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Mirrors > Home > MPE Home > Th. List > xaddcom | Unicode version |
Description: The extended real addition operation is commutative. (Contributed by NM, 26-Dec-2011.) |
Ref | Expression |
---|---|
xaddcom |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elxr 11354 | . 2 | |
2 | elxr 11354 | . . . 4 | |
3 | recn 9603 | . . . . . . 7 | |
4 | recn 9603 | . . . . . . 7 | |
5 | addcom 9787 | . . . . . . 7 | |
6 | 3, 4, 5 | syl2an 477 | . . . . . 6 |
7 | rexadd 11460 | . . . . . 6 | |
8 | rexadd 11460 | . . . . . . 7 | |
9 | 8 | ancoms 453 | . . . . . 6 |
10 | 6, 7, 9 | 3eqtr4d 2508 | . . . . 5 |
11 | oveq2 6304 | . . . . . . 7 | |
12 | rexr 9660 | . . . . . . . 8 | |
13 | renemnf 9663 | . . . . . . . 8 | |
14 | xaddpnf1 11454 | . . . . . . . 8 | |
15 | 12, 13, 14 | syl2anc 661 | . . . . . . 7 |
16 | 11, 15 | sylan9eqr 2520 | . . . . . 6 |
17 | oveq1 6303 | . . . . . . 7 | |
18 | xaddpnf2 11455 | . . . . . . . 8 | |
19 | 12, 13, 18 | syl2anc 661 | . . . . . . 7 |
20 | 17, 19 | sylan9eqr 2520 | . . . . . 6 |
21 | 16, 20 | eqtr4d 2501 | . . . . 5 |
22 | oveq2 6304 | . . . . . . 7 | |
23 | renepnf 9662 | . . . . . . . 8 | |
24 | xaddmnf1 11456 | . . . . . . . 8 | |
25 | 12, 23, 24 | syl2anc 661 | . . . . . . 7 |
26 | 22, 25 | sylan9eqr 2520 | . . . . . 6 |
27 | oveq1 6303 | . . . . . . 7 | |
28 | xaddmnf2 11457 | . . . . . . . 8 | |
29 | 12, 23, 28 | syl2anc 661 | . . . . . . 7 |
30 | 27, 29 | sylan9eqr 2520 | . . . . . 6 |
31 | 26, 30 | eqtr4d 2501 | . . . . 5 |
32 | 10, 21, 31 | 3jaodan 1294 | . . . 4 |
33 | 2, 32 | sylan2b 475 | . . 3 |
34 | pnfaddmnf 11458 | . . . . . . . 8 | |
35 | mnfaddpnf 11459 | . . . . . . . 8 | |
36 | 34, 35 | eqtr4i 2489 | . . . . . . 7 |
37 | simpr 461 | . . . . . . . 8 | |
38 | 37 | oveq2d 6312 | . . . . . . 7 |
39 | 37 | oveq1d 6311 | . . . . . . 7 |
40 | 36, 38, 39 | 3eqtr4a 2524 | . . . . . 6 |
41 | xaddpnf2 11455 | . . . . . . 7 | |
42 | xaddpnf1 11454 | . . . . . . 7 | |
43 | 41, 42 | eqtr4d 2501 | . . . . . 6 |
44 | 40, 43 | pm2.61dane 2775 | . . . . 5 |
45 | 44 | adantl 466 | . . . 4 |
46 | simpl 457 | . . . . 5 | |
47 | 46 | oveq1d 6311 | . . . 4 |
48 | 46 | oveq2d 6312 | . . . 4 |
49 | 45, 47, 48 | 3eqtr4d 2508 | . . 3 |
50 | 35, 34 | eqtr4i 2489 | . . . . . . 7 |
51 | simpr 461 | . . . . . . . 8 | |
52 | 51 | oveq2d 6312 | . . . . . . 7 |
53 | 51 | oveq1d 6311 | . . . . . . 7 |
