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Theorem omndmul 29045
Description: In a commutative ordered monoid, the ordering is compatible with group power. (Contributed by Thierry Arnoux, 30-Jan-2018.)
Hypotheses
Ref Expression
omndmul.0 𝐵 = (Base‘𝑀)
omndmul.1 = (le‘𝑀)
omndmul.2 · = (.g𝑀)
omndmul.o (𝜑𝑀 ∈ oMnd)
omndmul.c (𝜑𝑀 ∈ CMnd)
omndmul.x (𝜑𝑋𝐵)
omndmul.y (𝜑𝑌𝐵)
omndmul.n (𝜑𝑁 ∈ ℕ0)
omndmul.l (𝜑𝑋 𝑌)
Assertion
Ref Expression
omndmul (𝜑 → (𝑁 · 𝑋) (𝑁 · 𝑌))

Proof of Theorem omndmul
Dummy variables 𝑚 𝑛 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 omndmul.n . 2 (𝜑𝑁 ∈ ℕ0)
2 oveq1 6556 . . . 4 (𝑚 = 0 → (𝑚 · 𝑋) = (0 · 𝑋))
3 oveq1 6556 . . . 4 (𝑚 = 0 → (𝑚 · 𝑌) = (0 · 𝑌))
42, 3breq12d 4596 . . 3 (𝑚 = 0 → ((𝑚 · 𝑋) (𝑚 · 𝑌) ↔ (0 · 𝑋) (0 · 𝑌)))
5 oveq1 6556 . . . 4 (𝑚 = 𝑛 → (𝑚 · 𝑋) = (𝑛 · 𝑋))
6 oveq1 6556 . . . 4 (𝑚 = 𝑛 → (𝑚 · 𝑌) = (𝑛 · 𝑌))
75, 6breq12d 4596 . . 3 (𝑚 = 𝑛 → ((𝑚 · 𝑋) (𝑚 · 𝑌) ↔ (𝑛 · 𝑋) (𝑛 · 𝑌)))
8 oveq1 6556 . . . 4 (𝑚 = (𝑛 + 1) → (𝑚 · 𝑋) = ((𝑛 + 1) · 𝑋))
9 oveq1 6556 . . . 4 (𝑚 = (𝑛 + 1) → (𝑚 · 𝑌) = ((𝑛 + 1) · 𝑌))
108, 9breq12d 4596 . . 3 (𝑚 = (𝑛 + 1) → ((𝑚 · 𝑋) (𝑚 · 𝑌) ↔ ((𝑛 + 1) · 𝑋) ((𝑛 + 1) · 𝑌)))
11 oveq1 6556 . . . 4 (𝑚 = 𝑁 → (𝑚 · 𝑋) = (𝑁 · 𝑋))
12 oveq1 6556 . . . 4 (𝑚 = 𝑁 → (𝑚 · 𝑌) = (𝑁 · 𝑌))
1311, 12breq12d 4596 . . 3 (𝑚 = 𝑁 → ((𝑚 · 𝑋) (𝑚 · 𝑌) ↔ (𝑁 · 𝑋) (𝑁 · 𝑌)))
14 omndmul.o . . . . . 6 (𝜑𝑀 ∈ oMnd)
15 omndtos 29036 . . . . . 6 (𝑀 ∈ oMnd → 𝑀 ∈ Toset)
16 tospos 28989 . . . . . 6 (𝑀 ∈ Toset → 𝑀 ∈ Poset)
1714, 15, 163syl 18 . . . . 5 (𝜑𝑀 ∈ Poset)
18 omndmul.y . . . . . . 7 (𝜑𝑌𝐵)
19 omndmul.0 . . . . . . . 8 𝐵 = (Base‘𝑀)
20 eqid 2610 . . . . . . . 8 (0g𝑀) = (0g𝑀)
21 omndmul.2 . . . . . . . 8 · = (.g𝑀)
2219, 20, 21mulg0 17369 . . . . . . 7 (𝑌𝐵 → (0 · 𝑌) = (0g𝑀))
2318, 22syl 17 . . . . . 6 (𝜑 → (0 · 𝑌) = (0g𝑀))
24 omndmnd 29035 . . . . . . 7 (𝑀 ∈ oMnd → 𝑀 ∈ Mnd)
