Theorem ltaprg 7934

Description: Ordering property of addition. Proposition 9-3.5(v) of [Gleason] p. 123. (Contributed by Jim Kingdon, 26-Dec-2019.)

Assertion

Ref	Expression
ltaprg	⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) → (𝐴<P 𝐵 ↔ (𝐶 +P 𝐴)<P (𝐶 +P 𝐵)))

Proof of Theorem ltaprg

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ltaprlem 7933	. . 3 ⊢ (𝐶 ∈ P → (𝐴<P 𝐵 → (𝐶 +P 𝐴)<P (𝐶 +P 𝐵)))
2	1	3ad2ant3 1047	. 2 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) → (𝐴<P 𝐵 → (𝐶 +P 𝐴)<P (𝐶 +P 𝐵)))
3		ltexpri 7928	. . . . 5 ⊢ ((𝐶 +P 𝐴)<P (𝐶 +P 𝐵) → ∃𝑥 ∈ P ((𝐶 +P 𝐴) +P 𝑥) = (𝐶 +P 𝐵))
4	3	adantl 277	. . . 4 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ (𝐶 +P 𝐴)<P (𝐶 +P 𝐵)) → ∃𝑥 ∈ P ((𝐶 +P 𝐴) +P 𝑥) = (𝐶 +P 𝐵))
5		simpl1 1027	. . . . . . 7 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ (𝑥 ∈ P ∧ ((𝐶 +P 𝐴) +P 𝑥) = (𝐶 +P 𝐵))) → 𝐴 ∈ P)
6		simprl 531	. . . . . . 7 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ (𝑥 ∈ P ∧ ((𝐶 +P 𝐴) +P 𝑥) = (𝐶 +P 𝐵))) → 𝑥 ∈ P)
7		ltaddpr 7912	. . . . . . 7 ⊢ ((𝐴 ∈ P ∧ 𝑥 ∈ P) → 𝐴<P (𝐴 +P 𝑥))
8	5, 6, 7	syl2anc 411	. . . . . 6 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ (𝑥 ∈ P ∧ ((𝐶 +P 𝐴) +P 𝑥) = (𝐶 +P 𝐵))) → 𝐴<P (𝐴 +P 𝑥))
9		addassprg 7894	. . . . . . . . . . . 12 ⊢ ((𝐶 ∈ P ∧ 𝐴 ∈ P ∧ 𝑥 ∈ P) → ((𝐶 +P 𝐴) +P 𝑥) = (𝐶 +P (𝐴 +P 𝑥)))
10	9	3com12 1234	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ P ∧ 𝐶 ∈ P ∧ 𝑥 ∈ P) → ((𝐶 +P 𝐴) +P 𝑥) = (𝐶 +P (𝐴 +P 𝑥)))
11	10	3expa 1230	. . . . . . . . . 10 ⊢ (((𝐴 ∈ P ∧ 𝐶 ∈ P) ∧ 𝑥 ∈ P) → ((𝐶 +P 𝐴) +P 𝑥) = (𝐶 +P (𝐴 +P 𝑥)))
12	11	adantrr 479	. . . . . . . . 9 ⊢ (((𝐴 ∈ P ∧ 𝐶 ∈ P) ∧ (𝑥 ∈ P ∧ ((𝐶 +P 𝐴) +P 𝑥) = (𝐶 +P 𝐵))) → ((𝐶 +P 𝐴) +P 𝑥) = (𝐶 +P (𝐴 +P 𝑥)))
13		simprr 533	. . . . . . . . 9 ⊢ (((𝐴 ∈ P ∧ 𝐶 ∈ P) ∧ (𝑥 ∈ P ∧ ((𝐶 +P 𝐴) +P 𝑥) = (𝐶 +P 𝐵))) → ((𝐶 +P 𝐴) +P 𝑥) = (𝐶 +P 𝐵))
14	12, 13	eqtr3d 2267	. . . . . . . 8 ⊢ (((𝐴 ∈ P ∧ 𝐶 ∈ P) ∧ (𝑥 ∈ P ∧ ((𝐶 +P 𝐴) +P 𝑥) = (𝐶 +P 𝐵))) → (𝐶 +P (𝐴 +P 𝑥)) = (𝐶 +P 𝐵))
15	14	3adantl2 1181	. . . . . . 7 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ (𝑥 ∈ P ∧ ((𝐶 +P 𝐴) +P 𝑥) = (𝐶 +P 𝐵))) → (𝐶 +P (𝐴 +P 𝑥)) = (𝐶 +P 𝐵))
16		simpl3 1029	. . . . . . . 8 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ (𝑥 ∈ P ∧ ((𝐶 +P 𝐴) +P 𝑥) = (𝐶 +P 𝐵))) → 𝐶 ∈ P)
17		addclpr 7852	. . . . . . . . 9 ⊢ ((𝐴 ∈ P ∧ 𝑥 ∈ P) → (𝐴 +P 𝑥) ∈ P)
18	5, 6, 17	syl2anc 411	. . . . . . . 8 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ (𝑥 ∈ P ∧ ((𝐶 +P 𝐴) +P 𝑥) = (𝐶 +P 𝐵))) → (𝐴 +P 𝑥) ∈ P)
19		simpl2 1028	. . . . . . . 8 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ (𝑥 ∈ P ∧ ((𝐶 +P 𝐴) +P 𝑥) = (𝐶 +P 𝐵))) → 𝐵 ∈ P)
20		addcanprg 7931	. . . . . . . 8 ⊢ ((𝐶 ∈ P ∧ (𝐴 +P 𝑥) ∈ P ∧ 𝐵 ∈ P) → ((𝐶 +P (𝐴 +P 𝑥)) = (𝐶 +P 𝐵) → (𝐴 +P 𝑥) = 𝐵))
21	16, 18, 19, 20	syl3anc 1274	. . . . . . 7 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ (𝑥 ∈ P ∧ ((𝐶 +P 𝐴) +P 𝑥) = (𝐶 +P 𝐵))) → ((𝐶 +P (𝐴 +P 𝑥)) = (𝐶 +P 𝐵) → (𝐴 +P 𝑥) = 𝐵))
22	15, 21	mpd 13	. . . . . 6 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ (𝑥 ∈ P ∧ ((𝐶 +P 𝐴) +P 𝑥) = (𝐶 +P 𝐵))) → (𝐴 +P 𝑥) = 𝐵)
23	8, 22	breqtrd 4135	. . . . 5 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ (𝑥 ∈ P ∧ ((𝐶 +P 𝐴) +P 𝑥) = (𝐶 +P 𝐵))) → 𝐴<P 𝐵)
24	23	adantlr 477	. . . 4 ⊢ ((((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ (𝐶 +P 𝐴)<P (𝐶 +P 𝐵)) ∧ (𝑥 ∈ P ∧ ((𝐶 +P 𝐴) +P 𝑥) = (𝐶 +P 𝐵))) → 𝐴<P 𝐵)
25	4, 24	rexlimddv 2665	. . 3 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ (𝐶 +P 𝐴)<P (𝐶 +P 𝐵)) → 𝐴<P 𝐵)
26	25	ex 115	. 2 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) → ((𝐶 +P 𝐴)<P (𝐶 +P 𝐵) → 𝐴<P 𝐵))
27	2, 26	impbid 129	1 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) → (𝐴<P 𝐵 ↔ (𝐶 +P 𝐴)<P (𝐶 +P 𝐵)))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∧ w3a 1005   = wceq 1398   ∈ wcel 2203  ∃wrex 2521   class class class wbr 4109  (class class class)co 6050  Pcnp 7606   +_P cpp 7608  <_P cltp 7610

