Theorem 1lt2pq 3872

Description: One is less than two (one plus one).

Assertion

Ref	Expression
1lt2pq	⊢ 1Q <Q (1Q +Q 1Q)

Proof of Theorem 1lt2pq

Step	Hyp	Ref	Expression
1		1lt2pi 3826	. . . . 5 ⊢ 1o <N (1o +N 1o)
2		1pi 3805	. . . . . 6 ⊢ 1o ∈ N
3		mulidpi 3808	. . . . . 6 ⊢ (1o ∈ N → (1o ·N 1o) = 1o)
4	2, 3	ax-mp 6	. . . . 5 ⊢ (1o ·N 1o) = 1o
5	4, 4	opreq12i 3011	. . . . 5 ⊢ ((1o ·N 1o) +N (1o ·N 1o)) = (1o +N 1o)
6	1, 4, 5	3brtr4 2085	. . . 4 ⊢ (1o ·N 1o) <N ((1o ·N 1o) +N (1o ·N 1o))
7		oprex 3018	. . . . . 6 ⊢ (1o ·N 1o) ∈ V
8		oprex 3018	. . . . . 6 ⊢ ((1o ·N 1o) +N (1o ·N 1o)) ∈ V
9	7, 8	ltmpi 3825	. . . . 5 ⊢ (1o ∈ N → ((1o ·N 1o) <N ((1o ·N 1o) +N (1o ·N 1o)) ↔ (1o ·N (1o ·N 1o)) <N (1o ·N ((1o ·N 1o) +N (1o ·N 1o)))))
10	2, 9	ax-mp 6	. . . 4 ⊢ ((1o ·N 1o) <N ((1o ·N 1o) +N (1o ·N 1o)) ↔ (1o ·N (1o ·N 1o)) <N (1o ·N ((1o ·N 1o) +N (1o ·N 1o))))
11	6, 10	mpbi 164	. . 3 ⊢ (1o ·N (1o ·N 1o)) <N (1o ·N ((1o ·N 1o) +N (1o ·N 1o)))
12	2	elisseti 1355	. . . 4 ⊢ 1o ∈ V
13	12, 12, 8, 7	ordpipq 3850	. . 3 ⊢ ([⟨1o, 1o⟩] ~Q <Q [⟨((1o ·N 1o) +N (1o ·N 1o)), (1o ·N 1o)⟩] ~Q ↔ (1o ·N (1o ·N 1o)) <N (1o ·N ((1o ·N 1o) +N (1o ·N 1o))))
14	11, 13	mpbir 165	. 2 ⊢ [⟨1o, 1o⟩] ~Q <Q [⟨((1o ·N 1o) +N (1o ·N 1o)), (1o ·N 1o)⟩] ~Q
15		df-1q 3837	. 2 ⊢ 1Q = [⟨1o, 1o⟩] ~Q
16	15, 15	opreq12i 3011	. . 3 ⊢ (1Q +Q 1Q) = ([⟨1o, 1o⟩] ~Q +Q [⟨1o, 1o⟩] ~Q )
17	2, 2	pm3.2i 234	. . . 4 ⊢ (1o ∈ N ∧ 1o ∈ N)
18		addpipq 3848	. . . 4 ⊢ (((1o ∈ N ∧ 1o ∈ N) ∧ (1o ∈ N ∧ 1o ∈ N)) → ([⟨1o, 1o⟩] ~Q +Q [⟨1o, 1o⟩] ~Q ) = [⟨((1o ·N 1o) +N (1o ·N 1o)), (1o ·N 1o)⟩] ~Q )
19	17, 17, 18	mp2an 520	. . 3 ⊢ ([⟨1o, 1o⟩] ~Q +Q [⟨1o, 1o⟩] ~Q ) = [⟨((1o ·N 1o) +N (1o ·N 1o)), (1o ·N 1o)⟩] ~Q
20	16, 19	eqtr 1119	. 2 ⊢ (1Q +Q 1Q) = [⟨((1o ·N 1o) +N (1o ·N 1o)), (1o ·N 1o)⟩] ~Q
21	14, 15, 20	3brtr4 2085	1 ⊢ 1Q <Q (1Q +Q 1Q)

Syntax hints:   ↔ wb 127   ∧ wa 196   = wceq 1091   ∈ wcel 1092  ⟨cop 1810   class class class wbr 2054  (class class class)co 3001  1_oc1o 3099  [cec 3198  Ncnpi 3766   +_N cpli 3767   ·_N cmi 3768   <_N clti 3769   ~_Q ceq 3772  1_Qc1q 3774   +_Q cplq 3775   <_Q cltq 3778

This theorem is referenced by:  ltaddpq 3873  1pr 3911

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-un 1076  ax-pow 1077  ax-reg 1078  ax-inf 1079

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3or 582  df-3an 583  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-ne 1192  df-ral 1205  df-rex 1206  df-reu 1207  df-rab 1208  df-v 1349  df-sbc 1441  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-pss 1494  df-nul 1708  df-if 1777  df-pw 1799  df-sn 1811  df-pr 1812  df-tp 1814  df-op 1815  df-uni 1920  df-int 1966  df-iun 1996  df-tr 2042  df-br 2063  df-opab 2098  df-eprel 2122  df-id 2125  df-po 2128  df-so 2138  df-fr 2169  df-we 2186  df-ord 2202  df-on 2203  df-lim 2204  df-suc 2205  df-om 2373  df-xp 2424  df-rel 2425  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fun 2432  df-fn 2433  df-f 2434  df-f1 2435  df-fv 2438  df-rdg 2970  df-opr 3003  df-oprab 3004  df-1o 3104  df-oadd 3106  df-omul 3107  df-er 3200  df-ec 3202  df-qs 3205  df-ni 3794  df-pli 3795  df-mi 3796  df-lti 3797  df-plpq 3829  df-enq 3831  df-nq 3832  df-plq 3833  df-ltq 3836  df-1q 3837

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