐶) ⊆ (1^st ‘𝐵))
7	1, 6	eqssd 3109	. . . 4 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ (𝐴 +_P 𝐵) = (𝐴 +_P 𝐶)) → (1^st ‘𝐵) = (1^st ‘𝐶))
8		addcanprlemu 7416	. . . . 5 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ (𝐴 +_P 𝐵) = (𝐴 +_P 𝐶)) → (2^nd ‘𝐵) ⊆ (2^nd ‘𝐶))
9		addcanprlemu 7416	. . . . . 6 ⊢ (((𝐴 ∈ P ∧ 𝐶 ∈ P ∧ 𝐵 ∈ P) ∧ (𝐴 +_P 𝐶) = (𝐴 +_P 𝐵)) → (2^nd ‘𝐶) ⊆ (2^nd ‘𝐵))
10	4, 9	sylbi 120	. . . . 5 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ (𝐴 +_P 𝐵) = (𝐴 +_P 𝐶)) → (2^nd ‘𝐶) ⊆ (2^nd ‘𝐵))
11	8, 10	eqssd 3109	. . . 4 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ (𝐴 +_P 𝐵) = (𝐴 +_P 𝐶)) → (2^nd ‘𝐵) = (2^nd ‘𝐶))
12	7, 11	jca 304	. . 3 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ (𝐴 +_P 𝐵) = (𝐴 +_P 𝐶)) → ((1^st ‘𝐵) = (1^st ‘𝐶) ∧ (2^nd ‘𝐵) = (2^nd ‘𝐶)))
13		preqlu 7273	. . . . 5 ⊢ ((𝐵 ∈ P ∧ 𝐶 ∈ P) → (𝐵 = 𝐶 ↔ ((1^st ‘𝐵) = (1^st ‘𝐶) ∧ (2^nd ‘𝐵) = (2^nd ‘𝐶))))
14	13	3adant1 999	. . . 4 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) → (𝐵 = 𝐶 ↔ ((1^st ‘𝐵) = (1^st ‘𝐶) ∧ (2^nd ‘𝐵) = (2^nd ‘𝐶))))
15	14	adantr 274	. . 3 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ (𝐴 +_P 𝐵) = (𝐴 +_P 𝐶)) → (𝐵 = 𝐶 ↔ ((1^st ‘𝐵) = (1^st ‘𝐶) ∧ (2^nd ‘𝐵) = (2^nd ‘𝐶))))
16	12, 15	mpbird 166	. 2 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ (𝐴 +_P 𝐵) = (𝐴 +_P 𝐶)) → 𝐵 = 𝐶)
17	16	ex 114	1 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) → ((𝐴 +_P 𝐵) = (𝐴 +_P 𝐶) → 𝐵 = 𝐶))

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 ∧ w3a 962 = wceq 1331 ∈ wcel 1480 ⊆ wss 3066 ‘cfv 5118 (class class class)co 5767 1^st c1st 6029 2^nd c2nd 6030 Pcnp 7092 +_P cpp 7094

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-coll 4038 ax-sep 4041 ax-nul 4049 ax-pow 4093 ax-pr 4126 ax-un 4350 ax-setind 4447 ax-iinf 4497

This theorem depends on definitions: df-bi 116 df-dc 820 df-3or 963 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ne 2307 df-ral 2419 df-rex 2420 df-reu 2421 df-rab 2423 df-v 2683 df-sbc 2905 df-csb 2999 df-dif 3068 df-un 3070 df-in 3072 df-ss 3079 df-nul 3359 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-int 3767 df-iun 3810 df-br 3925 df-opab 3985 df-mpt 3986 df-tr 4022 df-eprel 4206 df-id 4210 df-po 4213 df-iso 4214 df-iord 4283 df-on 4285 df-suc 4288 df-iom 4500 df-xp 4540 df-rel 4541 df-cnv 4542 df-co 4543 df-dm 4544 df-rn 4545 df-res 4546 df-ima 4547 df-iota 5083 df-fun 5120 df-fn 5121 df-f 5122 df-f1 5123 df-fo 5124 df-f1o 5125 df-fv 5126 df-ov 5770 df-oprab 5771 df-mpo 5772 df-1st 6031 df-2nd 6032 df-recs 6195 df-irdg 6260 df-1o 6306 df-2o 6307 df-oadd 6310 df-omul 6311 df-er 6422 df-ec 6424 df-qs 6428 df-ni 7105 df-pli 7106 df-mi 7107 df-lti 7108 df-plpq 7145 df-mpq 7146 df-enq 7148 df-nqqs 7149 df-plqqs 7150 df-mqqs 7151 df-1nqqs 7152 df-rq 7153 df-ltnqqs 7154 df-enq0 7225 df-nq0 7226 df-0nq0 7227 df-plq0 7228 df-mq0 7229 df-inp 7267 df-iplp 7269

This theorem is referenced by: lteupri 7418 ltaprg 7420 enrer 7536 mulcmpblnr 7542 mulgt0sr 7579 srpospr 7584

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