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Theorem addcanprg 7646

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 7644	. . . . 5 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ (𝐴 +_P 𝐵) = (𝐴 +_P 𝐶)) → (1^st ‘𝐵) ⊆ (1^st ‘𝐶))
2		3ancomb 988	. . . . . . 7 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ↔ (𝐴 ∈ P ∧ 𝐶 ∈ P ∧ 𝐵 ∈ P))
3		eqcom 2191	. . . . . . 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 7644	. . . . . 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 3187	. . . 4 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ (𝐴 +_P 𝐵) = (𝐴 +_P 𝐶)) → (1^st ‘𝐵) = (1^st ‘𝐶))
8		addcanprlemu 7645	. . . . 5 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ (𝐴 +_P 𝐵) = (𝐴 +_P 𝐶)) → (2^nd ‘𝐵) ⊆ (2^nd ‘𝐶))
9		addcanprlemu 7645	. . . . . 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 3187	. . . 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 7502	. . . . 5 ⊢ ((𝐵 ∈ P ∧ 𝐶 ∈ P) → (𝐵 = 𝐶 ↔ ((1^st ‘𝐵) = (1^st ‘𝐶) ∧ (2^nd ‘𝐵) = (2^nd ‘𝐶))))
14	13	3adant1 1017	. . . 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 980 = wceq 1364 ∈ wcel 2160 ⊆ wss 3144 ‘cfv 5235 (class class class)co 5897 1^st c1st 6164 2^nd c2nd 6165 Pcnp 7321 +_P cpp 7323

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 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2162 ax-14 2163 ax-ext 2171 ax-coll 4133 ax-sep 4136 ax-nul 4144 ax-pow 4192 ax-pr 4227 ax-un 4451 ax-setind 4554 ax-iinf 4605

This theorem depends on definitions: df-bi 117 df-dc 836 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ne 2361 df-ral 2473 df-rex 2474 df-reu 2475 df-rab 2477 df-v 2754 df-sbc 2978 df-csb 3073 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-nul 3438 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-int 3860 df-iun 3903 df-br 4019 df-opab 4080 df-mpt 4081 df-tr 4117 df-eprel 4307 df-id 4311 df-po 4314 df-iso 4315 df-iord 4384 df-on 4386 df-suc 4389 df-iom 4608 df-xp 4650 df-rel 4651 df-cnv 4652 df-co 4653 df-dm 4654 df-rn 4655 df-res 4656 df-ima 4657 df-iota 5196 df-fun 5237 df-fn 5238 df-f 5239 df-f1 5240 df-fo 5241 df-f1o 5242 df-fv 5243 df-ov 5900 df-oprab 5901 df-mpo 5902 df-1st 6166 df-2nd 6167 df-recs 6331 df-irdg 6396 df-1o 6442 df-2o 6443 df-oadd 6446 df-omul 6447 df-er 6560 df-ec 6562 df-qs 6566 df-ni 7334 df-pli 7335 df-mi 7336 df-lti 7337 df-plpq 7374 df-mpq 7375 df-enq 7377 df-nqqs 7378 df-plqqs 7379 df-mqqs 7380 df-1nqqs 7381 df-rq 7382 df-ltnqqs 7383 df-enq0 7454 df-nq0 7455 df-0nq0 7456 df-plq0 7457 df-mq0 7458 df-inp 7496 df-iplp 7498

This theorem is referenced by: lteupri 7647 ltaprg 7649 enrer 7765 mulcmpblnr 7771 mulgt0sr 7808 srpospr 7813

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