Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > ltaprlem | Structured version Visualization version GIF version |
Description: Lemma for Proposition 9-3.5(v) of [Gleason] p. 123. (Contributed by NM, 8-Apr-1996.) (New usage is discouraged.) |
Ref | Expression |
---|---|
ltaprlem | ⊢ (𝐶 ∈ P → (𝐴<P 𝐵 → (𝐶 +P 𝐴)<P (𝐶 +P 𝐵))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ltrelpr 9699 | . . . . . 6 ⊢ <P ⊆ (P × P) | |
2 | 1 | brel 5090 | . . . . 5 ⊢ (𝐴<P 𝐵 → (𝐴 ∈ P ∧ 𝐵 ∈ P)) |
3 | 2 | simpld 474 | . . . 4 ⊢ (𝐴<P 𝐵 → 𝐴 ∈ P) |
4 | ltexpri 9744 | . . . . 5 ⊢ (𝐴<P 𝐵 → ∃𝑥 ∈ P (𝐴 +P 𝑥) = 𝐵) | |
5 | addclpr 9719 | . . . . . . . 8 ⊢ ((𝐶 ∈ P ∧ 𝐴 ∈ P) → (𝐶 +P 𝐴) ∈ P) | |
6 | ltaddpr 9735 | . . . . . . . . . 10 ⊢ (((𝐶 +P 𝐴) ∈ P ∧ 𝑥 ∈ P) → (𝐶 +P 𝐴)<P ((𝐶 +P 𝐴) +P 𝑥)) | |
7 | addasspr 9723 | . . . . . . . . . . . 12 ⊢ ((𝐶 +P 𝐴) +P 𝑥) = (𝐶 +P (𝐴 +P 𝑥)) | |
8 | oveq2 6557 | . . . . . . . . . . . 12 ⊢ ((𝐴 +P 𝑥) = 𝐵 → (𝐶 +P (𝐴 +P 𝑥)) = (𝐶 +P 𝐵)) | |
9 | 7, 8 | syl5eq 2656 | . . . . . . . . . . 11 ⊢ ((𝐴 +P 𝑥) = 𝐵 → ((𝐶 +P 𝐴) +P 𝑥) = (𝐶 +P 𝐵)) |
10 | 9 | breq2d 4595 | . . . . . . . . . 10 ⊢ ((𝐴 +P 𝑥) = 𝐵 → ((𝐶 +P 𝐴)<P ((𝐶 +P 𝐴) +P 𝑥) ↔ (𝐶 +P 𝐴)<P (𝐶 +P 𝐵))) |
11 | 6, 10 | syl5ib 233 | . . . . . . . . 9 ⊢ ((𝐴 +P 𝑥) = 𝐵 → (((𝐶 +P 𝐴) ∈ P ∧ 𝑥 ∈ P) → (𝐶 +P 𝐴)<P (𝐶 +P 𝐵))) |
12 | 11 | expd 451 | . . . . . . . 8 ⊢ ((𝐴 +P 𝑥) = 𝐵 → ((𝐶 +P 𝐴) ∈ P → (𝑥 ∈ P → (𝐶 +P 𝐴)<P (𝐶 +P 𝐵)))) |
13 | 5, 12 | syl5 33 | . . . . . . 7 ⊢ ((𝐴 +P 𝑥) = 𝐵 → ((𝐶 ∈ P ∧ 𝐴 ∈ P) → (𝑥 ∈ P → (𝐶 +P 𝐴)<P (𝐶 +P 𝐵)))) |
14 | 13 | com3r 85 | . . . . . 6 ⊢ (𝑥 ∈ P → ((𝐴 +P 𝑥) = 𝐵 → ((𝐶 ∈ P ∧ 𝐴 ∈ P) → (𝐶 +P 𝐴)<P (𝐶 +P 𝐵)))) |
15 | 14 | rexlimiv 3009 | . . . . 5 ⊢ (∃𝑥 ∈ P (𝐴 +P 𝑥) = 𝐵 → ((𝐶 ∈ P ∧ 𝐴 ∈ P) → (𝐶 +P 𝐴)<P (𝐶 +P 𝐵))) |
16 | 4, 15 | syl 17 | . . . 4 ⊢ (𝐴<P 𝐵 → ((𝐶 ∈ P ∧ 𝐴 ∈ P) → (𝐶 +P 𝐴)<P (𝐶 +P 𝐵))) |
17 | 3, 16 | sylan2i 685 | . . 3 ⊢ (𝐴<P 𝐵 → ((𝐶 ∈ P ∧ 𝐴<P 𝐵) → (𝐶 +P 𝐴)<P (𝐶 +P 𝐵))) |
18 | 17 | expd 451 | . 2 ⊢ (𝐴<P 𝐵 → (𝐶 ∈ P → (𝐴<P 𝐵 → (𝐶 +P 𝐴)<P (𝐶 +P 𝐵)))) |
19 | 18 | pm2.43b 53 | 1 ⊢ (𝐶 ∈ P → (𝐴<P 𝐵 → (𝐶 +P 𝐴)<P (𝐶 +P 𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ∃wrex 2897 class class class wbr 4583 (class class class)co 6549 Pcnp 9560 +P cpp 9562 <P cltp 9564 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-inf2 8421 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-oadd 7451 df-omul 7452 df-er 7629 df-ni 9573 df-pli 9574 df-mi 9575 df-lti 9576 df-plpq 9609 df-mpq 9610 df-ltpq 9611 df-enq 9612 df-nq 9613 df-erq 9614 df-plq 9615 df-mq 9616 df-1nq 9617 df-rq 9618 df-ltnq 9619 df-np 9682 df-plp 9684 df-ltp 9686 |
This theorem is referenced by: ltapr 9746 |
Copyright terms: Public domain | W3C validator |