addclpr - Intuitionistic Logic Explorer

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Theorem addclpr 7910

Description: Closure of addition on positive reals. First statement of Proposition 9-3.5 of [Gleason] p. 123. Combination of Lemma 11.13 and Lemma 11.16 in [BauerTaylor], p. 53. (Contributed by NM, 13-Mar-1996.)

**Assertion**
Ref	Expression
addclpr	⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝐴 +_P 𝐵) ∈ P)

Proof of Theorem addclpr Dummy variables 𝑥 𝑦 𝑧 𝑤 𝑣 𝑔 ℎ 𝑞 𝑟 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-iplp 7841	. . . 4 ⊢ +_P = (𝑤 ∈ P, 𝑣 ∈ P ↦ ⟨{𝑥 ∈ Q ∣ ∃𝑦 ∈ Q ∃𝑧 ∈ Q (𝑦 ∈ (1^st ‘𝑤) ∧ 𝑧 ∈ (1^st ‘𝑣) ∧ 𝑥 = (𝑦 +_Q 𝑧))}, {𝑥 ∈ Q ∣ ∃𝑦 ∈ Q ∃𝑧 ∈ Q (𝑦 ∈ (2^nd ‘𝑤) ∧ 𝑧 ∈ (2^nd ‘𝑣) ∧ 𝑥 = (𝑦 +_Q 𝑧))}⟩)
2	1	genpelxp 7884	. . 3 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝐴 +_P 𝐵) ∈ (𝒫 Q × 𝒫 Q))
3		addclnq 7748	. . . 4 ⊢ ((𝑦 ∈ Q ∧ 𝑧 ∈ Q) → (𝑦 +_Q 𝑧) ∈ Q)
4	1, 3	genpml 7890	. . 3 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → ∃𝑞 ∈ Q 𝑞 ∈ (1^st ‘(𝐴 +_P 𝐵)))
5	1, 3	genpmu 7891	. . 3 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → ∃𝑟 ∈ Q 𝑟 ∈ (2^nd ‘(𝐴 +_P 𝐵)))
6	2, 4, 5	jca32 310	. 2 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → ((𝐴 +_P 𝐵) ∈ (𝒫 Q × 𝒫 Q) ∧ (∃𝑞 ∈ Q 𝑞 ∈ (1^st ‘(𝐴 +_P 𝐵)) ∧ ∃𝑟 ∈ Q 𝑟 ∈ (2^nd ‘(𝐴 +_P 𝐵)))))
7		ltanqg 7773	. . . . 5 ⊢ ((𝑥 ∈ Q ∧ 𝑦 ∈ Q ∧ 𝑧 ∈ Q) → (𝑥 <_Q 𝑦 ↔ (𝑧 +_Q 𝑥) <_Q (𝑧 +_Q 𝑦)))
8		addcomnqg 7754	. . . . 5 ⊢ ((𝑥 ∈ Q ∧ 𝑦 ∈ Q) → (𝑥 +_Q 𝑦) = (𝑦 +_Q 𝑥))
9		addnqprl 7902	. . . . 5 ⊢ ((((𝐴 ∈ P ∧ 𝑔 ∈ (1^st ‘𝐴)) ∧ (𝐵 ∈ P ∧ ℎ ∈ (1^st ‘𝐵))) ∧ 𝑥 ∈ Q) → (𝑥 <_Q (𝑔 +_Q ℎ) → 𝑥 ∈ (1^st ‘(𝐴 +_P 𝐵))))
10	1, 3, 7, 8, 9	genprndl 7894	. . . 4 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → ∀𝑞 ∈ Q (𝑞 ∈ (1^st ‘(𝐴 +_P 𝐵)) ↔ ∃𝑟 ∈ Q (𝑞 <_Q 𝑟 ∧ 𝑟 ∈ (1^st ‘(𝐴 +_P 𝐵)))))
