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Theorem addcanprg 7990

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 7988	. . . . 5 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ (𝐴 +_P 𝐵) = (𝐴 +_P 𝐶)) → (1^st ‘𝐵) ⊆ (1^st ‘𝐶))
2		3ancomb 1017	. . . . . . 7 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ↔ (𝐴 ∈ P ∧ 𝐶 ∈ P ∧ 𝐵 ∈ P))
3		eqcom 2240	. . . . . . 7 ⊢ ((𝐴 +_P 𝐵) = (𝐴 +_P 𝐶) ↔ (𝐴 +_P 𝐶) = (𝐴 +_P 𝐵))
4	2, 3	anbi12i 464	. . . . . 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 7988	. . . . . 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 3265	. . . 4 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ (𝐴 +_P 𝐵) = (𝐴 +_P 𝐶)) → (1^st ‘𝐵) = (1^st ‘𝐶))
8		addcanprlemu 7989	. . . . 5 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ (𝐴 +_P 𝐵) = (𝐴 +_P 𝐶)) → (2^nd ‘𝐵) ⊆ (2^nd ‘𝐶))
9		addcanprlemu 7989	. . . . . 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 3265	. . . 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 7846	. . . . 5 ⊢ ((𝐵 ∈ P ∧ 𝐶 ∈ P) → (𝐵 = 𝐶 ↔ ((1^st ‘𝐵) = (1^st ‘𝐶) ∧ (2^nd ‘𝐵) = (2^nd ‘𝐶))))
14	13	3adant1 1046	. . . 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

This proof depends on syntax axioms: → wi 4 ∧ wa 104 ↔ wb 105 ∧ w3a 1009 = wceq 1402 ∈ wcel 2209 ⊆ wss 3220 ‘cfv 5380 (class class class)co 6088 1^st c1st 6376 2^nd c2nd 6377 Pcnp 7665 +_P cpp 7667

This proof depends on 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 4248 ax-sep 4251 ax-nul 4261 ax-pow 4313 ax-pr 4348 ax-un 4580 ax-setind 4686 ax-iinf 4737

This proof 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-ral 2533 df-rex 2534 df-reu 2535 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-pw 3690 df-sn 3715 df-pr 3716 df-op 3718 df-uni 3936 df-int 3971 df-iun 4014 df-br 4132 df-opab 4194 df-mpt 4195 df-tr 4232 df-eprel 4436 df-id 4440 df-po 4443 df-iso 4444 df-iord 4513 df-on 4515 df-suc 4518 df-iom 4740 df-xp 4782 df-rel 4783 df-cnv 4784 df-co 4785 df-dm 4786 df-rn 4787 df-res 4788 df-ima 4789 df-iota 5340 df-fun 5382 df-fn 5383 df-f 5384 df-f1 5385 df-fo 5386 df-f1o 5387 df-fv 5388 df-ov 6091 df-oprab 6092 df-mpo 6093 df-1st 6378 df-2nd 6379 df-recs 6580 df-irdg 6645 df-1o 6691 df-2o 6692 df-oadd 6695 df-omul 6696 df-er 6811 df-ec 6813 df-qs 6817 df-ni 7678 df-pli 7679 df-mi 7680 df-lti 7681 df-plpq 7718 df-mpq 7719 df-enq 7721 df-nqqs 7722 df-plqqs 7723 df-mqqs 7724 df-1nqqs 7725 df-rq 7726 df-ltnqqs 7727 df-enq0 7798 df-nq0 7799 df-0nq0 7800 df-plq0 7801 df-mq0 7802 df-inp 7840 df-iplp 7842

This theorem is used by: lteupri 7991 ltaprg 7993 enrer 8109 mulcmpblnr 8115 mulgt0sr 8152 srpospr 8157

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