2519, 20mndidcl 17131 . . . . . . 7 (𝑀 ∈ Mnd → (0g𝑀) ∈ 𝐵)
2614, 24, 253syl 18 . . . . . 6 (𝜑 → (0g𝑀) ∈ 𝐵)
2723, 26eqeltrd 2688 . . . . 5 (𝜑 → (0 · 𝑌) ∈ 𝐵)
28 omndmul.1 . . . . . 6 = (le‘𝑀)
2919, 28posref 16774 . . . . 5 ((𝑀 ∈ Poset ∧ (0 · 𝑌) ∈ 𝐵) → (0 · 𝑌) (0 · 𝑌))
3017, 27, 29syl2anc 691 . . . 4 (𝜑 → (0 · 𝑌) (0 · 𝑌))
31 omndmul.x . . . . 5 (𝜑𝑋𝐵)
3219, 20, 21mulg0 17369 . . . . . . . 8 (𝑋𝐵 → (0 · 𝑋) = (0g𝑀))
3332adantr 480 . . . . . . 7 ((𝑋𝐵𝑌𝐵) → (0 · 𝑋) = (0g𝑀))
3422adantl 481 . . . . . . 7 ((𝑋𝐵𝑌𝐵) → (0 · 𝑌) = (0g𝑀))
3533, 34eqtr4d 2647 . . . . . 6 ((𝑋𝐵𝑌𝐵) → (0 · 𝑋) = (0 · 𝑌))
3635breq1d 4593 . . . . 5 ((𝑋𝐵𝑌𝐵) → ((0 · 𝑋) (0 · 𝑌) ↔ (0 · 𝑌) (0 · 𝑌)))
3731, 18, 36syl2anc 691 . . . 4 (𝜑 → ((0 · 𝑋) (0 · 𝑌) ↔ (0 · 𝑌) (0 · 𝑌)))
3830, 37mpbird 246 . . 3 (𝜑 → (0 · 𝑋) (0 · 𝑌))
39 eqid 2610 . . . . 5 (+g𝑀) = (+g𝑀)
4014ad2antrr 758 . . . . 5 (((𝜑𝑛 ∈ ℕ0) ∧ (𝑛 · 𝑋) (𝑛 · 𝑌)) → 𝑀 ∈ oMnd)
4118ad2antrr 758 . . . . 5 (((𝜑𝑛 ∈ ℕ0) ∧ (𝑛 · 𝑋) (𝑛 · 𝑌)) → 𝑌𝐵)
4240, 24syl 17 . . . . . 6 (((𝜑𝑛 ∈ ℕ0) ∧ (𝑛 · 𝑋) (𝑛 · 𝑌)) → 𝑀 ∈ Mnd)
43 simplr 788 . . . . . 6 (((𝜑𝑛 ∈ ℕ0) ∧ (𝑛 · 𝑋) (𝑛 · 𝑌)) → 𝑛 ∈ ℕ0)
4431ad2antrr 758 . . . . . 6 (((𝜑𝑛 ∈ ℕ0) ∧ (𝑛 · 𝑋) (𝑛 · 𝑌)) → 𝑋𝐵)
4519, 21mulgnn0cl 17381 . . . . . 6 ((𝑀 ∈ Mnd ∧ 𝑛 ∈ ℕ0𝑋𝐵) → (𝑛 · 𝑋) ∈ 𝐵)
4642, 43, 44, 45syl3anc 1318 . . . . 5 (((𝜑𝑛 ∈ ℕ0) ∧ (𝑛 · 𝑋) (𝑛 · 𝑌)) → (𝑛 · 𝑋) ∈ 𝐵)
4719, 21mulgnn0cl 17381 . . . . . 6 ((𝑀 ∈ Mnd ∧ 𝑛 ∈ ℕ0𝑌𝐵) → (𝑛 · 𝑌) ∈ 𝐵)
4842, 43, 41, 47syl3anc 1318 . . . . 5 (((𝜑𝑛 ∈ ℕ0) ∧ (𝑛 · 𝑋) (𝑛 · 𝑌)) → (𝑛 · 𝑌) ∈ 𝐵)
49 simpr 476 . . . . 5 (((𝜑𝑛 ∈ ℕ0) ∧ (𝑛 · 𝑋) (𝑛 · 𝑌)) → (𝑛 · 𝑋) (𝑛 · 𝑌))
50 omndmul.l . . . . . 6 (𝜑𝑋 𝑌)
5150ad2antrr 758 . . . . 5 (((𝜑𝑛 ∈ ℕ0) ∧ (𝑛 · 𝑋) (𝑛 · 𝑌)) → 𝑋 𝑌)
52 omndmul.c . . . . . 6 (𝜑𝑀 ∈ CMnd)
5352ad2antrr 758 . . . . 5 (((𝜑𝑛 ∈ ℕ0) ∧ (𝑛 · 𝑋) (𝑛 · 𝑌)) → 𝑀 ∈ CMnd)