54 | 50, 52, 53 | 3eqtr4a 2524 | . . . . . 6 |
55 | xaddmnf2 11457 | . . . . . . 7 | |
56 | xaddmnf1 11456 | . . . . . . 7 | |
57 | 55, 56 | eqtr4d 2501 | . . . . . 6 |
58 | 54, 57 | pm2.61dane 2775 | . . . . 5 |
59 | 58 | adantl 466 | . . . 4 |
60 | simpl 457 | . . . . 5 | |
61 | 60 | oveq1d 6311 | . . . 4 |
62 | 60 | oveq2d 6312 | . . . 4 |
63 | 59, 61, 62 | 3eqtr4d 2508 | . . 3 |
64 | 33, 49, 63 | 3jaoian 1293 | . 2 |
65 | 1, 64 | sylanb 472 | 1 |
Colors of variables: wff setvar class |
Syntax hints: -> wi 4 /\ wa 369
\/ w3o 972 = wceq 1395 e. wcel 1818
=/= wne 2652 (class class class)co 6296
cc 9511 cr 9512 0 cc0 9513 caddc 9516 cpnf 9646 cmnf 9647
cxr 9648
cxad 11345 |
This theorem is referenced by: xaddid2 11468 xleadd2a 11475 xltadd2 11478 xposdif 11483 xadd4d 11524 hashunx 12454 xrsnsgrp 18454 xrs1cmn 18458 blcld 21008 xrsxmet 21314 metdstri 21355 vdgrf 24898 xaddeq0 27573 xlt2addrd 27578 xrge0npcan 27684 esumle 28065 esumlef 28070 measun 28182 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1618 ax-4 1631 ax-5 1704 ax-6 1747 ax-7 1790 ax-8 1820 ax-9 1822 ax-10 1837 ax-11 1842 ax-12 1854 ax-13 1999 ax-ext 2435 ax-sep 4573 ax-nul 4581 ax-pow 4630 ax-pr 4691 ax-un 6592 ax-cnex 9569 ax-resscn 9570 ax-1cn 9571 ax-icn 9572 ax-addcl 9573 ax-addrcl 9574 ax-mulcl 9575 ax-mulrcl 9576 ax-mulcom 9577 ax-addass 9578 ax-mulass 9579 ax-distr 9580 ax-i2m1 9581 ax-1ne0 9582 ax-1rid 9583 ax-rnegex 9584 ax-rrecex 9585 ax-cnre 9586 ax-pre-lttri 9587 ax-pre-lttrn 9588 ax-pre-ltadd 9589 |
This theorem depends on definitions: df-bi 185 df-or 370 df-an 371 df-3or 974 df-3an 975 df-tru 1398 df-ex 1613 df-nf 1617 df-sb 1740 df-eu 2286 df-mo 2287 df-clab 2443 df-cleq 2449 df-clel 2452 df-nfc 2607 df-ne 2654 df-nel 2655 df-ral 2812 df-rex 2813 df-rab 2816 df-v 3111 df-sbc 3328 df-csb 3435 df-dif 3478 df-un 3480 df-in 3482 df-ss 3489 df-nul 3785 df-if 3942 df-pw 4014 df-sn 4030 df-pr 4032 df-op 4036 df-uni 4250 df-br 4453 df-opab 4511 df-mpt 4512 df-id 4800 df-po 4805 df-so 4806 df-xp 5010 df-rel 5011 df-cnv 5012 df-co 5013 df-dm 5014 df-rn 5015 df-res 5016 df-ima 5017 df-iota 5556 df-fun 5595 df-fn 5596 df-f 5597 df-f1 5598 df-fo 5599 df-f1o 5600 df-fv 5601 df-ov 6299 df-oprab 6300 df-mpt2 6301 df-er 7330 df-en 7537 df-dom 7538 df-sdom 7539 df-pnf 9651 df-mnf 9652 df-xr 9653 df-ltxr 9654 df-xadd 11348 |
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