11		addnqpru 7903	. . . . 5 ⊢ ((((𝐴 ∈ P ∧ 𝑔 ∈ (2^nd ‘𝐴)) ∧ (𝐵 ∈ P ∧ ℎ ∈ (2^nd ‘𝐵))) ∧ 𝑥 ∈ Q) → ((𝑔 +_Q ℎ) <_Q 𝑥 → 𝑥 ∈ (2^nd ‘(𝐴 +_P 𝐵))))
12	1, 3, 7, 8, 11	genprndu 7895	. . . 4 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → ∀𝑟 ∈ Q (𝑟 ∈ (2^nd ‘(𝐴 +_P 𝐵)) ↔ ∃𝑞 ∈ Q (𝑞 <_Q 𝑟 ∧ 𝑞 ∈ (2^nd ‘(𝐴 +_P 𝐵)))))
13	10, 12	jca 306	. . 3 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (∀𝑞 ∈ Q (𝑞 ∈ (1^st ‘(𝐴 +_P 𝐵)) ↔ ∃𝑟 ∈ Q (𝑞 <_Q 𝑟 ∧ 𝑟 ∈ (1^st ‘(𝐴 +_P 𝐵)))) ∧ ∀𝑟 ∈ Q (𝑟 ∈ (2^nd ‘(𝐴 +_P 𝐵)) ↔ ∃𝑞 ∈ Q (𝑞 <_Q 𝑟 ∧ 𝑞 ∈ (2^nd ‘(𝐴 +_P 𝐵))))))
14	1, 3, 7, 8	genpdisj 7896	. . 3 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → ∀𝑞 ∈ Q ¬ (𝑞 ∈ (1^st ‘(𝐴 +_P 𝐵)) ∧ 𝑞 ∈ (2^nd ‘(𝐴 +_P 𝐵))))
15		addlocpr 7909	. . 3 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → ∀𝑞 ∈ Q ∀𝑟 ∈ Q (𝑞 <_Q 𝑟 → (𝑞 ∈ (1^st ‘(𝐴 +_P 𝐵)) ∨ 𝑟 ∈ (2^nd ‘(𝐴 +_P 𝐵)))))
16	13, 14, 15	3jca 1208	. 2 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → ((∀𝑞 ∈ Q (𝑞 ∈ (1^st ‘(𝐴 +_P 𝐵)) ↔ ∃𝑟 ∈ Q (𝑞 <_Q 𝑟 ∧ 𝑟 ∈ (1^st ‘(𝐴 +_P 𝐵)))) ∧ ∀𝑟 ∈ Q (𝑟 ∈ (2^nd ‘(𝐴 +_P 𝐵)) ↔ ∃𝑞 ∈ Q (𝑞 <_Q 𝑟 ∧ 𝑞 ∈ (2^nd ‘(𝐴 +_P 𝐵))))) ∧ ∀𝑞 ∈ Q ¬ (𝑞 ∈ (1^st ‘(𝐴 +_P 𝐵)) ∧ 𝑞 ∈ (2^nd ‘(𝐴 +_P 𝐵))) ∧ ∀𝑞 ∈ Q ∀𝑟 ∈ Q (𝑞 <_Q 𝑟 → (𝑞 ∈ (1^st ‘(𝐴 +_P 𝐵)) ∨ 𝑟 ∈ (2^nd ‘(𝐴 +_P 𝐵))))))
17		elnp1st2nd 7849	. 2 ⊢ ((𝐴 +_P 𝐵) ∈ P ↔ (((𝐴 +_P 𝐵) ∈ (𝒫 Q × 𝒫 Q) ∧ (∃𝑞 ∈ Q 𝑞 ∈ (1^st ‘(𝐴 +_P 𝐵)) ∧ ∃𝑟 ∈ Q 𝑟 ∈ (2^nd ‘(𝐴 +_P 𝐵)))) ∧ ((∀𝑞 ∈ Q (𝑞 ∈ (1^st ‘(𝐴 +_P 𝐵)) ↔ ∃𝑟 ∈ Q (𝑞 <_Q 𝑟 ∧ 𝑟 ∈ (1^st ‘(𝐴 +_P 𝐵)))) ∧ ∀𝑟 ∈ Q (𝑟 ∈ (2^nd ‘(𝐴 +_P 𝐵)) ↔ ∃𝑞 ∈ Q (𝑞 <_Q 𝑟 ∧ 𝑞 ∈ (2^nd ‘(𝐴 +_P 𝐵))))) ∧ ∀𝑞 ∈ Q ¬ (𝑞 ∈ (1^st ‘(𝐴 +_P 𝐵)) ∧ 𝑞 ∈ (2^nd ‘(𝐴 +_P 𝐵))) ∧ ∀𝑞 ∈ Q ∀𝑟 ∈ Q (𝑞 <_Q 𝑟 → (𝑞 ∈ (1^st ‘(𝐴 +_P 𝐵)) ∨ 𝑟 ∈ (2^nd ‘(𝐴 +_P 𝐵)))))))
18	6, 16, 17	sylanbrc 421	1 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝐴 +_P 𝐵) ∈ P)