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-coll 4225  ax-sep 4228  ax-nul 4236  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-setind 4659  ax-iinf 4710

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-ral 2525  df-rex 2526  df-reu 2527  df-rab 2529  df-v 2815  df-sbc 3043  df-csb 3139  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-nul 3509  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-int 3950  df-iun 3993  df-br 4110  df-opab 4172  df-mpt 4173  df-tr 4209  df-eprel 4410  df-id 4414  df-po 4417  df-iso 4418  df-iord 4487  df-on 4489  df-suc 4492  df-iom 4713  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-ima 4762  df-iota 5312  df-fun 5354  df-fn 5355  df-f 5356  df-f1 5357  df-fo 5358  df-f1o 5359  df-fv 5360  df-ov 6053  df-oprab 6054  df-mpo 6055  df-1st 6334  df-2nd 6335  df-recs 6536  df-irdg 6601  df-1o 6647  df-2o 6648  df-oadd 6651  df-omul 6652  df-er 6767  df-ec 6769  df-qs 6773  df-ni 7619  df-pli 7620  df-mi 7621  df-lti 7622  df-plpq 7659  df-mpq 7660  df-enq 7662  df-nqqs 7663  df-plqqs 7664  df-mqqs 7665  df-1nqqs 7666  df-rq 7667  df-ltnqqs 7668  df-enq0 7739  df-nq0 7740  df-0nq0 7741  df-plq0 7742  df-mq0 7743  df-inp 7781  df-iplp 7783  df-iltp 7785

This theorem is referenced by:  prplnqu  7935  addextpr  7936  caucvgprlemcanl  7959  caucvgprprlemnkltj  8004  caucvgprprlemnbj  8008  caucvgprprlemmu  8010  caucvgprprlemloc  8018  caucvgprprlemexbt  8021  caucvgprprlemexb  8022  caucvgprprlemaddq  8023  caucvgprprlem1  8024  caucvgprprlem2  8025  ltsrprg  8062  gt0srpr  8063  lttrsr  8077  ltsosr  8079  ltasrg  8085  prsrlt  8102  ltpsrprg  8118  map2psrprg  8120