5419, 28, 39, 40, 41, 46, 44, 48, 49, 51, 53omndadd2d 29039 . . . 4 (((𝜑𝑛 ∈ ℕ0) ∧ (𝑛 · 𝑋) (𝑛 · 𝑌)) → ((𝑛 · 𝑋)(+g𝑀)𝑋) ((𝑛 · 𝑌)(+g𝑀)𝑌))
5519, 21, 39mulgnn0p1 17375 . . . . 5 ((𝑀 ∈ Mnd ∧ 𝑛 ∈ ℕ0𝑋𝐵) → ((𝑛 + 1) · 𝑋) = ((𝑛 · 𝑋)(+g𝑀)𝑋))
5642, 43, 44, 55syl3anc 1318 . . . 4 (((𝜑𝑛 ∈ ℕ0) ∧ (𝑛 · 𝑋) (𝑛 · 𝑌)) → ((𝑛 + 1) · 𝑋) = ((𝑛 · 𝑋)(+g𝑀)𝑋))
5719, 21, 39mulgnn0p1 17375 . . . . 5 ((𝑀 ∈ Mnd ∧ 𝑛 ∈ ℕ0𝑌𝐵) → ((𝑛 + 1) · 𝑌) = ((𝑛 · 𝑌)(+g𝑀)𝑌))
5842, 43, 41, 57syl3anc 1318 . . . 4 (((𝜑𝑛 ∈ ℕ0) ∧ (𝑛 · 𝑋) (𝑛 · 𝑌)) → ((𝑛 + 1) · 𝑌) = ((𝑛 · 𝑌)(+g𝑀)𝑌))
5954, 56, 583brtr4d 4615 . . 3 (((𝜑𝑛 ∈ ℕ0) ∧ (𝑛 · 𝑋) (𝑛 · 𝑌)) → ((𝑛 + 1) · 𝑋) ((𝑛 + 1) · 𝑌))
604, 7, 10, 13, 38, 59nn0indd 11350 . 2 ((𝜑𝑁 ∈ ℕ0) → (𝑁 · 𝑋) (𝑁 · 𝑌))
611, 60mpdan 699 1 (𝜑 → (𝑁 · 𝑋) (𝑁 · 𝑌))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 195  wa 383   = wceq 1475  wcel 1977   class class class wbr 4583  cfv 5804  (class class class)co 6549  0cc0 9815  1c1 9816   + caddc 9818  0cn0 11169  Basecbs 15695  +gcplusg 15768  lecple 15775  0gc0g 15923  Posetcpo 16763  Tosetctos 16856  Mndcmnd 17117  .gcmg 17363  CMndccmn 18016  oMndcomnd 29028
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-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-inf2 8421  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892
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-nel 2783  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-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-riota 6511  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-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-n0 11170  df-z 11255  df-uz 11564  df-fz 12198  df-seq 12664  df-0g 15925  df-preset 16751  df-poset 16769  df-toset 16857  df-mgm 17065  df-sgrp 17107  df-mnd 17118  df-mulg 17364  df-cmn 18018  df-omnd 29030
This theorem is referenced by: (None)
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