Colors of variables: wff set class

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ∧ wa 104 ↔ wb 105 ∨ wo 720 ∧ w3a 1009 ∈ wcel 2209 ∀wral 2528 ∃wrex 2529 𝒫 cpw 3688 class class class wbr 4131 × cxp 4774 ‘cfv 5380 (class class class)co 6088 1^st c1st 6376 2^nd c2nd 6377 Qcnq 7653 +_Q cplq 7655 <_Q cltq 7658 Pcnp 7664 +_P cpp 7666

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 7677 df-pli 7678 df-mi 7679 df-lti 7680 df-plpq 7717 df-mpq 7718 df-enq 7720 df-nqqs 7721 df-plqqs 7722 df-mqqs 7723 df-1nqqs 7724 df-rq 7725 df-ltnqqs 7726 df-enq0 7797 df-nq0 7798 df-0nq0 7799 df-plq0 7800 df-mq0 7801 df-inp 7839 df-iplp 7841

This theorem is used by: addnqprlemfl 7932 addnqprlemfu 7933 addnqpr 7934 addassprg 7952 distrlem1prl 7955 distrlem1pru 7956 distrlem4prl 7957 distrlem4pru 7958 distrprg 7961 ltaddpr 7970 ltexpri 7986 addcanprleml 7987 addcanprlemu 7988 ltaprlem 7991 ltaprg 7992 prplnqu 7993 addextpr 7994 caucvgprlemcanl 8017 cauappcvgprlemladdru 8029 cauappcvgprlemladdrl 8030 cauappcvgprlemladd 8031 cauappcvgprlem1 8032 caucvgprlemladdrl 8051 caucvgprlem1 8052 caucvgprprlemnbj 8066 caucvgprprlemopu 8072 caucvgprprlemloc 8076 caucvgprprlemexbt 8079 caucvgprprlemexb 8080 caucvgprprlemaddq 8081 caucvgprprlem2 8083 enrer 8108 addcmpblnr 8112 mulcmpblnrlemg 8113 mulcmpblnr 8114 ltsrprg 8120 1sr 8124 m1r 8125 addclsr 8126 mulclsr 8127 addasssrg 8129 mulasssrg 8131 distrsrg 8132 m1p1sr 8133 m1m1sr 8134 lttrsr 8135 ltsosr 8137 0lt1sr 8138 0idsr 8140 1idsr 8141 00sr 8142 ltasrg 8143 recexgt0sr 8146 mulgt0sr 8151 aptisr 8152 mulextsr1lem 8153 mulextsr1 8154 archsr 8155 srpospr 8156 prsrcl 8157 prsradd 8159 prsrlt 8160 caucvgsrlemcau 8166 caucvgsrlemgt1 8168 mappsrprg 8177 map2psrprg 8178 pitonnlem1p1 8219 pitonnlem2 8220 pitonn 8221 pitoregt0 8222 pitore 8223 recnnre 8224 recidpirqlemcalc 8230 recidpirq 8231

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