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Theorem addcanprg 7771

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 7769	. . . . 5 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ (𝐴 +_P 𝐵) = (𝐴 +_P 𝐶)) → (1^st ‘𝐵) ⊆ (1^st ‘𝐶))
2		3ancomb 991	. . . . . . 7 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ↔ (𝐴 ∈ P ∧ 𝐶 ∈ P ∧ 𝐵 ∈ P))
3		eqcom 2211	. . . . . . 7 ⊢ ((𝐴 +_P 𝐵) = (𝐴 +_P 𝐶) ↔ (𝐴 +_P 𝐶) = (𝐴 +_P 𝐵))
4	2, 3	anbi12i 460	. . . . . 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 7769	. . . . . 6 ⊢ (((𝐴 ∈ P ∧ 𝐶 ∈ P ∧ 𝐵 ∈ P) ∧ (𝐴 +_P 𝐶) = (𝐴 +_P 𝐵)) → (1^st ‘𝐶) ⊆ (1^st ‘𝐵))
6	4, 5	sylbi 121	. . . . 5 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ (𝐴 +_P 𝐵) = (𝐴 +_P 𝐶)) → (1^st ‘𝐶) ⊆ (1^st ‘𝐵))
7	1, 6	eqssd 3221	. . . 4 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ (𝐴 +_P 𝐵) = (𝐴 +_P 𝐶)) → (1^st ‘𝐵) = (1^st ‘𝐶))
8		addcanprlemu 7770	. . . . 5 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ (𝐴 +_P 𝐵) = (𝐴 +_P 𝐶)) → (2^nd ‘𝐵) ⊆ (2^nd ‘𝐶))
9		addcanprlemu 7770	. . . . . 6 ⊢ (((𝐴 ∈ P ∧ 𝐶 ∈ P ∧ 𝐵 ∈ P) ∧ (𝐴 +_P 𝐶) = (𝐴 +_P 𝐵)) → (2^nd ‘𝐶) ⊆ (2^nd ‘𝐵))
10	4, 9	sylbi 121	. . . . 5 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ (𝐴 +_P 𝐵) = (𝐴 +_P 𝐶)) → (2^nd ‘𝐶) ⊆ (2^nd ‘𝐵))
11	8, 10	eqssd 3221	. . . 4 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ (𝐴 +_P 𝐵) = (𝐴 +_P 𝐶)) → (2^nd ‘𝐵) = (2^nd ‘𝐶))
12	7, 11	jca 306	. . 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 7627	. . . . 5 ⊢ ((𝐵 ∈ P ∧ 𝐶 ∈ P) → (𝐵 = 𝐶 ↔ ((1^st ‘𝐵) = (1^st ‘𝐶) ∧ (2^nd ‘𝐵) = (2^nd ‘𝐶))))
14	13	3adant1 1020	. . . 4 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) → (𝐵 = 𝐶 ↔ ((1^st ‘𝐵) = (1^st ‘𝐶) ∧ (2^nd ‘𝐵) = (2^nd ‘𝐶))))
15	14	adantr 276	. . 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 167	. 2 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ (𝐴 +_P 𝐵) = (𝐴 +_P 𝐶)) → 𝐵 = 𝐶)
17	16	ex 115	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 104 ↔ wb 105 ∧ w3a 983 = wceq 1375 ∈ wcel 2180 ⊆ wss 3177 ‘cfv 5294 (class class class)co 5974 1^st c1st 6254 2^nd c2nd 6255 Pcnp 7446 +_P cpp 7448

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 617 ax-in2 618 ax-io 713 ax-5 1473 ax-7 1474 ax-gen 1475 ax-ie1 1519 ax-ie2 1520 ax-8 1530 ax-10 1531 ax-11 1532 ax-i12 1533 ax-bndl 1535 ax-4 1536 ax-17 1552 ax-i9 1556 ax-ial 1560 ax-i5r 1561 ax-13 2182 ax-14 2183 ax-ext 2191 ax-coll 4178 ax-sep 4181 ax-nul 4189 ax-pow 4237 ax-pr 4272 ax-un 4501 ax-setind 4606 ax-iinf 4657

This theorem depends on definitions: df-bi 117 df-dc 839 df-3or 984 df-3an 985 df-tru 1378 df-fal 1381 df-nf 1487 df-sb 1789 df-eu 2060 df-mo 2061 df-clab 2196 df-cleq 2202 df-clel 2205 df-nfc 2341 df-ne 2381 df-ral 2493 df-rex 2494 df-reu 2495 df-rab 2497 df-v 2781 df-sbc 3009 df-csb 3105 df-dif 3179 df-un 3181 df-in 3183 df-ss 3190 df-nul 3472 df-pw 3631 df-sn 3652 df-pr 3653 df-op 3655 df-uni 3868 df-int 3903 df-iun 3946 df-br 4063 df-opab 4125 df-mpt 4126 df-tr 4162 df-eprel 4357 df-id 4361 df-po 4364 df-iso 4365 df-iord 4434 df-on 4436 df-suc 4439 df-iom 4660 df-xp 4702 df-rel 4703 df-cnv 4704 df-co 4705 df-dm 4706 df-rn 4707 df-res 4708 df-ima 4709 df-iota 5254 df-fun 5296 df-fn 5297 df-f 5298 df-f1 5299 df-fo 5300 df-f1o 5301 df-fv 5302 df-ov 5977 df-oprab 5978 df-mpo 5979 df-1st 6256 df-2nd 6257 df-recs 6421 df-irdg 6486 df-1o 6532 df-2o 6533 df-oadd 6536 df-omul 6537 df-er 6650 df-ec 6652 df-qs 6656 df-ni 7459 df-pli 7460 df-mi 7461 df-lti 7462 df-plpq 7499 df-mpq 7500 df-enq 7502 df-nqqs 7503 df-plqqs 7504 df-mqqs 7505 df-1nqqs 7506 df-rq 7507 df-ltnqqs 7508 df-enq0 7579 df-nq0 7580 df-0nq0 7581 df-plq0 7582 df-mq0 7583 df-inp 7621 df-iplp 7623

This theorem is referenced by: lteupri 7772 ltaprg 7774 enrer 7890 mulcmpblnr 7896 mulgt0sr 7933 srpospr 7938

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