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Theorem addcanprg 6806

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 6804	. . . . 5 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ (𝐴 +_P 𝐵) = (𝐴 +_P 𝐶)) → (1^st ‘𝐵) ⊆ (1^st ‘𝐶))
2		3ancomb 927	. . . . . . 7 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ↔ (𝐴 ∈ P ∧ 𝐶 ∈ P ∧ 𝐵 ∈ P))
3		eqcom 2083	. . . . . . 7 ⊢ ((𝐴 +_P 𝐵) = (𝐴 +_P 𝐶) ↔ (𝐴 +_P 𝐶) = (𝐴 +_P 𝐵))
4	2, 3	anbi12i 447	. . . . . 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 6804	. . . . . 6 ⊢ (((𝐴 ∈ P ∧ 𝐶 ∈ P ∧ 𝐵 ∈ P) ∧ (𝐴 +_P 𝐶) = (𝐴 +_P 𝐵)) → (1^st ‘𝐶) ⊆ (1^st ‘𝐵))
6	4, 5	sylbi 119	. . . . 5 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ (𝐴 +_P 𝐵) = (𝐴 +_P 𝐶)) → (1^st ‘𝐶) ⊆ (1^st ‘𝐵))
7	1, 6	eqssd 3016	. . . 4 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ (𝐴 +_P 𝐵) = (𝐴 +_P 𝐶)) → (1^st ‘𝐵) = (1^st ‘𝐶))
8		addcanprlemu 6805	. . . . 5 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ (𝐴 +_P 𝐵) = (𝐴 +_P 𝐶)) → (2^nd ‘𝐵) ⊆ (2^nd ‘𝐶))
9		addcanprlemu 6805	. . . . . 6 ⊢ (((𝐴 ∈ P ∧ 𝐶 ∈ P ∧ 𝐵 ∈ P) ∧ (𝐴 +_P 𝐶) = (𝐴 +_P 𝐵)) → (2^nd ‘𝐶) ⊆ (2^nd ‘𝐵))
10	4, 9	sylbi 119	. . . . 5 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ (𝐴 +_P 𝐵) = (𝐴 +_P 𝐶)) → (2^nd ‘𝐶) ⊆ (2^nd ‘𝐵))
11	8, 10	eqssd 3016	. . . 4 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ (𝐴 +_P 𝐵) = (𝐴 +_P 𝐶)) → (2^nd ‘𝐵) = (2^nd ‘𝐶))
12	7, 11	jca 300	. . 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 6662	. . . . 5 ⊢ ((𝐵 ∈ P ∧ 𝐶 ∈ P) → (𝐵 = 𝐶 ↔ ((1^st ‘𝐵) = (1^st ‘𝐶) ∧ (2^nd ‘𝐵) = (2^nd ‘𝐶))))
14	13	3adant1 956	. . . 4 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) → (𝐵 = 𝐶 ↔ ((1^st ‘𝐵) = (1^st ‘𝐶) ∧ (2^nd ‘𝐵) = (2^nd ‘𝐶))))
15	14	adantr 270	. . 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 165	. 2 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ (𝐴 +_P 𝐵) = (𝐴 +_P 𝐶)) → 𝐵 = 𝐶)
17	16	ex 113	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 102 ↔ wb 103 ∧ w3a 919 = wceq 1284 ∈ wcel 1433 ⊆ wss 2973 ‘cfv 4922 (class class class)co 5532 1^st c1st 5785 2^nd c2nd 5786 Pcnp 6481 +_P cpp 6483

This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 576 ax-in2 577 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-13 1444 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-coll 3893 ax-sep 3896 ax-nul 3904 ax-pow 3948 ax-pr 3964 ax-un 4188 ax-setind 4280 ax-iinf 4329

This theorem depends on definitions: df-bi 115 df-dc 776 df-3or 920 df-3an 921 df-tru 1287 df-fal 1290 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ne 2246 df-ral 2353 df-rex 2354 df-reu 2355 df-rab 2357 df-v 2603 df-sbc 2816 df-csb 2909 df-dif 2975 df-un 2977 df-in 2979 df-ss 2986 df-nul 3252 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-int 3637 df-iun 3680 df-br 3786 df-opab 3840 df-mpt 3841 df-tr 3876 df-eprel 4044 df-id 4048 df-po 4051 df-iso 4052 df-iord 4121 df-on 4123 df-suc 4126 df-iom 4332 df-xp 4369 df-rel 4370 df-cnv 4371 df-co 4372 df-dm 4373 df-rn 4374 df-res 4375 df-ima 4376 df-iota 4887 df-fun 4924 df-fn 4925 df-f 4926 df-f1 4927 df-fo 4928 df-f1o 4929 df-fv 4930 df-ov 5535 df-oprab 5536 df-mpt2 5537 df-1st 5787 df-2nd 5788 df-recs 5943 df-irdg 5980 df-1o 6024 df-2o 6025 df-oadd 6028 df-omul 6029 df-er 6129 df-ec 6131 df-qs 6135 df-ni 6494 df-pli 6495 df-mi 6496 df-lti 6497 df-plpq 6534 df-mpq 6535 df-enq 6537 df-nqqs 6538 df-plqqs 6539 df-mqqs 6540 df-1nqqs 6541 df-rq 6542 df-ltnqqs 6543 df-enq0 6614 df-nq0 6615 df-0nq0 6616 df-plq0 6617 df-mq0 6618 df-inp 6656 df-iplp 6658

This theorem is referenced by: lteupri 6807 ltaprg 6809 enrer 6912 mulcmpblnr 6918 mulgt0sr 6954 srpospr 6959

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